M.S. Thesis Proposal
Robust Modeling of Two-Dimensional Shapes
Massachusetts Institute of Technology
Jared Glover
11th March 2007
1 Introduction 1
1 Introduction
Shape modeling and analysis arises in many different disciplines, with numer- ous applications ranging from computer graphics to archeology to robotics to medicine. The specific problems one encounters in each field are as var- ied as they are important, ranging from questions such as “what is the best way to unload a dishwasher?” in robotics to “is this tumor cancerous?” in medical diagnosis. Thus, no one shape modeling technique will be able to answer all questions in every domain. However, within a specific field, one may formulate a set of guidelines which roughly capture the properties any shape modeling method should have in order to be most useful to problems in that domain. In computer graphics, these properties may be things like ease of use in calculating lighting effects and collisions, ability to represent scanned objects from the real world, and smoothness/resolution. In robotics and computer vision, on the other hand, we care mostly about accuracy and the ability to perform complicated inference tasks, such as shape recognition (“what is it?”) or shape completion (“what does the occluded part of that object look like?”).
In robotics, we also want to use these models to accomplish practical tasks such as navigation and object manipulation, which require action-effect estimation (e.g. “how likely is it that I will drop this object if I place my fingers here, here, and here?” or “what is the probability that I can fit through that small opening in the wall?”). Both of these questions would be trivial (dynamics aside) in a world with no uncertainty, such as inside a computer game. However, in the real world we must deal constantly with uncertainty in not only sensing, but also in modeling and prediction.
In this thesis much of the discussion and methodology will be motivated by problems encountered in robotics–specifically those dealing with uncer- tainty; however, the shape modeling framework and inference capabilities presented are applicable to a wide range of fields and as such should be taken as a tool to be used whenever the circumstances fit.
2 Background
The ability to easily infer the geometry of the world around us is some- thing that humans often take for granted, yet some of the most challenging problems in robotics and computer vision in recent years have dealt with precisely these issues. In mobile robotics, the problem of simultaneous local- ization and mapping (SLAM) has captivated researchers for decades, while the corresponding problem in computer vision–known as structure from mo-
2.1 Discriminative Shape Analysis 2
tion (SFM)–has also received a great deal of attention. In each domain, the goal is to accurately estimate the geometry of the environment around the robot (or camera) from noisy sensor data, while at the same time recovering the trajectory of the robot (camera) through the world. In computer vision, the problem is compounded by the sheer volume of sensory data which must be processed. In both the SLAM and SFM communities, recent breakthrough techniques have increasingly used statistical tools to handle the errors in sens- ing and estimation [32, 10]. By modeling not only the best estimate of the world geometry, but also the covariances of those estimates, robust solutions to SLAM and SFM are becoming more attainable.
Neither SLAM nor SFM are considered to be shape modeling problems in the traditional sense, as they deal with the geometry of whole environments, not individual objects. However, we believe that the lessons from these two problems can and should be applied to modeling the shapes of objects.
2.1 Discriminative Shape Analysis
In computer vision, most work on 2-D shape recognition has fallen short of modeling the uncertainty in sensing and modeling directly. Instead, shape dissimilarity metrics are formed, and one simply compares a new shape to a database of exemplars in order to find the best match. All the work then goes into finding the right shape distance, and typically a very non-trivial transformation is required on the raw geometric data (e.g. an object silhou- ette contour) in order to perform the classification in a more discriminative
“shape space”. In addition, care must be taken in choosing the exemplar shapes so that the size of the classification problem does not blow up with the number of object classes. Such techniques typically work well when the number of classes is small (e.g. in medical imaging), and the ability to dis- criminate between classes (i.e. classification) is the most important problem to be solved.
In general, there are two types of problems on which these discriminative approaches can fail. The first is when the shape of objects within a shape class varies in a non-trivial way (for example, in an articulated object or a biological object, as in Figure 1). In these cases, the same type of object can look very different depending on the instance, and either the shape distance must be modified in order to take these deformations into account, or multiple instances of the same object class must be stored in the shape database in order to encompass the full spectrum of variability within the class.
