Mathematics and Image Analysis

Paris, September 18-21, 2006

A high level scientific workshop entitled Mathematics and Image Analysis will be held in Paris in September 2006. This conference is organised by GDR MSPC with support of Universite Paris Dauphine, INRIA, Thales Air Defence and DGA. The scientific program will include invited conferences at the interface between researches in applied mathematics (PDE's, Statistical Methods, Wavelets, Level sets, Variational methods,...) and new developments in various areas of computer vision, related to mathematical topics including Shape, Deformations, Motion, Restoration, Invariants, Scale-space, Information Theory, ...

The workshop venue should be at University Paris Dauphine in the west part of Paris.
Registration information is available in french or english.
Talks will be given in either in English or French, according to preference of the speaker.

This document can be found on web page http://www.ceremade.dauphine.fr/~cohen/mia2006 with links to complete papers and slides for most speakers.

To Subscribe to the diffusion list for GDR MSPC
send email to "cohen - at - ceremade.dauphine.fr”

General Chair
Laurent Cohen

Organizing and Scientific Commitee

Frédéric Barbaresco
Laurent Cohen
Rachid Deriche
Alain Trouvé
Laurent Younes

Shape Tube Metric and Geodesic Characterisation

  (Pretty-print pdf file for) SCHEDULE: 

All talks will take place in Amphi 8, second floor.
Registration and Breakfast (on the first day) and Coffee breaks will be complimentary in "Bar des Etudiants" next to Amphi 8.

Monday, September 18th, 2006

09:30 - 10:00 Registration Breakfast

10:00 - 11:00 S. Kichenassamy The mathematical analysis of the Perona-Malik

equation and its practical impact

11:00 - 12:00 Jean Ponce Geometry and 3D computer vision: What we (kind of)

know how to do, what we don't, and why anyone should care

12:00 - 14:00 DEJEUNER - LUNCH

14:00 - 15:15 Baba Vemuri A novel mathematical model for the

diffusion weighted MR signal reconstruction

15:15 - 15:45 Maxime Descoteaux Processing High Angular Resolution Diffusion

Imaging Data to Recover Crossing Fibers

15:45 - 16:15 Pause Cafe - Coffee Break

16:15 - 16:45 Bernhard Burgeth Mathematical Morphology for Matrix-Fields: Ordering vs PDE

16:45 - 17:15 Nir Sochen Geometric flows over Lie Groups

17:15 - 18:15 Xavier Pennec Statistical Computing on Manifolds: From

Riemannian Geometry to Computational Anatomy

Tuesday, September 19th, 2006

09:00 - 11:00 Jean-P. Zolesio Shape Tube Metric and Geodesic Characterisation

11:00 - 11:15 Pause Cafe - Coffee Break

11:15 - 12:30 Stephane Lafon Diffusion geometries for dimensionality reduction in image analysis

12:30 - 14:00 DEJEUNER - LUNCH

14:00 - 15:30 D. Terzopoulos Deformable and Functional Models in Medical Image Analysis

15:30 - 15:45 Pause Cafe - Coffee Break

15:45 - 16:15 Leo Grady Computing Exact Discrete Minimal Surfaces: Extending and Solving

the Shortest Path Problem in 3D with Application to Segmentation

16:15 - 16:45 Jean-Ph. Pons Upgrading the level set method: point correspondence,

topological constraints and deformation priors

16:45 - 17:15 David Tschumperle Fast Anisotropic Smoothing of Multi-Valued

Images using Curvature-Preserving PDE's

17:15 - 17:45 Stanley Durrleman Definition of anisotropic de-noising operators through

sectional curvature, wide range of applications from

gray-level images to high resolution Doppler spectrum

Wednesday, September 20th, 2006

9:45 - 10:45 Kaleem Siddiqi Medial Representations

10:45 - 11:30 Pedro Felzenszwalb Representation and Detection of Shapes in Images

11:30 - 12:00 Samir Chafik Analysis of Facial Shapes

12:00 – 12:30 Tammy Riklin-Raviv Shape Based Segmentation

12:15 - 14:00 DEJEUNER - LUNCH

14:00 - 15:00 Luminita Vese Meyer’s models for image decomposition and

computational approaches

15:00 - 15:30 Pierre Weiss Some applications of L infinite norms in image processing.

15:30 - 16:15 Didier Auroux Image restoration and classification by topological

asymptotic expansion

16:15 – 16:45 He Lin MR Image reconstruction by using the iterative refinement method and nonlinear

inverse scale space methods

Thursday, September 21st, 2006

09:45 - 11:15 Ch. Schnoerr Variational Analysis of Fluid Flow Image Sequences

11:15 - 12:15 Tal Nir Over-Parameterized Variational Optical Flow

12:15 - 14:00 DEJEUNER - LUNCH

14:00 - 15:00 Michael Elad Sparse and Redundant Signal Representation,

and its Role in Image Processing

15:00 - 16:00 Jean-Luc Starck Morphological Component Analysis

16:00 - 17:00 To be confirmed

All talks will take place in Amphi 8, second floor.
Registration and Breakfast (on the first day) and Coffee breaks will be complimentary in "Bar des Etudiants" next to Amphi 8.
Lunch is not provided by the conference. Participants are free to get lunch from different places inside (Ground floor/Rez-de-Chaussée) or outside the university. Many restaurants can be found by taking the Bus PC1 (accross the street from the university) one or two stops away to Porte Maillot or Porte des Ternes or the Metro to Victor Hugo or Etoile one or two stations away. Across the street from university you can also find restaurant ``K'fe court'' with the tennis club.

