Image-based Water Surface Reconstruction with Refractive Stereo

Image-based Water Surface Reconstruction with Refractive Stereo
Image-based Water Surface Reconstruction with
Refractive Stereo
by
Nigel Jed Wesley Morris
A thesis submitted in conformity with the requirements
for the degree of Master of Science
Graduate Department of Computer Science
University of Toronto
c 2004 by Nigel Jed Wesley Morris
Copyright Abstract
Image-based Water Surface Reconstruction with Refractive Stereo
Nigel Jed Wesley Morris
Master of Science
Graduate Department of Computer Science
University of Toronto
2004
We present a system for reconstructing water surfaces using an indirect refractive stereo
reconstruction method. Our work builds on previous work on image-based water reconstruction that uses single view refractive reconstruction techniques. We combine this
approach with a stereo matching algorithm. Depth determination relies upon the refractive disparity of points on a plane below the water. We describe how the location of
points on the water surface can be determined by hypothesizing a depth from the refractive disparity of one camera view. Then the second camera view is used to verify the
depth. We compare two potential metrics for this matching process. We then present results from our algorithm using both simulated and empirical input, analyzing the results
to determine the primary factors that contribute toward accurate surface point determination. We also show how this process can be used to reconstruct sequences of dynamic
water and present several result sets.
ii
Acknowledgements
I would like to acknowledge the insightful support given to me by my supervisor Kiriakos
Kutulakos. I would also like to thank Allan Jepson for his thorough examination of
my work and helpful comments. Thanks to all the members of the DGP Lab group
for interesting discussions and for making my Masters experience enjoyable. Thanks
especially to Joe Laszlo, Paul Yang and Mike Wu for mulling over my ray tracing and
refraction problems and always being ready to lend a hand. Finally I wish to thank my
parents and my brothers for their consistent support and encouragement.
iii
Contents
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Related work
2.1
Appearance modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
The plenoptic function and light fields . . . . . . . . . . . . . . .
6
Plenoptic measurement . . . . . . . . . . . . . . . . . . . . . . . .
7
Matting and environment matting . . . . . . . . . . . . . . . . . .
8
Matting for composition . . . . . . . . . . . . . . . . . . . . . . .
8
Environment matting for transparent and reflective objects . . . .
8
Environment matting extensions . . . . . . . . . . . . . . . . . . .
10
Environment matting extended to multiple viewpoints . . . . . .
10
Stereo reconstruction of Lambertian scenes . . . . . . . . . . . . . . . . .
10
2.2.1
Basic stereo reconstruction . . . . . . . . . . . . . . . . . . . . . .
11
Dense stereo vs. feature-based and sparse reconstruction . . . . .
13
Global vs local/window disparity . . . . . . . . . . . . . . . . . .
13
2.2.2
Matching cost determination . . . . . . . . . . . . . . . . . . . . .
14
2.2.3
Cost Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1.2
2.2
5
iv
2.2.4
Computation and optimization of the disparity . . . . . . . . . . .
15
Reconstruction of opaque non-Lambertian scenes . . . . . . . . . . . . .
16
2.3.1
Stereo reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3.2
Shape from reflection . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3.3
Shape from polarization . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.4
Laser rangefinders . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Reconstruction of transparent media . . . . . . . . . . . . . . . . . . . .
19
2.4.1
Computerized Tomography . . . . . . . . . . . . . . . . . . . . . .
19
2.4.2
Multi-view reconstruction with transparency . . . . . . . . . . . .
19
2.4.3
Shape from distortion
. . . . . . . . . . . . . . . . . . . . . . . .
20
Reconstruction of water . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.5.1
Reconstruction using transparency . . . . . . . . . . . . . . . . .
21
2.5.2
Reconstruction using light reflection . . . . . . . . . . . . . . . . .
21
Shape from shading . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Shape from polarization . . . . . . . . . . . . . . . . . . . . . . .
22
Shape from refraction . . . . . . . . . . . . . . . . . . . . . . . . .
23
Shape from refractive distortion . . . . . . . . . . . . . . . . . . .
23
Shape from refractive irradiance . . . . . . . . . . . . . . . . . . .
23
Laser rangefinders . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.6
Simulation of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3
2.4
2.5
2.5.3
3 Imaged-based reconstruction of Water
3.1
3.2
29
Imaging water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.1.1
Physical and optical properties of water
. . . . . . . . . . . . . .
30
3.1.2
Imaging of water surfaces . . . . . . . . . . . . . . . . . . . . . .
31
The geometry of stereo water surface reconstruction . . . . . . . . . . . .
37
3.2.1
37
Deriving the surface normal from the incident and refracted rays .
v
3.2.2
3.3
The geometry of indirect stereo triangulation . . . . . . . . . . . .
40
Practical water surface reconstruction . . . . . . . . . . . . . . . . . . . .
44
3.3.1
Pattern specification for feature localization and correspondence .
47
3.3.2
Implementation of indirect stereo triangulation . . . . . . . . . . .
48
3.3.3
The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4 Results
4.1
4.2
4.3
54
Apparatus and Physical Setup . . . . . . . . . . . . . . . . . . . . . . . .
54
4.1.1
Apparatus and imaging system . . . . . . . . . . . . . . . . . . .
55
4.1.2
Camera calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Water surface reconstruction simulation . . . . . . . . . . . . . . . . . . .
58
4.2.1
Simulation implementation . . . . . . . . . . . . . . . . . . . . . .
58
4.2.2
Error metric analysis . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.3
Feature localization error and calibration error comparison . . . .
65
4.2.4
Analysis of localization error at varying depths . . . . . . . . . . .
65
4.2.5
Simulation data compared to real world data . . . . . . . . . . . .
70
Water surface sequences . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5 Conclusion
5.1
87
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
A Simulation algorithm
90
Bibliography
92
vi
List of Figures
2.1
Environment matting setup . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Stereo disparity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
The general shape from refraction setup . . . . . . . . . . . . . . . . . .
25
2.4
Two frames from water simulation results [26] . . . . . . . . . . . . . . .
27
3.1
Refraction of light at the air-water interface . . . . . . . . . . . . . . . .
31
3.2
Imaging of points beneath water surface . . . . . . . . . . . . . . . . . .
32
3.3
Solutions space of (normal, depth) pairs . . . . . . . . . . . . . . . . . .
34
3.4
Stereo Imaging constrains the normal . . . . . . . . . . . . . . . . . . . .
35
3.5
Stereo Imaging in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.6
Determining the surface normal . . . . . . . . . . . . . . . . . . . . . . .
38
3.7
Indirect stereo triangulation . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.8
Improved Error Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.9
Interpolation for hypothesis verification in 2D . . . . . . . . . . . . . . .
49
3.10 Bilinear interpolation of refractive disparities . . . . . . . . . . . . . . . .
51
4.1
Physical setup and apparatus . . . . . . . . . . . . . . . . . . . . . . . .
56
4.2
Trade-off between baseline length and reconstructable region size . . . . .
57
4.3
Calibration error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.4
Localization error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.5
Reconstruction gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
vii
4.6
Error metric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.7
Simulation graphs for varying calibration and localization errors . . . . .
67
4.8
Simulation graphs for varying calibration and localization errors . . . . .
68
4.9
Simulation graphs for varying calibration and localization errors . . . . .
69
4.10 Simulation graphs for varying depths and localization errors . . . . . . .
71
4.11 Simulation graphs for varying depths and localization errors . . . . . . .
72
4.12 Simulation graphs for varying depths and localization errors . . . . . . .
73
4.13 Simulation results compared to empirical results . . . . . . . . . . . . . .
75
4.14 Frame of sequence POUR-A . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.15 Frame of sequence POUR-A . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.16 Frame of sequence POUR-A . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.17 Frame of sequence POUR-A . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.18 Frame of sequence POUR-B . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.19 Frame of sequence POUR-B . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.20 Frame of sequence POUR-B . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.21 Frame of sequence POUR-B . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.22 Pattern distorted by splash . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.23 Reconstruction from RIPPLE sequence . . . . . . . . . . . . . . . . . . .
86
viii
Chapter 1
Introduction
“The world turns softly
Not to spill its lakes and rivers,
The water is held in its arms
And the sky is held in the water.
What is water,
That pours silver,
And can hold the sky?”
-Hilda Conkling
Water has fascinated mankind since the earliest times. It is more than just a necessity
of life; water has inspired art, poetry, myth and science. Thales the ancient Greek
philosopher described water as the primary principle, or the foundation of all matter. The
polymorphism of water and its optical magnificence demands awe and often trepidation
upon the high seas. It inspired the ancient Greek god Poseidon, ruler of the seas lending
him the ability to change shape at will.
This thesis engages the problem of capturing the shape of dynamic water from images.
We present a method for finding points on water surfaces using images from a stereo
1
Chapter 1. Introduction
2
camera rig. Our system is able to generate sequences of captured surface geometry of
flowing water.
1.1
Motivation
The goal of producing realistic imagery of water was long been sought after by the
computer graphics community [60, 46, 29, 28, 25]. Work in this area has taken the form
of simulating the flow of water by modeling approximations of the physical laws that
govern fluids. At best these are approximations and often subtle water surface effects are
missing.
The approach taken by our work and those in the computer vision community as
well as those in oceanography has been to extract the shape of water surfaces from
images of water [69, 78, 41, 56, 42]. This previous work has sought to take advantage
of water’s optical properties to reconstruct a surface. Techniques that have used water’s
reflectivity to reconstruct the surface have had less success than those that utilize the
refractive property of water [42]. Most of these refraction methods use a single viewpoint
and assume an orthographic projection [56, 47, 23]. This requires a relatively distant
camera to minimize the projection distortion. Our work, in contrast, utilizes a refractive
approach with stereo cameras. We thus avoid some of the assumptions and inaccuracies
of these previous methods. Our solution also requires no special sensors or equipment
such as laser rangefinders or external lenses.
Although the motivation for this reconstruction in the oceanography community is
often for analysis of wind-driven waves, we are also motivated by the possibilities of using
this data to obtain or enhance the appearance of novel computer generated images of
water. We expect that our work could contribute to any of the following applications:
• The capture of liquid phenomena for composition into animation or film footage.
• Creation of a library of liquid effects allowing the generation of arbitrary liquid
Chapter 1. Introduction
3
flows out from a composition of the library effects.
• Oceanographic studies of wind-driven waves.
• For precise measurement of transparent objects for Engineering.
• As a first step toward determination of internal fluid flow.
1.2
Contributions
Here is a summary of the primary contributions of this thesis:
• We provide a review of image and sensor-based reconstruction techniques for specular, transparent and refractive objects. We build up this by examining the viability
of using these techniques for water surface reconstruction.
• We present a novel reconstruction algorithm for refractive liquids that combines
stereo reconstruction with a shape from refractive distortion approach.
• We present and analyze two metrics for testing the validity of surface points on
refractive surfaces within the context of a multi-view system.
• We propose an experimental configuration for our algorithm. We present and discuss the results achieved from this setup and compare them to a simulation of our
algorithm.
1.3
Thesis outline
This thesis is organized into five chapters. Following this introduction, in Chapter 2, we
present background on reconstruction techniques in computer vision. We review stereo
reconstruction methods followed by techniques for determining the shape of transparent,
Chapter 1. Introduction
4
shiny and refractive objects. We also examine previous techniques for reconstructing
water surfaces. A review of water simulation is also included.
In Chapter 3 we present our algorithm for reconstructing water surfaces. We discuss issues involved in the implementation as well as our solutions and the resulting
implications.
