# <span class=redbold>Poster Session/Reception</span>

Monday, April 3, 2006 - 3:45pm - 5:15pm

Lind 400

**A Minimum Description Length Objective Function for Groupwise**

Non-Rigid Image Registration

Stephen Marsland (Massey University)

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.**Principal Component Geodesics for Planar Shape Spaces**

Stephan Huckemann (Georg-August-Universität zu Göttingen)

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 Newton-type Total Variation Diminishing Flow**

Wolfgang Ring (Karl-Franzens-Universität Graz)

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.**Segmentation of Ultrasound Images with Shape Priors - Application to Automatic Cattle Rib-eye Area Estimation**

Gregory Randall (University of the Republic)Pablo Sprechmann (University of the Republic)

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.**Local Feature Modeling in Image Reconstruction-segmentation**

Hstau Liao (University of Minnesota, Twin Cities)

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.**Model Selection for 2D Shape**

Kathryn Leonard (California Institute of Technology)

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.**A Variational Approach to Image and Video Super-resolution**

Todd Wittman (University of Minnesota, Twin Cities)

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.**A Demo on Shape of Curves**

Washington Mio (Florida State University)

I will present a brief demo on shape geodesics between curves in Euclidean

spaces and a few applications to shape clustering.**Shape Space Smoothing Splines for Planar Landmark Data**

Ian Dryden (University of Nottingham)

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.**Riemannian Metrics on the Space of Solid Shapes**

P. Thomas Fletcher (The University of Utah)

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.**First-Order Modeling and Analysis of Illusory Shapes/Contours**

Yoon Jung (University of Minnesota, Twin Cities)Jianhong Shen (University of Minnesota, Twin Cities)

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.)**Metric Curvatures and Applications**

Emil Saucan (Technion-Israel Institute of Technology)

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.**Bayesian Extraction of Contours in Images Using Gradient Vector Fields and Intrinsic Shape Priors**

Anuj Srivastava (Florida State University)

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.**Statistical Models for Contour Tracking**

Namrata Vaswani (Iowa State University)

(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.**3D Shape Warping based on Geodesics in Shape Space**

Martin Kilian (Technische Universität Wien)

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.**A Newton-type Total Variation Diminishing Flow**

Wolfgang Ring (Karl-Franzens-Universität Graz)

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.**Higher-order regularization of geometries and Mumford-Shah surfaces**

Marc Droske (University of California, Los Angeles)

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.**Using Shape Based Models for Detecting Illusory Contours, Disocclusion, and Finding Nonrigid Level-Curve Correspondences**

Sheshadri Thiruvenkadam (University of California, Los Angeles)

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.**Highly Accurate Segmentation Using Geometric Attraction-Driven Flow in Edge-Regions**

Chang-Ock Lee (Korea Advanced Institute of Science and Technology (KAIST))

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.**Application of PCA and Geodesic 3D Evolution of Initial Velocity**

in Assessing Hippocampal Change in Alzheimer's Disease

Lei Wang (Washington University School of Medicine)

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.**Robust Variational Computation of Geodesics on a Shape Space**

Daniel Cremers (Rheinische Friedrich-Wilhelms-Universität Bonn)

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.**Statistics and Metrology for Geometry Measuring Machine (GEMM)**

Z.Q. John Lu (National Institute of Standards and Technology)

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.**View-invariant Recognition Using Corresponding Object Fragments**

Evgeniy Bart (University of Minnesota, Twin Cities)

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.**Manifold-based Models for Image Processing**

Michael Wakin (Rice University)

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.**Generative Model and Consistent Estimation Algorithms for non-rigid**

Deformation Model

Stephanie Allassonniere (École Normale Supérieure de Cachan)

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.**A Metric Space of Shapes — The Conformal Approach**

Eitan Sharon (Brown University)

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.**Statistical Computing on Manifolds:**

From Riemannian Geometry to Computational Anatomy

Xavier Pennec (Institut National de Recherche en Informatique Automatique (INRIA))

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.**Mumford-Shah with A-Priori Medial-Axis Information**

Matthias Fuchs (Leopold-Franzens Universität Innsbruck)Otmar Scherzer (Leopold-Franzens Universität Innsbruck)

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.**Axial Representation of Shapes Based on Principal Curves**

Yan Cao (Johns Hopkins University)

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.