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# CISP "WARPSHOP" 2003

Monday 22 Sep 2003 - Wednesday 24 Sep 2003

## Location

The workshop will take place in room 053, in the basement of building 305/321,
Informatics and Mathematical Modelling (IMM),
Technical University of Denmark, Lyngby, Denmark.

## Program


Monday 22 Sep   Warp theory
======================================================================
13:00 - 14:30   Chris Glasbey, Edinburgh, UK
"Matching and distortion criteria for image warping"

14:30 - 15:00   Coffee

15:00 - 16:00   Bo Markussen, Berlin, DE.
"Minimal variation filtration of flows of diffeomorphisms"

16:00 - 17:00   Mads Nielsen, ITU, DK
"Brownian Warps"

Tuesday 23 Sep  Theory & Applications
====================================================================
09:30 - 11:00   Sarang Joshi, North Carolina, USA
"Large Deformation Diffeomorphisims and Gaussian
Random Fields for Statistical Characterization
of Anatomical Variability."

11:00 - 11:15   Coffee

11:15 - 12:00   Mikkel Stegmann, DTU, DK
"Generative Models of Faces and Medical Images"

12:00 - 13:30   Lunch (DTU Build 101, Faculty Club)

13:30 - 14:30   John Gustafsson, Chalmers, Goteborg, SE
"Warping 2-D gel electrophoresis images"

14:30 - 15:00   Coffee

15:00 - 15:30   Aleksandr Dubinskiy, DTU, DK
"A multiscale generative model for animating shapes
and parts"

15:30 - 16:00   Karl Skoglund, DTU, DK
"3-D Face Modelling and Applications"

16:00 - 16:45    Lars Kai Hansen, DTU, DK
"Statistical evaluation of brain warps"

19:00 -         Dinner downtown

Wednesday 24 Sep Brains and more
=======================================================
09:30 - 11:00   Monica Hurdal, Tallahassee, USA
"Discrete Conformal Mapping of Cortical Surfaces"

11:00 - 11:15   Coffee

11:15 - 12:00   Tron Darvann, Univ. Copenhagen, DK
"Applications of warping to visualization of asymmetry
and growth in subjects with craniofacial malformations."

12:00 - 13:30   Lunch (DTU Build 101, Faculty Club)

13:30 - 15:00   Olivier Faugeras, INRIA, FR
"Variational Methods for Multimodal image Matching:
theory and applications"

15:00 - 15:15   Coffee

15:15 - 16:15   Guy Orban, Leuven, BE
"Warping Monkey Cortex to Human Cortex"

16:15 - 17:15   Gregoire Malandain, INRIA, FR
"TBA"



### Organization

Prof. Lars Kai Hansen
Informatics and Mathematical Modelling
B321 Technical University of Denmark
email: lkh (a) imm. dtu. dk

Secretary Ms. Ulla Nørhave
Informatics and Mathematical Modelling
B321 Technical University of Denmark
email: un (a) imm. dtu. dk

## "Warpstracts"

### Matching and distortion criteria for image warping

Chris Glasbey, Biomathematics and Statistics, Scotland

The choice of warping function can be formulated statistically, as maximum penalised likelihood. The likelihood measures the similarity or match between images after warping and the penalty is a measure of distortion of a warping. We identify null-set distortion criteria, with each criterion uniquely minimised by a particular set of polynomial functions, and we construct matching criteria in the Fourier domain. Typically, high-frequency terms are dominated by noise, and we use a Fourier-von Mises image model, in which phase differences between Fourier-transformed images having von Mises distributions. However, if high-frequency terms are informative, then aliasing has to be taken into account. We propose a parametric model for the power- and cross-spectra, which leads to a powerful method for estimating, and correcting, the spatial mis-alignment between bands of multiband images.

### Minimal variation filtration of flows of diffeomorphisms.

Bo Markussen, Humboldt-University Berlin, Germany.

In this talk I will discuss various mathematical and statistical concepts related to the question of how one could make inference of the entire warp (mathematically described by a diffeomorphism) from prior knowledge and partial observations. Building a mathematical model we need a method of modelling our knowledge and ignorance simultaneously. To do this people familiar with statistical methods may propose a Baysian framework. This was done by Nielsen, Johansen et. al. (2001), and initiated my interest in the problem. The mathematical counterpart of our ignorance is here a Brownian motion of Jacobiants, which is mathematically difficult to describe due to the non-commutativity in dimension $d \ge 2$. Another approach by Joshi and Miller (2000) proposes the unknown coefficient in a deterministic differential equation (giving a flow of diffeomorphisms) as the mathematical counterpart of our ignorance. Yet another approach is to replace the unknown deterministic differential equation with a known stochastic differential equation. This is in some sense a combination of the two aforementioned approaches, and allow for a simultaneous modelling of both ignorance and prior knowledge. The statistical inference is replace by probabilistic inference, i.e. a filtration problem. Probabilistically this problem is easier (which may be a surprise), and I will argue that the concept of variation gives a natural optimization criteria. Having developed a stochastic model and formulated an inference principle, the next step is to derive the associated filtration algorithm. I will present an iterative optimization scheme using orthogonal projections, i.e. matrix operations. The partial observations can be of various kinds, e.g. observations of points, gradients or higher order derivatives at various points in time. Moreover, matching of e.g. lines and surfaces can also be handled to some extent. Nanna Glerup (ITU) have been working on an implementation of the algorithm, and I hope to able to present some practical examples.

### Brownian Warps

Non-rigid registration requires a smoothness or regularization term for making the warp field regular. Standard models in use here include b-splines and thin-plate splines. In this paper, we suggest a regularizer which is based on first principles, is symmetric with respect to source and destination, and fulfills a natural semi-group property for warps. We construct the regularizer from a distribution on warps. This distribution arises as the limiting distribution for concatenations of warps just as the Gaussian distribution arises as the limiting distribution for the addition of numbers. Through an Euler-Lagrange formulation, algorithms for obtaining maximum likelihood registrations are constructed. The technique is demonstrated using 2D examples. The methodology is compared to thin-plate splines.