Abstracts

Didier AUROUX
Image restoration and classification by topological asymptotic expansion.
Authors : Didier AUROUX, Mohamed MASMOUDI, Lamia BELAID
Laboratoire MIP , Université Paul Sabatier Toulouse 3
31062 Toulouse cedex 9, France
Emails: {auroux,masmoudi}@mip.ups-tlse.fr
and
ENIT-LAMSIN
BP37, 1002 Tunis Belvédère , Tunisia
Email: lamia.belaid@esstt.rnu.tn

Web page :
http://mip.ups-tlse.fr/~auroux

SLIDES
Abstract :
We present in this talk a new way for modeling and solving image restoration and classification problems, the topological gradient method. This method is considered in the frame of variational approaches and the minimization of potential energy with respect to conductivity. The numerical experiments show the efficiency of the topological gradient approach. The image is most of the time restored or classified at the first iteration of the optimization process. Moreover, the computational cost of this iteration is reduced drastically using spectral methods. We also propose an algorithm which provides the optimal classes (number and values) for the unsupervised regularized classification problem.

PAPER

A few references are availaible through this link:
http://mip.ups-tlse.fr/~auroux/prod_en.php

Bernhard Burgeth
Mathematical Morphology for Matrix-Fields: Ordering vs PDE
Authors: B. Burgeth(*), S. Didas, A. Bruhn, J. Weickert, W. Welk
Affiliation of all authors:
Faculty of Mathematics and Computer Science
Saarland University, Building E1 1 66041 Saarbrücken Germany

(*) Current affiliation of first author:
Eindhoven University of Technology
Department of Biomedical Engineering
WH 2.110 NL-5600 MB Eindhoven The Netherlands
Homepage: http://www.mia.uni-saarland.de/burgeth/index.shtml

SLIDES
Abstract: Data in the form of matrix-fields are becoming increasingly important in image processing. For instance, diffusion tensor magnetic resonance imaging (DT-MRI) is an modern medical image acquisition technique that results in such a data type. These data have to be filtered and analysed. Hence it is desirable to have the tools of mathematical morphology at one`s disposal. In this talk we explain how fundamental morphological operations, starting from dilation and erosion and continuing to morphological derivatives, can be extended to the setting of matrices. Two approaches will be presented, one is based on the Loewner ordering, the other one utilises variants of nonlinear PDEs. Finally we report on experiments performed with real 3D DT-MRI data.

A PAPER covering the material of this talk to some extend is
B. Burgeth et al., Morphology for Tensor Data: Ordering versus PDE-Based Approach. Technical Report No. 162, Department of Mathematics, Saarland University, Saarbrücken, Germany.
Accepted for publication in Image and Vision Computing.
http://www.mia.uni-saarland.de/Publications/burgeth_pp162.pdf

Samir Chafik chafik @ enic.fr

Geometrical Analysis of Facial Surfaces.

LIFL/ Universite de Lille 1, France

http://www.enic.fr/people/chafik/

Joint works with Anuj Srivastava (Florida State University) and Mohamed Daoudi (LIFL/ Univ Lille1 France).

SLIDES

In recent years, there has been an increasing interest in analyzing shapes of objects. This research is motivated in part by the fact that shapes of objects form an important feature for characterizing them, with application in recognition, tracking and classification. Since most objects of interest are 3D objects, and 3D observations of objects using laser scans are becoming readily available, an important goal is to analyze shapes of two-dimensional surfaces in R^3. In particular, given two facial surfaces (3D faces in this work ), the task is to constructing a path between their shapes showing the deformation between the surface reached to the target surface, and to quantify differences between them.

In this talk, We model a human face called facial surface, as a two-dimensional smooth connected manifold, and we present a differential-geometric technique for constructing a path between two different facial surfaces. We will also discuss our previous work on 3D face recognition using shapes of facial curves.

Maxime Descoteaux
Maxime.Descoteaux @ sophia.inria.fr
Processing High Angular Resolution Diffusion Imaging Data to  Recover Crossing Fibers
Maxime Descoteaux, Rachid Deriche
Odyssee Team, INRIA Sophia Antipolis / ENPC-Paris / ENS- ULM Paris, France
web page: http://www-sop.inria.fr/odyssee/team/Maxime.Descoteaux/index.en.html

SLIDES
Abstract:
Diffussion MRI is a Magnetic Resonance Imaging (MRI) modality able to  non invasively quantify in vivo the diffusion of water molecules in  biological tissues such as the white matter in the brain. This  relatively new imaging modality, pioneered twenty years ago by Denis  Le Bihan, acquires at each voxel, image intensities, referred to as  diffusion, related to the relative mobility of endogenous tissue  water molecules that reflect the structure of the underlying  biological tissues at a microscopic scale, well beyond the usual  image resolution.  In 1994, Peter Basser, together with J. Mattiello  and D.LeBihan, introduced the formalism of the Diffusion Tensor (DT)  and what is known as DT-MRI. P. Basser proposed to characterize the  orientation dependence of diffusion by an effective self-diffusion  tensor given by a 3x3 symmetric positive definite tensor and to  estimate it directly from the signal intensities.  Due the well-known  limitations of DT-MRI, high angular resolution diffusion imaging  (HARDI)is currently of great interest to characterize voxels  containing multiple fiber crossings. In particular, Q-ball imaging  (QBI) is now a popular reconstruction method to obtain the  orientation distribution function (ODF) of these multiple fiber  distributions. The latter captures all important angular contrast by  expressing the probability that a water molecule will diffuse into  any given solid angle. However, QBI and other high order spin  displacement estimation methods involve non-trivial numerical  computations and lack a straightforward regularization process. In  this talk, a simple linear and regularized analytic solution for the  Q-ball reconstruction of the ODF is developed.  First, the signal is  modeled with a physically meaningful high order spherical harmonic  series by incorporating the Laplace-Beltrami operator in the  solution. This leads to a mathematical simplification of the Funk- Radon transform using the Funk-Hecke theorem.  In doing so, a fast  and robust model-free ODF approximation is obtained.  Comparative  results are presented against Tuch's original QBI technique on a  biological phantom and on in-vivo human brain data.  Finally, we  discuss and explore interesting applications of the  spherical  harmonics description of spherica functions.

