Modeling by Example

Modeling by Example
Modeling by Example
Thomas Funkhouser,1 Michael Kazhdan,1 Philip Shilane,1 Patrick Min,2
William Kiefer,1 Ayellet Tal,3 Szymon Rusinkiewicz,1 and David Dobkin1
1 Princeton
University
2 Utrecht
University
3 Technion
- Israel Institute of Technology
Abstract
In this paper, we investigate a data-driven synthesis approach to
constructing 3D geometric surface models. We provide methods
with which a user can search a large database of 3D meshes to find
parts of interest, cut the desired parts out of the meshes with intelligent scissoring, and composite them together in different ways to
form new objects. The main benefit of this approach is that it is both
easy to learn and able to produce highly detailed geometric models
– the conceptual design for new models comes from the user, while
the geometric details come from examples in the database. The
focus of the paper is on the main research issues motivated by the
proposed approach: (1) interactive segmentation of 3D surfaces, (2)
shape-based search to find 3D models with parts matching a query,
and (3) composition of parts to form new models. We provide new
research contributions on all three topics and incorporate them into
a prototype modeling system. Experience with our prototype system indicates that it allows untrained users to create interesting and
detailed 3D models.
Keywords: databases of geometric models, 3D shape matching,
interactive modeling tools
1
Introduction
One of the most significant obstacles in computer graphics is providing easy-to-use tools for creating detailed 3D models. Most
commercial modeling systems are difficult to learn, and thus their
use has been limited to a small set of trained experts. Conversely,
3D sketching programs are good for novices, but practical for creating only simple shapes. Our goal is to provide a tool with which
almost anybody can create detailed geometric models quickly and
easily.
In this paper, we investigate “modeling by example,” a datadriven approach to constructing new 3D models by assembling
parts from previously existing ones. We have built an interactive tool that allows a user to find and extract parts from a large
database of 3D models and composite them together to create new
3D models. This approach is useful for creating objects with interchangeable parts, which includes most man-made objects (vehicles, machines, furniture, etc.) and several types of natural objects
(faces, fictional animals). Our current implementation employs a
database of more than 10,000 models, including multiple examples
of almost every type of household object.
The main motivation for this approach is that it allows untrained
users to create detailed geometric models quickly. Unlike previous
interactive modeling systems, our users must only search, select,
and combine existing parts from examples in the database – i.e.,
Figure 1: Modeling by example: geometric parts extracted from a
database of 3D models can be used to create new objects. The large
brown chair was built from the circled parts of the others.
they rarely have to create new geometry from scratch. As a result,
the user interface can be simpler and accessible to a wider range
of people. For example, when making the rocking chair shown in
Figure 1, the user started with a simple chair (top-left), and then
simply replaced parts. The commands were very simple, but the
result has all the geometric details created by the expert modelers
who populated the database. This approach provides a new way
to make 3D models for students, designers of virtual worlds, and
participants in on-line 3D games.
In the following sections, we address the main research issues in
building such a system: segmenting 3D surfaces into parts, searching a database of 3D models for parts, and compositing parts from
different models. Specifically, we make the following research
contributions: (1) an intelligent scissors algorithm for cutting 3D
meshes, (2) a part-in-whole shape matching algorithm, (3) a method
for aligning 3D surfaces optimally, and (4) a prototype system for
data-driven synthesis of 3D models. Experience with our prototype
system indicates that it is both easy to learn and useful for creating
interesting 3D models.
2
Related Work
This paper builds upon related work in several sub-fields of computer graphics, geometric modeling, and computer vision.
Geometric modeling: Our system is a 3D modeling tool. However, its purpose is quite different than most previous modeling systems (e.g., [Wavefront 2003]). It is intended for rapidly combining
existing geometry into new models, and not for creating new geometry from scratch. As such, it has a synergistic relationship with
other modeling systems: our tool will benefit from improvements
to existing modeling systems, since there will then be larger/better
databases of 3D geometry, while other modeling systems will likely
benefit from including the methods described in this paper to provide better utilization of existing models.
Sketch modeling tools: Our system shares many ideas with 3D
sketching systems, such as Sketch [Zeleznik et al. 1996] and
Teddy [Igarashi et al. 1999]. Like these systems, we follow the general philosophy of keeping the user interface simple by inferring the
intention of a few, easy-to-learn commands, rather than providing
an exhaustive set of commands and asking the user to set several
parameters for each one. However, previous systems have achieved
their simplicity by limiting the complexity and types of shapes that
can be created by the user. We achieve our simplicity by leveraging
existing geometry stored in a database.
Data-driven synthesis: Our work is largely inspired by the recent
trend towards data-driven synthesis in computer graphics. The general strategy is to acquire lots of data, chop it up into parts, determine which parts match, and then stitch them together in new and
interesting ways [Cohen 2000]. This approach has been demonstrated recently for a number of data types, including motion capture data (e.g., [Lee et al. 2002]). However, to our knowledge, it has
never been applied to 3D surface modeling. Perhaps this is because
3D surfaces are more difficult to work with than other data types:
they are harder to “chop up” into meaningful parts; they have more
degrees of freedom affecting how they can be positioned relative
to one another; they have no obvious metric for identifying similar
parts in the database; and, they are harder to stitch together. These
are the issues addressed in this paper.
3
System Overview
The input to our system is a database of 3D models, and the output
is a new 3D model created interactively by a user. The usual cycle
of operation involves choosing a model from the database, selecting
a part of the model to edit, executing a search of the database for
similar parts, selecting one of the models returned by the search,
and then performing editing operations in which parts are cut out
from the retrieved model and composited into the current model.
This cycle is repeated until the user is satisfied with the resulting
model and saves it to a file. The motivation for this work cycle is
that it requires the user to learn very few commands (open, save,
select, cut, copy, paste, undo, search, etc.), all of which are familiar
to almost every computer user.
A short session with our system is shown in Figure 2. Imagine
that a school child wants to investigate what the Venus de Milo
sculpture looked like before her arms were broken off. Although
there are several theories, some believe that she was holding an
apple aloft in her left hand, and her right arm was posed across her
midsection [Curtis 2003]. Of course, it would be very difficult for
a child to construct plausible 3D models for two arms and an apple
from scratch. So, we investigate extracting those parts from other
3D models available in our database.
