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Abstracts and Talk Materials
Shape Spaces
April 3 - 7, 2006

The link between Bayesian and variational approaches is well known in the image analysis community in particular in the context of deformable models. However, true generative models and consistent estimation procedures are usually not available and the current trend is the computation of statistics mainly based on PCA analysis. We advocate in this paper a careful statistical modeling of deformable structures and we propose an effective and consistent estimation algorithm for the various parameters (geometric and photometric) appearing in the models.

In this work, invariant object recognition is achieved by learning to compensate for appearance variability of a set of class-specific features. For example, to compensate for pose variations of a feature representing an eye, eye images under different poses are grouped together. This grouping is done automatically during training. Given a novel face in e.g. frontal pose, the model for it can be constructed using existing frontal image patches. However, each frontal patch has profile patches associated with it, and these are also incorporated in the model. As a result, the model built from just a single frontal view can generalize well to distinctly different views, such as profile.

By the "realistic biometric context" of my title, I mean an investigation of well-calibrated images from a moderately large sample of organisms in order to evaluate some nontrivial hypothesis about systematic form-factors (e.g., a group difference). One common approach to such problems today is "geometric morphometrics," a short name for the multivariate statistics of landmark location data. The core formalism here, which handles data schemes that mix discrete points, curves, and surfaces, applies otherwise conventional linear statistical modeling strategies to representatives of equivalence classes of these schemes under similarity transformations or relabeling maps. As this tradition has matured, algorithmic successes involving statistical manipulations and the associated diagrams have directed our community's attention away from a serious underlying problem: Most biological processes operate not on the submanifolds of the data structure but in the embedding space in-between. In that context constructs such as diffeomorphism, shape distance, and image energy are mainly metaphors, however visually compelling, that may have no particular scientific authority when some actual biometrical hypothesis is being seriously weighed. Instead of phrasing this as a problem in the representation of a signal, it may be useful to recast the problem as that of a suitable model for noise (so that signal becomes, in effect, whatever patterns rise above the amplitude of the noise). The Gaussian model of conventional statistics can be derived as an expression of the symmetries of a plausible physical model (the Maxwell distribution in statistical mechanics), and it would be nice if some equally compelling symmetries could be invoked to help us formulate biologically meaningful noise models for deformations.

We have had initial success with a new model of self-similar isotropic noise borrowed from the field of stochastic geometry. In this approach, a deformation is construed not as a deterministic mapping but as a distribution of mappings given by an intrinsic random process such that the plausibility of a meaningful focal structural finding is the same regardless of physical scale. Simulations instantiating this process are graphically quite compelling--their selfsimilarity comes as a considerable (and counterintuitive) surprise--and yet as a tool of data analysis, for teasing out interesting regions within an extended data set, the symmetries (and their breaking, which constitutes the signal being sought) seem quite promising.

My talk will review the core of geometric morphometrics as it is practiced today, sketch the deep difficulties that arise in even the most compelling biological applications, and then introduce the formalisms that, I claim, sometimes permit a systematic circumvention of these problems when the context is one of a statistical data analysis of a serious scientific hypothesis.

This work is joint with K. V. Mardia.

Generalized cylinders model uses hierarchies of cylinder-like modeling primitives to describe shapes. We propose a new definition of axis for cylindrical shapes based on principal curves. In a 2D case, medial axis can be generated from the new axis, and vice versa. In a 3D case, the new axis gives the natural (intuitive) curve skeleton of the shape instead of complicated surfaces generated as medial axis. This is illustrated by numerical experiments on 3D laser scan data.

No abstract.

Parametric shape representations are considered as orbits on an appropriate manifold. The distance between shapes is determined by computing geodesics between these orbits. We propose a variational framework to compute geodesics on a manifold of shapes. In contrast to existing algorithms based on the shooting method, our method is more robust to the initial parameterization, is less prone to self-intersections of the contour. Moreover computation times improve by a factor of about 1000 for typical resolutions.

Second Chances - Friday, April 7

Implicit (level set) representations of shape are known to have several advantages over explicit ones. In particular they do not rely on a specific choice of parameterization and they naturally allow for topological changes of the embedded shapes. In my presentation, I will summarize some recent advances regarding metrics on implicit representations, nonparametric and dynamical shape models for implicit representations, and statistical inference of shapes within a Bayesian framework for segmentation and tracking. These allow, for example, to infer temporally consistent segmentations of an image sequence by computing the most likely embedding function given an input image, and given the embedding functions computed for the previous images.