The second type of problem on which these discriminative approaches to shape analysis can fail is when we want to perform additional inference tasks other than shape recognition, such as shape completion or boundary
2.1 Discriminative Shape Analysis 3
Figure 1: Shape classes with high variance, due to articulation (left) and biology (right).
It is very difficult to construct a single shape metric which will enable robust, accurate discrimination of shape categories for classes such as these.
Figure 2: Boundary estimation. We want to predict the spatial location of every point on an object’s boundary (both hidden and observed).
estimation (Figure 2). In these cases, it is not enough to just recognize what an object is–we must also infer the correct geometry of the new object given the shape class it belongs to (and given a partial and/or noisy estimate of the new shape’s geometry). If shapes within the same shape class all look exactly the same, then shape completion and boundary estimation may be achievable, so long as the transformation from raw geometric data to the discriminative shape space is sufficiently straightforward (i.e. the gradients with respect to shape distance are well behaved). However, if the shape class has too much variation, or the transformation to shape space is too non-linear, accurate recovery of the geometry of objects can be very difficult.
Thus, we will seek primarily generative models for shape analysis, as we spell out next in the thesis statement.
3 Thesis Statement 4
3 Thesis Statement
In cases where shapes vary non-trivially within classes and one wishes to perform additional, geometric inference tasks such as shape completion and boundary estimation, we take the lessons learned from SLAM and SFM and apply statistical shape modeling to achieve robust solutions. In addition, since accurate recovery of object geometry is required, we argue the following:
1. The shape model should be generative, and the transformation from raw geometric data to the underlying shape model should preserve all geometric quantities of interest.
2. All of the raw geometric data, be it laser range measurements or pixel locations, should be used in the shape model.
By (1) we mean that the shape model should respect the geometry of the raw data (e.g. relative angles, distances, etc.), so that accurate geometric inference is easily achievable. By (2) we mean that a complete representation of observed shape should be formed, as this will enable the widest range of shapes to be modeled, and also (more importantly) since the inference tasks we are primarily interested in require precise knowledge of the complete object boundary. In the limiting case, using the raw data itself to represent object shape will satisfy both (1) and (2), although we must still find some way to capture the variability of shapes in order to perform robust shape inference.
4 2-D Shape Models
There are currently three major approaches to 2-D shape modeling. The first approach is to use a set of points, or a point cloud representation. With this model, shapes are represented as a collection of point locations in the plane.
Beyond that, each point cloud model handles inference differently. The most common inference tasks are recognition and registration (alignment). Some models directly compute point correspondences between point clouds, then use either a Euclidean distance (in the simplest cases), or more commonly a Procrustes distance [23] (which is invariant to position, scale, and orienta- tion) or a bending energy [7, 1] (required to warp one shape into the other), in order to compute a shape dissimilarity metric. Correspondences are com- puted either by estimating the best alignment and the point correspondences jointly, as in the Iterated Closest Point algorithm [3], or by forming a pairwise assignment cost function between each pair of points across the two point
4.1 A Simple Model for 2-D Shape 5
clouds and using either bipartite matching [1] or relaxation labeling [30] to find a good matching. Alternatively, an example of work where point corre- spondences are not directly computed is [17], where a Hausdorff distance is computed in order to perform both matching and registration simultaneously.
