minimum description length shape modelling

Informatics and Mathematical Modelling / Image Analysis

Minimum Description Length Shape Modelling

Hildur ÓlafsdóttirInformatics and Mathematical ModellingTechnical University of Denmark (DTU)

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Outline

Motivation Background Objective function Shape representation Optimisation methods Cases – 2D

– Head silhouettes (gender classification)– Corpus callosum

Extension to 3D– Case: Rat kidneys

Summary

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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

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Two sub-problems1. Define shape borders from the set of images2. 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

1 2

MDL

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A small example

Manual

Equidistant

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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)

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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

: shape parameter for shape k, mode m

: number of modes

: Eigenvector defining principal direction m

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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

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Objective Function III

Calculate the description length for a 1D Gaussian model

a) DL for coding of the data, using the model b) DL for coding of the parameters in the model

Total description length of a shape model (approximation)

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Shape Representation IParameterisation function

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Shape Representation IIParameterisation function

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Optimisation Procedure

Manipulate k

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Build shape model(PCA)

Evaluate DL

Procrustesalignment

END

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ns

Mode 2

Mode 1

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Optimisation strategies Davies 2001 – a) Genetic algorithms, b) Nelder-Mead downhill Simplex Thodberg 2003 (DTU)– Pattern Search algorithm

– Freely available code

Erikson 2003 (Lund University) – Steepest Descent algorithm

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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 C: Weighting

factorN: #markss: #shapeskir: Curvature in point i of shape r

T : Tolerance param. : Fractional distance of point i

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Silhouette Case1IData

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

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Silhouette Case IIIDemonstration of the optimisation process

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Silhouette Case IIAdding curvature

Before After

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Silhouette Case IVShape models

Equidistant landmarking MDL based landmarking

-3std mean +3std -3std mean +3std

Mode 1

Mode 2

Mode 3

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Silhouette Case VGender classification

Logistic regression model on a subset of PCA scores Leave-one-out cross validation

Correct [%]

Manual marks 65

MDL marks 85

Human scoring 65±5

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Silhouette Case VIGender classification

Best fit of logistic regression model Worst fit of logistic regression model

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Corpus callosum case1 I

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

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Corpus callosum case IIMDL-based landmarkingManual landmarking

VT = 0.0087 VT = 0.0038

VTOT=0.0087 VTOT=0.0038

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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

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Extension to 3D II

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Rat kidneys1 I

1From R.H. Davies et al. 3D Statistical Shape Models Using Direct Optimisation of Description Length. ECCV 2002.

MDL-based landmarking

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Rat kidneys IICompactness Generalisation ability

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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

minimum description length shape modelling

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