In this talk, we give an overview of a discrete exterior calculus and some of its multiple applications to computational modeling, ranging from geometry processing to physical simulation. We will focus on discrete differential forms (the building blocks of this calculus) and show how they provide differential, yet readily discretizable computational foundations for shape spaces — a crucial ingredient for numerical fidelity. Parameterization and quad meshing will be stressed as straightforward, yet powerful applications.

Active contours form a class of variational methods, based on nonlinear PDEs, for image segmentation. Typically these methods introduce a local smoothing of edges due to a length minimization or minimization of a related energy. These methods have a tendency to smooth corners, which can be undesirable for tasks that involve identifying man-made objects with sharp corners. We introduce a new method, based on image snakes, in which the local geometry of the curve is incorporated into the dynamics in a nonlinear way. Our method brings ideas from image denoising and simplification of high contrast images - in which piecewise linear shapes are preserved - to the task of image segmentation. Specifically we introduce a new geometrically intrinsic dynamic equation for the snake, which depends on the local curvature of the moving contour, designed in such a way that corners are much less penalized than for more classical segmentation methods. We will discuss further extensions that allow segmentation based on geometric shape priors.

Joint work with A. Bertozzi.

A method for fitting smooth curves through a series of shapes of landmarks in two dimensions is presented using unrolling and unwrapping procedures in Riemannian manifolds. An explicit method of calculation is given which is analogous to that of Jupp and Kent (1987, Applied Statistics) for spherical data. The resulting splines are called shape space smoothing splines. The method resembles that of fitting smoothing splines in Euclidean spaces in that: if the smoothing parameter is zero the resulting curve interpolates the data points, and if it is infinitely large the curve is the geodesic line. The fitted path to the data is defined such that its unrolled version at the tangent space of the starting point is a cubic spline fitted to the unwrapped data with respect to that path. Computation of the fitted path consists of an iterative procedure which converges quickly, and the resulting path is given in a discretized form in terms of a piecewise geodesic path. The procedure is applied to the analysis of some human movement data.

The work is joint with Alfred Kume and Huiling Le.

We formulate the space of solid objects as an infinite-dimensional Riemannian manifold in which each point represents a smooth object with non-intersecting boundary. Geodesics between shapes provide a foundation for shape comparison and statistical analysis. The metric on this space is chosen such that geodesics do not produce shapes with intersecting boundaries. This is possible using only information of the velocities on the boundary of the object. We demonstrate the properties of this metric with examples of geodesics of 2D shapes.

Joint work with Ross Whitaker.

We minimize the Mumford-Shah functional over a space of parametric shape models. In addition we penalize large deviations from a mean shape prior. This mean shape is the average of shapes obtained by segmenting a set of training images. The parametric description of our shape models is motivated by their medial axis representation.

The central idea of our approach to image segmentation is to represent the shapes as boundaries of a medial skeleton. The skeleton data is contained in a product of Lie-groups, which is a Lie-group itself. This means that our shape models are elements of a Riemannian manifold. To segment an image we minimize a simplified version of the Mumford-Shah functional (as proposed by Chan & Vese) over this manifold. From a set of training images we then obtain a mean shape (and the corresponding principal modes) by performing a Principal Geodesic Analysis.

The metric structure of the shape manifold allows us to measure distances from this mean shape. Thus, we regularize the original segmentation functional with a distance term to further segment incomplete/noisy image data.

Computational Anatomy (CA) introduces the idea that shapes may be transformed into each other by geodesic deformations on groups of diffeomorphisms. In particular, the template matching approach involves Riemannian metrics on the tangent space of the diffeomorphism group and employs their projections onto specific landmark shapes, or image spaces. A singular momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the geodesic equation. This isomorphism suggests a new dynamical paradigm for CA, as well as a new data representation.

