BRDF Acquisition with Basis Illumination

BRDF Acquisition with Basis Illumination
BRDF Acquisition with Basis Illumination
Shruthi Achutha
B.E, Visweshariah Technological University, 2003
Master of Science
The Faculty of Graduate Studies
(Computer Science)
The University Of British Columbia
July, 2006
© Shruthi Achutha 2006
To create a realistic image we need to characterize how light reflects off a surface.
In optics terminology, the complete Bidirectional Reflectance Distribution Function
is needed. At a given point on a surface the BRDF is a function of two directions,
one toward the light and one toward the viewer. Any device for measuring the 4 D
reflectance data has to obtain measurements over the hemisphere of incident and exitant
directions which can be quite tedious if done using the brute-force approach.
In this thesis, we describe an efficient image-based acquisition setup that has no
moving parts and can acquire reflectance data in a matter of a few minutes. We make
this possible by using curved reflective surfaces that eliminate the need to move either
the camera or the light source. The acquisition speedup mostly comes from the way
we optically sample the BRDF data into a suitable basis. This also saves us the postprocess compression of data. We then encode the data into a compact form that is
suitable for use in various rendering systems.
List of Tables
List of Figures
The Bidirectional Reflectance Distribution Function
Approaches to modeling BRDFs
Thesis Organization
Related Work
Analytical Reflectance Models
BRDF Acquisition
BRDF Representation and Storage
Physical Setup and Design
Final Design
Design Validation
Basis Functions
Acquisition and Results
Physical Calibration
Reflectance Calibration
Basis Validation
Preprocessing Basis Images
Data Capture
Postprocessing Measured Data
Rendering and Evaluation
Conclusions and Future Work
Appendix: Radiometric Terms
List of Tables
Design Parameters
List of Figures
BRDF Geometry
A typical gonioreflectometer
BRDF capture using a camera
Physical setup of our acquisition bench
Snapshot of a working prototype of our BRDF acquisition setup. . . .
Parabolic mirror and mounting beam
Role of the beam splitter
Depicting various parameters in the design of the optical components.
Depicting the design process
Path of the extreme rays through the projector aperture
The 2D prototype used for design validation
Angular plots of the zonal basis functions
3.10 Depicting the processing pipeline for zonal basis validation
3.11 Profile templates for parabola and dome surface verification
Projector Calibration
Camera Calibraion
18% diffuse gray card used as the reflectance standard
Images of the gray card captured by the camera
Acquisition Pipeline
An overview of the preprocessing process
An overview of the postprocessing pipeline
List of Figures
Buddha model rendered in Eucalyptus Grove environment
I would like to thank my supervisor, Wolfgang Heidrich, and my project partner, Abhijeet Ghosh, for all their ideas, guidance and encouragement. It was a pleasure to work
with them.
I want to thank my family for their loving support: my mother Tara Achutha, my
father A.K.Achutha and my brother Guru Pradeep. I also want to thank my friends
here at UBC, including Shantanu, Mugdha, Tanaya, Hagit, Dhruti and all the others,
for making my time here during the graduate program such an enjoyable experience
and for being with me at various stages of my thesis writing in particular and Masters
program in general.
I would also like to thank everyone in the PSM and Imager Lab for their help at
varied occasions.
Finally, I would like to thank everyone in Department of Computer Science and
at U B C who have helped me directly or indirectly in making my graduate experience
Chapter 1
The appearance of its material goes a long way in determining what we perceive an
object to be. This affects how we might interact with the object, how much value we
attach to it, how we use it and various other factors. In art and architecture, various materials and colors are used in a way as to create a specific look or effect. For the same
reason, synthesizing realistic images of objects and scenes has been one of the major
goals of computer graphics. We could use skilled artistry to create highly detailed and
realistic models. The other option is to capture the appearance of real-world materials and digitize it. The applications of such digitized data would range from image
synthesis, e-commerce, digital-libraries to cultural heritage.
To create such a realistic image we need to characterize how light reflects off a
surface. In optics terminology, the complete Bidirectional Reflectance Distribution
Function described by Nicodemus et al.
[27] is needed. At a given point on a sur-
face the BRDF is a function of two directions, one toward the light and one toward
the viewer. The characteristics of the BRDF will determine what "type" of material
the viewer thinks the displayed object is composed of, so the choice of BRDF model
and its parameters is important. There are a variety of basic strategies for modeling
BRDFs. This thesis describes a novel design that we built to capture the reflectance
of a material. Section 1.1 gives a more precise definition of BRDF. Section 1.2 gives
a brief description of the strategies used to model BRDF of a material and section 1.3
gives an outline of how the rest of the thesis is organized.
Chapter 1. Introduction
Figure 1.1: BRDF Geometry
The Bidirectional Reflectance Distribution
As light flows through a medium and reaches an opaque surface, part of it gets absorbed
by the surface and the rest gets reflected. How much of the incoming light gets reflected
depends on the material properties including its composition and structure. The simplest approach to characterize surface reflectance of a material is to describe how light
arriving at a point from an infinitesimal solid angle gets reflected in all directions. We
could then model the look of any material by integrating over the incident distribution. The bidirectional reflectance distribution function or BRDF takes this approach
to describe the reflectance of a surface.
The geometry of the bidirectional reflection process is depicted in Figure 1.1. Light
arriving at a differential surface dA from a direction (0,-,<)>,•) through a solid angle da, is
reflected in the direction (0 ,((v) centered within a cone of d(O . The BRDF is defined as
the ratio of the directionally reflected radiance to the directionally incident irradiance.
Chapter 1. Introduction
MX,<Di,(Or) =
dEx,i(l, co,)
' th incident spectral irradiance (i.e., the incident flux of a given waves
length per unit area of the surface) and dLx is the reflected spectral radiance (i.e., the
reflected flux of a given wavelength per unit area per unit solid angle). Since the BRDF
definition above includes a division by the solid angle (which has units steradians [sr]),
the units of a BRDF are inverse steradians [1 /sr]. We can drop the wavelength notation for simplicity. In a BRDF capture system, measurement is integrated over a finite
bandwidth of radiation. The BRDF now becomes,
/,(44) = ^
This model of reflectance assumes that the surface is homogenous. BRDF is a function
of four variables: two variables specify the incoming light direction, two other variables specify the outgoing light direction. Note that the BRDF is reciprocal, i.e., if the
incoming and outgoing directions are reversed, the function still has the same value.