Shape interpolation: Our work shares many ideas with “shape
by example” [Sloan et al. 2001] and other blending systems
whose goal is to create new geometric forms from existing ones
(e.g., [Lazarus and Verroust 1998]). However, our approach is quite
different: we focus on recombining parts of shapes rather than morphing between them. We take a combinatorial approach rather than
an interpolative one. Accordingly, the types of shapes that we can
create and the research issues we must address are quite different.
We believe that our approach is better suited for creating shapes
composed of many parts, each of which has a discrete set of possible forms (e.g., cars, tables, computers, etc.), while interpolation is
better for generating new shapes resulting from deformations (e.g.,
articulated motions).
Geometric search engines: Our system includes the ability to
search a large database of 3D models for matches based on keyword
and/or shape similarity. In this respect, it is related to 3D search
engines that have recently been deployed on the Web (e.g., [Chen
et al. 2003; Corney et al. 2002; Funkhouser et al. 2003; Paquet and
Rioux 1997; Suzuki 2001; Vranic 2003]). Several such systems
have acquired impressive databases and allow users to download
3D models for free. In our current implementation, we use the data
of the Princeton 3D Model Search Engine [Min et al. 2003]. That
system and ones like it employ text-based search methods similar
to ours. However, their shape-based matching algorithms consider
only whole-object shape matching. In this paper, we address the
harder problem of part-in-whole shape matching.
To our knowledge, this is the first time that a large database of
example 3D models and shape-based retrieval methods have been
integrated into an interactive modeling tool.
Figure 2: Screenshots of a ten-minute session demonstrating the
main features of our system being used to investigate what Venus
looked like before she lost her arms.
4
Intelligent Scissoring of 3D Meshes
The first issue we address is segmenting 3D models into parts. Our
goal is to provide simple and intuitive tools with which the user can
quickly and robustly specify a meaningful subset of a 3D model (a
part) to form a selection, a query for a search, and/or the target for
a surface editing operation.
Of course, many models come already decomposed into scene
graph hierarchies or multiple disconnected components, and several approaches have been proposed for automatic mesh segmentation (e.g., [Katz and Tal 2003]). We maintain these segmentation(s)
when they are available. However, often the provided segmentation
is not what the user needs for a particular editing operation, and
thus we must also consider interactive methods.
There are several possible approaches to interactive segmentation of 3D meshes. First, most commercial modeling tools allow
the user to draw a screen-space split line or “lasso” and then partition the mesh vertices according to their screen space projections
(Figure 3). This approach is only able to make cuts aligned with the
camera view direction, which is often not what the user wants (top
row of Figure 3). Second, some systems allow the user to select a
sequence of vertices on the surface mesh and cut along the shortest paths between them [Gregory et al. 1999; Wong et al. 1998;
Zöckler et al. 2000]. Although this method supports cuts of arbitrary shape, it requires great care when selecting points (because the
cut is constrained to pass exactly through those points), points must
be specified in order (which makes fixing mistakes difficult), and
the camera view must be rotated to get points on multiple sides of
the object (otherwise, the shortest path will not encircle the object).
Finally, other systems (e.g., [Lee and Lee 2002]) have used “active
contours” to adjust a user-drawn cut to follow the closest seams of
the mesh. While this approach allows the user to specify the cut
less precisely, we find active contours hard to control – i.e., it is difficult to find a set of parameters that prevent the snakes from getting
stuck on small bumps and irregularly sampled regions of the mesh
while ensuring that they find the natural seams of the mesh nearby
the user’s cut. Perhaps more sophisticated variational approaches
Naïve algorithm
Intelligent Scissors
In this example, a text search yields a 3D model of the armless Venus (a). Then, two boxes are drawn representing the desired
pose of a new left arm (b), and a part-in-whole shape search yields a
sculpture of Hebe with an arm in a similar pose (c) as the top match.
The matching arm is cut off Hebe using intelligent scissors with a
single, approximate mouse stroke (black line in (d)), producing a
cut along the natural seam over her shoulder and through her arm
pit separating her arm (yellow) from her body (purple). The arm is
then copied to the clipboard and pasted into the window with the
Venus sculpture. A good initial placement for the arm is found with
automatic optimal alignment to the two boxes (e), saving the user
most of the work of interactively moving, rotating, and scaling it
to the right place. Next, a hole is cut in Venus’ left shoulder and
stitched to the open boundary of the pasted arm to make a watertight junction with smooth blending (f). The second arm is found
with text search, and similar cut, copy, paste, and blend operations
transfer it onto Venus’ body (g). Finally, cutting the bowl out of her
left hand and replacing it with an apple found with text keyword
search yields the final model (h). While the result is certainly not
historically accurate, it took less than 10 minutes to make, and it
provides a plausible rendition suitable for school and entertainment
applications.
Although this example is mostly for pedagogy and fun, it demonstrates the main features of our system: (1) segmenting surface
models into parts, (2) searching a 3D model database for parts, and
(3) compositing multiple parts into a new model. The following
sections detail our research contributions and design decisions in
each of these areas.
(a) User stroke
(b) Front view
(c) Top view
Figure 3: A screen-space “lasso” (top row) produces an unexpected
segmentation when the camera view is not perfectly aligned with
the desired cut. In contrast, our intelligent scissors (bottom row)
finds the optimal cut through the stroke, which may or may not be
orthogonal to the view direction.
(e.g., coarse-to-fine refinement) can address this issue. Rather, we
decided to provide a more explicit method for the user to control
the placement of the cut while guaranteeing that an optimal seam is
found.