Paper A FAST AND ROBUST ODF ESTIMATION ALGORITHM IN Q-BALL IMAGING

TR: http://www.inria.fr/rrrt/rr-5768.html
Most recent articles on our research can be found here:
http://www-sop.inria.fr/odyssee/team/Maxime.Descoteaux/pages/most_recent.html

Stanley Durrleman
Definition of anisotropic de-noising operators through sectional curvature,

wide range of applications from gray-level images to high resolution Doppler spectrum
Authors : Stanley Durrleman, Frédéric Barbaresco

SLIDES

Abstract Removing noise in measured physical data is a task of great importance in a wide range of scientific fields : satellite imaging, medical imaging, radar signal processing... The known de-noising methods in the literature are often defined only for a particular application and, as far as we know, none are able to define a de-noising process that does not make any assumptions about the type of data. That's why we aim at defining differential operators that could be defined for data of any dimension in the real and complex spaces. These operators will be anisotropic in order to preserve the geometrical information contained in the data like edges or discontinuities. Moreover, there is rarely a canonical way to represent the data and since scientists are used to writing data in several coordinate systems, the operators will be invariant under a change of data parametrization as well.

Our approach is based on a geometrical model of noise resting on the sectional curvature. This geometrical growth enables us to distinguish points of noise from points of edges or discontinuities. Our method consists then in minimizing the total squared sectional curvature in the images. We first apply our ideas in the case of gray-level images for which the sectional curvature is the Gaussian curvature and have therefore well-known geometrical interpretations. We apply afterwards the method to de-noise radar Doppler spectrum, proving how generic it could be.

PAPER

References
F.Barbaresco, Information Intrinsic Geometric Flows, MAXENT'06 Conf. Paris, July 2006, http://djafari.free.fr/maxent2006
F.Barbaresco, Etude et extension des flots de Ricci, Kahler-Ricci et Calabi dans le cadre du traitement de l'image et de la geometrie de l'information, GRETSI'05, Louvain la Neuve, Septembre 2005
A.Sapira, N.Sochen, R.Kimmel, Geometric Filter, Diffusion Flows and Kernels in Image processing, Handbook of Geometric Computing, Springer-Verlag, Heidelberg, pp 203-230, February 2005

Michael Elad

elad @ cs.technion.ac.il

Sparse and Redundant Signal Representation, and its Role in Image Processing
The CS department, the Technion, Israel

http://www.cs.technion.ac.il/~elad

SLIDES

Abstract: In signal and image processing, we often use transforms in order to simplify
operations or to enable better treatment to the given data. A recent trend
in these fields is the use of overcomplete linear transforms that lead to a
sparse description of signals. This new breed of methods is more difficult
to use, often requiring more computations. Still, they are much more
effective in applications such as signal compression and inverse problems.
In fact, much of the success attributed to the wavelet transform in recent
years, is directly related to the above-mentioned trend. In this talk we
will present a survey of this recent path of research, and its main results.
We will discuss both the theoretic and the applicative sides to this field.
No previous knowledge is assumed (... just common sense, and little bit of
linear algebra).
References: see the above webpage.

Pedro Felzenszwalb
Representation and Detection of Shapes in Images
Assistant Professor
Department of Computer Science
University of Chicago
http://people.cs.uchicago.edu/~pff/
-----------------------------------  

SLIDES

The study of shape is a recurring theme in computer vision.  For
example, shape is one of the main sources of information that can be
used for object recognition.  In this talk I will describe how
two-dimensional shapes can be represented using triangulated polygons.
This representation has important properties both from a perceptual
and a computational point of view.  I will consider the problem of
matching deformable templates to images and give an efficient
algorithm to solve the problem in a wide variety of situations.  I
will also describe a stochastic grammar that can generate arbitrary
triangulated polygons while capturing Gestalt principles of shape
regularity.  Intuitively the grammar tends to generate shapes that
have smooth boundaries and a nice decomposition into almost symmetric
parts.  I will illustrate how this grammar can be used as a generic
model for detecting objects in images.

There is a paper covering some of the material in my talk at:
http://people.cs.uchicago.edu/~pff/papers/shapesJ.pdf

Leo Grady
Computing Exact Discrete Minimal Surfaces: Extending and Solving the Shortest Path Problem in 3D with Application to Segmentation
Siemens Corporate Research,
Department of Imaging and Visualization,
Princeton NJ 08540
http://cns.bu.edu/~lgrady/
SLIDES

Shortest path algorithms on weighted graphs have found widespread use in the computer vision literature. Although a shortest path may be found in a 3D weighted graph, the character of the path as an object boundary in 2D is not preserved in 3D. An object boundary in three dimensions is a (2D) surface. Therefore, a discrete minimal surface computation is necessary to extend shortest path approaches to 3D data in applications where the character of the path as a boundary is important. This minimal surface problem finds natural application in the extension of the intelligent scissors/live wire segmentation algorithm to 3D. In this paper, the discrete minimal surface problem is both formulated and solved on a 3D graph. Specifically, we show that the problem may be formulated as a linear programming problem that is computed efficiently with generic solvers. References:
1) Leo Grady and Eric L. Schwartz, "Isoperimetric Graph Partitioning for Image Segmentation", IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 28, no. 3, pp. 469-475, March 2006.
Paper
2) Leo Grady, "Random Walks for Image Segmentation", accepted to IEEE Trans. on Pattern Analysis and Machine Intelligence, 2006. Paper

Satyanad KICHENASSAMY
The mathematical analysis of the Perona-Malik equation and its practical impact

Laboratoire de Mathématiques (UMR 6056)
   CNRS and Universite de Reims Champagne-Ardenne
   Moulin de la Housse, B.P. 1039
   F-51687 Reims Cedex 2
   FRANCE
Web page:   http://perso.orange.fr/kichenassamy/

SLIDES
Abstract: The Perona-Malik equation (PM) is a familiar tool in image processing, which has proved useful after regularization. However, we show that it can also be efficient in some cases without regularization, with an appropriate choice of parameters. Our theory of generalized solutions may be used as a guide to a reliable implementation of PM in concrete situations. After reviewing the derivations leading to PDEs of PM type, we assess their practical impact on segmentation or denoising procedures.