We present our results in Chapter 4. We provide details on the experimental setup
and imaging procedure. This is followed by a description of our experimental simulation
algorithm. We present and discuss the simulation results, comparing them to experimental results. Finally we present results from reconstructed sequences of dynamic water.
The thesis is concluded in Chapter 5 where we discuss the implications of our work
and future avenues of research that build upon this foundation. We include additional
algorithmic details in the Appendix.
Chapter 2
Related work
“Let us have wine and women, mirth and laughter, sermons and water the day after.”
-Lord Byron
The complex dynamics of water present a major challenge when attempting to capture
the behaviour and appearance of water in a virtual environment. Two primary approaches
have been explored to reach this end. The first approach involves the simulation of both
the hydrodynamics and the optical properties of water in order to generate a virtual
model of water. Rendering techniques have been developed to produce realistic images
from these models. The second approach attempts to interpret and exploit images or
sensory data of water in order to infer the physical shape and behaviour of water, thus
allowing the creation of new images.
In fact the necessity of shape inference in this second path is not clear. Therefore we
initially present techniques that attempt to model phenomena directly from images and
then transition to techniques that infer the geometric shape. We examine reconstruction
methods that reconstruct scenes with simple lighting models proceeding to more complicated systems that handle specular reflections and transparency. We then discuss how
effective each technique is for water surface reconstruction.
Subsequently we present research that takes the approach of simulating liquid phe5
Chapter 2. Related work
6
nomena and how improved physical models have enhanced the accuracy of these techniques.
We conclude the chapter with a summary of the major obstacles and short-comings
of the current techniques for generating virtual models and imagery of water.
2.1
Appearance modeling
Appearance modeling seeks to generate novel views of scenes without attempting to
infer shape information from the scene itself. The focus is to capture the appearance of
the scene through images and then produce novel views of the scene from either a new
viewpoint, or by modifying another aspect of the scene, such as the background. We will
examine a number of techniques that follow this process.
2.1.1
The plenoptic function and light fields
One key concept in appearance modeling is the plenoptic function. It is a function that
fully describes all the light rays converging at a particular point from every direction
[1]. The plenoptic function is directionally parameterized by spherical coordinates θ
and φ. The light intensity of the rays is also dependent on wavelength (λ). Three
more parameters specify the location of the point in space (Vx , Vy , Vz ), and a temporal
parameter (t) can also be included when measuring a temporal sequence. Here is the full
description of the plenoptic function P :
P = P (θ, φ, λ, t, Vx, Vy Vz )
(2.1)
Typically, a camera view of a scene captures a pencil of rays converging on the centre
of projection of the camera. If the plenoptic function were to be known for every point
in a scene, then it would be possible to view the scene from any position and angle.
Knowing the plenoptic function at every point allows us to compute the plenoptic function
Chapter 2. Related work
7
directionally parameterized by the camera’s field of view and located at the camera’s
centre of projection.
Plenoptic measurement
One branch of research in computer vision has developed around utilizing the idea of the
plenoptic function sampling for recreating novel views of a scene. This work leverages
on the idea that the plenoptic function is redundant in ‘free-space’, where there are no
occluding objects. In other words, a ray through a scene has the same intensity at every
point as long as it does not strike any occluding object. Thus any light ray in a scene can
be parameterized by two points on two parallel planes, rather than the five parameters
described above [48, 34].
Images are used to sample the light rays converging on the centre of projection of
a camera. The CCD elements of the camera record the light intensity converging from
a particular directional footprint, rather than individual rays. Thus the image pixels
represent the average intensity of bundles of rays. So measurement of the plenoptic
function starts with many images or samples of the scene. Then new views of the scene
are generated by interpolating between sampled light rays collected from the set of images. Interestingly, Chai et al. have shown that fewer images are necessary when some
geometric properties of the scene are known [20].
Sampling the plenoptic function for water is especially problematic. Water is not
static, so the plenoptic function may change at each time instant. This means that sampling must be done instantaneously from all expected angles. Water’s optical properties
present another challenge, as its appearance is predominantly a reflection or refraction
of light emitted from the rest of the scene. This means the plenoptic sampling may need
to be performed within the desired scene, rather than a controlled laboratory environment. The reflective and transmissive properties of water may also cause elements of the
sampling rig to appear in the images.
Chapter 2. Related work
2.1.2
8
Matting and environment matting
Matting for composition
Matting is a technique for separating the background from the foreground in images,
usually so that the foreground can be composited over a new background. Typically a
matte is formed that is opaque over the background, partially transparent at the edges
of the foreground and fully transparent over the rest of the foreground.
Environment matting for transparent and reflective objects
Matting can be used to approximate the appearance of transparent objects by blending
the matte with the background in those areas that are transparent. This technique breaks
down when the foreground object significantly refracts or reflects light, since a direct
blend with the background is insufficient to describe the distortion actually occurring.
A technique called environment matting attempts to resolve these issues by determining
what background footprint best maps to a particular pixel in the refracting or reflecting
foreground object [81]. The end result is a function for every foreground pixel that
includes the traditional matte, as well as the contribution of light from refraction or
reflection of the surrounding environment.
The general approach is to take a series of images of the foreground object with structured textures on screens surrounding the object. The texture set consists of a hierarchy
of vertical and horizontal stripes of varying thickness and are used to determine the best
axis aligned rectangular region whose average pixel value maps through a particular foreground pixel. This rectangular region is computed by optimizing over the set of images
that were collected with the set of environment textures.
9
Chapter 2. Related work
B
A
q
Figure 2.1: Environment matting setup. For a given pixel q that is part of the image
of the object, the colour of the pixel is composed of a reflected region of the pattern on
the side A and a refracted region of the background pattern B. Several images are taken
with varying stripe thickness and orientations for the patterns.
Chapter 2. Related work
10
Environment matting extensions
The original technique for environment matting requires static objects, since multiple
images must be captured to determine the background mapping as described above.
This limitation can be overcome by simplifying the model and capture setup [21]. This
is achieved by capturing the object in a darkened room, with a single colour gradient
map background. A background smoothness constraint is used to reduce the complexity
and the lack of ambient light allows the foreground colouring to be discarded. Given
these assumptions, sequences of dynamic refractive objects, such as water pouring into
a glass, can be captured and matted against arbitrary backgrounds. Unfortunately this
reconstruction is only suitable for a single viewpoint.
Environment matting extended to multiple viewpoints
Another major restriction of the environment matting technique is its fixed viewpoint.
Several methods that capture the reflectance field of objects from multiple viewpoints
have been proposed [52, 24]. One of these focuses on reconstructing transparent objects
using an extension of the environment matting technique. They utilize a rotating camera
and lighting rig that captures the visual hull of objects as well as the environment matte
from multiple viewpoints. A form of unstructured light field interpolation is used to
determine the lighting for every visible point of the object. Although this technique is
only suitable for static objects, it effectively captures many optical effects not previously
attempted. Since objects are captured from multiple viewpoints, the data is much more
useful in terms of animation and visualization.
2.2
Stereo reconstruction of Lambertian scenes
Appearance modeling avoids making inferences about the shape of the scene by relying on
large numbers of images of the scene to generate novel views. By extracting information
11
Chapter 2. Related work
about the scene geometry, fewer images are needed to construct new views. This is
especially important for reconstructing dynamic phenomena such as water.
Before we can examine the complex case of water, we will first look at some of the
background of stereo work for simple Lambertian scenes. We will cover previous work on
stereo reconstruction techniques that extract geometric scene information from binocular
parallax. We then outline several different approaches to the problem and the issues
involved with each.
2.2.1
Basic stereo reconstruction
Stereo reconstruction is one of the most common techniques for determining geometric
information about a scene. It is derived from the human visual system, and works by
leveraging the parallax between corresponding points in two views of the scene (see Figure
2.2). The relative displacement of the corresponding points in the two views is known
as their disparity. Conventional stereo vision determines the depth (z) of the point from
the stereo baseline or the line connecting the centre of projection of both views [59]. The
formulation for the depth given an binocular view with parallel optical axes is:
z=
BF
,
d
(2.2)
where B is the length of the stereo baseline, F is the focal length of the cameras and d
is the disparity between the images of the point.
Most stereo algorithms have at least a subset of the following stages:
• Matching cost determination - Determination of point correspondences between
views and the assignment of a cost to each candidate
• Cost aggregation - Aggregation of the costs of all points
• Computation and optimization of the disparity
12
Chapter 2. Related work
d
z
d
B
F
Figure 2.2: The figure shows two cameras separated by a baseline B. Both cameras image
an object at depth z away from the baseline on their image planes. Both cameras have a
focal depth of F . The dotted circles on the image planes indicate the image location of
the object in the other view. The disparity of the image of the object between the two
views is d.
Chapter 2. Related work
13
The rest of this section will discuss various techniques for solving parts of the stereo
reconstruction problem as well as some of the difficulties faced in stereo reconstruction.
We examine whether stereo is appropriate for extracting geometric shape from images
of water surfaces and how the challenges of conventional stereo approaches apply to our
problem area. An extensive review of dense, binocular stereo algorithms can be found in
[66].
Dense stereo vs. feature-based and sparse reconstruction
Stereo reconstruction techniques can be divided into dense and sparse point matching
methods. Dense methods attempt to find a correspondence between every pixel in the
stereo images [27, 14]. Often optical flow or incremental algorithms are used in this case
and the displacement between corresponding pixels constitutes the stereo disparity [13].
Sparse stereo methods often rely upon feature matching between images such as edges
or corners that can be accurately localized [54, 49, 79]. Correspondences between image
features in the images are then determined.
Although water surfaces are smooth and relatively featureless, sparse techniques can
be used for indirect stereo reconstruction. The water’s reflectivity or refractivity can be
utilized to redirect a sparse pattern that can be used in turn for reconstruction purposes.
Global vs local/window disparity
Most techniques can also be divided into global or local reconstruction algorithms. Global
techniques typically compute disparity values for every point and then minimize an energy
function based on the sum of costs for every point along with a smoothness term. Local
or window approaches attempt to optimize each point separately by aggregating the cost
of the neighbourhood or support region around the point, given some disparity estimate.
Chapter 2. Related work
2.2.2
14
Matching cost determination
The most basic methods for matching pixels between stereo views measure the squared
[59, 37] or absolute difference between pixels [44]. In order to reduce the impact of
mismatched pixels, several techniques have been developed. These include robust estimators using truncated quadrics and contaminated Gaussians that help to eliminate
outliers [13, 14]. Another matching technique is normalized cross-correlation that is similar to the sum of squared differences but also normalizes the matching window before
comparison.
Intensity gradients are sometimes used for matching, having the benefit of being
insensitive to camera bias and gain. Often these camera artefacts are removed in a preprocessing stage [22]. Sparse reconstruction techniques sometimes use a binary matching
technique when seeking to match detected edges or other features [4, 35, 19].
An important innovation suggested by Birchfield and Tomasi is to match pixels in one
image with interpolated sub-pixel offsets in the other image, rather than merely seeking
matches at integral offsets [12]. Matching can be especially problematic when there are
objects with repeated textures or edges. This can easily lead to mismatches, although
reconstruction with multiple views can help to alleviate this [59].