The second major approach to 2-D shape modeling is to represent the shape of an object by a continuous curve. These models can then be sub- divided into those which represent only closed curves (contours) and those which represent either closed or open curves. The majority of widely-used work in this area has been done on closed contours, as most of the physical objects people wish to model have 2-D extent (although handwriting is a good counter-example). Many popular closed contour models in recent work have used variational methods (e.g. active contours or level sets) to perform image segmentation and tracking tasks [4, 9, 18, 31]. The basic idea in both active contours (snakes) and level sets is to use an energy minimization to evolve an initial curve to fit image intensity data, where energy terms include smoothness and degree to which the contour fits the image intensities. More general curves (both open and closed) have been modeled by convex poly- gon parts [25], fourier descriptors [26], algebraic curves [40, 42], and turning functions [41], among others. Another closed curve model which deserves mention is curvature scale space [29], where shape is modeled by the singu- larities of a curve evolution process as the curve is smoothed by a Gaussian kernel. Latecki’s convex polygon parts model [25] also models shape across curvature scales, and this general idea is a motivating factor for some of the work done in this thesis.
The third approach to 2-D shape modeling is found in shape skeletoniza- tion. Shape skeletons are limited to modeling closed contours, and in many cases can be thought of as simply an approximate dual representation for the object boundary. The advantage to skeleton models is that they can present a hierarchical decomposition of shape (e.g. the spine to the limbs to the fingers and toes), and are widely used on both articulated shapes and on shapes with distinct parts. Some examples of skeleton-based shape models are medial axis [5], shock graphs [36, 38], and shape axis trees [12].
Several good survey articles of 2-D shape modeling exist. For a good start, see [37, 44, 43].
4.1 A Simple Model for 2-D Shape
In order to accomplish the thesis statement goals set out in section 3, one finds great difficulty in applying many of the existing shape representations.
Some methods require a transformation which makes it very difficult or even impossible to recover the original geometry (e.g. fourier descriptors, curva-
4.2 Kendall’s Shape Space 6
ture scale space). Other methods require a great deal of hard decisions to be made in the modeling process (e.g. skeletons, part-based models), mak- ing it very difficult to develop statistical variants of these models. Many of the rest are either extremely unwieldy in practice or have not been demon- strated to perform well on basic shape inference tasks such as classification and retrieval.
In this thesis we will instead use a seemingly simple, yet powerful model–
representing object shape as a set of discrete points around its boundary. The drawbacks to this model are obvious–we are (for now) restricted to closed contours, we cannot represent continuous curves, and as with point clouds we must solve a correspondence problem in order to compare shapes. However, the advantages are (i) it can be made entirely non-parametric (i.e. we can incorporate the pixel or range measurement locations in the model), (ii) we have a natural cyclic ordering of points around the contour, and (iii) most importantly, we can draw upon a vast amount of established literature on the statistical shape modeling of sets of ordered landmarks.
The most influential pioneer of statistical shape modeling was David Kendall, who developed a rigorous mathematical understanding of the topol- ogy of shape spaces, which we discuss next.
4.2 Kendall’s Shape Space
Let us represent a shape z as a set of n points z1, z2, . . . znin some Euclidean space. We will restrict ourselves to two-dimensional points (representing shapes in a plane) such that zi = (xi, yi), although extensions to three di- mensions are feasible. We will assume these points are ordered (so that z can be defined as a vector), and represent a closed contour (such as the letter
”O”, as opposed to the letter “V”). In general, the shape model can be made de facto invariant to point ordering if we know correspondences between the boundary points of any two shapes we wish to compare. Even in the case of closed contours, however, care must still be taken to choose the correct starting point. In order to make our model invariant to changes in position and scale, we can normalize the shape so as to have unit length with centroid at the origin; that is,
z′ = {z′i = (xi− ¯x, yi− ¯y)} (1)
τ = z′
|z′| (2)
4.3 Statistical Shape Modeling 7
We call τ the pre-shape of z. Since τ is a unit vector, the space of all possible pre-shapes of n points is a unit hyper-sphere, S2n−3∗ , called pre-shape space 1. Any pre-shape is a point on the hypersphere, and all rotations of the shape lie on an orbit, O(τ ), of this hypersphere. If we wish to compare shapes using some distance metric between them, the spherical geometry of the shape space requires a geodesic distance rather than Euclidean distance.