The main references for this talk are Soliton Dynamics in Computational Anatomy, D. D. Holm, J. T. Ratnanather, A. Trouvé, L. Younes, http://arxiv.org/abs/nlin.SI/0411014 Momentum Maps and Measure-valued Solutions for the EPDiff Equation, D. D. Holm and J. E. Marsden, In The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, 203-235,Progr. Math., 232, J.E. Marsden and T.S. Ratiu, Editors, Birkhäuser Boston, Boston, MA, 2004. Also at http://arxiv.org/abs/nlin.CD/0312048 D. D. Holm and M. F. Staley, Interaction Dynamics of Singular Wave Fronts, at Martin Staley's website, under "Recent Papers" at http://cnls.lanl.gov/~staley/

Currently, principal component analysis for data on a manifold such as Kendall's landmark based shape spaces is performed by a Euclidean embedding. We propose a method for PCA based on the intrinsic metric. In particular for Kendell's shape spaces of planar configurations (i.e. complex projective spaces) numerical methods are derived allowing to compare PCA based on geodesics to PCA based on Euclidean approximation.

Joint work with Herbert Ziezold (Universitaet Kassel, Germany).

A primary goal of Computational Anatomy is the statistical analysis of anatomical variability. A natural question that arises is how dose one define the image of an "Average Anatomy" given a collection of anatomical images. Such an average image must represent the intrinsic geometric anatomical variability present. Large Deformation Diffeomorphic transformations have been shown to accommodate the geometric variability but performing statistics of Diffeomorphic transformations remains a challenge. Standard techniques for computing statistical descriptions such as mean and principal component analysis only work for data lying in a Euclidean vector space. In this talk, using the Riemannian metric theory the ideas of mean and covariance estimation will be extended to non-linear curved spaces, in particular for finite dimensional Lie-Groups and the space of Diffeomorphisms transformations. The covariance estimation problem on Riemannian manifolds is posed as a metric estimation problem. Algorithms for estimating the "Average Anatomical" image as well as for estimating the second order geometrical variability will be presented.

In visual cognition, illusions help elucidate certain intriguing but latent perceptual functions of the human vision system, and their proper mathematical modeling and computational simulation are therefore deeply beneficial to both biological and computer vision. Inspired by existent prior works, the current paper proposes a first-order energy-based model for analyzing and simulating illusory shapes and contours. The lower complexity of the proposed model facilitates rigorous mathematical analysis on the detailed geometric structures of illusory shapes/contours. After being asymptotically approximated by classical active contours (via Lebesgue Dominated Convergece), the proposed model is then robustly computed using the celebrated level-set method of Osher and Sethian with a natural supervising scheme. Potential cognitive implications of the mathematical results are addressed, and generic computational examples are demonstrated and discussed. (Joint work with Prof. Jackie Shen; Partially supported by NSF-DMS.)

In the context of Shape Spaces a warp between two objects becomes a curve in Shape Space. One way to construct such a curve is to compute a geodesic joining the initial shapes. We propose a metric on the space of closed surfaces and present some morphs to illustrate the behavior of the metric.

We propose a highly accurate segmentation algorithm for objects in an image that has simple background colors or simple object colors. There are two main concepts, "geometric attraction-driven flow" and "edge-regions," which are combined to give an exact boundary. Geometric attraction-driven flow gives us the information of exact locations for segmentation and edge-regions helps to make an initial curve quite close to an object. The method can be successfully done by a geometric analysis of eigenspace in a tensor field on a color image as a two-dimensional manifold and a statistical analysis of finding edge-regions.

There are two successful applications. One is to segment aphids in images of soybean leaves and the other is to extract a background from a commercial product in order to make 3D virtual reality contents from many real photographs of the product. Until now, those works have been done by a manual labor with a help of commercial programs such as Photoshop or Gimp, which is time-consuming and labor-intensive. Our segmentation algorithm does not have any interaction with end users and no parameter manipulations in the middle of process.

We derive an intrinsic, quantitative measure of suitability of shape models for any shape bounded by a simple, twice-differentiable curve. Our criterion for suitability is efficiency of representation in a deterministic setting, inspired by the work of Shannon and Rissanen in the probabilistic setting. We compare two shape models, the boundary curve and Blum's medial axis, and apply our efficiency measure to chose the more efficient model for each of 2,322 shapes.

Given some local features (shapes) of interest, we produce images that contain those features. This idea is used in image reconstruction-segmentation tasks, as motivated by electron microscopy .

In such application, often it is necessary to segment the reconstructed volumes. We propose approaches that directly produce, from the tomograms (projections), a label (segmented) image with the given local features.

Joint work with Gabor T. Herman, CUNY.

NIST is developing the Geometry Measuring Machine (GEMM) for precision measurements of aspheric optical surfaces. Mathematical and statistical principles for GEMM will be presented. We especially focus on the uncertainty theory of profile reconstruction from GEMM using nonparametric local polynomial regression. Newly developed metrology results in Machkour-Deshayes et al (2006) for comparing GEMM to NIST Moore M-48 Coordinate Measuring Machine will also be presented.