Also, when there is no scattering along the ray, the radiance does not vary along the
direction of propagation, allowing us to measure the reflected radiance at any distance
from the reflecting surface.
Approaches to modeling BRDFs
In this section we will provide a brief description of the strategies used to model BRDF
of a material. They will be covered in more detail in the next chapter.
Early attempts to characterize realistic reflection involved deriving a mathematical
representation for light transport. These include physically inspired analytical reflection models or empirical models that provide closed form solutions of the reflectance
function. These models provide only an approximation to the reflectance of real-world
materials. Most of them describe only a particular subclass of materials. Over the
years, such models have evolved to include many of the complex physical phenomena,
but obtaining intuitive material parameters for these models remain a difficult task.
Chapter 1. Introduction
Another approach that is similar in spirit is the measure-and-fit approach. This
involves measuring the BRDF for different combinations of incident and exitant directions and then fitting this measured data to an analytical model using some form of
optimization. There are several problems with this approach including salient features
of measured data being eliminated due to modeling errors in the analytical model and
the choice of the optimization parameter being non-intuitive.
The other alternative is to actually measure the reflectance data from real world
materials. The earliest attempts in this direction was the construction of a device called
Here a sample is placed at the center of the device, a light source
and a photometer are moved about the hemisphere above the sample and measurements
of the reflectance are taken every few degrees. Such devices measure a single radiance
value at a time rendering the acquisition process very time-consuming.
Advances in digital photography and high dynamic range imaging techniques have
led to efforts involving image-based BRDF acquisition techniques. There has been
considerable development in this area. But most of them require highly controlled
lighting environment with multiple light sources over the hemisphere of incoming directions and mechanical parts for moving the camera and/or the test sample. Such data
acquisitions, though a cheaper and faster alternative to gonioreflectometers, can easily
take upto several hours to obtain a dense sample set. Also, the huge amount of data
obtained needs to be processed in order to remove noise and to compact them in a way
so that they can be used in real-time rendering applications.
In this thesis, we describe an efficient image-based acquisition setup that has no
moving parts and can acquire reflectance data in a matter of a few minutes. We make
this possible by using curved reflective surfaces that eliminate the need to move either
the camera or the light source. The acquisition speedup mostly comes from the way
we optically sample the BRDF data into a suitable basis. This also saves us the postprocess compression of data. We then encode the data into a compact form that is
suitable for use in various rendering systems.
Chapter 1. Introduction
Thesis Organization
This thesis describes the design, principles and fabrication of a working prototype of
our optical design for efficient image based BRDF acquisition. The next chapter, Chapter 2 provides information regarding the related work in the field of BRDF acquisition
and representation. Chapter 3 provides a discussion of the mechanical design and physical setup of the optical bench, the geometrical optics involved in measurement and a
brief description of the basis functions that was developed and used for efficient acquisition. Chapter 4 contains details of calibration, acquisition and some preliminary
measurement results of various materials. Chapter 5 presents the conclusions and future work. A brief description of various radiometric terms is provided in the appendix.
The concept and design of the acquisition setup is a joint work in collaboration
with Abhijeet Ghosh and Wolfgang Heidrich. I worked on creating the geometry of
the various optical components, validating the design and basis functions. The basis
functions were developed by Abhijeet. The various calibrations were done again in
collaboration with Abhijeet. I worked on the various stages involved in the acquisition
while Abhijeet worked on rendering the acquired data.
Chapter 2
Related Work
Before we present our proposed algorithm for the acquisition we give an overview of
the relevant previous work in this chapter. We begin with a review of various analytical
BRDF models that give a mathematical representation for light reflection. Then we
review various BRDF acquisition techniques used in computer graphics. Finally we
discuss about the various basis representations used for BRDFs.
Analytical Reflectance Models
Early attempts at approximating surface reflectance provided a compact mathematical
representation based on a small number of parameters. The parameters are either obtained by fitting to measured data or by manually tweaking around to approximate the
desired material. These analytical models mostly fall under two categories. (1) Empirical models that are not based on the underlying physics, but provide a class of functions
that can be used to approximate reflectance. (2) Physics based models that take into
account the physical properties of the material while modeling a specific phenomenon
or class of material.
Empirical Models
One of the earliest models which is still widely used today was proposed by Phong [28].
The model is a sum of diffuse component and a cosine weighted specular lobe. It can
be expressed as
f (l,v)
(v r)i
= p + p ±-j-,
Chapter 2. Related Work
where / is the normalized vector towards light; f is the reflected light direction; q is the
specular reflection exponent; p</ and p are the diffuse and specular coefficients. This
model is based on adhoc observation of reflectance and is neither energy conserving
nor reciprocal.
Blinn [3] adopted the model for a more physically accurate reflection by computing
the specular component based on the halfway vector h:
/ r
(/,v) = ^
p , ^ f
Lafortune [18] presented a more elaborate model based on the Phong model that
accounted for off-specular peaks, retro-reflection and anisotropy:
f (l,v) = ^+L [C , (l ,v )+C j(l , )
x i
+C -&,vz)]*
y Vy
Ward [35] proposed a model for anisortopic reflection based on an elliptical gaussian distribution o f normals. It is both energy conserving and reciprocal. It can be
expressed as:
e x p
t a n 2 s
^ +^)l
where 5 is the angle between the half vector and the normal; (>
| is the azimuth angle of
the half vector projected into the surface plane; a ,a
are the standard deviations of
the surface slope in x,y directions, respectively. This does not model Fresnel effects or
Physically-based Models
Initially these models have been developed by applied physicists. The Torrance-Sparrow
model [34] derives the specular component by assuming the reflecting surface to be
composed of microfacets based on a gaussian distribution. It includes a Fresnel term
for off-specularity and also accounts for shadowing and masking with respect to the
microfacet distribution. Ashikhmin [2] proposed an expression for the shadowing and
masking for any arbitrary distribution of microfacet normals. Poulin and Fournier [29]
Chapter 2. Related Work
Source Driver Hoop
Transmittance Detector
Figure 2.1: A typical gonioreflectometer with movable light source and photometer
presented an anisotropic reflection model assuming a microgeometry of oriented cylindrical grooves. Several extensions have been made to these models, for example He et
[14, 15] developed a model to account for arbitrary polarization of incident light
to describe effects like interference. Numerous other models of reflection have been
developed in computer graphics and vision communities.