4.1
Painting Strokes
We allow the user to paint “strokes” on the mesh surface to specify where cuts should be made (Figure 4a). Each stroke has a
user-specified width (r) representing a region of uncertainty within
which the computer should construct the cut to follow the natural
seams of the mesh (our current system considers only cuts along
edges of the mesh). From the user’s perspective, the meaning of
each paint stroke is “I want to cut the surface along the best seam
within here.” From the system’s perspective, it specifies a constraint
that the cut must pass within r pixels of every point on the stroke,
and it provides parameters for computing the cost of cutting along
every edge (e) in the mesh:
cost(e) = clen (e) × cang (e)α × cdist (e)β × cvis (e)γ × cdot (e)δ
where clen (e) is the edge’s length, and α , β , γ , and δ trade off between factors based on the dihedral angle (θe ) of its adjoining faces
(cang (e) = θe /2π ), its visibility at the time the stroke was drawn
(cvis (e) = 1 if e was visible, and 0.5 otherwise), the orientation of
its surface normal (~N) with respect to the view direction (~V ) when
the stroke was drawn (cdot (e) = (1 + ~V · ~N)/2), and the maximum
distance (d) from the centerline of the stroke to the screen space
projection of the edge (cdist (e) = r−d
r ). Default values for α , β , γ ,
and δ are all one.
Intuitively, this definition imposes less cost for edges that are
shorter, along more concave seams of the mesh, closer to the center of the stroke, and invisible or back-facing when the stroke was
drawn. The first three terms are self-explanatory. The fourth term
(cvis (e)) is motivated by the observation that a user probably would
have painted on a visible edge if a cut were desired there – i.e., not
drawing on a visible portion of the mesh implies “don’t cut here.”
This term combines with cdot (e) to encourage the least cost closed
contour to traverse the “back-side” of the mesh. Without these two
terms, the least cost closed contour through the stroke would most
likely fold back along itself, traveling from the end of the stroke
back to the beginning along a path close to the stroke on the front
side of the mesh. However, with either of these terms, the cost of
traversing the back side of the mesh diminishes (Figure 4f), and the
closed contour usually encircles the mesh, making it possible to cut
the mesh with one stroke (Figure 4g-h). In particular, with sufficient weighting of these cost terms (γ >> α and/or δ >> α ), the
C1 to all vertices in C2 (or vice-versa) while constrained to traverse
only edges within the stroke (yellow dotted lines in Figure 4c), and
(2) find the least cost path connecting all vertices in C2 back to all
vertices in C1 without any constraints (red dotted lines in Figure 4c).
The optimal cut is the pair of paths, one from each sub-problem,
that forms a closed contour with least total cost [Mitchell 2003].
These sub-problems are solved with an execution of Dijkstra’s algorithm for each vertex in C1 .
The computational complexity of this algorithm for a single
stroke is O(k1 · n log n), where n is the number of edges in the mesh,
and k1 is the number of vertices in C1 . Since k1 is usually small,
and least cost path searches usually cover a subset of the mesh (Figure 4d-f), running times are interactive in practice. This algorithm
took under a second for all examples shown in this paper.
4.3
Refining the Cut
Our system also allows the user to refine the cut interactively with
“over-draw” strokes. Immediately after the first stroke, the system
can partition the mesh according to the computed optimal cut (the
default). Alternatively, it can display a “proposed cut” for verification. If the user is not satisfied, she can draw new strokes that
refine the cut incrementally (Figure 5). This feature encourages the
user to draw broad strokes quickly, in any order, and then iteratively
refine the details only where necessary.
Figure 4: Cutting the bunny with intelligent scissoring: (a) the user
draws a wide paint stroke; (b) the system identifies all vertices in
the caps of the stroke, C1 and C2 ; (c) it then finds the least cost
paths from every vertex in C1 to every vertex in C2 twice, once
constrained to lie within the stroke (yellow dotted lines) and once
without any constraints (red dotted lines), and forms the proposed
cut out of the pair of paths with least total cost; (d-f) Since the edges
traversed by the algorithm (wireframe gray) have less cost (lighter
gray values) in concave seams and on the back-side of the mesh,
(g-f) the least cost cut partitions the mesh into two parts (red and
green) along a natural seam of the mesh.
closed contour is guaranteed to traverse the back side of the mesh
whenever the sum of costs to travel from the stroke endpoints to the
closest silhouette boundaries is less than the cost to travel between
the endpoints along the front side of the mesh.
4.2
Finding the Cut
We solve a constrained least cost path problem to find the optimal
cut specified by the user’s stroke. More formally, we find the sequence of mesh edges with least cost that passes within r pixels of
every point on the user’s stroke in sequence. This is a graph version
of the “safari problem,” where the “cages” to visit are the sequence
of overlapping circular neighborhoods defined by the centerline and
width of the user’s stroke. Since we do not have explicit start and
stop vertices, we cannot find the least cost path directly with Dijkstra’s algorithm. Rather, we must search over many possible start
and stop vertices to find the least cost path anywhere within the
stroke. While this approach is computationally more expensive, it
allows the user to cut meshes quickly with partial and approximate
strokes, while the computer fills in the details of the cut automatically.
The key to solving this least cost path problem efficiently is observing that the cut must pass through at least one vertex in the
“cap” at each end of the stroke, C1 and C2 – i.e., the set of vertices projecting within r pixels of the first and last points on the
stroke (Figure 4b). Thus, we can divide-and-conquer by solving
two sub-problems: (1) find the least cost path from all vertices in
Figure 5: Cutting the face of Athena with intelligent scissoring:
(a) the user draws an imprecise first stroke (gray); (b) the system
proposes a cut (yellow curve); (c) the user draws an overdraw stroke
(gray) to refine the cut; (d) the system splices in the least cost path
traveling from V1 first to C1 (red) then to C2 within the stroke (blue)
and finally to V2 (green); (e) the proposed cut contour is updated; (f)
the final result is a segmentation of the mesh into two parts (green
and red) separated by natural seams of the mesh.
For each over-draw stroke, S, the system splices a new, locally
optimal path through S into the proposed cut contour. The system
splices out the portion of the proposed contour between the previously painted vertices, V1 and V2 , closest to the two endpoints of
the over-draw stroke (Figure 5c). It then splices in a new contour
with least cost from V1 to V2 traveling within r of every pixel in the
over-draw stroke. This path is found in three stages (Figure 5c).
First, a single execution of Dijkstra’s algorithm finds the least cost
paths from V1 to all vertices in the over-draw stroke’s first cap, C1 .
Then, those paths are augmented with the least cost paths within
the stroke from all vertices within C1 to all vertices in C2 , the second cap of the over-draw stroke. Finally, those paths are augmented
with the least cost paths from all vertices in C2 to V2 , the point at
which the new cut contour connects back to the original proposed
cut (Figure 5d). The path found with overall least cost is spliced
into the proposed cut (Figure 5e).