Stereo reconstruction on water surfaces has some apparent advantages over general
scenes. Typically a water surface is smooth and exhibits few occlusions when viewed from
overhead as long as splashes are discounted. Thus many of the matching cost techniques
for handling discontinuities are unnecessary. On the other hand, stereo matching with
water is non-trivial due to its specular reflectance and its refractive nature. Matching
would have to be performed indirectly using either a reflection or a refracted image which
may be discontinuous or warped by the water surface.
Chapter 2. Related work
2.2.3
15
Cost Aggregation
Stereo reconstruction techniques have used a wide range of support regions to enhance
matching. Two-dimensional support regions can be aggregated over square windows [59],
shiftable windows [16, 2], and windows with adaptive size [58, 45]. Three-dimensional
aggregation techniques attempt to match surfaces with areas of similar disparity or a
similar disparity gradient [61, 62]. This permits sloping surfaces to be more accurately
detected.
Aggregation on fixed windows can be performed by convolution or box filters. Another
method that is used is iterative-diffusion, where the weighted cost of neighbouring pixels
are added to the local pixel [65, 68].
2.2.4
Computation and optimization of the disparity
Local methods typically just take the disparity associated with the minimal cost as
determined by the aggregation stage. This has the problem that points in the reference
image may not have a one-to-one mapping to points in the second image.
Global methods tend to concentrate on this stage. They typically minimize an energy
function as follows:
E(d) = Edata (d) + λEsmooth (d)
(2.3)
The data function, Edata (d), measures how the disparity function d matches the reference image to the second image using some aggregate matching cost function. The
smoothness term Esmooth (d) measures the energy associated with smoothness or discontinuity in the support region around the point. Previous work has focussed on robust
smoothing functions that handle smooth surfaces as well as discontinuities [73, 14, 65, 33].
Colour and intensity discontinuities have also been used to predict surface discontinuity
[18, 30, 16].
Chapter 2. Related work
16
Another area of research has looked at how best to minimize the energy function
defined above. Some traditional energy minimization routines are continuation [15],
simulated annealing [33, 50, 6], highest confidence first [19] and mean-field annealing
[31].
Another class of global optimization algorithms use dynamic programming to minimize Equation (2.3) on a scanline basis by finding the minimum-cost path through a
matrix of matching costs of pixels in the two corresponding scanlines [8, 7, 32, 22, 16, 12].
2.3
Reconstruction of opaque non-Lambertian scenes
Many reconstruction algorithms make the implicit or explicit assumption of view independent lighting, or a Lambertian shading model. This assumption breaks down for shiny or
specularly reflective surfaces. This is particularly relevant to water reconstruction, since
water surfaces are highly specular.
In this section we examine a variety of reconstruction methods that attempt to reconstruct shape in non-Lambertian, opaque scenes. We look at several stereo-based
techniques that model general non-Lambertian scenes. We then examine methods that
focus on purely specular or mirroring surfaces. The first technique of this type infers
shape from distortions in the reflected images of curved specular surfaces. This is followed by a description of a voxel-based technique for mirroring objects. Then we discuss
how polarization sensors and laser rangefinders may be used to determine the shape of
reflective objects.
2.3.1
Stereo reconstruction
Recently there has been a concerted focus on reconstruction of specular surfaces with
stereo [10, 11]. One approach is to remove specular highlights in a pre-processing stage
before reconstructing as before [57]. Another technique leverages Helmholtz reciprocity
Chapter 2. Related work
17
to capture the shape of objects with arbitrary reflectance properties. An image pair
of the object is taken with a reciprocating camera and light. This guarantees that the
pixel intensities in both images of corresponding points on the object depend only on the
surface shape and not on the object’s reflectance properties [80].
More recently Treuille et al. also proposed a method for capturing the shape of objects
with general reflectance properties [74]. Their technique avoids the reciprocity constraint
of the camera and light set up. Instead they rely on observations of the target object
along with a known example object that exhibits the same reflectance properties. They
use the known normals and observations of the reference object to determine orientation
consistency in the target object. In addition they describe a technique for handling
self-shadowing on the objects.
2.3.2
Shape from reflection
Another technique that has been applied to determine the shape of objects that mirror
light has been to leverage distortion and non-linearity that occurs during this redirection.
Although reflected images project linearly across flat mirrors, distortions in the mirror
will in turn distort the reflected image allowing shape inference of the mirror surface.
Curved specular surfaces have been reconstructed by inferring shape from the distortion of lines and line intersections [64]. A formula was derived to determine the tangent
normal of the specular object for a calibrated point defined by the intersection of two
lines. The formula utilized the curvature of the images of the intersecting lines to determine the surface normal and then the surface location at that point. Other methods
have used the distortion of patterns to infer surface slope [36].
Another approach to reconstructing purely specular surfaces is to model the surface
by localizing features or a pattern in the reflected image. One technique that seeks to do
this uses a multi-view voxel carving technique with a normal consistency check [17]. The
technique reconstructs mirror-like surfaces, discretizing the space around the surface into
Chapter 2. Related work
18
voxels. Next, each voxel is assigned a normal from each camera view that would account
for the reflected image had the specular surface passed through that voxel. Voxels which
have inconsistent normal sets are then eliminated, leaving the voxels that best represented
the true surface.
2.3.3
Shape from polarization
When light reflects off of a surface, some of the light becomes polarized in the direction
of the surface normal. The phase image of the object encodes the orientation of the
reflection plane which is defined as the plane spanned by the surface normal and the
incident ray.
Several methods exist for determining the surface normal once the reflection plane
is determined. One technique is to use a second view to constrain the normal to an
epipolar line and then use a global minimization approach to solve the surface normals
as well as depths [63]. Another approach assumes surface smoothness and the normal at
object boundaries is perpendicular to the viewing angle. Once the normal is determined
at these edge points, degree of polarization images are used to propagate the solutions
over the rest of the object [53].
2.3.4
Laser rangefinders
An alternative method to determining shape of objects from images it to use laser rangefinders. These typically work by projecting laser light onto the object surface and measuring this reflected light at a known receiver. The process accurately triangulates points on
the object surface but usually requires a Lambertian surface. Recently, laser rangefinders have been developed that are able to effectively reconstruct the shape of specular
objects as well [3]. This is done by restricting the angle of the incident light to a single
direction by attaching several parallel plates at an angle in front of the CCD elements.
The vertical plates, along with a horizontal slit block incident light except from the one
Chapter 2. Related work
19
expected angle, allowing surface triangulation. Clearly this technique would not be suitable for scanning fluctuating water, yet it is a certainly an advance for reconstructing
static specular surfaces.
2.4
Reconstruction of transparent media
For many years, transparency reconstruction has been common in medical imaging systems. These approaches are meant for purely transparent scenes and do not deal well with
occlusions. Recently methods have been developed to integrate common computer vision
techniques with scenes containing opaque and transparent objects. Instead of treating
transparent objects as an obstacle, another approach has been to utilize the refractive
properties of transparent objects as a means to reconstruct the surface of the objects.
2.4.1
Computerized Tomography
Transparent media have long been reconstructed with medical imaging systems using
Computerized Tomography. CT techniques use density images to reconstruct slices of
the structure of a volumetric object [43]. Each image records the density of the object
within the each projected pixel cone. This information is all compiled together and then
interpreted to produce a density map of the transparent object. One of the primary
methods for this compilation is called back-propagation [43].
2.4.2
Multi-view reconstruction with transparency
Voxel carving techniques have been applied to transparent objects. One such volumetric
carving technique seeks to deal with transparent objects through a modified version of
voxel carving. For each ray through each pixel the voxels along the ray are assigned
weights that govern how much that particular voxel contributes to the pixel colour. The
weights translate to transparency values. The algorithm uses an iterative approach to
Chapter 2. Related work
20
find the most consistent set of voxels and weights given all the views of the object. These
weighted voxels are known as ‘Roxels’. In this technique, uncertainty is modelled by
transparency, so if the precise location of a surface edge is uncertain it will appear to be
blurry.
Tsin et al. provide a method for handling stereo reconstruction in the presence of
translucency and reflections [75]. They describe how a scene can be reconstructed with
multiple layers under these conditions when computing depth. So reflected objects are
assigned a depth layer as well as the reflector. The work also describes a method of
extracting the correct colours of the component layers.
2.4.3
Shape from distortion
Shape from distortion techniques can also be applied to transparent objects by inferring
surface shape of a refractive object from the distortion it causes to light transmitted
through it.
One recent method has been presented for inferring shape and pose of transparent
objects from a moving camera’s image sequence [9]. Features are tracked throughout the
sequence and an objective function that characterizes the shape and pose of the transparent object is minimized. The work restricts the target objects to be parameterized by
a single parameter such as super-quadrics. This is a clear step forward in reconstructing
shape from transparent media, although the low-dimensional parameterizations reduce
the generality of the method and makes it inappropriate for dynamic transparent objects
such as water.
2.5
Reconstruction of water
In order to reconstruct water surfaces, its optical properties must be exploited to infer the
surface shape. When a light ray strikes a water surface from air, part of it is mirrored and
Chapter 2. Related work
21
reflects off of the surface. The rest of the light is refracted and transmitted through the
water. Water can be considered as a transparent, reflective or refractive object, leading
to a multitude of reconstruction approaches. In this section we examine the feasibility of
techniques that attempt to reconstruct water in each of these ways.
2.5.1
Reconstruction using transparency
Most of the methods for reconstructing transparent surfaces break down when they are
applied to water in a similar way to plenoptic sampling. CT techniques would require
many simultaneous images of the water and it would be difficult to avoid imaging the
capture equipment at the same time. The imaging technique also presents a problem.
Neither direct optical images nor ultrasound will work due to the surface refraction.
Magnetic resonance is also unusable due to the slow rate of capture. The Roxel algorithm
has the same problems as CT techniques since it does not consider refraction and must
be simultaneously imaged from multiple views.
2.5.2
Reconstruction using light reflection
Shape from shading
Shape from shading techniques attempt to infer geometric shape from the shading of a
surface given some expected or known reflectance and lighting model [40]. If the object’s
reflectance properties are known and the light source location is known, then the surface
shading depends only on the surface normal. Thus from an image of the object, it is
possible to infer surface normals from the pixel intensities.
Traditional shape from shading algorithms assume a Lambertian reflectance model
as it is isotropic and the shading is independent of the viewing angle. Reconstructing
purely specular surfaces, such as water, presents several challenges. Using a single point
light source is often insufficient, as the surface will only receive a highlight where the
Chapter 2. Related work
22
viewing angle and the light incident angle on the surface are equal. With a point light
source only part of surface will be lit, where the surface normal is such that the incident
light reflects directly toward the camera.
Several attempts have been made to reconstruct water surfaces using reflected light.
One approach utilizes stereo images taken under natural lighting conditions and then
uses traditional stereo image matching for Lambertian surfaces [69]. The resolution of the
reconstruction appears to be insufficient for the determination of small wavelength waves.
The second problem is that of correspondence error resulting from specular bias between
the binocular views. Other specular artefacts plague waves with limited amplitude.
Another technique directly uses the specular highlight falloff to compute shape [67].
Several images of the surface from different orientations are used to determine surface
slope at various points on the surface. Once these seed slopes are found, solutions are
grown around these points by searching for the best surface orientation that minimizes
the difference between the expected irradiance given that orientation and the observed
irradiance.
Reconstruction of water surfaces typically do not have some of the common problems of occlusions or discontinuities found in many reconstruction scenarios except when
splashing occurs. There is high degree of non-linearity when determining surface slope
from irradiance due to the transparent nature of water [42]. Reflectivity on the water
surface is governed by Fresnel’s coefficients, causing a high degree of reflection at grazing angles but very little at acute angles. Also a very large light source is required for
reconstruction at grazing angles.