BRDF Acquisition
Despite the complexity of recent analytical models, there are still some situations where
they do not capture the reflectance of some real world materials. Also, the parameters
required by these models are not always easily obtainable. Measurement is the most
straightforward approach to obtain BRDF data for a broad class of materials. This
section presents a survey of such measurement techniques.
A device which measures BRDF is called a gonioreflectometer (gonio refers to the
capability to measure data in multiple directions). A typical arrangement as shown in
Chapter 2. Related Work
Figure 2.2: Setup to simultaneously capture reflectance in all outgoing directions
Figure 2.1 consists of a photometer that moves with respect to a surface sample which
in turn moves with respect to a light source. There are mechanical elements to ensure
the four degrees of freedom required to measure the complete reflectance function. The
main problem with using a gonioreflectometer is its cost which can be attributed to its
inefficiency; it measures a single value at a time and hence a dense set of measurements
can take a large amount of time. Also, it requires a highly controlled environment to
prevent data corruption by noise.
Dana et al [7] developed such an equipment for measuring Bidirectional Texture
Functions (BTFs). Their system comprised of a robot, lamp, PC, photometer and a
video camera. The planar sample and camera are moved to obtain 205 different measurements over the entire hemisphere of directions. They compiled the data for 60
different materials into a publicly available CUReT database [5]. This sparse data set
requires a function fitting to arrive at a useful model.
Chapter 2. Related Work
Image Based B R D F Acquisition
Ward's imaging gonioreflectometer
[35] as shown in Figure 2.2 was a significantly fast
and inexpensive measurement device. He used a semisilvered hemispherical mirror,
a C C D camera with a fisheye lens and a movable collimated light source to capture
the reflectance data. This setup enables every outgoing direction to be measured with
a single image, thereby eliminating the need for 2 degrees of movement out of 4 that
is required for BRDF measurement. This greatly reduces the acquisition time. The
drawback of the design is the difficulty in measuring BRDF values near grazing angles.
Since most specular materials are specular in the grazing directions this setup cannot
be used to measure highly specular BRDFs.
Marschner et al. [23] constructed an efficient BRDF measurement device based on
two cameras, a light source and a spherical test sample of homogenous material. Their
method works by taking images of the sample under illumination from an orbiting light
source and generates densely spaced samples.
When the measured sample set is sparse, they can be fit to an analytical BRDF
model using some optimization strategy. Lensch et al. [20] presented a clustering procedure to model spatially varying BRDFs andfiteach cluster to a Lafortune model [18].
They then used principal
component analysis to compute basis BRDFs for material
Dana [6] built a BRDF/BTF measurement device that used the approach of curved
mirrors to remove the need for hemispherical positioning of the camera and light
source. Simple planar translations are used to vary the illumination direction. The
device consists of optical components such as a beam splitter, concave parabolic mirror, C C D camera and translation stages. It allows multiple viewing directions to be
measured simultaneously. This arrangement is very similar to our design. But as we
will describe later, we can capture a much wider range of directions.
Matusik et al [25] presented a radically different approach to modeling BRDFs.
They proposed a generative data-driven reflectance model for isotropic BRDFs based
on acquired reflectance data. They acquired BRDF data for a large representative set
of materials using a device similar to that built by Marschner et al [23]. They used
Chapter 2. Related Work
linear and non-linear dimensionality reduction techniques to obtain a low-dimensional
manifold that characterizes the BRDFs. They let users define intuitive parameters for
navigating within BRDF models and each of this movement in the low-dimensional
space produces a novel but valid BRDF. The main drawback of this approach is its
Han and Perlin [13] developed a unique BTF measuring device using a kaleidoscope that could simultaneously illuminate and image a sample from various directions
using a single camera and light source. The advantage includes low-cost, no moving
parts, insltu measurement capability and portability. This can be used with data-driven
surface models. The number of simultaneous captures is definitely the main advantage,
but this is still much smaller that what we can capture with our setup.
Recently Ngan et al [26] built a new setup for high resolution acquisition of anisotropic
BRDFs. Their setup is similar to Lu et al [21] , but instead of a spherical sample they
use strips of sample material on a rotating cylinder, obtained from planar samples of the
material at various orientations. The acquisition takes upto 16 hours for each sample.
They sample isotropic and anisotropic data with intervals of 1 and 2 degrees respectively, and about 85% and 25% of the sample set bins have measured data.
BRDF Representation and Storage
The reflectance data directly obtained through measurements is unwieldy to store due to
its high dimensionality and size. Also, there could be inherent noise in the data which
makes it unsuitable for direct rendering. Researchers have tackled this problem in a
number of ways. Some have resorted to representing the data in some basis; whereas
others have tried fitting an analytical function to approximate the data. In this section
we will look at some of the ways in which measured data can be represented using
basis functions.
Chapter 2. Related Work
Spherical Harmonics
Spherical Harmonics have been a pretty popular basis for representing BRDF data [30,
36]. They are the 2D analogue of Fourier series on a sphere. The BRDF function is
approximated by a finite number of terms of the spherical harmonic series defined as
/(e,(t>) = ir= £L-< V / , ( e ^ ) ,
is the spherical harmonic basis function of order and degree m, and Q
some constant. The main problem in using spherical harmonics to represent BRDFs is
that a large number of terms is required to represent asymmetric and high frequency
features. This is due to the fact that BRDF is a function defined on a hemisphere
whereas spherical harmonic basis functions are defined on the whole sphere.
Spherical Wavelets
Wavelets are suitable for representing functions containing high frequency content in
some regions and low frequency in others. Schroeder et al. [31] extended wavelets to
spherical domain to efficiently represent spherical functions. They used these spherical wavelets to represent a 2D slice of BRDF by keeping the viewing direction constant. Lalonde [19] proposed solutions to representing the four-dimensional BRDF
with wavelets. The main problem with wavelets is that when a small number of wavelet
coefficients are used to represent a smooth function, it can lead to aliasing problems.
This affects the quality of rendering adversely.
Zernike Polynomials
The approximation described above are sphere-based, i.e., they represent a hemispherical function as a special case of a spherical function. The other approach is to map
points on a hemisphere onto a disk. Keondrink [17] takes this approach of representing
BRDFs using orthonormal basis functions on the unit disk. They project the Zemike
polynomial functions onto a hemisphere and use tensor-products of these functions
to represent functions defined on a pair of hemispheres. The CUReT BRDF database
Chapter 2. Related Work
represents measured BRDFs using these basis functions. Unfortunately, there are problems associated even with this approach. The truncation of high-frequency coefficients
is likely to cause "ringing". Further, the evaluation of BRDFs at a particular incident
and exitant directions requires computation time that is proportional to the number of
nonzero coefficients and hence the problem of high storage and computation costs.