Additionally, in order to ensure this distance is invariant to rotation, we define the distance between two shapes τ1 and τ2 as the smallest distance between their orbits:
dp[τ1, τ2] = inf[d(φ, ψ) : φ ∈ O(τ1), ψ ∈ O(τ2)] (3)
d(φ, ψ) = cos−1(φ · ψ) (4)
We call dp the Procrustean metric [22] where d(φ, ψ) is the geodesic dis- tance between φ and ψ. Since the inverse cosine function is monotonically decreasing over its domain, it is sufficient to maximize φ · ψ, which is equiv- alent to minimizing the sum of squared distances between corresponding points on φ and ψ (since φ and ψ are unit vectors). For every rotation of φ there exists a rotation of ψ which will find the global minimum geodesic distance. Thus, to find the minimum distance, we need only rotate one pre- shape while holding the other one fixed. We call the rotated ψ which achieves this optimum (θα∗(τ2)) the orthogonal Procrustes fit of τ2 onto τ1, and the angle α∗ is called the Procrustes fit angle.
Representing the points of τ1 and τ2 in complex coordinates, which nat- urally encode rotation in the plane by scalar complex multiplication, the Procrustes distance minimization can be solved:
dp[τ1, τ2] = cos−1|τ2Hτ1| (5)
α∗ = arg(τ2Hτ1), (6)
where τ2H is the Hermetian, or complex conjugate transpose of the complex vector τ2.
4.3 Statistical Shape Modeling
From the Procrustean metric, which is essentially a least squares distance, one can then proceed to compute an estimate of the mean shape of a set of pre-shapes τ1, . . . , τn by minimizing the sum of squared (Procrustean) dis- tances from each pre-shape to the mean shape, µ∗:
1Following [39], the star subscript is added to remind us that S2∗p−3 is embedded in R2p, not the usual R2p−2.
4.3 Statistical Shape Modeling 8
−0.3 −0.2 −0.1 0 0.1 0.2 0.3
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
(a) Partial contour to be com- pleted
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25
(b) Completion using fork model
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25
(c) Completion using spoon model
−0.3 −0.2 −0.1 0 0.1 0.2 0.3
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
(d) Completion using tool model
Figure 3: Shape classification of partial contours. In the second, third and fourth image, the mean shape of the shape class is shown in blue on the left, and the approximate maximum likelihood completion to the partial contour is shown on the right.
µ∗ = arg inf
kµk=1
X
i
[dp(τi, µ)]2 (7)
In 2-D, this minimization can be done in closed form, and in higher di- mensions iterative algorithms exist to compute the mean shape [2, 15].
In order to estimate the covariance of the shape distribution from the sample pre-shapes τ1, . . . , τn, there are several ways to proceed. Simplest is to rotate each τi to fit the mean shape µ∗ (in the Procrustes sense), and then project the rotated pre-shapes into the tangent space of the pre-shape hy- persphere at the mean shape. Once in tangent space, Principle Components Analysis (PCA) can be used to estimate the linear covariance structure of the shape distribution. However, if the sample shapes vary significantly from each other, and the variance is not well modeled with this linear approxima- tion, one can use a complex Bingham distribution [11], which is essentially a unit-conditioned, zero-mean Gaussian, defined directly on the hypersphere.
Various other hyper-sphere shape distributions exist of increasing complexity,
4.4 Shape Completion 9
which are discussed at length in [39].
4.4 Shape Completion
We now turn to the problem of estimating the complete geometry of an object from an observation of part of its contour (Figure 3). We phrase this as a maximum likelihood estimation problem, estimating the missing points of a shape with respect to the Gaussian tangent space shape distribution defined in the previous section.
Let us represent a shape as:
z=z1 z2
(8) where z1 = m contains the p points of our partial observation of the shape, and z2 contains the n − p unknown points that complete the shape. Given a shape distribution D on n points with mean µ and covariance matrix Σ, and given z1 containing p measurements (p < n) of our shape, our task is to compute the last n − p points which maximize the joint likelihood, PD(z).