We discuss some new statistical methods for matching configurations of points in space where the points are either unlabelled or have at most a partial labelling constraining the match. The aim is to draw simultaneous inference about the matching and the transformation. Various questions arise: how to incorporate concommitant information? How to simulate realistic configurations? What are the implementation issues? What is the effect of multiple comparisons when a large data base is used?, and so on. Applications to protein bioinformatics, and image analysis will be described. We will also discuss some open problems and suggest directions for future work.

Groupwise non-rigid registration aims to find a dense correspondence across a set of images, so that analogous structures in the images are aligned. For purely automatic inter-subject registration the meaning of correspondence should be derived purely from the available data (i.e., the full set of images), and can be considered as the problem of learning correspondences given the set of example images. We demonstrate that the Minimum Description Length (MDL) approach is a suitable method of statistical inference for this problem, and give a brief description of applying the MDL approach to transmitting both single images and sets of images, and show that the concept of a reference image (which is central to defining a consistent correspondence across a set of images) appears naturally as a valid model choice in the MDL approach. This poster provides a proof-of-concept for the construction of objective functions for image registration based on the MDL principle.

The L2 or H0 metric on the space of smooth plane regular closed curves induces vanishing geodesic distance on the quotient Imm(S1,R2)/Diff(S1). This is a general phenomenon and holds on all full diffeomorphism groups and spaces Imm(M,N)/Diff(M) for a compact manifold M and a Riemanninan manifold N. Thus we have to consider more complicated Riemannian metrics using lenght or curvature, and we do this is a systematic Hamiltonian way, we derive geodesic equation and split them into horizontal and vertical parts, and compute all conserved quantities via the momentum mappings of several invariance groups (Reparameterizations, motions, and even scalings). The resulting equations are relatives of well known completely integrable systems (Burgers, Camassa Holm, Hunter Saxton).

Second Chances - Monday, April 3

I will present a brief demo on shape geodesics between curves in Euclidean spaces and a few applications to shape clustering.

There are so many Riemannian metrics on the space of curves, it is worthwhile to compare them. I will take one fixed shape and contrast the shape of the unit ball in 5 of these metrics. After that, I want to discuss in more detail one particular Riemannian metric which was first proposed by Younes and has recently been investigated by Mio-Srivastava and by Shah.

The mathematical foundations of invariant signatures for object recognition and symmetry detection are based on the Cartan theory of moving frames and its more recent extensions developed with a series of students and collaborators. The moving frame calculus leads to mathematically rigorous differential invariant signatures for curves, surfaces, and moving objects. The theory is readily adapted to the design of noise-resistant alternatives based on joint (or semi-)differential invariants and purely algebraic joint invariants. Such signatures can be effectively used in the detection of exact and approximate symmetries, as well as recognition and reconstruction of partially occluded objects. Moving frames can also be employed to design symmetry-preserving numerical approximations to the required differential and joint differential invariants.

Based on a Riemannian manifold structure, we have previously develop a consistent framework for simple statistical measurements on manifolds. Here, the Riemannian computing framework is extended to several important algorithms like interpolation, filtering, diffusion and restoration of missing data. The methodology is exemplified on the joint estimation and regularization of Diffusion Tensor MR Images (DTI), and on the modeling of the variability of the brain. More recent developments include new Log-Euclidean metrics on tensors, that give a vector space structure and a very efficient computational framework; Riemannian elasticity, a statistical framework on deformations fields, and some new clinical insights in anatomic variability.

Automatic ultrasound (US) image segmentation is a difficult task due to the important amount of noise present in the images and to the lack of information in several zones produced by the acquisition conditions. In this paper we propose a method that combines shape priors and image information in order to achieve this task. This algorithm was developed in the context of quality meat assessment using US images. Two parameters that are highly correlated with the meat production quality of an animal are the under-skin fat and the rib eye area. In order to estimate the second parameter we propose a shape prior based segmentation algorithm. We introduce the knowledge about the rib eye shape using an expert marked set of images. A method is proposed for the automatic segmentation of new samples in which a closed curve is fitted taking in account both the US image information and the geodesic distance between the evolving and the estimated mean rib eye shape in a shape space. We think that this method can be used to solve many similar problems that arise when dealing with US images in other fields. The method was successfully tested over a data base composed of 600 US images, for which we have two expert manual segmentations.