Stephane Lafon
Diffusion geometries for dimensionality reduction in image analysis.
Google Inc., Mountain View, CA 94043, USA
http://www.math.yale.edu/~sl349/

SLIDES

Various questions related to image analysis involve large inverse problems and high-dimensional data sets. Consequently, the typical complexity of these problems is very high, and the common way to deal with this issue is to use regularization (e.g. functional minimization) or enforcement of a prior (e.g. of a statistical model). Direct dimensionality reduction is traditionally performed using global linear methods such as Principal Component Analysis or Multi-Dimensional Scaling. The impact of such techniques is generally limited as one tries to fit a global linear representation to the data. Although there is some empirical evidence that many "real-life" data sets have a low intrinsic dimensionality (for instance in collections of image patches), the nonlinearity of these data structures needs to be taken into account for an efficient dimensionality reduction. In this talk, I present a powerful framework for dimension reduction, based on the spectral properties of certain Markov chains constructed on the data. This set of data-driven techniques unifies ideas from recent advances in nonlinear dimension reduction, as well as from spectral clustering and harmonic analysis. I explain how to construct a system of coordinates to parametrize data sets, and to obtain meaningful low-dimensional representations. These diffusion coordinates organize data sets based on their intrinsic geometry, hence capturing the underlying nonlinear structures. I also introduce an explicit metric induced by the diffusion coordinates on the data, namely, the diffusion distance. This distance generalizes PageRank and the notion of proximity that it defines proves extremely useful in the context of classification and pattern recognition. I show that diffusion geometries can be used for data fusion and integration of sensors. The multiscale perspective and the problem of out-of-sample extension are also addressed. All these ideas are illustrated with examples from image analysis (inpainting, denoising, segmentation), pattern recognition, graph matching and document classification.
References available at: http://www.math.yale.edu/~sl349/publications/publications.htm

PAPER1 to appear
PAPER2 to appear

He Lin

MR Image reconstruction by using the iterative refinement method and nonlinear

inverse scale space methods

UCLA

www.math.ucla.edu/~helin

Authors: Lin He, Ti-Chiun Chung, Stanley Osher, Tong Fang and Peter Speier

Affiliation: UCLA Mathematics Department, for Stanely Osher, Lin He

Siemens Corporate Research, Princeton, for Ti-Chiun Chang, Tong Fang

Siemens AG Med, Erlangen, Germany for Peter Speier

Paper: ftp://ftp.math.ucla.edu/pub/camreport/cam06-35.pdf

SLIDES

Abstract: Magnetic resonance imaging (MRI) reconstruction from sparsely sampled

data has been a dificult problem in medical imaging field. We approach this problem by

formulating a cost functional that includes a constraint term that is imposed by the raw

measurement data in k-space and the L1 norm of a sparse representation Aof the reconstructed image. The

sparse representation is usually realized by total variational regularization and/or wavelet

transform. We have applied the Bregman iteration to minimize this functional to recover ¯ner scales in our

recent work. Here we propose nonlinear inverse scale space methods in addition to the iterative

refinement procedure.

Numerical results from the two methods are presented and it shows that the

nonlinear inverse scale space method is a more efficient algorithm than the iterated refinement method.

Refrences: 1. E. J. Candès and J. Romberg (2004). Practical Signal Recovery from

Random Projections

http://www.acm.caltech.edu/~emmanuel/papers/PracticalRecovery.pdf

2. E. J. Candès, J. Romberg and T. Tao (2005). Stable Signal Recovery from

Incomplete and Inaccurate Measurement,

http://www.acm.caltech.edu/~emmanuel/papers/StableRecovery.pdf

3. S. Osher, M. Burger, D. Goldfarb, J.-J. Xu, and W. Yin, An iterative

regularization

method for total variation based image restoration, Multiscale Modeling and

Simulation, 4

(2005), pp. 460-489.

ftp://ftp.math.ucla.edu/pub/camreport/cam04-13.pdf

4. M. Burger, G. Gilboa, S. Osher, and J.-J. Xu, Nonlinear inverse scale space

methods,

Comm. Math. Sci., 4 (2006), pp. 175-208.

ftp://ftp.math.ucla.edu/pub/camreport/cam05-66.pdf

Tal Nir.

Over-Parameterized Variational Optical Flow
Joint works with Alfred M. Bruckstein and Ron Kimmel.
Computer Science Department, Technion, Haifa 32000, Israel

SLIDES
Abstract:
We introduce a novel optical flow estimation process based on a spatio-temporal model with varying coefficients multiplying a set of basis functions at each pixel. Previous optical flow estimation methodologies did not use such an over-parameterized representation of the flow field as the problem is ill-posed even without introducing any additional parameters: Neighborhood based methods like Lucas-Kanade determine the flow in each pixel by constraining the flow to be constant in a small area. Modern variational methods represent the optic flow directly via its x and y components at each pixel. The benefit of over-parameterization becomes evident in the smoothness term, which instead of directly penalizing for changes in the optic flow, integrates a cost on the deviation from the assumed optic flow model. Previous variational optic flow techniques are special cases of the proposed method, used in conjunction with a constant flow basis function. Experimental results with the novel flow estimation process yielded significant improvements with respect to the best results published so far.

Xavier Pennec.

Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy
Joint works with Pierre Fillard, Vincent Arsigny and Nicholas Ayache.