In order for us to transform our completed vector, z = (z1, z2)T, into a pre-shape, we must first normalize translation and scale. However, this can- not be done without knowing the last n − p points. Furthermore, the Pro- crustes minimizing rotation from z’s pre-shape to µ depends on the missing points, so any projection into the tangent space (and corresponding likeli- hood) will depend in a highly non-linear way on the location of the missing points. We can, however, compute the missing points z2 given an orientation and scale. This leads to an iterative algorithm that holds the orientation and scale fixed, computes z2 and then computes a new orientation and scale given the new z2. The translation term can then be computed from the completed contour z.
4.5 Correspondences
One potential drawback to contour-based shape analysis (compared to other representations such as point cloud) is that we require correspondences be- tween all points of any two shapes we wish to compare using the Procrustean metric2. While this caveat is clearly the major source of power to our model, it can also be a major difficulty in real-world problems where correspondences can be difficult to identify.
2We also require that the number of points on each shape be the same.
4.5 Correspondences 10
In order to find correspondences between the points of two contours, one typically gives a cost function, C(i, j), for assigning point i on the first con- tour to point j on the second contour, and these costs are usually assumed to all be independent. In addition, we may have constraints on the matching;
for example, we may require a one-to-one matching or an order-preserving matching. Scott and Nowak [34] define the Cyclic Order-Preserving Assign- ment Problem (COPAP) as the problem of finding an optimal matching such that the assignment of corresponding points preserves the cyclic ordering in- herited from the contours. Alternatively, if we don’t require the assignment to preserve contour ordering, yet we do desire a one-to-one matching, the problem may be formulated as a bipartite matching [1] or as a relaxation labeling [30].
The point assignment cost function, C(i, j), can incorporate either local or global shape information, and in certain applications may include richer data, such as appearance in images. The shape context [1] is an example of a cost function which uses global shape information–it compares log-polar histograms of point displacements from point i on one contour and point j on the other contour. Difference in curvature would be an example of a local cost function.
We have developed a novel point assignment cost function, which we call the Procrustean Local Shape Distance (PLSD). The PLSD is a local cost func- tion which uses the geometry of the contour itself, rather than using derived statistics such as curvature or medial axis skeletons, in order to compute a similarity measure between the local shapes of contour points. PLSD is similar in nature to Belongie’s shape contexts, in that both methods directly use the observed geometry of the contour in order to compute the shape signature of a point. However, shape contexts consider the relative global shape with respect to each point, while the PLSD captures only local shape information.
Another attribute of our approach is that the PLSD is a multi-resolution similarity measure–it compares local shapes of varying sizes in order to de- termine the overall match quality. The only other shape models we are aware of which use such a multi-resolution technique are the curvature scale space model [29] and Latecki’s convex polygon parts model [25], where contours are convolved with Gaussian kernels of varying sizes. Also similar to our method is the SIFT feature representation [28], which also use multi-resolution infor- mation to compare features, although SIFT features are computed on images, while the PLSD applies to contours.
Given a point xi on contour x, we define the local neighborhood of size k (for k odd) as:
4.5 Correspondences 11
(a) (b) (c)
(d) (e) (f)
Figure 4: Procrustes shape distance over varying local shape neighborhood sizes, k. dP
is plotted vs. k for neighborhoods around the point marked by “*” of size k = 3...21 for (a)-(c) a good match, and (d)-(f) a bad match.
ηk(xi) = hxi−(k−1)/2, ..., xi, ..., xi+(k−1)/2i (9) The Procrustean Local Shape Distance, dP LS, between two points, xi and yj is the minimum Procrustean shape distance over varying neighborhood sizes, k:
dP LS(xi, yj) = min
k dP[ηk(xi), ηk(yj)] (10)
In Figure 4, we see the Procrustes distance, dP plotted vs. neighborhood size, k, for both good and bad matches. For good matches, this local shape distance curve tends to slope downward initially until the local shape is com- pletely encompassed in the neighborhood, and then rises again as neighboring features are encountered. For bad matches, the curve may still start at a low value, since at the smallest neighborhood sizes the measurement noise may dominate the shape distance, but as the neighborhood grows and more of the geometry is captured, the shape distance will increase.