Joint work with P. Arias, A. Pini, G. Sanguinetti, P. Cancela, A. Fernandez, and A.Gomez.

A new type of geometric flow is derived from variational principles as a steepest descent flow for the total variation functional with respect to a variable, Newton-like metric. The resulting flow is described by a coupled, non-linear system of differential equations. Geometric properties of the flow are investigated, the relation to inverse scale space methods is discussed, and the question of appropriate boundary conditions is addressed. Numerical studies based on a finite element discretization are presented.

A new type of geometric flow is derived from variational principles as a steepest descent flow for the total variation functional with respect to a variable, Newton-like metric. The resulting flow is described by a coupled, non-linear system of differential equations. Geometric properties of the flow are investigated, the relation to inverse scale space methods is discussed, and the question of appropriate boundary conditions is addressed. Numerical studies based on a finite element discretization are presented.

Variational methods are presented which allow to correlate pairs of implicit shapes in 2D and 3D images, to morph pairs of explicit surfaces, or to analyse motion pattern in movies. A particular focus is on joint methods. Indeed, fundamental tasks in image processing are highly interdependent: Registration of image morphology significantly benefits from previous denoising and structure segmentation. On the other hand, combined information of different image modalities makes shape segmentation significantly more robust. Furthermore, robustness in motion extraction of shapes can be significantly enhanced via a coupling with the detection of edge surfaces in space time and a corresponding feature sensitive space time smoothing.

The methods are based on a splitting of image morphology into a singular part consisting of the edge geometry and a regular part represented by the field of normals on the ensemble of level sets. Mumford-Shah type free discontinuity problems are applied to treat the singular morphology both in image matching and in motion extraction. For the discretization a multi scale finite element approach is considered. It is based on a phase field approximation of the free discontinuity problems and leads to effective and efficient algorithms. Numerical experiments underline the robustness of the presented approaches.

A geometric framework for comparing manifolds given by point clouds is first presented in this talk. The underlying theory is based on Gromov-Hausdorff distances, leading to isometry invariant and completely geometric comparisons. This theory is embedded in a probabilistic setting as derived from random sampling of manifolds, and then combined with results on matrices of pairwise geodesic distances to lead to a computational implementation of the framework. The theoretical and computational results described are complemented with experiments for real three dimensional shapes. In the second part of the talk, based on the notion Minimizing Lipschitz Extensions and its connection with the infinity Laplacian, a computational framework for surface warping and in particular brain warping (the nonlinear registration of brain imaging data) is presented. The basic concept is to compute a map between surfaces that minimizes a distortion measure based on geodesic distances while respecting the boundary conditions provided. In particular, the global Lipschitz constant of the map is minimized. This framework allows generic boundary conditions to be applied and allows direct surface-to-surface warping. It avoids the need for intermediate maps that flatten the surface onto the plane or sphere, as is commonly done in the literature on surface-based non-rigid brain image registration. The presentation of the framework is complemented with examples on synthetic geometric phantoms and cortical surfaces extracted from human brain MRI scans.

Joint works with F. Memoli and P. Thompson.

Various notions of metric curvature, such as Menger, Haantjes and Wald were developed early in the 20-th Century. Their importance was emphasized again recently by the works of M. Gromov and other researchers. Thus metric differential geometry was revived as thriving field of research.

Here we consider a number of applications of metric curvature to a variety of problems. Amongst them we mention the following:

(1) The problem of better approximating surfaces by triangular meshes. We suggest to view the approximating triangulations (graphs) as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. Here intrinsic, discrete, metric definitions of differentiable notions such as Gauss, mean and geodesic curvatures are considered.

(2) Employing metric differential geometry for the analysis weighted graphs/networks. In particular, we employ Haantjes curvature, i.e. as a tool in communication networks and DNA microarray analysis.

This represents joint work with Eli Appleboim and Yehoshua Y. Zeevi.

We introduce a metric hyperbolic space of shapes that allows shape classification by similarities. The distance between each pair of shapes is defined by the length of the shortest path continuously morphing them into each other (a unique geodesic). Every simple closed curve in the plane (a "shape") is represented by a 'fingerprint' which is a differentiable and invertible transformation of the unit circle onto itself (a 1D, real valued, periodic function). In this space of fingerprints, there exists a group operation carrying every shape into any other shape, while preserving the metric distance when operating on each pair of shapes. We show how this can be used to define shape transformations, like for instance 'adding a protruding limb' to any shape. This construction is the natural outcome of the existence and uniqueness of conformal mappings of 2D shapes into each other, as well as the existence of the remarkable homogeneous Weil-Petersson metric.