Shape from polarization
Shape from polarization algorithms typically cannot handle internal reflections although
some predict general internal reflections and reduce the polarization images accordingly. This is only an approximate solution and error is still accumulated from the
Chapter 2. Related work
23
inter-reflections.
Theoretically, a multi-view or binocular stereo shape from polarization approach could
reconstruct water, although according to our knowledge this has not been attempted.
Polarization methods tend to deviate from our motivation to design a simpler, pure
image-based system for accurate water surface reconstruction.
2.5.3
Shape from refraction
Water reconstruction techniques that have treated water as a refractive medium have
produced the most promising results and avenues of research. Determining shape from
refraction techniques avoid many of the problems associated with shape from reflection
algorithms. Refraction non-linearities are much lower than those for reflection, allowing
a smaller light source or pattern and since most refraction techniques light the surface
from below, specular artefacts do not occur.
Shape from refractive distortion
Water surfaces have also been reconstructed through refractive distortion [56]. One
algorithm for reconstruction has four parts: First optical flow is computed on the image
of the pattern as it is distorted by the water. Then the average of the optical flow
displacements is taken to be the true location of a particular pattern point. Then the
surface normal for every point in every frame is computed given the displacement from
the computed ‘true’ location. Finally a surface is integrated from the surface normals.
This technique assumes a distant camera and only works on low amplitude waves and
the surface is reconstructed up to some unknown scale factor.
Shape from refractive irradiance
Several image intensity based techniques for recovering surface shape from transparent
media using refraction have been presented in the past [78, 41, 47, 23]. Most techniques
Chapter 2. Related work
24
have been designed to determine the slope of water surfaces. The classic imaging setup
is show below in Figure 2.3.
Light rays from a screen pass through the lens that collimates them so that certain
intensities or colours correspond to parallel light ray columns and are then refracted by
the water surface to the distant camera. This has the result of associating colour or
intensity with particular surface slopes.
There are several techniques for generating the screen, some using an attenuated light
source from one end, some using an HSV coloured gradient and others just a lit monotone
intensity gradient [78].
An important assumption in all these techniques is that of an infinitely distant camera.
This is to assure parallel incoming rays from the water surface. Yet distortions are still
going to affect results as this assumption cannot be modeled precisely. Also error is
bound to be introduced by the collimating lens. Light attenuation from the water will
also affect the slope intensities differently in different parts of the image as the underwater
path lengths will vary.
Laser rangefinders
Laser rangefinders have been developed to measure water surfaces typically by projecting
a laser ray through the water and measuring the ray’s deflection due to refraction. This
has been done both by firing the ray from beneath the surface and detecting it’s projection
on a screen above the water [77], or the reverse where the deflected ray projects on to a
screen beneath the water. Geometrically the surface normal can be determined by the
detection of the refracted ray, and an iterative method can be used to determine the
water surface intersection point.
25
Chapter 2. Related work
Camera
Camera Image
Water
surface
n
n
n
k
k
k
Collimating
lens
g1
g2
g3
Gradient
Figure 2.3: Rays of light (gi ) from a point on the gradient radiate out and are collimated
by the lens into a common direction (k). These rays strike the water surface and only
one certain surface normal (n) will refract them toward the distant camera. Thus in the
camera’s image, pixel colours correspond to surface normals.
Chapter 2. Related work
2.6
26
Simulation of water
Early work in fluid simulation typically focused on wave generation and used simple hydrodynamic models for sinusoidal waves [60, 51]. Splines were used to simulate wave
refraction [76]. Detailed fluid pressure and viscous effects were largely ignored or approximated by particle systems for splashing or breaking waves.
The progression in fluid simulation in computer graphics has been to more closely
approximate physical models and the result has been increased realism. Fluid advection
and pressure flow are governed by the Navier-Stokes equations and many papers have
attempted to approximate these non-linear equations to capture the desired realism.
Some early attempts, such as the work by Kass and Miller [46], simplified the equations for
shallow water and used them to generate animated height fields. This work did not take
into account rotational or pressure based effects, preventing the characteristic eddying
and swirling effects of fluid. Following this work, Foster and Metaxes [29] utilized work
done in the Computation Fluid Dynamics field by Harlow and Welch [38] who described
the full characterization of the Navier-Stokes equations. The fluid was discritized into
a grid of cubes. The Navier-Stokes equations were then solved explicitly and the fluid
advected. Further attempts to improve efficiency and robustness of the system were
examined [28]. Stam presented an improvement in his ‘Stable Fluids’ [71] to implicitly
solve the system with much larger time steps while still maintaining robustness.
An important aspect of fluid simulation research is to improve the visualization of the
fluid effects. Work on liquid surface representation using level sets introduced the most
realistic looking examples seen so far. Level sets were combined with particles to allow
for splashing [25]. Liquid rendering was then further improved by focusing on accurately
representing and rendering the liquid surface using an improved particle and level set
approach [26]. Photorealistic results have been produced by such simulations, yet these
methods are computationally intense and by nature simplifications of the actual physical
processes, potentially losing secondary motion and subtle effects (Figure 2.4).
Chapter 2. Related work
27
Figure 2.4: Two frames from water simulation results [26]
Significant work has gone into simulating fluids with particle systems, often to simulate waterfalls or other dynamic effects [70]. Recently this work has begun to generate
fluid effects at interactive rates. One effective method has been to simulate a liquid with
particles but to render the surface using an interpolation method known as Smooth Particle Hydrodynamics to achieve interactive simulation rates [55]. The method computes
a Navier-Stokes simulation for each particle and interpolates between particles using a
radial basis function to determine the fluid surface.
2.7
Summary
The simulation of water in computer graphics has received a good deal of attention in
recent years and impressive images have been developed. Despite this, simulations still
rely on simplified physics models and complex phenomena such as breaking waters are
difficult to produce.
Water simulation must deal with complex hydrodynamics and surface tension, solving
or approximating non-linear partial derivative systems in order to generate believable
images and flows. In contrast to this, water capture techniques manipulate the relatively
simple optical properties of water to capture the shape of water, without the need for
Chapter 2. Related work
28
hydrodynamic models.
The decision for what optical property to use is vital to accurate reconstruction.
Techniques that use specular reflection suffer from water’s non-linear Fresnel reflection
coefficient. This results in very little reflection when viewing a surface perpendicularly
but much greater reflection at grazing angles. The inverse is true of refraction. In view
of this, it is not surprising that refraction based techniques have been more successful at
reconstruction water surfaces.
Although the sensor-based techniques appear to produce effective results, we are more
interested in the more accessible image-based approaches. Of the image-based refraction
based techniques, shape from refractive irradiance and shape from distortion techniques
seem to be the most effective. Despite this success, these techniques often suffer from
inaccuracies in their image modelling assumptions, such as a distant orthographic camera
and a collimating lens.
These inaccuracies could be improved by combining the refractive reconstruction approach with the well developed stereo techniques seen earlier. In light of this, we propose
that a multi-view stereo approach that uses an indirect matching technique similar to
the shape from distortion technique in [56] could improve reconstruction accuracy and
remove some of the imaging assumptions.
Chapter 3
Imaged-based reconstruction of
Water
“Only a fool tests the depth of the water with both feet.”
-African Proverb
In this chapter we discuss the physical properties of water, and how those properties
influence our design for a system to reconstruct water surfaces from images. We will
present a system that addresses many of the concerns with previous techniques outlined
in the last chapter.
Our design attempts to fulfill the following goals:
• Physically-consistent water surface reconstruction,
• Reconstruction of rapid sequences of flowing, shallow water,
• High reconstruction resolution,
• Use of a minimal number of viewpoints and props.
Our work focuses on recreating a precise definition of the water surface from images.
We consider the problem of reconstructing internal flow as beyond the scope of this work,
although accurate knowledge of the surface can be considered an important first step.
29
Chapter 3. Imaged-based reconstruction of Water
30
We present a sparse multi-view approach to determine the water surface. Multi-view
reconstruction approaches have been used before for water surfaces, but only within
the context of shade-from-shading [69]. Instead we propose that stereo, combined with
shape-from-distortion, is an effective and accurate approach to the problem, gaining from
the benefits of refraction over reflection reconstruction. Previous work has utilized water
surface distortion but only viewed from a single camera [56]. Also, our stereo technique
does not assume distant, orthographic views of the surface, making our model more
physically consistent. Having a stereo system also negates the need to have an extra
collimating lens under the water, as used by some previous single camera techniques
[78, 41, 47].
We also describe how our system is capable of accurately reconstructing very shallow
water.
3.1
3.1.1
Imaging water
Physical and optical properties of water
Light is refracted or bent when there is a density change in the media it is traveling
through. The well known Snell’s law governs light refraction; its general form is as
follows:
r1 sin θi = r2 sin θr
(3.1)
Where r1 is the refractive index of the first medium, r2 is the refractive index of the
second and θi and θr are the incident and refracted angles. At the interface between water
and air, there is a significant change in density and light rays are noticeably refracted. We
can simplify Snell’s law in this case, since the refractive index of air is 1 as in Equation
(3.2). It is important to understand that the incident and refracted rays always lie on
Chapter 3. Imaged-based reconstruction of Water
31
a plane, regardless of the surface normal. Thus it is valid to visualize refraction at an
interface in two dimensions (Figure 3.1). Snell’s law is written as
sin θi = rw sin θr .
(3.2)
n
air
water
θi
p
θr
Figure 3.1: A ray is refracted at a surface point between water and air with a surface
normal n.
When light strikes the water-air interface, part of the light is reflected and part is
refracted. The ratio of reflected to refracted light increases as the angle of incidence
increases. If we continue to increase the incident angle, the refracted angle approaches
90 degrees. At this point we say that the incident angle has reached the critical angle.
Any further increase in the incident angle results in total internal reflection, with no light
refracted.
Refraction of light also depends on the wavelength of the light. So red light has a
higher refractive index than blue light. This property is commonly utilized in prism light
dispersion experiments.
3.1.2
Imaging of water surfaces
Water tends to exhibit slight absorption primarily in the green and red spectra, thus resulting in its typical blue hue. It would be possible to determine depth from absorption,
but the absorption rates are so low (approximately 0.005 cm−1 for red light [72]) that
32
Chapter 3. Imaged-based reconstruction of Water
accurate measurements of shallow water would be difficult with typical imaging equipment. Thus, rather than directly attempting to image water, we examine constraints for
indirect surface measurement.
c
I
q'
q
α
z
n
S
p
θδ - α θδ
T
f
f'
Figure 3.2: Imaging of points beneath the water surface. Feature f is refracted at point
p toward the camera c and is imaged on the image plane at q′ . When no water is in the
tank, f is directly imaged at q. Feature f ′ is the projection of the refracted image point
q′ .
Consider the imaging setup in Figure 3.2. The figure shows rays traced from points
beneath the water surface to an ideal camera, with its centre of projection located at
c. The points are imaged on the image-plane (I) where the rays intersect it (q and
q′ ). q corresponds to the image of the point f without water and q′ is the image of the
point f with water. We have two unknowns, the distance of the surface point from the
camera (z) and the surface normal (n) that define our solution space. Figure 3.3 shows a
Chapter 3. Imaged-based reconstruction of Water
33
solution space for surface normal, depth pairs (n1 , z1 ), (n2 , z2 )...(nm , zm ). Note that for
every depth value, we have a unique surface normal that could account for the refractive
disparity. As the depth value is increased, the slope of the normal must also increase to
compensate until the physical limits of refraction are reached. Depth is computed from
the points as follows1 :
zi = kpi − ck.