Asclepios Team, INRIA Sophia Antipolis.
2004 Route des Lucioles, BP93, F-06902 Sophia Antipolis Cedex
http://www-sop.inria.fr/asclepios/personnel/Xavier.Pennec/

SLIDES
Computational anatomy is an emerging discipline that aim at analysing and modeling the biological variability of the human anatomy. The goal in not only to model the normal variations among a poulation, but also discover morphological differences between normal and pathological populations, and possibly to dtect, model and classify the pathologies from structural anomalities. To reach this goal, the method is to identify anatomically representative geometric features (points, tensors, curves, surfaces, volume transformations), and to describe their statistical distribution. This can be done for instance via a mean shape and covariance structure after a group-wise matching. Then, in order to compare populations, we need to compare feature distributions and to test for statistical differences.

Unfortunately, geometric features often belong to manifolds that are not vector spaces.  Based on a Riemannian manifold structure, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and test. We also provide an efficient algorithm to compute the mean value, and tractable approximations (with their limits) of the generalized normal law for small variances. This show that we can effectively implement and work with these definitions.

Then, we extend the Riemannian computing framework to PDEs for smoothing and interpolation of fields of features with the example of positive define symmetric matrices (tensors). These covariance matrices are used to in  Diffusion Tensor Imaging or to describe the joint anatomical variability at different places (Green function) in shape variability analysis. As symmetric positive definite matrices constitute a convex half-cone in the vector space of matrices, many usual operations (like the mean) are stable in this space. However, negative eigenvalues appear when one estimates the tensors from original data or when one uses standard numerical schemes for smoothing these data. We show that the choice of a convenient Riemannian metric allows to generalize consistently to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data.

The methodology is exemplified on two important applications: the joint estimation and regularization of Diffusion Tensor MR Images (DTI), and the modeling of the variability of the brain from a dataset of precisely delineated anatomical structures (sulcal lines) in the cerebral cortex. In this context, we obtain a dense 3D variability map which proves to be in accordance with previously published results on smaller samples subjects. We also propose statistical tests which demonstrate that our model is globally able to recover the missing information and innovative methods to analyze the asymmetry of brain variability.

The talk will also be illustrated by recent developments, including new Log-Euclidean metrics on tensors, that give a vector space structure to this manifold and a very efficient computational framework; Riemannian elasticity, a statistical framework on deformations fields for estimating the anatomical variability and regularizing accordingly in dense non-linear registration algorithms, and new clinical insights in scoliosis thanks to the statistical analysis of the anatomic variability of the spine.

** /X. Pennec./ Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements*. /JMIV/, 2006. In press. Preprint: ftp://ftp-sop.inria.fr/epidaure/Publications/Pennec/Pennec.JMIV06.pdf
*
* /X. Pennec et al. /A Riemannian Framework for Tensor Computing*. /IJCV/, 66(1):41-66, 2006.
Preprint: ftp://ftp-sop.inria.fr/epidaure/Publications/Pennec/Pennec.IJCV05.pdf
*
* /V. Arsigny et al./*/ /*Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors*. /Magnetic Resonance in Medicine/, 2006. In press. Preprint: ftp://ftp-sop.inria.fr/epidaure/Publications/Arsigny/arsigny_mrm_2006.pdf

** /P. Fillard et al. /Extrapolation of Sparse Tensor Fields: Application to the Modeling of Brain Variability*. In /Proc. of IPMI'05/, LNCS 3565 of /LNCS/, pages 27-38, July 2005. ftp://ftp-sop.inria.fr/epidaure/Publications/Fillard/Fillard.IPMI05.pdf

** /X. Pennec et al./ Riemannian Elasticity: A statistical regularization framework for non-linear registration*. In /Proc. of MICCAI 2005, Part II/, LNCS 3750, pages 943-950, 2005. ftp://ftp-sop.inria.fr/epidaure/Publications/Pennec/Pennec.MICCAI05.pdf

** J. Boisvert et al. 3D Anatomic Variability Assesment of the Scoliotic Spine Using Statistics on Lie Groups*. In /Proc. of ISBI 2006/, April 2006. ftp://ftp-sop.inria.fr/epidaure/Publications/Pennec/Boisvert.ISBI06.pdf

Jean Ponce
Geometry and 3D computer vision: What we (kind of) know how to do, what we don't, and why anyone should care
Département d'Informatique, Ecole Normale Supérieure, Paris
http://www.di.ens.fr/~ponce
Joint work with Yasutaka Furukawa, Akash Kushal, Svetlana Lazebnik, Kenton McHenry, Fred Rothganger, and Cordelia Schmid.

SLIDES
I will present my views on the role of geometry in computer vision, a domain concerned with the automated interpretation of digital imagery. I will focus on two challenging problems, namely the acquisition of three-dimensional (3D) object and scene models from multiple pictures --- a process known as 3D photography, and the identification of previously observed objects (or object categories) in new images --- a process known as object recognition. In the first part of the talk, I will show that an essential part of the relationship between the shape of solids bounded by smooth surfaces amd their image outlines is inherently projective. This observation leads to a better qualitative understanding of the image formation process, as well as effective image-based algorithms for high-fidelity 3D photography. In the second part of my presentation, I will argue that object recognition is probably the most challenging and exciting problem in computer vision today, but that, despite exciting recent progress, several key representational issues (including, but not limited to, geometric ones) have yet to be addressed. I will illustrate this point by discussing some recent results and open issues. I will conclude with a discussion of potential applications of 3D photography and object recognition to non-traditional domains such as archaeology, anthropology, cultural heritage preservation, film post-production and special effects, and forensics.

References

Jean Ponce, Martial Hebert, Cordelia Schmid, Andrew Zisserman (eds.).
*Toward category-level object recognition.*
Springer-Verlag Lecture Notes in Computer Science, 2006. In press.