This incremental refinement approach has several desirable
properties. First, it provides local control, guaranteeing that previously drawn strokes will not be overridden by new strokes unless
they are in close proximity. Second, it is fast to compute, since
most of the least cost path searches are constrained to lie within
the stroke. Finally, it allows the user to specify precisely and naturally where the splice should be made both by simply starting and
stopping the over-draw stroke with the cursor near the proposed
contour.
5
Search for Similar Parts
The second issue we address is searching a large database of 3D
models. The challenge is to develop effective mechanisms whereby
users can enter a query specifying the objects they want to retrieve
from the database and the system quickly returns the best matches
and suggests them as candidates for future editing operations. In
general, we would like to support two types of queries. Initially,
we expect the user to search for whole objects that represent the
general shape they would like to construct – requiring an interface
supporting whole-object matching. As the user progresses through
her editing operations, we expect that she will want to replace individual parts of the model with parts from other models – requiring
an interface supporting partial object matching. In either case, it is
important that the interface be easy to use and capable of finding
similar parts efficiently.
Perhaps the simplest and most common query interface is textual
keywords. Most users are familiar with this type of query and it
is well suited for finding 3D models based on their semantics if
models are well-annotated. However, recent research [Min 2004]
has indicated that many models are poorly annotated. This problem
is further exacerbated if we are interested in finding parts of models,
which tend to have even less annotation. For this reason, our system
augments text search with shape-based methods.
Traditional methods for matching shapes [Besl and Jain 1985;
Loncaric 1998; Tangelder and Veltkamp 2004] have focused on
whole-object matching, providing methods for finding models
whose overall shape is similar to a query. We provide this feature
in our system, but would also like to support “part-in-whole” shape
searches. This type of query matches whole objects, but with special emphasis on a selected part. It is very useful for finding specific
parts within a class of objects. For instance, consider a situation in
which a user has a chair and would like to replace some of the parts
(Figure 6). If she performs a shape similarity query with only the
legs of the chair selected, then the system should retrieve chairs
with similarly shaped legs, independent of the presence or shapes
of their arms. On the other hand, if she selects only the arms of
the chair and performs a shape similarity query, the system should
find chairs with arms. Figure 6 shows the results achieved by our
system in this case.
Traditional shape matching methods that represent each 3D
model by a multi-dimensional feature vector (a shape descriptor)
and define the dissimilarity between a pair of models as the distance (L1 , L2 , L∞ , etc.) between them are not well-suited for this
type of query. They consider two models the same if and only if
their descriptors are equal. In this work, we would like to represent
a single model by many different descriptors, depending on the features selected by the user. Since it is imperative that a model with
Figure 6: Results of shape similarity queries where the query provided to the system is (top) the chair with the legs selected, and
(bottom) the chair with the arms selected.
features selected always match itself, we must use an alternative
approach.
The notion of shape similarity that we use is based on the sum
of squared distances for models aligned in the same coordinate system. Specifically, we define the distance between two models as the
sum of the squares of the distances from every point on one surface
to the closest point on the other, and vice-versa. This definition
of shape similarity is intuitive, since it approximates the amount
of work required to move points on one surface to the other (as in
[Rubner et al. 2000; Tangelder and Veltkamp 2003]), and it implies
that two shapes should be subsets of one another to achieve a good
match. It is also well suited for feature based matching, since we
can associate a weight to each point and then scale the contribution
of each summand accordingly. Then, selected parts (points with
higher weight) can contribute more to the measure of shape similarity than others.
While a direct approach for computing the sum of squared distances would require a complex integration over the surfaces of the
models, we present a new method for computing this distance that
is easy to implement (Figure 7). For each model A in the database,
we represent the model by two voxel grids, RA and EA . The first
voxel grid, RA , is the rasterization of the boundary, with value 1 at
a voxel if the voxel intersects the boundary, and value 0 if it does
not. The second voxel grid, EA is the squared Euclidean Distance
Transform of the boundary, with the value at a voxel equal to the
square of the distance to the nearest point on the boundary. In order
to compare two models A and B we simply set the distance between
the two of them to be equal to:
d(A, B) = hRA , EB i + hEA , RB i,
the dot product of the rasterization of the first model with the square
distance transform of the second, plus the dot product of the rasterization of the second model with the squared-distance transform of
the first. The dot product hRA , EB i is equal to the integral over the
surface of A of the square distance transform of B. Thus, it is equal
precisely to the minimum sum of square distances that points on the
surface of A need to be moved in to order to lie on the surface of B.
For matching parts of models, the process is very similar. Given
a triangulated model A with a subset of triangles, S ⊂ A, selected by
the user as features with some weight w, we can compute the feature
weighted descriptor of A, {RA,S , EA }, by setting RA,S to be the rasterization of A with value w for points on S, a value of 1 for points
on A that are not in S, and a value of 0 everywhere else. When we
compute the dot product of the weighted rasterization with the distance transform of another model, hRA,S , EB i, we get the weighted
sum of square distances, with the contribution of square distances
from points on S scaled by weighting factor w. This approach allows us to compute just a single shape descriptor for each model
in the database, while using it for matching queries with arbitrary
feature weighting.
Overall, the advantages of this shape descriptor are two-fold.
First, its underlying notion of similarity is based on the sum of
squared distances between points on two surfaces, which is intuitive to users. Second, it works for matching both whole objects
and objects with specific parts within the same framework, requiring just a single descriptor for every object in the database. Moreover, this descriptor has the non-trivial property that the distance
between two different feature weighted representations of the same
model is always equal to zero.
we find that the subtleties of 3D placement are difficult to master for
most novice users. So, we are motivated to find the best possible
automatic strategies.
In particular, our users are often faced with the problem of replacing one part with another. For instance, they may have created
a simple version of a part and then queried the database for similar ones to replace it (Figure 2b). Or, they may have started with
a plain version of an object selected from the database, and then
want to replace parts from it with better versions. In either case, we
would like to provide a simple automatic command with which the
new part (the query) can be placed in the same coordinate frame as
the one it is replacing (the target). Specifically, we would like to
solve for the translation, scale, rotation, and mirror that minimizes
the sum of squared distances from each point on one surface to the
closest point on the other, and vice-versa.