### Large Deformation Diffeomorphisims and Gaussian Random Fields for Statistical Characterization of Anatomical Variability

Sarang Joshi, Department of Radiation Oncology, Biomedical Engineering and Computer Science, University of North Carolina at Chapel Hill

In this talk I will present mathematical and computational techniques for the representation of complex anatomical structures and the inherent biological variability of these structures as manifested in imagery. To represent the complex structures and accommodate their inherent variabilities, pattern theoretic global shape modeling is used. In the context of global shape modeling the complex anatomical structures are represented by a collection $\Omega$ of coordinatized sub-manifolds ${\cal M}$, while their biological variability is accommodated via the definition of vector field transformations ${\cal H}$ preserving the global characteristics of the structures.

Within this framework, the single most important component is the generation of diffeomorphic registration transformations between different anatomies. Landmark Matching, first championed by Bookstein has been used extensively for generating registration transformations and for the study of anatomical variation.

One of the fundamental limitations of classical landmark matching is that it is based on small deformation kinematics, implying that it dose not necessarily produce a diffeomorphism. The focus of this talk is the extension of landmark matching to the large deformation setting insuring the generation of diffeomorphisms.

For understanding the inherent biological variability a Bayesian approach is taken. Methods for the empirical construction of probability measures on anatomical variation, from a set of vector field transformations mapping an ensemble of anatomical images to a common coordinate system are also presented. Gaussian measures are induced on the space of transformations of sub-manifolds via the empirical estimation of the means and the covariances. The induced probability measures are employed for the construction of the typical'' or the average anatomy representing the ensemble.

Resent results of the application of these methods to the study of anatomical variation during radiation treatment of cancer in the lung and prostrate will also be presented.

### Generative Models of Faces and Medical Images

Mikkel Stegmann, IMM, DTU

This talk will demonstrate applications of active apperance models using simple piece-wise affine warping. Applications include unsupervised interpretation and quantification of face images, cardiac MRI, brain MRI, lung radiographs et cetera.

### Warping 2-D gel electrophoresis images

John Gustafsson, Mathematical Statistics, Chalmers University of Technology, SE- 412 96 Göteborg Sweden

Two-dimensional gel electrophoresis is still the only method currently available which is capable of simultaneously separating and quantifying thousands of proteins from cell and tissue samples. A crucial step in this technique is image analysis where the protein spot pattern in different images are to be matched. Standard work-flow includes an initial spot detection step and a subsequent point pattern matching step. Due to incomplete spot detection and non-linear spatial distortions, this approach requires extensive manual guidance. This is referred to as the "image analysis bottle-neck" by the 2-D gel community. With the availability of fast desktop computers, which allows matching at the pixel level, there has been increasing interest in methods from the image warping field.

In this talk I will introduce 2-D gels, review how image warping has been used for 2-D gels images and describe my two contributions: (i) correcting for spatial distortions due to current leakage during the experimental procedure by performing singleton image warping with a mapping function derived from a physiochemical model, and (ii) penalising the introduction of severe warpings when using standard techniques to register gel images.

### A multiscale generative model for animating shapes and parts

Aleksandr Dubinskiy, IMM, DTU, DK

### 3-D Face Modelling and Applications

Karl Skoglund, IMM, DTU, DK

### Statistical evaluation of brain warps

Lars Kai Hansen, IMM, DTU, DK

A wide range of techniques and software tools are available with which to process functional neuroimaging data sets. The necessary performance metrics or benchmark data sets with which to evaluate and compare these tools are only beginning to appear. I will discuss the use of predictive methods, learning curves, and resampling -components of the so-called "NPAIRS" framwork- for evalutation of data analytic pipelines.

U. Kjems et al.: The Quantitative Evaluation of Functional Neuroimaging Experiments: Mutual Information Learning Curves. NeuroImage 15(4): 772-786 (2002).

S.C. Strother et al.: The Quantitative Evaluation of Functional Neuroimaging Experiments: The NPAIRS Data Analysis Framework. NeuroImage 15(4):747-771 (2002).

### Discrete Conformal Mapping of Cortical Surfaces

Monica K. Hurdal, Department of Mathematics, Florida State University

The variability in the shape, location and size of the gyri and sulci of the human brain make it very difficult to compare functional and anatomical brain data across subjects. The belief that most of the cortical processing occurs in the thin layer of grey matter has lead to the idea of surface-based coordinate systems. "Unfolding" and "flattening" a cortical surface may make it easier to impose coordinate systems; comparing "flat" maps across subjects may lead to improved localization of functional foci. In this presentation I will discuss a mathematical technique called Circle Packing that I am using to compute discrete conformal (angle-preserving) maps of the brain. A circle packing is a collection of circles with a specified pattern of tangencies and yields an approximation to a conformal map. We are interested in conformal mappings because the Riemann Mapping Theorem states that conformal mappings exist and are mathematically unique. In contrast, maps that attempt to preserve area or length will always have distortion. I will also discuss some of the topological issues that arise in computing maps of the brain and describe some of the conformal metrics we are exploring such as conformal modulus.

### Applications of warping to visualization of asymmetry and growth in subjects with craniofacial malformations

Tron Darvann, Univ. Copenhagen, DK

### Warping Monkey Cortex to Human Cortex

Guy Orban, KU Leuven, BE

### Variational Methods for Multimodal image Matching: theory and applications

Olivier Faugeras, INRIA, FR

### (TBA)

Gregoire Malandain, INRIA, FR

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