Moreover, rotation matrices are not available for them.
Hemispherical Basis
Makhotkin's hemispherical harmonics [22], derived from shifted adjoint Jacobi polynomials, can be computed easily using a simple recurrence relation. But they lacked
rotation matrices. Gautron et al [10] proposed a novel hemispherical basis derived from
associated Legendre polynomials. The hemispherical functions have to be converted to
spherical harmonics for arbitrary rotations.
Although Hemispherical functions provide a natural basis for the representation of
BRDF data, they are not suitable for our, purpose. This is because our measurement
space comprises of a hemispherical region with a 9° hole at the top. Ghosh et al [11]
have developed a novel basis for this zone of directions on the hemisphere where we actually capture the BRDF data. This zonal basis is a generalization to the hemispherical
basis. They also provide a method to project zonal basis data onto spherical harmonic
basis. This allows data captured by our system to be easily integrated into existing
rendering algorithms.
Chapter 3
Physical Setup and Design
A BRDF is a 4D function, where for each incoming illumination direction, we have
to measure the outgoing reflectance in every direction over the hemisphere. Figure 2.1
shows the setup that would be required. The light source illuminates aflatsample from
a single direction and the photometer measures the reflectance distribution over the
hemisphere sequentially. This is repeated for each incoming direction over the hemisphere. Sampling this high dimensional space sequentially is impractical and hence the
need to look at approaches to measure multiple points simultaneously.
Now suppose we place a mirrored dome over the sample and photograph it as
shown in Figure 2.2. We can simultaneously capture all outgoing directions in a single photograph using a fisheye lens camera [35]. This speeds up the capture process
However, there still remains the issue of point sampling every incoming direction.
If we think about the problem from a mathematical perspective, the incoming illumination can be thought of as a function which can be expressed as a linear combination
of a set of basis functions defined over the hemisphere. To enable this kind of basis
illumination we introduced a parabolic mirror as shown in Figure 3.1. The basis image
from the projector illuminates the parabola and the geometric optics work in a way that
the sample gets illuminated from a wide range of directions over the hemisphere.
Our acquisition setup is efficient and is less error prone since it has no moving
parts. Everything including the camera, the projector (light source), sample material
and mirrored components remain fixed. By projecting and capturing the response to a
set of basis functions, we achieve the effect of providing the four degrees of freedom
required by any BRDF measurement device. We optically sample the illumination and
Chapter 3. Physical Setup and Design
^ ^ \
* Camera
• Sample Material
Figure 3.1: Physical setup of our acquisition bench.
Chapter 3. Physical Setup and Design
reflectance data as a part of the capture procedure. This in turn has added benefits in
terms of noise reduction, avoiding data redundancy and post processing. Our setup
consists of mirrored components (dome and parabola) that cover approximately 90%
of incident and exitant directions over the hemisphere; a camera which is positioned
in a way as to capture the full zone of reflecting directions measurable by the system;
and a projector that acts as the light source, projecting a sequence of basis images that
cover the range of incident directions.
The set of basis functions that are projected can be used to approximate any illumination environment. When we project a particular basis function onto the sample, what
we capture is the response of the sample to that particular basis function. Once we have
the response of the sample to a suitable number of basis illumination, we have enough
information to compute the basis coefficients, which in turn can be used to compute the
sample's response to any illumination environment. This acquisition process turns out
to be orders of magnitude faster than point sampling and measuring for every incident
direction at resolutions that we achieve (i.e., atleast 1 measurement per degree). Also,
what we capture is a low pass filtered version of the reflectance data which is compact and doesn't require any further application of data compression techniques. The
amount of averaging that happens depends on the maximum order of the basis images
for which we acquire data. This can be varied depending on the specific type of BRDF
we are acquiring.
A novel basis was developed over the zone of directions where we can project
and acquire data. We also carried out extensive simulations to design, optimize and
validate our setup. In the sections that follow, we will describe these simulation tests,
the fabrication process" and the setup and working of our optical bench. We will also
provide a brief overview of the zonal harmonics.
Figure 3.2 shows a snapshot of our BRDF measurement device. Here is a detailed
description of the various components.
Chapter 3. Physical Setup and Design
Figure 3.2: Snapshot of a working prototype of our B R D F acquisition setup.
Figure 3.3: Parabolic mirror and mounting beam shown seperately. In the actual setup,
the parabola is mounted on the beam. The beam can be inserted at the centre of the
dome and supported at the notches on the outer surface of the dome.
Chapter 3. Physical Setup and Design
Figure 3.4: Left: Beam splitter reflects the light from the projector onto the parabola.
Right: Light from the parabola passes through to the camera.
1. The mirrored components, i.e., the dome and the parabola were custom designed
and manufactured at our lab. Their design and fabrication will be discussed in
detail in later sections of this chapter.
2. DLP Projector (BenQ PB6210) that acts as the source of light by projecting basis images onto the parabola. The parabola reflects the image onto the dome
which in turn projects light onto the sample material from a wide range of directions. Resolution of the projector is 1024X768 and it has a 2000 Lumens peak
illumination intensity.
3. Prosilica EC 1350C is a firewire machine vision camera that captures images of
the parabola which actually reflects light coming from the dome which in turn is
illuminated by the reflectance of the sample material. Resolution of the camera
is 1360 x 1024 and it can acquire 12-bits per color channel.
4. A beam splitter is used so as to reflect the light from the projector onto the
parabola and at the same time to allow light reflected from the parabola to pass
through so that it can be captured by the camera behind it as shown in Figure 3.4.
5. The sample material is mounted onto a cylinder that can be inserted into the base
plate attached to the base of the dome. Height of the cylinder can be adjusted
Chapter 3. Physical Setup and Design
Camera/ Projector
Mirrored Surface
Opaque Surtace
Figure 3.5: Depicting various parameters in the design of the optical components.
based on the material thickness so as to ensure that the top surface of the material
is at the right position for optimum focus.
6. A lens is used to focus the projector at the required distance onto the parabolic
7. A narrow suspension beam shown in Figure 3.3 supports the parabola at the right
height. It spans the width of the dome and is supported by a notch in the dome.