Fred Rothganger, Svetlana Lazebnik, Cordelia Schmid, and Jean Ponce.
Segmenting, Modeling, and Matching Video Clips Containing Multiple Moving Objects <paper/pami06.pdf>.
IEEE Transactions on Pattern Analysis and Machine Intelligence, accepted, 2006.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/pami06.pdf

Svetlana Lazebnik, Yasutaka Furukawa, and Jean Ponce.
Projective Visual Hulls <paper/ijcv06a.pdf>.
Submitted to International Journal of Computer Vision, 2006.
://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/ijcv06a.pdf

Svetlana Lazebnik, Cordelia Schmid, and Jean Ponce.
Beyond Bags of Features: Spatial Pyramid Matching for Recognizing Natural Scene Categories <paper/cvpr06b.pdf>.
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, New York, June 2006, vol. II, pp. 2169-2178.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/cvpr06b.pdf

Yasutaka Furukawa and Jean Ponce.
High-Fidelity Image-Based Modeling <tr/cvr_tr_2006_02.pdf>.
CVR Technical Report 2006-02 <tech_reports.html>, May 2006.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/tr/cvr_tr_2006_02.pdf

Yasutaka Furukawa and Jean Ponce.
Carved Visual Hulls for Image-Based Modeling <paper/eccv06b.pdf>.
Proceedings of the European Conference on Computer Vision, Graz, Austria, May 2006.
Springer-Verlag Lecture Notes in Computer Science 3951, A. Leonardis, H. Bischof, and A. Prinz (eds.), volume 1, pp. 564-577.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/eccv06b.pdf

Akash Kushal and Jean Ponce.
Modeling 3D Objects from Stereo Views and Recognizing them in Photographs <paper/eccv06a.pdf>.
Proceedings of the European Conference on Computer Vision, Graz, Austria, May 2006.
Springer-Verlag Lecture Notes in Computer Science 3952, A. Leonardis, H. Bischof, and A. Prinz (eds.), volume 2, pp. 563-574.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/eccv06a.pdf

Fred Rothganger, Svetlana Lazebnik, Cordelia Schmid, and Jean Ponce.
3D Object Modeling and Recognition Using Local Affine-Invariant Image Descriptors and Multi-View Spatial Constraints <paper/ijcv04d.pdf>.
International Journal of Computer Vision, vol. 66, no. 3, March 2006, pp. 231-259.
http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/ijcv04d.pdf

Jean-Philippe Pons
Upgrading the level set method: point correspondence, topological constraints and deformation priors
Authors: Jean-Philippe Pons, joint work with Guillaume Charpiat, Florent Ségonne, Renaud Keriven, Olivier Faugeras.
Affiliations:
Jean-Philippe Pons, Florent Ségonne, Renaud Keriven: CERTIS, École Nationale des Ponts et Chaussées, Marne-la-Vallée, France.
Guillaume Charpiat: Odyssée Laboratory, École Normale Supérieure, Paris, France.
Olivier Faugeras: Odyssée Laboratory, INRIA, Sophia-Antipolis, France.

Webpage: http://www.enpc.fr/certis/people/pons.html
SLIDES
Abstract:
Our work tackles some limitations of the level set representation and of the minimization by geometric gradient flows, with the objective of increasing the applicability and efficiency of deformable models.
Specifically, we overcome the loss of the point correspondence and the inability to control topology changes with the level set method. We propose two associated applications in the field of medical imaging: the generation of unfolded area preserving representations of the cerebral
cortex, and the segmentation of several head tissues from anatomical magnetic resonance images. With regard to the minimization procedure, we show that the robustness to local minima can be improved by introducing deformation priors via a modification of the metric of the deformation space.

References:
J.-P. Pons, G. Hermosillo, R. Keriven and O. Faugeras. Maintaining the point correspondence in the level set framework. To appear in Journal of Computational Physics, 2006.
http://www.enpc.fr/certis/Papers/0Xjcp.pdf

G. Charpiat, R. Keriven, J.-P. Pons and O. Faugeras. Designing spatially coherent minimizing flows for variational problems based on active contours. In International Conference on Computer Vision, volume 2, pages 1403-1408, 2005.
http://www.enpc.fr/certis/Papers/05iccv_a.pdf

F. Ségonne, J.-P. Pons, E. Grimson and B. Fischl. A novel level set framework for the segmentation of medical images under topology control. In Workshop on Computer Vision for Biomedical Image Applications: Current Techniques and Future Trends, 2005.

Tammy Riklin-Raviv

Shape based segmentation

URL: http://www.eng.tau.ac.il/~tammy

Joint work with Nahum Kiryati and Nir Sochen from Tel-Aviv University

SLiDES

Abstract

Challenging object detection and segmentation tasks can be

facilitated by the availability of a reference object.

However, accounting for possible transformations between the different

object views, as part of the segmentation process, remains difficult.

The talk is divided into three parts.

First, I will present a variational approach to shape-based segmentation,

using a single reference object, that accounts for planar projective transformation.

Then I will introduce a variational framework for mutual segmentation of an

image pair, in which the emerging segmentation of the object in each view

provides a dynamic prior for the segmentation of the other image.

Finally, I will present segmentation of symmetrical object using the replicative

form of the object induced by the symmetry to facilitate the segmentation.

The proposed frameworks are exemplified on

various images and their superiority over state of the art variational segmentation

techniques is demonstrated.

This research was supported by MUSCLE:

Multimedia Understanding through Semantics, Computation and

Learning, a European Network of Excellence funded by the EC 6th

Framework IST Programme.

References:

T. Riklin-Raviv, N. Kiryati and N. Sochen, Prior-based Segmentation and Shape Registration in the Presence of Perspective Distortion International Journal of Computer Vision (IJCV). Online publication June 2006.

T. Riklin-Raviv, N. Kiryati and N. Sochen, Segmentation with Level Sets and Symmetry. In Proc. of IEEE Conference on Computer Vision and Pattern Recognition. (CVPR). June 2006.

T. Riklin-Raviv, N. Sochen and N. Kiryati, Mutual Segmentation with Level Sets. In the 5th IEEE Workshop on Perceptual Organization in Computer Vision (POCV) in conjunction with the CVPR. June 2006.