Based on the work of [Horn 1987; Horn et al. 1988], we can
solve for the optimal translation by moving the query part so that
its center of mass aligns with the center of mass of the target. Similarly, we can solve for the optimal scale by isotropically rescaling
the query part so that its mean variance (sum of squared distances of
points from the center of mass) is equal to the mean variance of the
target. However, solving for the optimal rotation using the method
of [Horn 1987; Horn et al. 1988] would require the establishment
of point-to-point correspondences which is often a difficult task.
A commonly used method for aligning two models is the ICP
algorithm [Besl and McKay 1992] which, given an initial guess,
will converge to a locally optimal solution minimizing the sum of
squared distances between the two models. In our approach we
provide a voxel space implementation of ICP that does not require
an initial guess and is guaranteed to give the globally optimal solution. Specifically, we use the recently developed signal processing
techniques of [SOFT 1.0 2003; Kostelec and Rockmore 2003] that
efficiently solve for the correlation of two functions – giving a 3D
array, indexed by rotations, whose values at each entry is the dot
product of the first function with the corresponding rotation of the
second. In particular, using the shape descriptors from Section 5
as the input functions for correlation, we obtain a 3D array whose
value at a given entry is the minimum sum of squared distances between the two models, at the corresponding rotation. Thus, searching the 3D correlation array for the entry with smallest value gives
the optimal rotation for aligning the two models. This process takes
O(N 4 ) for a N × N × N voxel grid, rather than the O(N 6 ) that would
be required for brute force search.
6
6.2
Figure 7: Two models are compared by computing the voxel rasterization and square distance transform of each one and defining
the distance measure of model similarity as the dot product of the
rasterization of the first with the distance transform of the second,
plus the dot product of the distance transform of the first with the
rasterization of the second. The resultant value is equal to the minimum sum of square distances that points on each model need to be
moved in order to lie on the other model.
Composition of Parts
The third issue we address is how to assemble parts from different
sources into a single model. Our goal is to provide interactive tools
for the user to position and orient parts when they are added to a
new model and possibly to stitch the surfaces together at the joints.
Commercial 3D modeling programs often provide interactive
tools to perform these functions. So, our goal here is not new. However, our system is intended for novice users, and thus we aim to
find simpler interfaces and more automatic methods than are typically found in commercial software. There are two challenges we
must address: part placement and part attachment.
6.1
Part Placement
When inserting a part into a model, the first challenge is to find the
transformation that places it into the appropriate coordinate frame
with respect to the rest of the model. We have investigated several options, including most of the interactive direct manipulation
(translation, rotation, anisotropic scale, etc.) and alignment commands (align centers, align tops, align anchors, align moments, etc.)
commonly found in 3D commercial modeling programs. However,
Part Attachment
Of course, the model resulting from simply placing parts next to one
another does not produce a connected, manifold surface. Although
disconnected meshes are common in computer graphics, especially
for man-made objects, our system provides simple methods based
on [Kanai et al. 1999] with which the user can stitch parts together
to form smooth, connected surfaces from multiple parts. While
these methods are not a research contribution of this paper, we describe them briefly for completeness.
When a user wishes to join two parts, she first selects an open
boundary contour on each mesh, C1 and C2, possibly first cutting
holes with intelligent scissors (Figure 8a). Then, she executes a
”join” command, which uses heuristics to establish vertex correspondences for filleting and blending automatically. Specifically,
the closest pair of vertices, V 1 and V 2, are found, with V 1 on C1
and V 2 on C2 (Figure 8b). Then, the orientation for C1 with respect
to C2 is found by checking the dot product of the vector from V 1 to
V 10 , a vertex 10% of the way around the length of C1, and the vector from V 2 to V 20 , defined similarly. If the dot product is negative,
the orientation of C1 is switched. Then, vertex correspondences
are established with a greedy algorithm that iteratively increments
the ”current vertex” on either C1 or C2, choosing the one such that
the Euclidean (or parametric) distance between the two current vertices is least. Finally, fillet edges are constructed between corresponding vertices (Figure 8c), and any resulting quadrilaterals are
split into triangles. Additionally, the join command can smooth vertices within a user-specified distance of the fillet by averaging their
positions with all their neighbors with user-specified weights for a
user-specified number of iterations (Figure 8d).
Although these filleting and blending methods are not particularly sophisticated, they always produce a watertight junction, and
they do make it possible to create natural objects with smooth surfaces (e.g., Figure 8e). In future work, we plan to include more
general and automatic approaches based on boolean operators, level
sets, and/or morphing (e.g., [Alexa 2001; Museth et al. 2002]).
Figure 8: Attaching the head of a cow to the body of a dog: (a) a
boundary contour is selected on each part (C1 and C2); (b) the pair
of closest points (V 1 and V 2) is found and the local direction near
those points is used to determine the relative orientation of the contours; (c) a fillet is constructed attaching the contours; (d) the mesh
is smoothed in the region nearby the seams of the fillet. (e) the
result is a smooth, watertight seam.
7
Implementation
We have implemented a prototype system with all the features described in this paper. In this section, we provide several implementation details and discuss limitations.
7.1
Hardware and Software
Our system’s architecture consists of a client running the user interface and two servers running the shape and text matcher respectively. The client PC has one 2.8 GHz Pentium IV processor, 1
GB memory, a GeForce4 graphics card, and is running Windows
XP. The server PCs each have two 2.2 GHz Xeon processors, 1 GB
memory, and are running Red Hat Linux 9.0.
7.2
Shape Descriptors
When computing shape descriptors, we rasterize the surfaces of
each 3D model into a 64 × 64 × 64 voxel grid, translating the center
of mass to its center, and rescaling the model so that twice its mean
variance is 32 voxels. Every model is normalized for rotation by
aligning its principal axes to the x-, y-, and z-axes. Finally, the descriptors are rescaled so that the L2 -norm of both the rasterization
and the Euclidean Distance Transform are equal to 1, and w = 10 is
used for feature weighted matching.