(3.3)
The solution space is restricted to surface normal and depth pairs that refract the
ray of light coming from f to the image point q′ . The physical properties constrain this
solution, as light cannot be refracted beyond the critical angle. The other restriction is
the maximum surface normal.
We also note that the distance to the water is not linearly related to the surface
normal as can be seen in Equation (3.4) as a result of the non-linearity of Snell’s law
(Equation (3.2)). Equation (3.4) relates depth (z) to the angular difference between the
incident and refracted rays (θδ ) as well as the refractive displacement angle (α). This
equation results from applying the sine law, given the geometric arrangement of Figure
3.2 as follows,
kf − ck
,
sin(θδ − α)
sin(π − θδ )
sin(θδ − α)
z = kf − ck
.
sin θδ
z
=
(3.4)
Bearing in mind that water is a highly dynamic liquid, we are unable to obtain
multiple views of the surface from a single camera. So if we consider an imaging setup
with a second camera as in Figure 3.4, we can use the second refractive displacement
information to triangulate the common surface point and surface normal. Note that
1
In contrast, conventional stereo depth is determined as distance to the projection of the point onto
the optical distance.
34
Chapter 3. Imaged-based reconstruction of Water
c
I
q'
n1
n2
nm
p1
p2
S
pm
T
f
f'
Figure 3.3: The Figure shows how a set of surface points (p1 , p2 ...pm ) at different depths
with corresponding normals (n1 , n2 ...nm ) could all refract f to the camera c through q′ .
although the rays c1 pf1 and c2 pf2 are shown on the same plane, this is not a necessary
requirement for our algorithm. In Figure 3.5, we illustrate how the cameras may be
oriented to one another in three dimensions. For clarity all further figures are consistently
presented in two dimensions even though the rays may not be coplanar. Also note that
all points of intersection are marked on the figures.
35
Chapter 3. Imaged-based reconstruction of Water
c1
c2
I1
I2
n
S
p
T
f2
f1
Figure 3.4: Points f1 and f2 on the plane T are both refracted at point p and imaged
in camera c1 and c2 respectively. Since both rays c1 pf1 and c2 pf2 intersect the water
surface at p, they share the common surface normal n. So, when two points are imaged
through a common surface point, they also share a common surface normal. This gives
us our stereo normal constraint for determining true surface points.
36
Chapter 3. Imaged-based reconstruction of Water
c2
c1
n
S
p
T
f2
f1
Figure 3.5: This figure shows the imaging system in three dimensions. Point f1 on T is
refracted at p toward camera c1 and point f2 is also refracted at p toward camera c2 .
These points make up two intersecting planes: points c1 , f1 , p lie on one plane and points
c2 , p and f2 lie on another plane. Notice that p lies on the intersection of the planes.
Chapter 3. Imaged-based reconstruction of Water
3.2
37
The geometry of stereo water surface reconstruction
In this section we examine the theory involved in determining surface points in our ideal
imaging model. First we will look at how the surface normal can be determined given a
known surface location. Then we will discuss our indirect stereo triangulation algorithm
for determining depth and surface normals given stereo imagery of an arbitrary surface.
3.2.1
Deriving the surface normal from the incident and refracted rays
We can determine the surface normal n that would cause the refraction of the incident
ray if we know the location of the surface point p. Refer to Figure 3.6 to see the imaging
setup. Using Snell’s law (Equation (3.2)) and our knowledge of the angle between the
incident and refracted rays (θδ ), we are able to derive a solution to the surface normal.
We define θδ as
θδ = θi − θr .
(3.5)
But θi and θr are both unknown. In contrast, the following points are known: the
surface point p, the image of the feature point q′ and the feature point f. Thus we can
determine the vectors of the incident (u) and refracted (v) rays:
u = q′ f ′ ,
(3.6)
v = pf.
(3.7)
Both u and v are normalized to obtain û and v̂:
u
,
kuk
v
v̂ =
.
kvk
û =
(3.8)
(3.9)
38
Chapter 3. Imaged-based reconstruction of Water
c
I
q'
q
α
n
S
p
θδ
u
v
T
f
f'
Figure 3.6: The figure shows the imaging of a point f with water (q′ ) and without water
(q). If the vectors defined by q′ f ′ and pf are known, we can determine the ray vectors u
and v, and hence, the surface normal n.
Chapter 3. Imaged-based reconstruction of Water
39
The inner product of û and v̂ gives us θδ :
θδ = û · v̂.
(3.10)
In order to find the surface normal, we need the incident angle θi (the angle between
the incident ray u and the normal n). We substitute (3.5) into Snell’s law (3.2) and
apply trigonometric identities to find an equation for the incident angle θi :
sin θi = rw sin(θi − θδ ),
sin θi = rw (sin θi cos θδ − cos θi sin θδ ),
rw sin θδ
,
rw cos θδ − 1
!
rw sin θδ
−1
= tan
.
rw cos θδ − 1
tan θi =
θi
(3.11)
So, given θδ and the refractive index for water (rw ), we can determine θi . The surface
normal n is then determined by rotating û by θi about the axis defined by û × v̂:
n = R(θi , û × v̂)û,
(3.12)
where R(β, x̂) is the rotation matrix of an angle β about an axis x̂.
The size of the incident angle (θi ) is strictly increasing as θδ is increased (within the
physical constraints), so there cannot be multiple values of θi for a particular θδ . This
supports our proposition that there is a unique normal for every depth.
Theorem 3.2.1 (Unique normal) For every refractive disparity of a point f imaged
in a camera c1 and hypothesized surface point p there is at most one normal n such that
the ray from the c1 to p is refracted to f.
Proof Without loss of generality, we will show that there can be at most one incident
angle which implies one surface normal.
The physics of refraction constrain the range of the incident angle, such that
0 ≤ θi < π/2.
40
Chapter 3. Imaged-based reconstruction of Water
By Snell’s Law, the refracted angle is also constrained to the following range,
0 ≤ θr < sin−1 (1/rw ).
Thus the difference between these angles, θδ , is physically constrained such that
0 ≤ θδ < π/2 − sin−1 (1/rw ).
The incident angle is computed in Equation 3.11. If we can show that this function
is monotonically increasing, then there can be at most one incident angle for any given
refraction.
Equation 3.11 can be written as follows:
tan θi =
rw sin θδ
.
rw cos θδ − 1
(3.13)
We know that the numerator is monotonically increasing within the specified range
for θδ . We also know that the denomenator is monotonically decreasing and approaches
zero when θδ approaches π/2 − sin−1 (1/rw ).
This means that the right hand side of Equation 3.13 is monotonically increasing.
The arctangent of this function is again monotonically increasing.
3.2.2
The geometry of indirect stereo triangulation
Figure 3.7 shows the geometric setup for indirect stereo triangulation in an ideal scene.
Suppose that two cameras with their centres of projection at c1 and c2 image a water
surface S above a plane T . The image planes of the cameras are denoted as I1 and I2 .
Moreover, suppose that we take two pairs of images of the plane T , first without water
and then with water. From these images reconstruction can proceed.
In order to determine a point on the surface, we use both cameras to triangulate the
surface point. We designate one of the two cameras to be the reference camera and the
other to be the verification camera. We present two metrics for measuring the correctness
Chapter 3. Imaged-based reconstruction of Water
41
of a surface point. Both of the metrics require us to determine a surface point from the
reference camera and then match the expected surface point against the image data from
the verification camera.
The basic reconstruction algorithm first selects a point f1 on the plane T . The images
of this point are found in the image plane I1 and are denoted as q1 without water and
q′1 with water. We know from Section §3.1.2 that the water surface intersection point
must lie along the ray traced through c1 and q′1 (u). The next step of the algorithm is
to traverse this ray, looking for the solution to p that best fits the image data.
We begin this search by hypothesizing a depth from c1 that gives us some surface
point (p′ ). Given this surface point and the location of the imaged feature point (f1 ), we
can determine the incident (u) and refracted rays (v) from Equations (3.6) and (3.7).
This allows us to compute θδ as in Equation (3.10). Next we substitute (3.10) into
(3.11) to get θi and then compute the normal n1 , that would refract u to f1 , from
Equation (3.12).
Since we hypothesized p′ , we need some way to verify whether p′ is close to the actual
surface location p. This is where we utilize our second camera. We trace a ray from p′
back to c2 , finding the image of a feature (f3 ) at q3 . This gives us a new set of incident
(uv ) and refracted (vv ) rays and difference angle (θvδ ):
uv = c2 q′3
(3.14)
vv = pf3
(3.15)
θvδ = uˆv · vˆv
(3.16)
We then use Equations (3.10), (3.11) and (3.12) to compute a second normal n2 for p′ .
At this point we apply our error metrics to determine the validity of the hypothesized
point p′ .
Chapter 3. Imaged-based reconstruction of Water
42
The algorithms for computing the metrics begin in the same way. They take a given
surface depth (z) as input for a particular feature. Figure 3.2 shows z as the distance
between the camera c and the surface point p. The feature imaged with and without
water determines a solution set of depths with corresponding normals. Since a depth
is given as input, the corresponding normal n1 is also determined (Figure 3.7). The
surface point associated with this depth is viewed from the verification camera and has
an associated refractive displacement. This refractive displacement also has a solution
set of depths and normals. Since the depth is already constrained by the specification of
the surface point p′ , we can compute a second normal n2 .
The first metric, which we call the normal collinearity metric, matches the normals
computed by the reference and verification cameras. The value (Enormal ) of the metric is
determined as follows:
Enormal = cos−1 (n1 · n2 )
(3.17)
The intuition for this matching differs from the classical stereo problem where points
are matched and the stereo disparity corresponds directly to the depth of the surface
point. In this case, we cannot directly image the surface point due to the refraction.
Instead we must use the view dependent refracted images to find the position. The
refraction is dependent on the orientation and depth of the water surface point and since
we hypothesize a depth we must try to account for the refraction with the surface normal.
Recall that surface normals translate to a unique water depth, so if the surface normals
that explain the refraction from both views are collinear, then this is a strong indication
that we have the true water depth. If the normals are not collinear, then angle between
normals should give a smooth estimate of the depth error.
We call the second metric the disparity difference metric. This metric measures the
difference in disparity that occurs when n1 is swapped for n2 and the incident rays from
the respective cameras are refracted. Figure 3.8 shows the disparities between f1 and f1′
43
Chapter 3. Imaged-based reconstruction of Water
c2
c1
I1
q1
I2
q3
z
n1
q2
n2
S‘
p'
n
S
p
T
f3
f2
f1
Figure 3.7: The Figure shows the reference camera (c1 ) viewing a point f1 on the plane
T . Then a depth z is hypothesized, giving a surface point p′ and a normal n1 . The
verification camera (c2 ) is used to verify the hypothesized surface point p′ , generating
a second normal n2 . Point p′ coincides with the actual surface point p if and only if
the normals computed from both cameras are identical. When p′ is not equal to surface
point p, we therefore obtain two normals, n1 and n2 .