Christoph Schnoerr
Variational Analysis of Fluid Flow Image Sequences
Joint work with: Jing Yuan, Paul Ruhnau, Annette Stahl, Gabriele Steidl
University of Mannheim, Dept. Math. and Computer Science
http://www.cvgpr.uni-mannheim.de/schnoerr/
Image processing plays a major role in experimental fluid mechanics. The analysis of image measurements from highly non-rigid and turbulent flows poses scientific challenges and has a high industrial impact. In my talk, I will focus on the importance of variational approaches in this connection in multiple ways: higher-order regularization, variational characterization of stable discretizations, variational flow decomposition, and physically consistent flow estimation from image sequences employing distributed parameter control.
Paper from VLSM'05
Paper from DAGM'06

 Kaleem Siddiqi

Medial Representation

McGill University, School of Computer Science and Centre for Intelligent Machines

Montreal, Canada

web page http://www.cim.mcgill.ca/~shape

SLIDES

Abstract In the late 60's Blum developed the notion of axis-morphologies for describing 2D and 3D forms. He proposed an interpretation of the local reflective symmetries of an object as as a "medial graph" and suggested that the implied part structure could could be used for object categorization and recognition. In this talk I will discuss a type of integral performed on a vector field defined as the gradient of the Euclidean distance function to the bounding curve (or surface) of an object. The limiting behavior of this integral as the enclosed area (or volume) shrinks to zero reveals an invariant which can be used to compute the Blum skeleton as well as to reveal the geometry of the object that it describes. I will also discuss our work on algorithms for computing medial loci using these ideas as well as applications of this technique to the problem of 2D and 3D object retrieval.

Nir Sochen

Geometric flows over Lie Groups

Authors:Yaniv Gur and Nir Sochen

Department of Applied Mathematics

School of Mathematical Sciences

Tel-Aviv University, Tel-Aviv 69978, ISRAEL

www.math.tau.ac.il/~sochen

SLIDES

The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to matrix Lie groups. We formulate the problem of these Lie group flows in terms of the Beltrami framework via a minimization of an action.

This action is defined over a Lie group manifold. By minimizing with respect to the group element, we obtain the equations of motion for the group element (or the corresponding connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows to regularize in the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N), the simplectic group Sp(N) and more.

This type of regularization preserves group properties, e.g. the orthogonality and the determinant for SO(N).

A special numerical scheme that preserves the Lie group structure is used. We apply our formalism various synthetic examples. Real example of regularization of DT-MRI idata is treated as well.

PAPER

This work is a continuation of the following paper:

Y. Gur and N. Sochen, , ``Denoising Tensors via Lie Group Flows", in Proceedings of the International Conference on Variational, Geometry and level-Sets methods in Computer Vision , Beijing, China, October 2005. link: http://www.math.tau.ac.il/~sochen/publications.html

Jean-Luc Starck

Morphological Component Analysis

CEA, Service d'astrophysique

http://jstarck.free.fr

SLIDES

The Morphological Component Analysis (MCA) is a
a new method which allows us to separate features
contained in an image when these features present
different morphological aspects. We show that MCA
can be very useful for decomposing images into texture
and piecewise smooth (cartoon) parts or for inpainting
applications. We extend MCA to a multichannel MCA (MMCA)
which leads to a new approach for blind source separation,
based on the morphological diversity concept instead of the
statistical independence of the source.

References:
http://jstarck.free.fr/IEEE04.pdf
http://jstarck.free.fr/AIEP04.pdf
http://jstarck.free.fr/ACHA05_Inpaint.pdf
http://jstarck.free.fr/IEEE06_MMCA.pdf

David Tschumperle

Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE's
GREYC / Image, 6 Bd du Maréchal Juin, 14050 Caen Cedex.

http://www.greyc.ensicaen.fr/~dtschump/

Demo page : http://www.greyc.ensicaen.fr/~dtschump/greycstoration/

SLIDES

We are interested in PDE's (Partial Differential Equations) in order to smooth
multi-valued images in an anisotropic manner. We point out the pros and cons
of the different equations proposed in the literature, then, we introduce a
tensor-driven PDE, regularizing images while taking the curvatures of specific
integral curves into account. We show that this constraint is particularly well suited for
the preservation of thin structures in an image restoration process.
A direct link is made between our proposed equation and a continuous formulation of the
LIC's (Line Integral Convolutions by [Cabral & Leedom:93]).
It leads to the design of a very fast and stable algorithm that implements our regularization method,
by successive integrations of pixel values along curved integral lines.
The scheme numerically performs with a sub-pixel accuracy and preserves then
thin image structures better than classical finite-differences discretizations.
We finally illustrate the efficiency of our generic curvature-preserving approach - in terms of speed and
visual quality - with different comparisons and various applications requiring image smoothing :
color images denoising, inpainting and image resizing by nonlinear interpolation.

References :
IJCV'06 article : http://www.greyc.ensicaen.fr/~dtschump/data/ijcv2006.pdf
ICIP'05 article : http://www.greyc.ensicaen.fr/~dtschump/data/icip2005.pdf

Technical Report Greyc

Demetri Terzopoulos
Deformable and Functional Models in Medical Image Analysis
University of California, Los Angeles
www.cs.ucla.edu/~dt

Demetri Terzopoulos will give also OTHER TALKS AT CEREMADE

SLIDES

The modeling of biological structures and the model-based interpretation of medical images present many challenging problems. I will present a powerful paradigm known as deformable models, which combines geometry, computational physics, and estimation theory. Deformable models evolve in response to simulated forces as dictated by the continuum mechanical principles of flexible materials, expressed mathematically via variational principles and PDEs. The talk will focus on several biomedical applications, including image segmentation using dynamic finite element and topologically adaptive deformable models, as well as recent work on "deformable organisms" which aims more fully to automate the segmentation process by augmenting deformable models with behavioral and cognitive control mechanisms. I will also discuss the recent trend towards functional modeling, such as craniofacial models that include the biomechanical modeling of facial tissues and muscles of facial expression.
references availaible through this link