The supporting clamps, beams and plates are the remaining components in the
The mirrored components specification and other design parameters were obtained by
carrying out simulations of the geometric optics involved. We simulated the camera and
projector by thin lens optics, considering various optical parameters like focal distance,
aperture size , etc. We wrote a ray tracer taking into account the camera and projector
resolutions and modeled everything in 2D since the optics follow automatically to 3D
due to symmetry around the optical axis.
Chapter 3. Physical Setup and Design
Figure 3.6: Depicting the design process.
The first step in the design process was to parameterize the 2D parabola, its distances from the sample, projector and camera. Due to Helmholtz reciprocity, the camera and projector can be positioned interchangeably without affecting the results of the
simulation. So we simply modeled them both at the same position as shown in Figure
3.5. The beam splitter takes care of this in the actual setup.
Given these parameters we iteratively built the polyline (which eventually described
the dome in 3D) by Euler integration. Figure 3.6 shows the steps in detail. The first
point is given by the intersection of the reflected ray of the ray incident tangential to the
lowest point on the parabolic mirror and the horizontal line through the surface of the
sample material. Let a be the angle between the horizontal line and the reflected ray.
Consider placing a planar mirror perpendicular to the bisector of angle a. The ray CP\
gets reflected along P\D\ by the parabolic mirror and at D\ the planar mirror reflects it
along D\ O. This ray would arrive at the sample surface at grazing angle.
Chapter 3. Physical Setup and Design
Consider the next ray CPi. This when reflected by the parabolic mirror intersects
the planar mirror at point Di- As described above, the angle bisector between this
ray P2D2 and D2O gives the surface normal of the next planar mirror segment starting
at D2. This process continues with the incident rays getting successively closer to
normal incidence, until they get occluded by the parabolic mirror as shown in Figure
3.6. That is when the iteration concludes. What we now have is a set of points that when
connected by mirrored segments generate reflections such that every ray originating
from the projector, incident on the parabola (from various directions) gets reflected
onto the mirror segments and again gets reflected to finally converge to a point on the
sample material. The set of points generated from the simulation happen to resemble
an arc which when rotated about the optical axis, takes the shape of a dome.
The various geometric parameters of this design were manually optimized according to the following design goals:
• Maximize the range of measurable directions.
• Maximize the number of measurements (pixels) possible per direction.
• Convenient spacing of various components.
• Robust to minor miscalibration errors of the optical components.
• Robust to minor fabrication errors of the dome and parabola.
F i n a l Design
We carried out numerous simulations and finally came up with a design that was a
good tradeoff between the various geometric constraints and robust to errors due to
misalignments and miscalibrations. Table 3.1 lists the design parameters.
With these parameters we were able to project about 100 pixels between the vertex
and tangent of the parabola and hence obtain about 1 pixel/degree measurement.
Chapter 3. Physical Setup and Design
9° to 90°
Measurable 9 range
Camera(COP) - parabola distance d
27 cm
Sample - parabola distance d
13.5 cm
Parabola vertical extent d
2.25 cm
Parabola tangent angle %
Dome dimensions
11" x l l " x 10"
Table 3.1: Design Parameters
Design Validation
In this section we will discuss in detail the steps that we carried out to ensure that our
design was optimal and robust before we went ahead with the manufacturing. Our
design validation process involved 2 major steps:
• Software simulations of real camera and projector optics with finite apertures
taking into account minor misalignments of various parts.
• Physical validation of the optics process by manufacturing a 2D prototype of the
Geometric Optics
We extended our raytracer to simulate camera and projector as thin lens devices. We
assumed an aperture of size 0.5 cm and traced the left and right extreme rays for every
ray that we had used in our pin-hole camera design as shown in Figure 3.7. The left
and right rays were incident at a slightly different angles on the parabola and hence
arrived at different angles on the sample material. Our objective was to minimize the
average of these angular offsets, a; and a , over all the rays sampled from the projector.
We had to test for various design parameters (as described in the previous section) and
also the focal lengths to obtain a good balance. We had a manufacturing constraint
with respect to the dome size imposed by our rapid prototyping machine. To test the
robustness of our design we simulated the optics by adding minor misalignments for
various components as follows:
Chapter 3. Physical Setup and Design
Figure 3.7: Tracing the path of the leftmost and the rightmost rays through the projector aperture. Here focal distance is the average of the total path length of the ray
starting from the projector until it reaches the sample surface after reflections at the
parabola and dome.
Chapter 3. Physical Setup and Design
Figure 3.8: The 2 D prototype used for design validation
• Horizontal and vertical offsets for the camera.
• Horizontal and vertical offsets for the projector.
• Horizontal offset for the parabola.
• Horizontal offset for the dome
This caused the rays to either not converge at all or converge at a point away from the
sample surface. We had to look for a design that produced minimal convergence error,
given minor misalignments. It was a difficult optimization process and we proceeded
by prioritizing the design parameters, fixing the desirable range for most of them and
used a systematic trial and error process to find the rest.
2D Prototype Design
In addition to carrying out software simulations to validate our design, we also built
a 2D prototype of a scaled down version of the final design as shown in Figure 3.8.
We built the prototype in our lab using a STRATASYS Vantage i rapid prototyping
machine. We used a reflective foil for the mirrored parts and sample surface. Using a
laser beam we were able to recreate the geometric optics that we had simulated thus
validating our design.
Chapter 3. Physical Setup and Design
Basis Functions
Abhijeet et al [11] developed a set of basis functions similar to the Spherical Harmonic
basis, to sample the reflectance over the measurable range of directions. In this section,
we will describe the orthonormal zonal basis and also discuss a way to convert them
from the zonal space to spherical space. The zonal basis were derived from shifted
Associated Legendre Polynomials (ALPs). These zonal basis (ZB) functions Z'"(Q, <())
, where m e {0,...,/}, are orthogonal over the interval [a, P] x [0, 2JT]. They can be
constructed from shifted ALPs P'" defined over the interval i f
[a, b] where
a = J^sinSc/S = cosP, and
b = f£ smBd&
= cos a .
Given K'", the zonal normalization constant,
KT = \Ll
+ \m\)\
the zonal basis function Z™(0, (j)) can be defined as
V^^'"cos(m(|))^ (cos0)
zT(e,4>) =
V2Kpsin{-m<b)Pf (cosQ)
Our reflectance acquisition setup can measure data in the interval 9 £ [7t/20,7i/2],
i.e. 9 ° , . . . ,90° from the surface normal. Thus, the zonal basis functions are orthogonal
in the interval [TI/20, 7t/2] x [0, 2TC] where a and b that define the shifted ALPs Pj" are
a = cos$ = cosn/2 = 0, and
b = cosa = cosn/20.