This is a joint work with David Mumford.

Joint work with Shantanu Joshi and Chunming Li.

A novel method for incorporating prior information about typical shapes in the process of object extraction from images, is proposed. In this approach, one studies shapes as elements of an infinite-dimensional, non-linear, quotient space. Statistics of shapes are defined and computed intrinsically using differential geometry of this shape space. Prior probability models are constructed implicitly on tangent bundle of shape space, using past observations. In past, boundary extraction has been achieved using curve-evolution driven by image-based and smoothing vector fields. The proposed method integrates a priori shape knowledge in form of vector fields in the evolution equation. The results demonstrate a significant advantage in segmentation of objects in presence of occlusions or obscuration.

Our previous work developed techniques for computing geodesics on shape spaces of planar closed curves, first with and later without restrictions to arc-length parameterizations. Using tangent principal component analysis (TPCA), we have imposed probability models on these spaces and have used them in Bayesian shape estimation and classification of objects in images. Extending these ideas to 3D problems, I will present a "path-straightening" approach for computing geodesics between closed curves in R3. The basic idea is to define a space of such closed curves, initialize a path between the given two curves, and iteratively straighten it using the gradient of an energy whose critical points are geodesics. This computation of geodesics between 3D curves helps analyze shapes of facial surfaces as follows. Using level sets of smooth functions, we represent any surface as an indexed collection of facial curves. We compare any two facial surfaces by registering their facial curves, and by comparing shapes of corresponding curves. Note that these facial curves are not necessarily planar, and require tools for analyzing shapes of 3D curve.

(This work is in collaboration with E. Klassen, C. Samir, and M. Daoudi)

Illusory contours are intrinsic phenomena in human vision. In this work, we present two different level set based variational models to capture a typical class of illusory contours such as Kanizsa triangle. The first model is based on the relative locations between illusory contours and objects as well as known shape information of the contours. The second approach uses curvature information via Euler's elastica to complete missing boundaries. We follow this up with a short summary of our current work on disocclusion using prior shape information.

Next, we look at the problem of finding nonrigid correspondences between implicitly represented curves. Given two level-set functions, we search for a diffeomorphism between their zero-level sets that minimizes a shape-similarity measure. The diffeomorphisms are generated as flows of vector fields, and curve-normals are chosen as the similarity criterion. The resulting correspondences are symmetric and the energy functional is invariant with respect to rotation and scaling of the curves. We also show how this model can be used as a basis to compare curves of different topologies.

Joint Work with: Tony Chan, Wei Zhu, David Groisser, Yunmei Chen.

The link between Bayesian and variational approaches is well known in the image analysis community in particular in the context of deformable models. However, the current trend is the computation of statistics mainly based on PCA analysis or non-linear extension on manifold using local linearization through the exponential mapping. We will try to show in talk that going from statistics to statistical modelling in the context of deformable models leads to interesting new questions, mainly unsolved, about the statistical modelling itself but also about the derivation of consistent and effective estimation algorithms.

(based on joint work with Yogesh Rathi, Allen Tannenbaum, Anthony Yezzi)

We consider the problem of sequentially segmenting an object(s) or more generally a "region of interest" (ROI) from a sequence of images. This is formulated as the problem of "tracking" (computing a causal Bayesian estimate of) the boundary contour of a moving and deforming object(s) from a sequence of images. The observed image is usually a noisy and nonlinear function of the contour. The image likelihood given the contour (observation likelihood") is often multimodal (due to multiple objects or background clutter or partial occlusions) or heavy tailed (due to outliers or low contrast). Since the state space model is nonlinear and multimodal, we study particle filtering solutions to the tracking problem.

If the contour is represented as a continuous curve, contour deformation forms an infinite (in practice, very large), dimensional space. Particle filtering from such a large dimensional space is impractical. But in most cases, one can assume that for a certain time period, "most of the contour deformation" occurs in a small number of dimensions. This effective basis" for contour deformation can be assumed to be fixed (e.g. space of affine deformations) or slowly time varying. We have proposed practically implementable particle filtering algorithms under both these assumptions.

