1. Minimum Description Length Shape Modelling Hildur Ólafsdóttir Informatics and Mathematical Modelling Technical University of Denmark (DTU)

2. Outline • Motivation • Background • Objective function • Shape representation • Optimisation methods • Cases – 2D • Head silhouettes (gender classification) • Corpus callosum • Extension to 3D • Case: Rat kidneys • Summary

3. Motivation I • Statistical shape models have shown considerable promise for image segmentation and interpretation • Require a training set of shapes, annotated so that marks correspond across the set • Manual annotation is tedious, subjective and almost impossible in 3D • MDL automatically establishes point correspondences in an optimisation framework

4. 1 2 MDL Two sub-problems • Define shape borders from the set of images • Annotate the shapes so that points correspond across the set MDL shape modelling solves sub-problem 2 => a semi-automatic approach to training set formation

5. A small example Manual Equidistant

6. Background • Introduced by Davies et al. in 2001 • Properties of a good shape model • Generalisation ability • Specificity • Compactness • Ockham’s razor paraphrased: • Simple descriptions interpolate/extrapolate best • Quantitative measure of simplicity – Description Length (DL) • In terms of shape modelling: Cost of transmitting the PCA coded model parameters (in number of bits)

7. : number of modes : shape parameter for shape k, mode m : Eigenvector defining principal direction m Objective Function I • The Shape model • Goal: Calculate the Description Length (DL) of the model • Mean shape and eigenvectors are assumed constant for a given training set => Calculate the DL of the shape space coordinates

8. Objective Function II • Eigenvectors are mutually orthonormal • Total DL can be decomposed to • Where is the DL of • How do we generally calculate description lengths?? • Shannon’s codeword length

9. Objective Function III • Calculate the description length for a 1D Gaussian model • DL for coding of the data, using the model • DL for coding of the parameters in the model • Total description length of a shape model (approximation)

10. Shape Representation IParameterisation function

11. Shape Representation IIParameterisation function

12. 1 2 ns Optimisation Procedure Manipulate k Evaluate DL END Mode 1 Procrustes alignment Mode 2 Build shape model (PCA)

13. Optimisation strategies • Davies2001 – a) Genetic algorithms, b) Nelder-Mead downhill Simplex • Thodberg 2003 (DTU)– Pattern Search algorithm • Freely available code • Erikson 2003 (Lund University) – Steepest Descent algorithm

14. Thodbergs implementationExtensions to the standard framework • A mechanism which prevents marks from piling up • A curvature term added to the objective function in the final iterations T: Tolerance param. : Fractional distance of point i C:Weighting factor N:#marks s:#shapes kir: Curvature in point i of shape r

15. Silhouette Case1IData 1From H.H. Thodberg et al. Adding Curvature to Minimum Description Length Shape Models. BMVC 2003

16. Silhouette Case IIIDemonstration of the optimisation process

17. Silhouette Case IIAdding curvature Before After

18. -3std mean +3std -3std mean +3std Mode 1 Mode 2 Mode 3 Silhouette Case IVShape models Equidistant landmarking MDL based landmarking

19. Silhouette Case VGender classification • Logistic regression model on a subset of PCA scores • Leave-one-out cross validation

20. Silhouette Case VIGender classification Best fit of logistic regression model Worst fit of logistic regression model

21. Corpus callosum case1 I 1From M. B. Stegmann et al.Corpus Callosum Analysis using MDL-based Sequential Models of Shape and Appearance. SPIE 2004

22. Corpus callosum case II Manual landmarking MDL-based landmarking VTOT=0.0087 VTOT=0.0038 VT = 0.0038 VT = 0.0087

23. Extension to 3D I • Each surface is represented as a triangular mesh – topologically equivalent to a sphere • Initialised by mapping each surface mesh to a unit sphere • Parameterisation of a given surface is manipulated by altering the mapped vertices on the sphere

24. Extension to 3D II

25. Rat kidneys1 I MDL-based landmarking 1From R.H. Davies et al. 3D Statistical Shape Models Using Direct Optimisation of Description Length. ECCV 2002.

26. Rat kidneys II Compactness Generalisation ability

27. Summary • MDL is a semi-automatic approach to a training set formation • A theoretically justified objective function is used in an optimisation framework as a quantitative measure of the quality of a given shape model • The method extends to 3D • Practical optimisation methods have been introduced • Freely available code from Thodberg (www.imm.dtu.dk/~hht) • Impressive results