Figure 3.9 gives the angular plots of the first few zonal basis functions, as copied
from [11]. For a more detailed discussion please refer to [11].
Chapter 3. Physical Setup and Design
x f
Figure 3.9: The plots of zonal basis functions Z
defined over the measurement space
[7i/20, JX/2] x [0, 2n], for / < 2 [copied from [11]].
Projecting zonal basis to spherical harmonics
We need a way to project zonal basis into spherical harmonics for the following
• SH basis works well with rendering algorithms since it supports 3D rotations.
• There is a cap of directions near the surface normal where we cannot measure
reflectance data and hence need a way to extrapolate the measured data.
Dual Basis
The SH functions are not orthogonal over the zone of measured directions. So, they
first define a set of dual basis functions that are orthogonal to SH over the zone. Let
ZB be defined over the space [a, b]. Given F™, the primal SH bases, they define Yp, the
corresponding dual bases over the zone as follows,
if / = p and m = q
[ Y, ?^dw
Chapter 3. Physical Setup and Design
Basis Image"...
Input Illumination
Zonal Basisllmages
BRDF Model', Model Parameters
-.Materia'rSpecific Parameters
(Output! Reflectance
in Zonal Basis
Projector-TofGameia Ray Tracer:
Zonal-BasisfeSpherical Harmonics •
, Transformation Module
Output Reflectance
in Spherical Basis !
Reflectance Data / * —
Data Extrapolation;
Model to Render
Rendering: Systeml
^Rendered Image . • ' ^
Environment Map (-
Figure 3.10: Depicting the processing pipeline for zonal basis validation.
^ = 1^'.
The zonal coefficients ZJ" are then transformed by a basis change matrix C to obtain
the corresponding SH coefficient fp. The elements of this matrix are given by
[ Z"Y«d<o.
Basis Validation
We took a systematic and thorough approach to test the zonal basis functions. Figure 3.10 gives theflowchartof the basis validation process. What follows is a brief
description of the various modules,
1. The basis image generator outputs a set of images corresponding to the first few
Chapter 3. Physical Setup and Design
zonal harmonics. This depends on the material that we would like to simulate,
higher the specularity or anisotropy, higher is the order of harmonics required.
For a typical matte material 4
order harmonics would suffice whereas for a
highly anisotropic material like velvet or specular material like metal, it may go
upto 6 order.
2. The BRDF simulator basically implements various analytical BRDF models as
• Anisotropic Ward Model [35]
• Ashikmin Shirley Microfacet Model [2]
• Cook Torrance Model [4]
• Ashikmin Shirley Phong BRDF Model [1]
Given the choice of BRDF model, the model parameters and the material specific
parameter values, the simulator generates BRDF value for any pair of incident
and exitant directions.
3. The projector-to-camera ray tracer simulates the ray optics for every projector
pixel in each basis image and uses the BRDF simulator to determine the value for
every pixel as captured by the camera. Hence, for every basis image it generates
the corresponding reflectance map as would have been generated by the camera.
These reflectance values are in zonal basis.
4. The zonal to SH transformation module converts the reflectance values for zonal
basis to spherical basis using the transformation matrices as described by equation 3.7.
5. This encoded data is then extrapolated in the region of the missing zone. The
resulting data is used in a physically based ray tracer to render a given model in
an illumination environment using the material selected in step 2.
Chapter 3. Physical Setup and Design
Figure 3.11: Profile templates for parabola and dome surface verification.
The 3D geometry for the dome, parabola and support structures were mostly developed
using software we wrote for the purpose. We used SolidWorks 3D C A D [32] package
for some minor geometry editing. We built these parts using a STRATASYS Vantage i
rapid prototyping machine. The vantage machine builds objects in ABS plastic material. These parts required about five days of build time in total. It works is as follows:
the program that controls the machine first slices the geometry into a number of layers,
each of a particular thickness. Within each layer it generates toolpaths that describe the
movement of the printhead that deposits the plastic and support material. Within each
layer the build precision of the print head is very high, thus enabling high precision
geometric details in the radial direction. But across layers this precision drops to about
7/1000'' of an inch. This causes tiny grooves of about 0.2mm thickness on a smooth
surface across layers. This was a big disadvantage to us since we required the parts to
be built with optical precision for our acquisition setup. This meant that the parts had
to be carefully polished to obtain a smooth surface for mirroring.
We opted for a local commercial service to do the polishing and mirroring. Here is
a brief description of the process involved.
1. The grooves on the curved surfaces were leveled by wet sanding with successively fine grid sand papers.
Chapter 3. Physical Setup and Design
2. It was then painted with black base paint.
3. This was followed by a coat of polyurethene based automotive primer.
4. A coat of polyurethene based clearcoat was then applied. This is a hardening
5. An automotive grade polishing compound was then used to polish the surface.
This was the final stage of polishing during which the polished surface was constantly measured using a profile template that we built for testing surface accuracy. These templates are shown in Figure 3.11.
6. MirraChrome [33] paint was used for mirroring the surface. MirraChrome plating provides 95% reflectivity of that provided by true chrome plating and is much
thicker than other mirroring mechanisms such as vacuum deposition. Thus it is
better suited for our purpose as it smoothes out minor surface inaccuracies in
addition to providing a reflective surface.
The thickness added to the surface by various paint coatings was about 5 — 6 mils
and was accounted for when building the parts.
Chapter 4
Acquisition and Results
Once the apparatus was set up, we carried out various calibration procedures. This involved camera and projector calibrations and reflectance calibration each of which will
be discussed in detail in section 4.1. The acquisition process then involved some initial
processing of basis data and of course the actual measurement. The measured data
was then post processed to encode it in a format suitable for rendering. The measured
reflectance data was then tested by rendering some scenes using a physically based ray
The acquisition, post-processing and rendering procedures will be explained in detail in sections 4.2 and 4.3, respectively. A brief discussion of fabrication problems is
given in section 4.4.
As in any data capture setup, we carried out certain calibration steps in order to minimize measurement errors and to standardize the captured data. Our calibration procedures involved not only physical alignment of the optical devices, but also photometric
calibration in order to obtain the relative scaling factors for our captured data with
respect to a known reflectance standard.