When matching 3D models, we resolve the ambiguity between
positive and negative principal axes by using the efficient axial
search method described in [Kazhdan 2004]. In particular, the coefficients of the voxel grids are decomposed into eight different component vectors and the measure of similarity at each of the eight
axial flips is computed by summing the dot products of the component vectors with appropriate sign. The dot product minimized over
the different axial flips is used as the measure of similarity between
models.
In order to facilitate efficient storage and matching of the shape
descriptors, we use standard SVD techniques to project the shape
descriptors onto a low-dimensional subspace. Specifically, we compute the covariance matrices of the eight component vectors of each
descriptor in the database, solve for the Eigenvalues of the matrices, and project each of the component vectors onto the subspace
spanned by the Eigenvectors associated with the top 100 Eigenvalues. This allows us to compress the initial 2 × 64 × 64 × 64 =
524, 288 dimensional representation down to a 2 × 8 × 100 = 1600
dimensional vector. This compression approach is well-suited for
our application because we define the distance between models in
terms of the dot product between two functions. Any part of the
query descriptor that is not well represented in the compressed domain is mostly orthogonal to the subspace repesented by the top
Eigenvectors and hence orthogonal to most of the target models
and does not contribute much to the dot product.
We have found that these choices maintained sufficient resolution to allow for discriminating matching, while limiting the size of
the descriptor and allowing for efficient computation and retrieval.
In particular, the descriptor of each model is computed in two seconds, on average, and a database of over 10,000 models returns
retrieval results in under one second. Our compressed descriptors
provide the same retrieval precision as the full ones, though they
are easier to store and more efficient to compare.
7.3
Database and Preprocessing
Our test database contains 11,497 polygonal models [Min et al.
2003], including 6,458 free models collected from the World Wide
Web during crawls in October 2001 and August 2002, and 5,039
models provided by commercial vendors (Viewpoint, Jose Maria
De Espona, and CacheForce). We limited our database to include
only models representing objects commonly found in the real world
(e.g., chairs, cars, planes, and humans). Other classes, including
molecules, terrains, virtual environments, terrains, and abstract objects were not included. The average triangle count is 12,940, with
a standard deviation of 33,520.
For all models in the test database, we executed a series of preprocessing steps. First, we converted all file formats to VRML 2.0
and PLY to ease processing by subsequent steps. Then, we analyzed the scene graph structure of the VRML files to create a default set of segments. This step took 8 seconds per model and produced 15.4 parts on average. Second, we extracted text keywords
for each model and built an index for them, which took 0.3 seconds per model. Third, we computed shape descriptors for every
model in the database, which took less than 2 seconds, on average,
per model. Finally, we produced a 160×120 “thumbnail” image of
each object, which took 4 seconds on average. The total preprocessing time was about 50 hours, and the cumulative storage of all
data on disk requires 20 gigabytes.
7.4
Limitations
Our system is a first prototype and has several limitations. First, it
includes a small subset of commands that normally would be found
in a 3D modeling system. Specifically, we provide only two commands for creating faces from scratch (insert cube and join parts)
and only two commands for moving vertices (affine transformation
and blend). This particular design decision was made to test modeling by example in its purest form. While it adds to the simplicity
of the system, it limits the types of objects that can be created. For
example, without surface deformation tools, it is difficult to put the
door of a Ford onto a Chevrolet. We expect that others implementing systems based on this approach in the future will include a more
complete set of modeling features.
Second, it allows segmentation and attachment of parts only
along existing vertices and edges of their meshes (our algorithms
do not split triangles adaptively), which can prevent certain operations on coarsely tessellated models (e.g., splitting a table in half
when the original mesh contains only one or two large polygons for
the entire tabletop). While edges of the mesh usually occur along
seams in man-made objects, and meshes representing natural objects are often highly tessellated, we plan to extend our system to
allow splitting and merging triangles in the future.
Third, our intelligent scissoring algorithm is not well-suited for
all types of segmentation tasks. For instance, if a surface is occluded from all viewpoints, the user cannot paint on it. Although
we do provide a “laser” mode in which all surfaces under the mouse
get cut (Figure 3), we think other user interface metaphors would be
better (e.g., [Owada et al. 2003]). Similarly, our painting interface
metaphor can produce unexpected results when the user paints over
a silhouette boundary. The problem is that our system makes sure
to cut through every part of the user’s stroke, which may connect
points adjacent in screen space but distant on the surface (Figure 9).
(a) User Stroke
(b) Proposed Cut
8
Results
In this section, we evaluate the main research contributions of the
paper, comparing our results to previous work where possible. We
first present results of experiments with each component of our system, and then we show models created for a number of different
applications (Section 8.4).
8.1
Scissoring Results
In order to help the reader evaluate our intelligent scissoring algorithm, we show several segmentations produced with our algorithm
and compare the resulting cuts and processing times with respect to
previous interactive and automatic systems.
Figure 10 shows screenshots from a session (left-to-right) during
which the user segmented a statue of Mercury into parts using intelligent scissors. Each image shows a stroke drawn by the user and
its segmentation result. Note how all the strokes are approximate,
and yet the cuts lie along natural seams of the 3D model. Note also
that every cut was made with a single stroke from the same camera
viewpoint. This entire segmentation sequence can be performed in
under one minute with our system. The same segmentation takes
over 8 minutes with our implementation of the method described
in [Gregory et al. 1999; Zöckler et al. 2000].
Figure 11 compares our intelligent scissoring results for a cheetah and a hand with results reported by [Katz and Tal 2003]. While
this comparison is not altogether fair, since our algorithm is interactive and theirs is automatic, it highlights the main disadvantage
of automatic approaches: they do not understand semantics. For instance, the cheetah segmentation produced by [Katz and Tal 2003]
includes portions of the animal’s back with the tail and neck and
contains an unnatural boundary between the right-hind leg and body
(Figure 11a). As a result, the parts cannot be simply pasted into another model without re-cutting them. Similarly, their hand segmentation does not separate all the bones. In contrast, our interactive
approach allows users to cut the models quickly along semantic
boundaries (only as needed). Our segmentation of the hand (Figure 11d) took 13 minutes (of user time), while the automatic segmentation (Figure 11c) took 28 minutes (of computer time) for the
same model [Katz and Tal 2003]. Our segmentation of the cheetah
took under thirty seconds.