5 Future Work 12
−0.25−0.2−0.15−0.1−0.05 0 0.05 0.1 0.15 0.2 0.25
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25
−0.25−0.2−0.15−0.1−0.05 0 0.05 0.1 0.15 0.2 0.25
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25
Figure 5: An example of overlapping object contours. Left to right: The original image, the measured contour, the correct contour segmentation.
We compute the PLSD as the minimum of one of these local shape dis- tance curves, although better results may be achieved using machine learning to compute a more discriminative statistic.
In previous work [13], we have developed heuristic-based search algo- rithms for solving the Cyclic Order-Preserving Assignment Problem (CO- PAP) using the PLSD cost function. However, in that work the emphasis was on algorithmic efficiency, not on the quality of the resulting correspon- dences. Thus, future work is needed to focus on an empirical evaluation of PLSD vs. shape contexts for point correspondences.
A fuller discussion of our current methods for statistical shape modeling can be found in [14].
5 Future Work
Although many of the basic algorithms needed for robust shape modeling have been developed in our previous work, we have yet to stitch everything together–that is, starting from sensory data (contours), autonomously solve the correspondence problem, generate statistical shape models, and use these models (in conjunction with additional correspondence algorithms) to per- form shape classification and completion of partial shape contours. Specif- ically missing is a correspondence algorithm for partial contours, as is a solution to the data association/segmentation problem found in Figure 5. In addition, no comparison has been done of any of our algorithms to existing methods in the literature.
For my master’s thesis, I would like to achieve the following goals:
• A. Correspondences
1. Evaluate the PLSD point assignment cost function and variants; com- pare to shape contexts, ICP, etc.
2. Develop and evaluate PLSD-based correspondence algorithms for par- tial contours.
5.1 Timeline 13
• B. Shape Modeling
1. Generate statistical shape models from images of 2-D shapes.
2. Generate statistical shape models from laser range data.
3. Generate view-dependent statistical shape models from images of 3-D shapes.
• C. Shape Recognition / Shape Completion
1. Evaluate full contour shape recognition on all three types of data sets listed above; compare to published performance of existing algorithms (e.g. shock graphs, shape contexts, etc.) on publicly-available shape data sets.
2. Evaluate several iterative methods for shape completion (with respect to tangent space principle components shape models) according to two measures: (i) completion accuracy as a function of occlusion size, and (ii) partial shape classification rate.
• D. Data Association / Segmentation
1. Use shape priors to segment image data.
2. Use shape models to separate overlapping shape contours.
5.1 Timeline
3/12: (A1,B1) Complete initial comparison (bullseye test) of correspon- dence algorithms for shape retrieval on MPEG7 Shape B data set. Collect additional image data sets of 2-D shapes.
3/16: (B1,C1) Complete full evaluation of shape retrieval/recognition of shape models (built with best correspondence algorithm variant from 3/12 task) on all image data sets of 2-D shapes.
3/23: (B3) Complete generation of view-dependent shape models of 3-D objects.
3/30: (Spring Break) (C1) Finish evaluation of shape recognition of view- dependent shape models of 3-D objects.
4/13: (A2) Finish development and implementation of algorithms for par- tial contour correspondences.
REFERENCES 14
4/20: (C2) Complete evaluation of partial shape completion on 2-D shape image data.
4/27: (C2) Complete evaluation of partial shape completion (and view recovery) on 2-D views of 3-D shape data.
* 5/4: (D1,D2) Segment images and overlapping contours with shape pri- ors. (B2) Generate shape models from laser data of chairs, couches, etc.
