Minimum Description Length Shape Modelling

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Minimum Description Length Shape Modelling

• Hildur Ólafsdóttir
• Informatics and Mathematical Modelling
• Technical 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-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
• gt a semi-automatic approach to training set
  formation

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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
• Ockhams 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 gt Calculate the DL of
  the shape space coordinates

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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??
• Shannons codeword length

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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)

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Shape Representation IParameterisation function
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Shape Representation IIParameterisation function
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Optimisation Procedure
Manipulate ?k
Evaluate DL
END
Mode 1
Procrustes alignment
Mode 2
Build shape model (PCA)
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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

T Tolerance param. Fractional distance
of point i
C Weighting factor N marks s
shapes kir Curvature in point i of shape
r
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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
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Silhouette Case VGender classification

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

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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 II
Manual landmarking
MDL-based landmarking
VTOT0.0087
VTOT0.0038
VT 0.0038
VT 0.0087
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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
MDL-based landmarking
1From R.H. Davies et al. 3D Statistical Shape
Models Using Direct Optimisation of Description
Length. ECCV 2002.
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Rat kidneys II
Compactness
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