Figure 9: The intelligent scissors algoritm ensures that the cut contour (blue and yellow line on right) visits all regions painted by the
user (left), which may be problematic when the stroke crosses an
interior silhouette boundary. Yellow portions of the cut are painted,
and blue ones are not.
Fourth, our part-in-whole shape matching method is only able
to find parts if they reside in the same relative location within their
respective whole objects. Consequently, it is not able to find all
possible instances of a part in the database (e.g., a bell in a church
does not match a bell on a fire engine). However, in many cases, the
geometric context of a part within the whole object is the defining
feature of a part - e.g., searching with the wheel of an airplane selected retrieves airplanes with wheels, and not all objects that have
disk shaped parts (plates, frisbees, round table tops, etc.). Further
research is required to characterize the space of partial shape matching problems and algorithms.
Finally, our system provides little user feedback about how
its segmentation, matching, alignment, and attachment algorithms
work, and thus the user can only hit the “undo” button and try again
when the system produces an unexpected result. In the future, we
plan to investigate how to include fine-grained controls for expert
users to handle difficult cases while maintaining a simple look and
feel for novice users.
[Katz et al, 2003]
Intelligent scissors
Figure 11: Comparison of segmentations produced by the automatic algorithm of [Katz et al, 2003] (left) and the interactive intelligent scissors algorithm described in this paper (right) for a model
of a cheetah and a hand (654,666 polygons). Note that the cuts are
better aligned with the semantic seams of the object with intelligent
scissors.
Figure 10: Example session in which a user segmented a model with intelligent scissoring. Gray strokes show the user’s input. Different
colored surfaces indicate separate parts computed by the intelligent scissoring algorithm. Note how the cuts between parts are smooth, wellplaced, and encircle the entire object, even though the user’s strokes were not very careful and all performed from the same viewpoint (the
rightmost image shows the cuts as seen from the backside of the mesh). This sequence took under a minute with our system.
8.2
Search Results
In order to measure the empirical value of the shape descriptor presented in Section 5, we designed experiments aimed at evaluating
the retrieval performance of the descriptor in whole-object and partin-whole matching.
For the whole-object matching experiment, we used the Princeton Shape Benchmark test data set [Shilane et al. 2004] to compare
the matching performance of our new descriptor with two state-ofthe-art descriptors: the Spherical Harmonic Representation of the
Gaussian Euclidean Distance Transform [Funkhouser et al. 2003]
and the Radial Extent Function [Vranic 2003]. We evaluated the
performance of retrieval using a precision vs. recall plot, which
gives the accuracy of retrieval as a function of the number of correct
results that are returned. The results of the experiment are shown in
Figure 12. Note that the precision achieved with the sum of squared
distances is higher than the precision of the other two methods at
every recall value. This result suggests that our descriptor is well
suited for whole-object matching.
Figure 12: Whole-object matching experiment: Precision versus recall curves showing that our sum of squared distances shape matching method (black curve) compares well with two state-of-the-art
methods for matching whole objects.
For the part-in-whole matching experiment, we compared the
retrieval performance of feature weighted matching with wholeobject matching (unweighted) using our sum of squared distances
framework. Specifically, we gathered a database of 116 chairs
from the Viewpoint Household Collection and labeled their presegmented parts (scene graph nodes) according to membership in
one of 20 classes (e.g. “T-Shaped Arms”, “Solid Back”, etc.). There
were 473 parts. We then ran two tests in which we would query with
a model and, for every part in that model, measure how quickly the
method would retrieve other models having parts within the same
class. During the first test, we queried with the whole-object descriptor (unweighted). During the second, we weighted the descriptor (w = 10) to emphasize the query part. As before, we measured
retrieval performance with precision-recall analysis. The results are
shown in Figure 13. They show that feature weighting can help our
matching algorithm retrieve models with specific parts, and suggest
that the user’s ability to selectively specify parts will, in general,
enhance her ability to retrieve models with similar components.
Figure 13: Part-in-whole matching experiment: Precision vs. recall curves showing that feature weighted matching (red curve) performs better than unweighted whole-object matching (blue curve)
for finding chairs with particular types of parts.
8.3
Alignment Results
In order to measure the quality of our optimal alignment approach
(Section 6), we ran a test to compare its performance with a traditional method (PCA) that aligns a pair of models by aligning their
principal axes and resolves directions of the principal axes with
heaviest axis flips [Elad et al. 2001].
For our test, we gathered 326 models and decomposed them into
45 classes. A human manually aligned all the models within a class
in a common coordinate frame. We then computed the aligning
rotations for every pair of models within the same class using both
PCA and our optimal alignment method. We measured error as
the angle of the obtained rotation, where zero angle indicates the
identity rotation – a perfect match of the human alignment.
Figure 14 shows results of the experiment, expressed in terms of
cumulative rotational error. For each angle of rotation α , the graph
shows the percentage of models that were correctly aligned within
α degrees. We show curves for the PCA method (blue) and our
algorithm (red). Note that the optimal alignment approach provides
much better alignments, closely matching the human’s for the most
model pairs. For example, if we consider alignments within 10
degrees of the human’s, we find that PCA aligns only 14% of the
models correctly, while the optimal alignment method aligns 75%.
Figure 14: A plot showing the percentage of models aligned within
a certain tolerance angle, α (horizontal axis). The graph shows
results for PCA alignment with heaviest axis flip and our optimal
alignment. Note that for a small error tolerance (e.g., within 10◦ ),
the optimal alignment algorithm succeeds for a much greater percentage of the models.
8.4
Applications
Our final challenge is to evaluate whether the “modeling by example” approach is useful: “Is it indeed easier and quicker to make
interesting models with our prototype than with other systems?”
Of course, this question is difficult to answer without a formal user
study, which is beyond the scope of this paper. In lieu of that, we
show several models created with our system by a user with no
prior experience with any 3D modeling tool (Figures 15-19). For
most examples, we show a photograph depicting the user’s target
(above/left) and describe the process by which the user proceeded
to make the model. Our goal is to demonstrate that it is possible
to construct new models from existing parts quickly and easily for
several applications:
• Virtual Worlds: Figure 15 shows a 3D model created to
mimic a person’s home office. The user was presented with
the photograph on the top and given the task of making a virtual replica. The resulting model (shown on the bottom) contains 143,020 polygons and took one hour to construct. While
it is not an exact copy, we are not aware of any other way that
novice users could create such a model so quickly.