44
Chapter 3. Imaged-based reconstruction of Water
and between f2 and f2′ . We define the distances between these paired points as e1 and e2
respectively:
e1 = kf1 − f1′ k
(3.18)
e2 = kf2 − f2′ k
(3.19)
Then we define the error metric to be the sum of these distances:
Edisp = e1 + e2
(3.20)
The disparity difference metric merges indirect stereo refraction with conventional
stereo. We expect the disparity difference to provide a deeper error surface in shallow
water where the surface normal has less bearing on the displacement. This metric builds
on the same intuition as the first metric, since it also penalizes mismatched normals.
When the water depth is shallow and feature localization errors become comparable to
the water depth, the effect of the normal on refraction becomes insignificant. The metric
models this by relating the error to the depth, so large normal differences at low depths
aren’t given as high an error as the same normal difference at a higher depth.
The theoretic process for verifying a hypothesized depth is shown in Algorithm 1.
Once the problem is broken down like this, we can perform a simple error minimization
routine to discover the actual depth of the water and the surface point p.
3.3
Practical water surface reconstruction
In the previous section we presented a method for determining points on the water
surface given binocular stereo views of the water. The process relies upon pairs of images
of points on the plane T with and without water. In this section we present our method
for localizing points and determining the correspondence between the points imaged with
and without water (refractive correspondence).
45
Chapter 3. Imaged-based reconstruction of Water
c1
c2
I1
I2
q1
q2
n1
n2
p'
f2
e2
f2'
f1'
e1
f1
Figure 3.8: The Figure shows a point f1 image by the reference camera (c1 ) and a point
f2 imaged by the verification camera (c2 ) generating normals n1 and n2 respectively.
The normal collinearity metric measures the angle between n1 and n2 . In contrast, the
disparity difference error metric is then determined by swapping normal n1 for n2 and
tracing rays from each camera and refracting them by the swapped normals to get f2′ and
f1′ . As in Equation (3.20), the metric is the sum of the distances e1 and e2 .
Chapter 3. Imaged-based reconstruction of Water
46
Algorithm 1: TheoreticDepthVerification
Input: Hypothesized depth z ′ , point on T : f1 , image of the point: f1′ , camera
centres of projection c1 and c2 , the refractive index of water rw
Output: Error E
1. Compute p′ from hypothesized depth z ′ along ray c1 f ′ ;
2. Compute u and v using Equations (3.6) and (3.7);
3. Find θδ from u and v using Equation (3.10);
4. Given θδ and rw , compute θi from Equation (3.11);
5. Given θi , u and v, compute n1 as from Equation (3.12);
6. Intersect p′ c2 with I2 to get the image of a point q′3 . This image point corresponds
to a point f3 on T ;
7. Compute uv and vv using Equations (3.14) and (3.15);
8. In the same manner as before, compute θvδ and n2 using uv and vv ;
9. Compute the error E from the disparity difference metric or normal collinearity
metric using Equations (3.20) or (3.17) respectively;
10. Return E;
Chapter 3. Imaged-based reconstruction of Water
47
So far we have described a system for determining single surface points at a particular
instant in time. Since we would like to be able to capture sequences of dynamic water
surfaces, we also require our system to track the points on T between frames.
Finally, we present the implementation of our water surface reconstruction algorithm
that uses a finite set of feature points on T . We also present our algorithm for reconstructing captured sequences.
3.3.1
Pattern specification for feature localization and correspondence
In order to locate points on T we require feature points that can be reliably localized in
images. In our system we place a pattern with sharp features onto T in full view of both
cameras. For reconstruction, the pattern must be fully visible, especially when covered
by water.
There are several challenges to localizing the features on the pattern. We require both
feature localization at particular frames and feature tracking of the apparent movement
of the features over time. Note that it is not the features that move, but their refracted
images that shift due to changes in the water surface between frames. We also need
to compute two correspondences. First, we must match features between our binocular
views of the pattern in order to determine refractive stereo disparity. Secondly, we need
to be able to find correspondences between the images of the pattern and images of
pattern through water.
The choice of pattern is crucial for our reconstruction algorithm and its accuracy. Our
system is implemented to use a monotone chequered pattern that provides hard edges
and distinct corners. The density of the pattern also affects reconstruction. If the pattern
is too dense, localization may suffer since the support region for the corners is smaller.
Also, a dense pattern is subject to a greater degree of feature elimination and separation
due to refraction of opposing normals. Elimination occurs when a feature point becomes
Chapter 3. Imaged-based reconstruction of Water
48
invisible to the camera due to refraction limits and separation occurs when two adjacent
features appear separated after refraction. These effects are also more pronounced in
deeper water.
In order to determine both the frame to frame correspondence and the corner localization, our system utilizes a Lucas-Kanade type template matching technique [5]. Template
images are generated around the checker corners from an image of the pattern without
water. We then match these templates against the corners in subsequent frames. The
support region around the corners allows for high localization precision. The templates
are locally specific and will not match against any of the four nearest corners since those
corners have reciprocated black and white checkers. This makes the algorithm more
robust to some elimination.
Finally our system is designed to handle two cases of refractive correspondence. For
reconstruction sequences that begin with no water, corner localization at the start of the
sequence is used to locate the feature positions on T and subsequent images are used for
reconstruction. The other case we handle is for sequences beginning from calm water.
In this case we require an image of the T to locate the features without water. We then
detect the features from the calm water images at the start of the sequence. Since the
water is calm, we assume there is no elimination or separation and the only distortion
is the monotonic refractive distortion. We are then able to locate the corners accurately
by stretching the grid to match at the boundaries and relocalizing each corner, giving a
correspondence for the refractive disparity.
3.3.2
Implementation of indirect stereo triangulation
Here we will outline the process for computing point locations and normals on the water
surface given a finite set of feature points on T .
We implemented the surface point triangulation algorithm as a one dimensional minimization problem. The cost function (C) takes in a hypothesized depth z ′ and returns
49
Chapter 3. Imaged-based reconstruction of Water
an error associated with that depth:
Edepth = C(z ′ )
(3.21)
Recall that in our error metrics, we utilize the second camera to verify the hypothesis.
Section §3.2.2 describes how verification works in theory, where a feature point exists at
the end of the verification ray, allowing direct verification of p. In practice, we only have
a finite number of spaced out features on T . We must therefore interpolate between the
nearest features in order to perform the verification. Figure 3.9 shows the scenario in
two dimensions with linear interpolation.
c1
c2
I2
I1
n1 n n2
p
f1
f2
f
Figure 3.9: When verifying a surface point p, due to the discrete placement of features,
we cannot assume that there will be a feature projecting through p to c2 . So we find
the refractive disparity of the nearest features and interpolate to get an approximate
disparity that we use to find the verification normal at p.
Chapter 3. Imaged-based reconstruction of Water
50
Verification is computed through the determination of the surface normal along the
verification ray. As can be seen in Figure 3.9, the normal of the verification ray itself
is not known. Instead we must compute the refractive disparities of the features that
project closest to the desired point. We then perform an interpolation step to find the
approximate refractive disparity along the verification ray (c2 p). This is then used to
find the verification normal. Although Figure 3.9 shows the scenario in two dimensions,
our implementation had to be three-dimensional. Thus we implemented a bilinear interpolation to approximate the normal, interpolating the normals at the four nearest
non-collinear corners. The interpolation is computed as shown in Figure 3.10.
Since the Snell’s law (Equation (3.2)) is non-linear and our surface is not necessarily
linear, the bilinear interpolation is not absolutely accurate. Despite this, water’s inherent
smoothness and a dense feature set with features located every few pixels means that we
can reasonably approximate the verification surface normal.
3.3.3
The algorithm
We utilized some of the physical constraints of the system in our depth estimation routine.
We assumed spatial smoothness of the water by limiting our depth search to values close
to the depth of neighbouring points.
We can put together all the pieces described previously to form an algorithm for
determining the error for a particular hypothesized water depth shown below:
Our global algorithm processes frame sequences and uses the DepthVerification algorithm to determine the water surfaces. The process cycles through each frame, tracking
the feature points as they are distorted by the water. It passes the tracked features
and a hypothesized depth into the DepthVerification algorithm which returns the error
associated with the hypothesis. This process is repeated and the error is minimized in
order to determine the best depth estimate and thus the location of the surface point. A
surface mesh is then generated from all the surface points in each frame.
Chapter 3. Imaged-based reconstruction of Water
51
t3
b
x
t4
c
t1
a
t2
Figure 3.10: Bilinear interpolation on imaged feature points in the reference camera.
Since our features are not dense and the refractive disparity is only known at these
features, we must interpolate to get disparity values for points lying in between the
feature points. The refractive disparity of a point x is approximated from the known
disparities of four localized feature points t1 , t2 , t3 and t4 . x is projected onto t1 t2 to
get a and onto t3 t4 to get b. Then x is projected onto ab to get c and the disparities at
the end points of t1 t2 and t3 t4 are interpolated to get disparities for a and b. Then the
disparities of a and b are interpolated to get a final disparity for c, which is the bilinear
approximation of x.
Chapter 3. Imaged-based reconstruction of Water
52
Algorithm 2: DepthVerification
Input: Hypothesized depth z ′ , feature f1 , feature image f1′ , set of all pattern
features F, set of features F2 imaged from the reference camera c2 , camera
centres of projection c1 and c2
Output: Error E
1. Compute p′ from hypothesized depth z ′ ;
2. Compute u and v using Equations (3.6) and (3.7);
3. Given p′ , u and v compute n1 from Equations (3.10), (3.11) and (3.12);
4. Find the four non-collinear features in F2 that project closest to the hypothesized
surface point from the view of the verification camera;
5. Bilinearly interpolate the refractive disparity of four features to get the approximate
refractive disparity of the verification ray. Then compute the verification normal
n2 ;
6. Swap n1 and n2 to compute the error distances e1 and e2 ;
7. Return E = e1 + e2 ;
Chapter 3. Imaged-based reconstruction of Water
Algorithm 3: SequenceReconstruction
Data : Binocular frame sequence of pattern through water. Calibrated camera
system. Initial feature locations. Start and end frames.
Result : Water mesh sequence
i ← startF rame ;
while i < endF rame do
foreach feature point f do
Minimize DepthVerification using Golden section algorithm to give bestDepth;
Determine surface point from bestDepth;
Generate mesh from set of surface points;
Perform Lucas-Kanade localization on each feature in the next image i + 1
using the previous feature location as a seed point;
53
Chapter 4
Results
“If you wish to drown, do not torture yourself with shallow water.”
-Bulgarian Proverb
In this chapter we describe the apparatus and physical setup for our system. We then
analyze the performance of our reconstruction system. We begin by selecting several
parameters that govern the error in our reconstruction. In order to measure this error
we present a set of metrics that allow us to examine the effect of our parameters. We
then explain how we designed a simulation of our algorithm to test the error parameters.
Subsequently, we present results from the simulation and compare them to results from
real world data. Finally we present results of reconstructed water sequences.
4.1
Apparatus and Physical Setup
We are also forced to constrain our system due to physical limitations of our apparatus.
Since our imaging system is not a perfect pinhole camera and nor does it produce an
orthographic projection we were careful to calibrate our system to take into account a
reasonable approximation of these imperfections. In this section we describe our physical
apparatus and setup. We describe the assumptions we make and the constraints we
54
Chapter 4. Results
55
employ.
4.1.1
Apparatus and imaging system
We decided to use a glass tank to constrain the water we were reconstructing. The tank
was raised on a frame (Figure 4.1) to allow an image to be projected onto the tank
bottom. We placed a back-lit chequered screen on the tank bottom to allow the image
to be viewed from above. The screen was in direct contact with the water to avoid any
other refraction. During our experiments, the only lighting of the scene came from the
lighting below the surface of the water.