Test shape recognition, shape completion, and segmentation on laser data.
5/11: Finish thesis draft.
5/18: Submit thesis.
* The tasks for 5/4 are marked as optional, since image segmentation and modeling laser range data are not the main focus of this thesis.
References
[1] Serge Belongie, Jitendra Malik, and Jan Puzicha. Shape matching and object recognition using shape contexts. IEEE Trans. Pattern Analysis and Machine Intelligence, 24(24):509–522, April 2002.
[2] Jos M. F. Ten Berge. Orthogonal procrustes rotation for two or more matrices. Psychometrika, 42(2):267–276, June 1977.
[3] P.J. Besl and N.D. McKay. A method for registration of 3-d shapes.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2):239–256, 1992.
[4] A. Blake and M. Isard. Active Contours. Springer-Verlag, 1998.
[5] H. Blum. Biological shape and visual science. Theoretical Biology, 38:205–287, 1973.
[6] F.L. Bookstein. A statistical method for biological shape comparisons.
Theoretical Biology, 107:475–520, 1984.
[7] Fred L. Bookstein. Principal warps: Thin-plate splines and the de- composition of deformations. IEEE Trans. Pattern Anal. Mach. Intell., 11(6):567–585, 1989.
[8] Norman Cliff. Orthogonal rotation to congruence. Psychometrika, 31(1):33–42, March 1966.
REFERENCES 15
[9] Daniel Cremers, Timo Kohlberger, and Christoph Schnorr. Shape statis- tics in kernel space for variational image segmentation. Pattern Recog- nition, 36(9):1929–1943, 2003.
[10] F. Dellaert, S.M. Seitz, C. Thorpe, and S. Thrun. EM, MCMC, and chain flipping for structure from motion with unknown correspondence.
Machine Learning, 50(1-2), 2003.
[11] I. Dryden and K. Mardia. Statistical Shape Analysis. John Wiley and Sons, 1998.
[12] Davi Geiger, Tyng-Luh Liu, and Robert V. Kohn. Representation and self-similarity of shapes. IEEE Trans. Pattern Anal. Mach. Intell., 25(1):86–99, 2003.
[13] Jared Glover, Christian Uldall Pedersen, and Erik Taarnhøj. Solving the cyclic order preserving assignment problem. Advanced Algorithms (6.854) Final Project, December 2006.
[14] Jared Glover, Daniela Rus, Nicholas Roy, and Geoff Gordon. Robust models of object geometry. In Proceedings of the IROS Workshop on From Sensors to Human Spatial Concepts, Beijing, China, 2006.
[15] J. C. Gower. Generalized procrustes analysis. Psychometrika, 40(1):33–
51, March 1975.
[16] John R. Hurley and Raymond B. Cattell. The procrustes program: Pro- ducing direct rotation to test a hypothesized factor structure. Behavioral Science, 7:258–261, 1962.
[17] D.P. Huttenlocher, G.A. Klanderman, and W.A. Rucklidge. Comparing images using the hausdorff distance. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(9):850–863, 1993.
[18] M. Isard and A. Blake. Contour tracking by stochastic propagation of conditional density. In European Conf. Computer Vision, pages 343–356, 1996.
[19] V. Jain and H. Zhang. Robust 2d shape correspondence using geodesic shape context. Pacific Graphics, pages 121–124, May 2005.
[20] M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz. Rotation invariant spherical harmonic representation of 3d shape descriptors, 2003.
REFERENCES 16
[21] D.G. Kendall. The diffusion of shape. Advances in Applied Probability, 9:428–430, 1977.
[22] D.G. Kendall. Shape manifolds, procrustean metrics, and complex pro- jective spaces. Bull. London Math Soc., 16:81–121, 1984.
[23] D.G. Kendall, D. Barden, T.K. Carne, and H. Le. Shape and Shape Theory. John Wiley and Sons, 1999.