Figure 15: Virtual mockup of a home office. The user was presented with a photograph (top) and asked to make a virtual replica
(bottom). Colors are used to visualize how the scene is decomposed
into parts from the database.
• Education: Motivated by a fifth grade assignment after a field
trip to the A.J. Meerwald, a schooner in the National Register
of Historic Places, a user created a virtual replica mimicking what a child could have done with our system. Figure 16
shows an image of the real schooner (top-left), the resulting
model (middle), and some of the parts that were used to construct it (around). Note that no boat in our database has more
than one or two parts in common with our final result. This
model took 90 minutes to construct, and yet contains details
far beyond what our user could have created in any other tool
without significant training (Figure 16).
• Entertainment: Figure 19 shows a 3D model of the Classic Radio Flyer Tricycle that could be incorporated in a computer game or digital video sequence. In this case, the closest
tricycle in our database (left) had only one part in common
with the Radio Flyer (the back plate). It was used mainly as a
template for searches and alignment. All other parts are new,
sometimes found in the least expected places. For instance,
the curved bar connecting the handle bars to the rear wheels
was extracted from the handle bar of a motorcycle, and the
tassels hanging from the handgrips were cut from the string
of a balloon.
• Digital Mockup: In several fields, it is common for people
to build a quick mockup by combining parts from existing
objects. Motivated by a police sketch application, a user decided to experiment with this approach for 3D modeling of
faces. The result (shown in the top-row of Figure 17) is a
plausible face created in 15 minutes. It combines five different parts (color-coded in the left-most image of the top row),
highlighting the value of our intelligent scissoring, alignment,
and composition methods. Note that the final face bears little
resemblance to any of the faces from which parts were extracted.
• Art: Motivated by a drawing of science fiction artist Shannon
Dorn (upper left of Figure 18), a user created a 3D model of a
flying centaur with the upper body of a woman, the body of a
horse, and wings from a bird. This example, which could be
included in a video game or animation sequence, leverages all
the features of our system. Yet, it also demonstrates a drawback – we do not provide the deformation tools required to
adjust the horse’s rear legs to match the pose in the image.
This is an issue for future work.
Figure 16: A.J. Meerwald: Creation of this model began with a text search for “ship,”
which produced a row boat (green, bottom) that was used for the hull. Then, to create
the sail, a box was inserted and anisotropically scaled to approximate a sail over the hull,
and a part-in-whole shape search resulted in the triangular sail (red, left), which was
automatically aligned with the box. The keel was found through a similar shape search
with a box positioned below the hull. The model with a good keel (pink, bottom left) was
one big triangle mesh, so the keel was cut with the intelligent scissors. Searching with
the entire shape as the query resulted in sailboats that provided the other sails and masts
(top center and right). The steering wheel (yellow, bottom right) was found through a
text search. The photograph shown in the top-left is courtesy of The Bayshore Discovery
Project.
Figure 17: Police Sketch: Here we show the 3D model of a fictitious person’s head
(green) created by combining parts from other heads in the database (top-left image is
color-coded by parts). Shape-based search was used to find the different models from
which the lips, nose, ears, face, and head were selected. All parts were cut with the
intelligent scissors and joined with blending to create a smooth, watertight model.
Figure 18: Sagittarius: We began with a text search for horse and wings. The wings
were cut off of a griffin (purple, left) with the intelligent scissors, then joined to the body
of the horse (yellow, top right). The head was cut off the horse, and its body was joined
to the upper torso of a woman (red, bottom). Wings were also joined to the woman at
the shoulder. The horse’s mane came from a lion (green, top). The rings on the horse’s
leg were produced by searching for the term belt and selecting one off of a Pilgrim’s hat
(right bottom). The image in the top-left is courtesy of Shannon Dorn.
Figure 19: Radio Flyer Tricycle: Making this model began with a text search for
“tricycle,” which produced one tricycle model (left) which was used as a template. Eventually, all parts of that model were replaced except for the back plate (yellow). A bicycle
seat (bottom) became the seat of the tricycle. The wheel guard came from a motorcycle
(right). The handle bar region was selected and used in a shape based search, which generated motorcycle handle bars (top) that more closely matched the Radio Flyer’s shape.
The tassels came from the string of a balloon (top). The image in the top-left is courtesy
of Radio Flyer, Inc.
9
Conclusions and Future Work
In this paper, we investigate modeling by example, a new paradigm
for creating 3D models from parts extracted from a large database.
We present algorithms for segmenting models into parts interactively, finding models in a database based on the shapes of their
parts, and aligning parts automatically. Experience with our prototype system indicates that it is easy to learn and useful for creating
interesting 3D models.
While this paper takes a small step down a new path, there
are many avenues for future work. An immediate area of possible research is to investigate how sketching interfaces can be
combined with our approach to produce new modeling tools for
novices. In our current system, users can “sketch” new geometry
only with boxes. In the future, it would be interesting to investigate more powerful sketch interfaces and study their interaction
with databases of models. We imagine many types of new tools
in which a user sketches coarsely detailed geometry and then the
system suggests replacement parts from the database.
Another related topic of possible research is to investigate which
sets of 3D modeling commands are best suited for novice users. In
our system, we use a database to simplify the user interface. This
leads us to ask in what other ways 3D modeling systems can become simpler while retaining or replacing their expressive power
as much as possible. This is a big question whose answer will have
impact on whether 3D models become a media for the masses in
the future.
Acknowledgements
We would like to thank Josh Podolak, Benedict Brown, Paul
Calamia, Nathaniel Dirksen, Joseph Mitchell, and the Tiggraph reviewers, who all provided assistance in the preparation of this paper. Viewpoint, Cacheforce, and Jose Maria De Espona donated
commercial databases of polygonal models. The National Science Foundation funded this project under grants CCR-0093343,
CCR-03-06283, IIS-0121446, and DGE-9972930. Ayellet Tal
and Patrick Min were supported in part by the Information Society Technologies ”[email protected]” Network of Excellence Grant
(506766) by the Commission of the European Communities.
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