Physical Calibration
This calibration procedure involved alignment of the camera and projector to the optical
axis. We began by calibrating the projector to the optical axis and followed this up with
camera calibration since it involved using the projector. We will discuss each of these
Chapter 4. Acquisition and Results
Figure 4.1: Aligning the projector to the optical axis. Left: Pictorial depiction of the
calibration setup with the crosses and the backplate. Right: Image of the crosses.
calibration procedures separately.
Projector C a l i b r a t i o n
The initial alignment of the projector was obtained by fixing 2 calibration crosses in a
position such that when light is projected on them in such a way that the shadows of
the 2 crosses on the backplate coincide, the projector is aligned to the optical axis. This
is evident from Figure 4.1. The crosses were designed in a way that when fixed on the
mounting plate, they are at the same height from the base as is the optical axis. They
were manufactured in ABS plastic using the rapid prototyping machine.
To obtain the correspondence between the projector pixels and points on the parabola
where these pixels get projected, we generated an image with a bright circle in the centre. We then projected this image onto the parabola and covered the base of the dome
with a semitransparent film marked at the central optical point. The projected circle
formed a circular image in the film after multiple reflections. We moved the circle in
Chapter 4. Acquisition and Results
Figure 4.2: Camera calibration. Left: Image of the crosses captured by the camera
when it is aligned to the optical axis. Right: Parabola image captured to recover the
camera pixel-to-parabola points correspondence. Note the checkerboard pattern in the
the image until it was symmetric with respect to the point marked on the film.
then adjusted the radius of the circle until the reflected rays converged to a point on the
film. This happens when the circle is projected accurately to match the circumference
of the parabola base. The position and size of the circle in the image gave us the required correspondence between projector pixels and points on the parabola. We used
this information in generating the projected basis images for data capture.
Camera Calibration
In order to align the camera to the optical axis of the dome, we suspended a cross at
the centre of the dome using a suspension beam in the same way as we suspend the
parabolic mirror. We put a second cross at the centre of the base plate covering the
We adjusted the camera until the 2 crosses overlapped symmetrically in the
captured image as shown in Figure 4.2. We also adjusted the camera focus until the
first cross was in focus. This is the required focal length as determined during optics
Next we determined the correspondence between the camera pixels and parabola
points in a way similar to the projector case. We pasted a checkerboard pattern on
the base plate covering the dome. We then projected the image with a bright circle (as
obtained above after projector calibration) and captured the image of the parabola using
the camera. The location and size of the parabola in the captured image gave us the
Chapter 4. Acquisition and Results
Figure 4.3: 18% diffuse gray card used as the reflectance standard
Figure 4.4: Images of the gray card captured by the camera at exposure times 62.5 ms,
125 ms and 250 ms.
required correspondence. This information is used in post-processing basis response
images captured by the camera as described by the acquisition process.
Reflectance Calibration
Due to inaccuracies in optical process, the measured reflectance values will not be
exact but some scaled version of the actual value. Our objective to carry out reflectance
calibration is to obtain the scaling factors (at each pixel) by measuring the reflectance
of a known standard material. We chose an 18% diffuse gray card for this purpose. It
has a uniform reflectance of
0 18
/r,0.!8(0),) = —
Ramoorthi et al [30] have shown that Lambertian diffuse reflectance can be encoded completely using the first 2 orders of spherical harmonics basis function. So
we projected upto 2
order zonal harmonic basis images and captured the correspond-
ing responses for the diffuse gray card. After projecting these measured coefficients
Chapter 4. Acquisition and Results
Basis Images
• Preprocessing'.
Basis Images
Sample Material
Data Capture
Response Images
Post processing •
Reflectance Data
Figure 4.5: A n overview of the acquisition pipeline.
into spherical harmonics, we recovered the hemispherical reflectance of the gray card
(fr,gray)- From this, we computed the scale factors (A,) for each co,- e Q as follows
f r ^ 2 d
The gray card and some of the captured responses are shown in Figures 4.3 and 4.4.
In this section we will discuss the preprocessing stage, data capture and postprocessing
stage given by the flowchart in Figure 4.5. Prior to the actual measurement, the basis
images are processed so as to convert then into a format suitable for projection. Then
the projection and data capture is carried out. In the postprocessing stage, the captured
data is processed so as to convert them into a format suitable for rendering. Each of
these stages will be discussed in detail.
Chapter 4. Acquisition and Results
••TV Basis Images.; ; *
(.pfm files)
Scale, each;i
max pixel - >
,' Convert 32-bit float"
values to 8-bit integers
V Sepaiate positives
<»and negative values.!-
Projector Images
' ? ( ppm files)
Figure 4.6: A n overview of the preprocessing process.
Preprocessing Basis Images
Basis Images are 2D latitude-longitude maps generated using the mathematical expression given by equation 3.3, where each pixel gives the value of the basis function in one
particular direction in the zone. These are floating point values including both positive
and negative numbers. For the projector to be able to project these images, they have
to be processed as shown in Figure 4.6.
The maximum pixel value amongst all images is used to scale pixels in every image.
Then the 32-bit floating point values are converted into 8-bit integers. The negative and
positive value pixels are stored separately as positive valued images with zeros in the
remaining positions. Later in the post-processing pipeline the responses of the negative
image is subtracted from that of the corresponding positive image. The images are now
in a format that can be projected.
Data Capture
Our data capture procedure involves projecting illumination in the form of zonal basis
function defined over the space [TC/20, K/2] x [0, In] obtained after preprocessing and
imaging the response of a sample to this illumination. The captured reflectance data
gives the coefficients of the zonal basis. The captured response image corresponds to a
Chapter 4. Acquisition and Results
Crop,& HDR'/Or
• ^generation jhd^gen)J*|
Captured Images < , I
'1 '
(jpg file's) ,f *
*."'• ''Subtract negatjve images from'
• •> .corresponding positive images &. '.
' ,'
fr-.i','- .
*"*• Basis^esponse^lmages ,)
•. in SphencalBasis";(.pfm.files)ij
Basis Response-Images .
jihrZortaf Basis' (.pfm files);
.HDR Images
( exr files)
"': Transformation .-"t
Figure 4.7: An overview of the postprocessing pipeline,
sampling of the outgoing directions. The optical process can be summarized as
z/"(e<j>) =
/ /
This approach is similar to that of Kautz et al [ 16]. For every basis input, we acquire
the response image at multiple exposures (mostly 3) in order to generate high dynamic
range (HDR) data [8].