The information contained in an image ("What does the image represent?") also has a geometric interpretation ("Where does the image reside in the ambient signal space?"). It is often enlightening to consider this geometry in order to better understand the processes governing the specification, discrimination, or understanding of an image. We discuss manifold-based models for image processing imposed, for example, by the geometric regularity of objects in images. We present an application in image compression, where we see sharper images coded at lower bitrates thanks to an atomic dictionary designed to capture the low-dimensional geometry. We also discuss applications in computer vision, where we face a surprising barrier -- the image manifolds arising in many interesting situations are in fact nondifferentiable. Although this appears to complicate the process of parameter estimation, we identify a multiscale tangent structure to these manifolds that permits a coarse-to-fine Newton method. Finally, we discuss applications in the emerging field of Compressed Sensing, where in certain cases a manifold model can supplant sparsity as the key for image recovery from incomplete information.

This is joint work with Justin Romberg, David Donoho, Hyeokho Choi, and Richard Baraniuk.

In large-deformation diffeomorphic metric mapping (LDDMM), the diffeomorphic matching of given images are modeled as evolution in time, or a flow, of an associated smooth velocity vector field V controlling the evolution. The geodesic length of the path in the space of diffeomorphic transformations connecting the given two images defines a metric distance between them. The initial velocity field v0 parameterizes the whole geodesic path and encodes the shape and form of the target image (1). Thus methods such as principal components analysis (PCA) of v0 leads to analysis of anatomical shape and form in target images without being restricted to small-deformation assumption (1, 2). Further, specific subsets of the principal components (eigenfunctions) discriminate subject groups, the effect of which can be visualized by 3D geodesic evolution of the velocity field reconstructed from the subset of principal components. An application to Alzheimer's disease is presented here.

Joint work with: Laurent Younes, M. Fais.

1. Vaillant, M., Miller, M. I., Younes, L. & Trouve, A. (2004) Neuroimage 23 Suppl 1, S161-9. 2. Miller, M. I., Banerjee, A., Christensen, G. E., Joshi, S. C., Khaneja, N., Grenander, U. & Matejic, L. (1997) Statistical Methods in Medical Research 6, 267-299.al Beg, J. Tilak Ratnanather.

Super-resolution seeks to produce a high-resolution image from a set of low-resolution, possibly noisy, images such as in a video sequence. We present a method for combining data from multiple images using the Total Variation (TV) and Mumford-Shah functionals. We discuss the problem of sub-pixel image registration and its effect on the final result.

Following the observation first noted by Michor and Mumford, that H0 metrics on the space of curves lead to vanishing distances between curves, Yezzi and Mennucci proposed conformal variants of H0 using conformal factors dependent upon the total length of a given curve. The resulting metric was shown to yield non-vanishing distance at least when the conformal factor was greater than or equal to the curve length. The motivation for the conformal structure, was to preserve the directionality of the gradient of any functional defined over the space of curves when compared to its H0 gradient. This desire came in part due to the fact that the H0 metric was the consistent choice of metric in all variational active contour methods proposed since the early 90's. Even the well studied geometric heat flow is often referred to as the curve shrinking flow as it arises as the gradient descent of arclength with respect to the H0 metric.

Changing strategies, we have decided to consider adapting contour optimization methods to a choice of metric on the space of curves rather than trying to constrain our metric choice in order to conform to previous optimization methods. As such, we reformulate the gradient descent approach used for variational active contours by utilizing gradients with respect to H1 metrics rather than H0 metrics. We refer to this class of active contours as "Sobolev Active Contours" and discuss their strengths when compared to more classical active contours based on the same underlying energy functionals. Not only due Sobolev active contours exhibit more regularity, regardless of the choice of energy to minimize, but they are ideally suited for applications in computer vision such as tracking, where it is common that a contour to be tracked changes primarily by simple translation from frame to frame (a motion which is almost free for many Sobolev metrics).

(Joint work with G. Sundaramoorthi and A. Mennucci.)

We present a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform feasible gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, with its projections on shape manifolds, and focus in particular on an implementation of the associated equations using iterated Jacobi fields.

We introduce the TUBE connection for domains with finite perimeters; Then a metric and we characterise the necessary condition for the geodesic tube. We obtain a complete metric space of Shapes with non prescribed toplogy. That metric extends the Courant metric developed in the book Shape and Geometry (Delfour and Z) , SIAM 2001.

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