We viewed the water from above with two cameras aiming from opposite ends of the
tank. A trade off exists between baseline length and the size of the reconstructable area.
A longer baseline produces greater disparity between refracted features, but the result is
a smaller overlap between the refracted images and thus the reconstructable area, as can
be seen in Figure 4.2. The overlap is necessary for our stereo triangulation as described
in Section §3.2.2.
We used twin Sony DXC-9000 3CCD cameras in progressive scan mode to feed synchronized image data into two Matrox Meteor II video capture boards. The images were
captured with a resolution of 640x480 pixels at 60 frames per second.
4.1.2
Camera calibration
In order to enhance the accuracy of our technique we wanted an accurate model for our
cameras and physical setup. To this end we performed intrinsic and extrinsic calibration.
We performed intrinsic camera calibration according to the technique described in
[39]. This allowed us to estimate the focal length, centre of projection and lens distortion.
We then extrinsically calibrated our stereo camera pair by imaging a common pattern
on the bottom of the tank. This gave us the transformation for both cameras to a new
coordinate system originating at the calibration pattern on the tank bottom. We per-
56
Chapter 4. Results
Camera 1
Camera 2
Back
Lighting
Figure 4.1: Physical setup and apparatus
57
Chapter 4. Results
Area with features
Reconstructable region
Figure 4.2: Trade-off between baseline length and reconstructable region size
Chapter 4. Results
58
formed the rest of the implementation using this coordinate system. Figure 4.1 shows the
calibration pattern on the tank bottom, ready for extrinsic calibration. The calibration
pattern was subsequently used in the reconstruction phase.
We calibrated the cameras using a short exposure time (1/500 s) so that motion blur
would not affect the reconstruction process. Another important step was to make sure
the camera was focused precisely on the pattern and calibrated well around that depth
range. We required bright lighting to compensate for the quick shutter speed and to allow
for as small an aperture size as possible. The small aperture was necessary to reduce
depth of field blurring.
4.2
Water surface reconstruction simulation
We performed several experiments using a simulation of our system in order to analyse
the expected performance and behaviour of the system on real data. First, we describe
how the simulation was created and how it approximates real world results. We then
analyze the performance of our two error metrics, selecting the disparity difference metric
as more effective. The remainder of our results are all computed using this metric. Then
we present and discuss results that compare the main error contributing factors in the
system. Finally we compare our simulated results to real world data.
4.2.1
Simulation implementation
Since our system is image-based and all our measurements are computed from the images,
reconstruction errors occur from the calibration of the cameras and the ability to localize
features within the images. We selected two parameters to quantify the error in the
system. These error parameters cover the two primary aspects noted and can readily
be estimated in our experiments on real data. We also selected a third parameter that
affects the system performance, the height of the water.
Chapter 4. Results
59
The first parameter is the calibration error. This is the error caused by misalignment
of the homographies of feature points in the images of both cameras projected onto an
extrinsic plane. The calibration error is caused by imperfect intrinsic calibration as well
as errors in the calibration of the extrinsic plane for both cameras. Figure 4.3 shows how
the calibration error affects the computation of the verification normal n2 .
The calibration error parameter is incorporated into our simulation by perturbing
the feature homography of the verification camera by some amount (∆fi ), normally
distributed around a mean. This mean is our input parameter and we label it as the
calibration error (ρ), measured in millimetres.
Secondly, our system cannot perfectly localize the features in the images due to camera
noise and limited resolution. Figure 4.4 shows how the localization error affects point
reconstruction. Since the reconstruction relies on the vectors formed from the imaged
feature points, error in those points translates in to reconstruction error for the surface
point. It is important to note that a drastic error in the localization may results in a
physically impossible reconstruction scenario, where the surface normal or depth cannot
achieve the refractive displacement. Our system disregards such points.
We incorporated this error into our localization error (ψ) parameter. The localization
error parameter is the mean of a Gaussian perturbation on the image plane applied to
all imaged feature points (∆q), measured in pixels.
Our third parameter, the height of the water h, affects reconstruction as it affects the
distance of the surface from the cameras, as well as the refractive disparity. In order to
simplify the simulation, the interpolation step and error associated with it is ignored.
The simulation works in a similar way to the global sequence reconstruction algorithm
as described in Section §3.3.3. Instead of tracking features through a sequence of images,
we generate feature points and compute the refractive displacement given the simulation
input parameters. The simulation works under the assumption of a flat water surface.
The simulation algorithm is designed to compute a set of behaviour and performance
60
Chapter 4. Results
c1
c2
I1
I2
q1
q2
n2 n
p
h
f2
f2+Δf2
f1
Δf2
Figure 4.3: Calibration error. When c2 is used to verify a surface point p, the point
is projected into the image plane of c2 . The feature imaged at the projection point q2
is used to compute the verification normal as described in §3.2.1. Due to calibration
error, the feature imaged at q2 may in fact by offset from the feature f2 imaged from
the reference camera c1 by some amount ∆f2 . This causes the verification normal n2 to
become slightly skewed.
61
Chapter 4. Results
c1
I1
n'
q′
∆q′
n
q
∆q
p'
p
h
f
f+Δ
Figure 4.4: Localization error. Feature f is imaged on the image plane I1 without water
at q and with water at q′ but due to noise and finite resolution, cannot be precisesly
localized. Thus there is some perturbation ∆qi in our image point. This perturbation in
turn causes a shift in the reconstruction point from p to p′ . The surface normal is also
affected.
62
Chapter 4. Results
gauges given varying heights and system errors.
c1
c2
I1
I2
q1
q2
n
ω
n1
n2
γ
p'
λ
p
f2
e2
f2'
f1'
e1
f1
Figure 4.5: Reconstruction gauges. The Figure shows a reconstructed point p′ a distance
λ away from the true location p and a distance ω from c1 . The reference camera c1
produces a normal n1 and the verification camera generates n2 . The error metric Edisp =
e1 + e2 computed as described in §3.2.2. The angle between the true normal n and the
reconstructed normal n1 is γ.
The simulation measures the following behaviour gauges. The measured quantities
are displayed in Figure 4.5.
• The average error metric (E) returned by Algorithm DepthVerificationSimulation
4.2.1. The error metric is computed as E = e1 + e2 , where e1 and e2 are determined
as described in §3.2.2.
• The standard deviation and the mean distance (λ) between reconstructed and ac-
Chapter 4. Results
63
tual surface points. The distance is computed as λ = kp′ − pk.
• The reconstruction system accuracy, defined as the average distance between the
reconstructed point and the actual surface point divided by the distance to the
camera (λ/ω).
• The mean and the standard deviation of the normal error, defined as the size of the
angle (γ) between the reconstructed normal and the actual normal. It is computed
as γ = cos−1 (n1 · n).
We implemented a slightly simpler version of the DepthVerification algorithm described in Section §3.3.3. This algorithm computes the error associated with a given
hypothesized depth but uses the input features, rather than searching for the closest
verification features and interpolating.
Algorithm 4: DepthVerificationSimulation
Input: Hypothesized depth z ′ , shifted feature f1 + ∆f1 , image of shifted feature
q1 , shifted feature f2 + ∆f2 , Camera centres of projection c1 and c2
Output: Error E
1. Compute surface point p′ = c1 + z ′ kq1 - c1 k;
2. Compute incident ray u1 = p′ − c1 ;
3. Compute refracted ray v1 = f1 + ∆f1 − p′ ;
4. Given p′ , u1 and v1 compute n1 from Equations (3.10), (3.11) and (3.12);
5. Compute incident ray u2 = p′ − c2 ;
6. Compute refracted ray v2 = f2 + ∆f2 − p′ ;
7. Given p′ , u2 and v2 compute n2 from Equations (3.10), (3.11) and (3.12);
8. Return error metric value E;
64
Chapter 4. Results
Given the DepthVerificationSimulation algorithm, we implemented a simulation algorithm that would take a given height and compute the behaviour gauges outlined above
for a range of localization errors and calibration errors. We generated the appropriate
refractive distortion given the input height and camera locations. Then we perturbed the
features for the localization error and we shifted the feature homographies to approximate
the calibration error. The algorithm is outlined in detail in Appendix A.
We implemented a second version of the Simulation algorithm that compared the
localization error to varying heights, while maintaining a constant calibration error. The
purpose of this was to examine the effect of water height upon the results.
4.2.2
Error metric analysis
In Section §3.2.2 we discussed two methods for matching features between the reference
and verification cameras. We presented the normal collinearity error metric which measured the angle between the normals computed by the reference and verification cameras
(4.1). The second metric, the disparity difference, measured the difference in disparity
between features viewed through the surface point from both cameras and the corresponding projected features computed when the normals are swapped (4.2),
Enormal = cos−1 (n1 · n2 ),
(4.1)
Edisp = kf1 − f1′ k + kf2 − f2′ k,
Edisp = e1 + e2 .
(4.2)
We ran a set of simulation experiments using both metrics to determine the behaviour
of each as seen in Figure 4.6. Both metrics showed similar behaviour above water heights
of 1mm. It is in the relatively shallower water that differences can be seen. The key
difference is in the distance error gauge (λ) where the error and error deviation for the
normal collinearity rises sharply as the depth drops below 1mm. The disparity difference
Chapter 4. Results
65
in contrast exhibits a relatively slight peak at depths below 0.3mm. Both metrics produce
a similar normal error (γ). The remaining experiments all employ the disparity difference
error metric.
4.2.3
Feature localization error and calibration error comparison
Our comparison between the localization error and calibration error suggests that localization affects the reconstruction to a much greater degree than the calibration (Figures
4.7, 4.8 and 4.9). Although the calibration and localization errors are measured in different units, in our set up 1 pixel distance projected to approximately 1mm in the tank
bottom. The results are all computed for a constant height of 5mm.
The calibration error causes the misalignment of the projected features from the
cameras. This means that the verification test does not occur at precisely the correct
location. Since we are dealing with flat water, the surface normal is constant over the
water and the only difference is the angle of the incident ray. Our cameras are not oblique
to the water surface and there is only a small change in the incident angle so only a small
change in the refractive displacement occurs. The refractive displacement is what is used
to determine the surface normal for verification, explaining why the calibration error has
little effect on the reconstruction depth.
We can see that the error metric results closely match the depth error gauges, suggesting that it is a valid error metric.
4.2.4
Analysis of localization error at varying depths
We used the second version of the simulation algorithm to generate graphs comparing the
effect of the localization error at varying depths (Figures 4.10, 4.11 and 4.12). We fixed
the calibration error to be 0.55mm, comparable to the calibration error determined from
66
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Chapter 4. Results
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1
Mean Normal E rror (γ) (radians )
1
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Mean E rror Metric (Edisp)
Mean Normal E rror (γ) (radians )
1. 4
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2. 5
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0
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1
1. 5
2
Water Height (h) (mm)
Figure 4.6: Error metric analysis. The normal collinearity metric is shown above the
disparity difference metric.