[24] Walter Kristof and Bary Wingersky. Generalization of the orthogonal procrustes rotation procedure to more than two matrices. In Proceedings, 79th Annual Convention, APA, pages 89–90, 1971.
[25] L. J. Latecki and R. Lak¨amper. Contour-based shape similarity. In Proc.
of Int. Conf. on Visual Information Systems, volume LNCS 1614, pages 617–624, June 1999.
[26] C. C. Lin and R. Chellappa. Classification of partial 2-d shapes using fourier descriptors. IEEE Trans. Pattern Anal. Mach. Intell., 9(5):686–
690, 1987.
[27] Haibin Ling and David W. Jacobs. Using the inner-distance for clas- sification of articulated shapes. IEEE Computer Vision and Pattern Recognition, 2:719–726, June 2005.
[28] David G. Lowe. Object recognition from local scale-invariant features.
In Proc. of the International Conference on Computer Vision ICCV, pages 1150–1157, 1999.
[29] F. Mokhtarian and A. K. Mackworth. A theory of multiscale curvature- based shape representation for planar curves. In IEEE Trans. Pattern Analysis and Machine Intelligence, volume 14, 1992.
[30] S. Ranade and A. Rosenfeld. Point pattern matching by relaxation.
Pattern Recognition, 12:269–275, 1980.
[31] Tammy Riklin-Raviv, Nahum Kiryati, and Nir Sochen. Prior-based seg- mentation by projective registration and level sets. In Proc. of the Inter- national Conference on Computer Vision ICCV, volume 1, pages 204–
211, Los Alamitos, CA, USA, 2005. IEEE Computer Society.
[32] W. Burgard S. Thrun and D. Fox. Probabilistic Robotics. MIT Press, Cambridge, MA, 2005.
REFERENCES 17
[33] Sch¨onemann. A generalized solution of the orthogonal procrustes prob- lem. Psychometrika, 31(1):1–10, March 1966.
[34] C. Scott and R. Nowak. Robust contour matching via the order pre- serving assignment problem. IEEE Transactions on Image Processing, 15(7):1831–1838, July 2006.
[35] Thomas Sebastian, Philip Klein, and Benjamin Kimia. On aligning curves. PAMI, 25(1):116–125, January 2003.
[36] Thomas Sebastian, Philip Klein, and Benjamin Kimia. Recognition of shapes by editing their shock graphs. volume 26, 2004.
[37] Thomas B. Sebastian and Benjamin B. Kimia. Curves vs skeletons in object recognition. Signal Processing, 85(2):247–263, February 2005.
[38] Kaleem Siddiqi, Ali Shokoufandeh, Sven J. Dickinson, and Steven W.
Zucker. Shock graphs and shape matching. In ICCV, pages 222–229, 1998.
[39] C. G. Small. The statistical theory of shape. Springer, 1996.
[40] J. Subrahmonia, D. B. Cooper, and D. Keren. Practical reliable bayesian recognition of 2d and 3d objects using implicit polynomials and algebraic invariants. IEEE Trans. on Pattern Analysis and Machine Intelligence, 18(5), May 1996.
[41] Mirela Tanase and Remco C. Veltkamp. Part-based shape retrieval. In MULTIMEDIA ’05: Proceedings of the 13th annual ACM international conference on Multimedia, pages 543–546, New York, NY, USA, 2005.
ACM Press.
[42] Jean-Philippe Tarel and David B. Cooper. The complex representa- tion of algebraic curves and its simple exploitation for pose estimation and invariant recognition. IEEE Trans. Pattern Anal. Mach. Intell., 22(7):663–674, 2000.
[43] R. C. Veltkamp and L. J. Latecki. Properties and performance of shape similarity measures. In 10th IFCS Conf. Data Science and Classification, 2006.
[44] Remco C. Veltkamp. Shape matching: Similarity measures and algo- rithms. Technical Report UU-CS-2001-03, Utrecht University, 2001.