Postprocessing Measured Data
The acquired images have to undergo some processing before we can get them in a
format suitable for rendering as shown in Figure 4.7. The images are first cropped
according to the correspondence parameters obtained during calibration.
H D R Image Generation
The multiple exposures are then combined using HDRgen [9], a software used to
generate high dynamic range images. The camera response curve required by hdrgen is obtained before camera calibration by taking 6-7 images of a scene at different
exposures. Once, we have the HDR images, the images corresponding to the initial
negative basis images are subtracted from the response for the corresponding positive
basis image to obtain the basis response images encoded in zonal basis.
Basis Projection and Extrapolation
Chapter 4. Acquisition and Results
The response images are then projected onto the spherical basis using the basis
transformation matrix C, as described in the previous chapter. Spherical harmonics is
also used to extrapolate data in the missing zone. Thus, what we have at this stage is
SH coefficients fp for every tabulated exitant direction (0 , <|)).
Rendering and Evaluation
During rendering the reflectance data is reconstructed from the tabulated SH coefficients. The reflected radiance in the viewing direction L is computed as follows:
L (QoA>)
/ / (0 ,<t)„,e,-,(t),)cose,L/(co/)y(co/)rfco
/n/r,(6 ,fe)(e/,0/)^(<a/)V(a)/)d(o
We acquired reflectance data for various materials including red velvet, red giftwrapping paper, golden brown chocolate box cover and glossy dark brown resin material. We used the Physically Based Ray Tracer [24] for rendering some models using
our acquired data. Figure 4.8 gives snapshots of the Buddha model rendered in the
Eucalyptus Grove environment. These were acquired and rendered by using upto 4thorder Spherical Harmonics. The images appear noisy only because the environment
map has been point-sampled and then the model has been rendered using ray-tracing.
If we were to project the environment map onto Spherical Harmonics prior to rendering, it would give us better quality results.
Even though we we're able to acquire some preliminary data, the precision of the curved
mirrors was not good enough to carry out extensive acquisitions. The errors accumulated during the multiple polishing and painting procedures caused these artifacts. The
artifacts were mostly in the form of variable bumpiness on the surface. Due to coupling
Chapter 4. Acquisition and Results
Figure 4.8: Buddha model rendered in Ecalyptus Grove environment, using reflectance
data we captured for (a) glossy brown resin material, (b) golden-brown chocolate box
paper and (c) glossy red gift-wrapping paper.
Chapter 4. Acquisition and Results
effect of the two mirrors, noise reduction techniques don't help in recovering the right
results from the acquired data. If we were to obtain the curved components using C N C
Machining or electroplating, then the surface prior to mirroring would be smooth and
not have the grooves that we obtained in the plastic models printed using our Rapid
Prototyping Machine. This would avoid the multiple polishing and painting procedures. In fact, electroplating is one of the methods employed to manufacture industry
standard mirrors.
The results that we obtained, though not very accurate, still provide proof of concept and design. With the new optical components, we would be able to recreate the
optics that we simulated in our design process. Hence, we would be able to acquire
reflectance data for a wide range of materials.
Chapter 5
Conclusions and Future Work
To conclude, the main contributions of our work include:
• A new image-based BRDF acquisition setup that is efficient and has no moving
• A set of zonal basis functions that are orthogonal over the zone of data capture,
that enables us to encode both input illumination and captured response. This
speeds-up the acquisition process in addition to avoiding data redundancy.
The setup that we currently have provides us with a working proof of concept and design. What we have provided is a novel optical design that is not only efficient, but
also is easily extensible. With a few minor changes we can use the setup in a number
of different ways. For example, by adding two degrees of freedom, we could acquire
spatially-varying BRDF with our setup. Next we discuss the proposed fabrication process and possible future work.
As discussed in the previous chapter, we are looking at alternatives for fabricating
precise mirror components (dome and the parabolic mirrors) namely, electroforming
or machining stainless steel components; followed by reflective coating and polishing.
We also plan to use the Mitsubishi PocketProjector instead of the DLP Projector
we are now using since it offers various advantages including ease of mounting and
shorter focus that we need for our setup. With our new optical setup, there are a few
things we would like to do including,
• Acquire reflectance data for an extensive set of materials and create a publicly
available database. We plan to capture materials like fabrics, paper, metals and
Chapter 5. Conclusions and Future Work
various interesting BRDFs that otherwise cannot be modeled well using analytical models.
• Test the capability of our basis acquisition setup and determine its limits in terms
of the types of materials we can measure. With the current design and basis
functions we might not be able to acquire some high frequency BRDFs. We
would like to see how far we can go in terms of capture capability.
• Explore alternative basis functions for highly specular materials. As discussed
above, we might have to look for alternative basis functions to be able to acquire
high specularity or anisotropy in BRDF data. This might entail using a basis that
is dependent on the type of BRDF data we are acquiring.
• Point sample reflection data and follow up by an analytical fitting procedure.
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Appendix A
Radiometric Terms
In computer graphics, the interaction of light with matter is often modeled geometrically using ray-optics. In this section we provide a brief description of some of the
radiometric terms used in this thesis.
Radiant Energy, denoted by Q, is the most basic radiometric unit, measured in Joules
[J]. For a photon of wavelength X, the particle model of light gives the energy Q in
terms of Planck's constant h and speed of light in vacuum c , as
Radiant Flux (or Radiant Power), denoted by <>
| , is the energy flowing through a surface
per unit time and is measure in Watts [W],
Radiant Flux Area Density, denoted by u, is a measure of energy flow given by radiant
flux per unit area, measured in [W/m ].
If the energy flow is toward the surface, it is referred to as irradiance (denoted by E)
and if the energy flow is away from the surface, it is referred to as radiosity or radiant
exitance (denoted by B).
Intensity, /, is the measure of flux with respect to solid angle instead of area and is
measured in [W/sr]
This is useful in describing point light sources, since the area goes to zero.
Appendix A. Appendix: Radiometric Terms
Radiance, denoted by L , is a measure o f radiant flux per unit projected area per unit
solid angle. Its unit is [W/m sr].
d 4>
L =
dA don cos 6
where 0 is the angle between the normal N o f the surface area element dA and the
direction of the flux {(). The cosine term represents the foreshortening with respect to
the flux direction. Spectral radiance is the radiance per unit wavelength interval and is
measured in [W/m ,sr| units.
For a more detailed description please refer to Glassner [12].
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