Baba C. Vemuri

A novel mathematical model for the diffusion weighted MR signal reconstruction

UFRF Professor and Director

Center for Vision, Graphics and Medical Imaging

Department of Computer & Information Science & Engineering

University of Florida, Gainesville, Florida 32611

joint work with Bing Jian, Evren Ozarslan, Paul Carney and Thomas Mareci

SLIDES

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecular

diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the

connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for

various clinical applications. In this talk, a novel statistical model for diffusion-attenuated MR signal

is presented which involves a continuous probability distribution over the space of symmetric positive

definite tensors Pn. This model is general enough to explain water molecular diffusion in a variety of

situations involving complex tissue geometry including single and multiple fiber occurrences. The signal

at each voxel is represented as a continuous mixture of second order tensors, i.e. the weights in the

mixture are represented by a continuous mixing distribution f(D), where D 2 Pn. In this work, the

MR signal is expressed as the Laplace transform of f(D) on Pn. I will present a closed form expression

for this Laplace transform, when f(D) is a Wishart or a mixture of Wishart distributions that can be

used to represent a large class of distributions on Pn. In this case, the MR signal behavior is given by a

Rigaut-type asymptotic fractal expression. I will show that our model is more accurate in representing

the diffusion-attenuated MR signal than the classic diffusion tensor models currently in vogue. Using this

new model in conjunction with a deconvolution approach, I will present an efficient estimation scheme for

the number of fibers and the water molecular displacement probability functions at each lattice point in

a HARDI data set. Both synthetic and real data sets are used to depict the performance of the proposed

algorithms.

Luminita Vese
Meyer's models for image decomposition and computational approaches
joint work with : Triet Le, Linh Lieu, John Garnett, Peter Jones and Yves Meyer
Affiliation of authors:
TL, LL, JG, LV: UCLA Department of Mathematics, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA.
PJ: Department of Mathematics, Yale University, PO Box 208283, New Haven, CT 06520-8283.
YM: CMLA E.N.S. de Cachan, 61 Avenue du President Wilson, 94235 Cachan, Cedex France.
http://www.math.ucla.edu/~lvese

SLIDES part1

SLIDES part2
In 2001, in an American Mathematical Society lecture series entitled "Oscillating Patterns in Image Processing and Nonlinear Evolution Equations", Yves Meyer has analyzed and questioned the total variation minimization model (Rudin, Osher, Fatemi) for separating cartoon from texture. He therefore proposed several refined variants of the TV model, by substituting the L^2 norm to the square of the fidelity term v=f-u by weaker norms of generalized functions. In particular, he proposed to model the texture component v in a u+v model by one of the spaces of generalized functions G=div(Linfinity), F=div(BMO) and E=Laplacian(B^1_{infinity,infinity}).
David Mumford and Basilis Gidas also show that images can be seen as samples from probability distributions of random variables supported on spaces of generalized functions, not on spaces of functions.
However, it is not easy to solve such models in practice. In this talk, I will review Meyer's models and present some computational approaches for image restoration and image decomposition into cartoon and texture.

A few references availaible online:
"Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures", Yves Meyer, University Lecture Series, AMS, 2001, Vol. 22. http://www.ams.org/bookstore?fn=20&arg1=ulectseries&item=ULECT-22
"Stochastic models for generic images", by D. Mumford and B. Gidas, http://www.dam.brown.edu/people/mumford/Papers/Generic5.pdf
"Image Decomposition Using Total Variation and div(BMO)", T.M. Le and L.A. Vese http://www.math.ucla.edu/~lvese/PAPERS/BMO_2005.pdf
Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing, L.A. Vese and S.J. Osher, http://www.math.ucla.edu/~lvese/PAPERS/JSC_VeseOsher.pdf

Pierre WEISS :
Some applications of L infinite norms in image processing.
Authors : Pierre WEISS, Laure Blanc-Féraud, Gilles Aubert
Affiliation of authors :
Pierre Weiss : Doctorant, Projet ARIANA commun I3S/INRIA Sophia Antipolis
Laure Blanc-Féraud : CNRS, Projet ARIANA commun I3S/INRIA Sophia Antipolis
Gilles Aubert : Université de Nice-Sophia Antipolis, Laboratoire J-A Dieudonné.
web page : http://www-sop.inria.fr/ariana/fr/lequipe.php?name=Weiss

SLIDES
abstract
Energies involving a L infinite norm appear in some recent applications of image
processing. Examples include the denoising of bounded noises, the decomposition
of an image into a textured and a geometric component proposed by Y. Meyer, or
a stable way to interpolate datas on curves and points proposed by V. Caselles and al..

Such a norm is difficult to minimize because it is non differentiable.
We will present some ways to handle that norm, and solve the above mentionned problems.
PAPER

Jean-Paul Zolesio
Shape Tube Metric and Geodesic Characterisation
INRIA Sophia Antipolis

We present recent mathematical developments on shape metrics but we shall escape many mathematical technicalities in order to give a self contain presentation of old and new results. -We revisit the classical diffeomorphism parametrization of shapes under eulerian point of view. - We extend it to non smooth setting where the flow mapping does not exists. - We introduce the tube distance metric for smooth domains - we develop the geodesic characterisation using intrinsic geometry and time-space curvature like terms. - We derive existence for metric and geodesic for the non smooth case. In doing so we need to recall new compactness results for "parabolic BV" boundedness - We derive the geodesic characterisation for the general case. That steps needs first to recall the "Transverse field Z" concept. It is an evolution equation governed by the Lie brackets of the main field V and the pertubation W. We need to extend that concept to the non smooth situation. The transverse field Z enables us to characterise the family of fields V whose generalized flow mapping will map a shape onto a given one. - We analyse the new Euler equation associated with the geodesic and give some applications using level set techniques.