67
Mean dis tance from actual location (λ) (mm)
Chapter 4. Results
1
0. 8
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6
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4
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2
S tandard Deviation of the Reconstruction Accuracy (λ/ω)
C alibration E rror (ρ) (mm)
x 10
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0
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0
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-4
6
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4
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2
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0. 2
0
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0
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Figure 4.7: Simulation graphs showing the mean distance between reconstructed points
and the actual points (top) and the standard deviation the depth reconstruction accuracy
(bottom) for varying calibration and localization errors
68
Chapter 4. Results
Mean E rror Metric (e)
0. 5
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4
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Localiz ation E rror (ψ) (pixels )
Figure 4.8: Simulation graphs showing the mean error metric (top) and the standard
deviation of the reconstructed depths (bottom) for varying calibration and localization
errors
69
S tandard Deviation of the Normal E rror (γ) (radians )
Chapter 4. Results
0. 2
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Figure 4.9: Simulation graphs showing the mean and standard deviation of the normal
error for varying calibration and localization errors
Chapter 4. Results
70
our real world apparatus. Depth slightly affects the reconstruction error but to a much
lesser extent than the localization error (Figures 4.10 and 4.11). These results suggest
that our algorithm robustly reconstructs a range of depths. Figure 4.12 demonstrates
the degeneration of the surface normal as the water depth approaches zero.
4.2.5
Simulation data compared to real world data
Next, we performed a set of experiments, reconstructing flat water surfaces at varying
water heights. We attempted to gauge the results in a similar manner to our simulation
gauges. The error metric gauge is directly comparable, but the true location of the
surface is unknown so the other gauges must be approximated. Since we were dealing
with flat water, we approximated the true surface by a best fit plane through all our data
points. This was achieved with Single Value Decomposition on the point set to determine
a planar basis.
We then measured the distance of each point from the plane for our distance gauge
λ and we compared the point normals to the plane normal for the normal error γ. Plots
of the results are shown in Figure 4.13.
In order to compare our empirical results with our simulation results, we needed to
determine appropriate values for the calibration and localization simulation parameters.
The calibration error in our empirical system can be estimated by projecting the
detected features from both cameras onto the tank bottom plane and measuring the mean
correspondence error between the two homographies of feature points. We obtained a
mean error of 0.55 mm and used this as our calibration error parameter in our simulation.
The localization error is not as readily available for measurement as the true location
of the features cannot be accurately known. We ran tests on our system, where we
localized corners for a sequence of twenty frames of our pattern without disturbance.
We determined an average position for the corner from these samples and then found
the mean perturbation of the samples around the average position. This test gauges the
71
Mean dis tance from actual location (λ) (mm)
Chapter 4. Results
1. 5
1
0. 5
0
100
0. 5
0. 4
50
0. 3
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S tandard Deviation of the Reconstruction Accuracy (λ/ω)
Water Height (h) (mm)
x 10
0
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0
Localiz ation E rror (ψ) (pixels )
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6
4
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100
0. 5
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50
0. 3
0. 2
Water Height (h) (mm)
0
0. 1
0
Localiz ation E rror (ψ) (pixels )
Figure 4.10: Simulation graphs showing the mean distance between reconstructed points
and the actual points (top) and the standard deviation the depth reconstruction accuracy
(bottom) for varying depths and localization errors
72
Chapter 4. Results
Mean E rror Metric (e)
0. 5
0. 4
0. 3
0. 2
0. 1
0
100
0. 5
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50
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Water Height (h) (mm)
0
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1
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100
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50
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0. 2
Water Height (h) (mm)
0
0. 1
0
Localiz ation E rror (ψ) (pixels )
Figure 4.11: Simulation graphs showing the mean error metric (top) and the standard
deviation of the reconstructed depths (bottom) for varying depths and localization errors
73
S tandard Deviation of the Normal E rror (γ) (radians )
Chapter 4. Results
0. 4
0. 3
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80
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0
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100
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80
0
100
Water Height (h) (mm)
Figure 4.12: Simulation graphs showing the mean and standard deviation of the normal
error for varying depths and localization errors
Chapter 4. Results
74
precision of our system, but is not able to determine the accuracy. Our experiments
revealed a precision of ˜0.1 pixels.
We ran our simulation using the computed calibration error parameter for several
values of localization error. We varied the localization error from 0.6 pixels to 1.2 pixels
in 0.2 pixel increments. Figure 4.13 shows the comparison between simulation results and
our results from observation. Our empirical results closely match the simulation results
in every category. However the localization error appears to be roughly 0.6-1.2 pixels
larger than the precision of 0.1 pixels. The same characteristic increase in normal error
is found as the depth decreases. The distance gauges show a similar robustness to water
height and the error range to the corresponding simulation results.
4.3
Water surface sequences
We reconstructed several sequences of captured flowing water. For each of these sequences
the input to our algorithm was a stereo view of a chequered pattern over which water
was poured.
The first two sequences were captured during the actual pouring of the water onto
the pattern area. In both cases the water depths were low, beginning at approximately
1-2mm deep and rising as more water was added. We label these sequences: POUR-A
and POUR-B.
Figures 4.14 to 4.17 show four frames from sequence POUR-A, along with the corresponding input images of the pattern from both cameras. This sequence used a pattern
checker size of approximately 4mm.
Figures 4.18 to 4.21 show four frames from the second sequence POUR-B, along with
the corresponding input images of the pattern as before. This sequence was rendered
with ray traced refraction and reflection, with a textured plane beneath the water so
that the results can be compared more closely to input. This sequence used a pattern
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Chapter 4. Results
ψ=1.0
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ψ=1.2
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Figure 4.13: Simulation results are shown above the corresponding empirical results for
four values of the localization error (ψ)
76
Chapter 4. Results
checker size of approximately 3mm. Notice that this sequence has some bubbles on the
water surface (Figure 4.19). The bubbles cause indentations in the water surface and the
reconstruction correctly models this.
Often the subtleties of the reconstruction cannot be seen without viewing animations
of the resulting sequences. In some of our reconstructions, low amplitude waves are
seen to propagate through the reconstructed surfaces that cannot be detected in single
images2 .
Our next reconstructed sequence is labelled as RIPPLE. It consists of the reconstruction of the surface after a few drops are dripped into water several centimetres deep.
We were unable to reconstruct the initial splash as the pattern was too distorted for the
corners to be matched correctly (shown in Figure 4.22). Had the water depth been lower
the initial splash would have been easier to reconstruct since less elimination would have
occurred. The reconstruction checker size was 3mm for this sequence.
We present one frame of the RIPPLE sequence in Figure 4.23. This figure shows
the set of reconstructed points as well as a rendered mesh of the frame. Notice the
sparse areas on the left and right edges of the reconstructed point set. These areas are
the results of overlapping as described in Section §4.1.1. Although these areas cannot be
reconstructed as accurately, their locations can be estimated using the nearest verification
features as shown.
2
We refer the reader to the resulting
http://www.dgp.toronto.ed/˜nmorris/thesis/
animations
that
are
available
here:
Chapter 4. Results
77
Figure 4.14: Frame of sequence POUR-A. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
78
Figure 4.15: Frame of sequence POUR-A. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
79
Figure 4.16: Frame of sequence POUR-A. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
80
Figure 4.17: Frame of sequence POUR-A. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
81
Figure 4.18: Frame of sequence POUR-B. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
82
Figure 4.19: Frame of sequence POUR-B. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
83
Figure 4.20: Frame of sequence POUR-B. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
84
Figure 4.21: Frame of sequence POUR-B. The top two rows are the stereo views of the
water. The bottom row is the reconstructed surface.
Chapter 4. Results
85
Figure 4.22: Image of the pattern distorted by a splash in the water. This pattern has
too much elimination for our reconstruction algorithm to localize enough of the corners
for a reasonable reconstruction.
Chapter 4. Results
86
a)
b)
Figure 4.23: a) Shows the reconstructed set of points from one frame of the RIPPLE
sequence. b) Shows the rendered mesh of the above point set.
Chapter 5
Conclusion
“The cure for anything is salt water - sweat, tears, or the sea.”
-Isak Dinesen
We have presented a new system for reconstructing the surface of water, utilizing
stereo images of a pattern refracted through the water. Our system builds upon work
that utilizes refractive distortion as well as stereo reconstruction research. We have
provided a theoretical outline of the algorithm that combines these two methods. An
implementation of our system was also presented. The implementation only requires a
simple stereo camera setup with no additional equipment.
We generated input data from a simulation and showed that the simulation results
were consistent with our empirical data. We also proposed two matching metrics for
determining points of the water surface. We showed that our disparity difference metric
outperformed the normal collinearity metric when the water depth approached the size
of the localization error.
We discovered that the localization error of pattern feature points contributes the
most to the error in water surface point determination, especially when the water is still.
The calibration error is expected to affect reconstruction accuracy to a greater extent
when the water is disturbed.
87
Chapter 5. Conclusion
88
Our system is built to allow the reconstruction of sequences of flowing water and our
results show that it is especially effective at reconstructing shallow flows. At greater
water depths the trade-off between the pattern density and the surface roughness that
can be captured is more noticeable.
While our system is described specifically for water, the technique described here can
readily be applied to other liquids by specifying different refractive indices.
There are several avenues available for improving and extending our system. We
outline them in the next section.
5.1
Future Work
Currently our system is based upon finding individual points on the water surface. In
order to improve the overall smoothness we propose that a global method could be applied
so that the surface is determined by global minimization of the whole set of points. It
may also be feasible to attach a temporal smoothness term to our surface generation, to
eliminate outliers that suddenly appear in a sequence.
Our system currently cannot handle splashing water. While it would be beneficial
to enhance the robustness of our surface determination to splashes, it would also be
interesting to capture such effects. We propose that a volume carving approach could be
applied to the splashing water in order to incorporate it with the generated surface.
Another enhancement to our system would be to remove the constraint of a planar
surface underneath the water. We believe that it would be possible to reconstruct the
ground surface below the water as well as the water surface given sufficient views of the
surfaces.
We foresee that this work may be used as a key piece in several larger bodies of work.
First, the determination of internal fluid flow from images would certainly require precise
knowledge of the surface topology, presenting a vital application for our work. Another
Chapter 5. Conclusion
89
use for this thesis may be in the collection of a library of liquid flows that may be used
as a tool to compose arbitrary flows.
Appendix A
Simulation algorithm
Here we present the details for our simulation algorithm. This algorithm takes in parameters for the calibration error range and localization error range and generates the
appropriate inputs for the depth verification algorithm. It returns a result set for the
input parameter ranges.
90
Appendix A. Simulation algorithm
Algorithm 5: Simulation with constant height
Input: Reconstruction height (z), Calibration error range (calibErrMin,
calibErrStep, calibErrMax), Localization error range (localErrMin,
localErrStep, localErrMax), Virtual camera centres of projection c1 and
c2 , Tank bottom plane T , numIterations
Result : Behaviour gauges
for ρ ← calibErrMin; ρ < calibErrMax; ρ+ = calibErrStep do
for ψ ← localErrMin; ψ < localErrMax; ψ+ = localErrStep do
Pick image coordinates q1 of feature f1 from Camera c1 ;
Determine actual surface location p = c1 + zkq1 - c1 k;
i←0
while i < numIterations do
Shift q1 by a random amount around a mean of ψ to get q1 + ∆q1 ;
Determine the adjusted surface point p + ∆p;
Intersect p + ∆p − c2 with T to find the virtual feature f2 ;
Shift f2 by a random amount around a mean of ρ to get f2 + ∆f2 ;
Project f2 + ∆f2 to c2 to get image coordinates q2 ;
Shift q2 by a random amount around a mean of ψ to get q2 + ∆q2 ;
Compute the shifted images of the features without water;
Minimize DepthVerificationSimulation to find the expected best depth
and error metric result;
i = i + 1;
Average expected best depths and error metric results;
Return data structure of averaged expected best depths, error metrics, expected
normals, actual depths and actual normals;
91
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