Statistical Shape Models - PowerPoint PPT Presentation

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Statistical Shape Models

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We will represent the shape using a set of points ... Example : Hip Radiograph. Spine Model. Distribution of Parameters. Learn p(b) from training set ... – PowerPoint PPT presentation

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Title: Statistical Shape Models


1
Statistical Shape Models
  • Eigenpatches model regions
  • Assume shape is fixed
  • What if it isnt?
  • Faces with expression changes,
  • organs in medical images etc
  • Need a method of modelling shape and shape
    variation

2
Shape Models
  • We will represent the shape using a set of points
  • We will model the variation by computing the PDF
    of the distribution of shapes in a training set
  • This allows us to generate new shapes similar to
    the training set

3
Building Models
  • Require labelled training images
  • landmarks represent correspondences

4
Suitable Landmarks
  • Define correspondences
  • Well defined corners
  • T junctions
  • Easily located biological landmarks
  • Use additional points along boundaries to define
    shape more accurately

5
Building Shape Models
  • For each example

x (x1,y1, , xn, yn)T
6
Shape
  • Need to model the variability in shape
  • What is shape?
  • Geometric information that remains when location,
    scale and rotational effects removed (Kendall)

Same Shape
Different Shape
7
Shape
  • More generally
  • Shape is the geometric information invariant to a
    particular class of transformations
  • Transformations
  • Euclidean (translation rotation)
  • Similarity (translationrotationscaling)
  • Affine

8
Shape
9
Statistical Shape Models
  • Given a set of shapes
  • Align shapes into common frame
  • Procrustes analysis
  • Estimate shape distribution p(x)
  • Single gaussian often sufficient
  • Mixture models sometimes necessary

10
Aligning Two Shapes
  • Procrustes analysis
  • Find transformation which minimises
  • Resulting shapes have
  • Identical CoG
  • approximately the same scale and orientation

11
Aligning a Set of Shapes
  • Generalised Procrustes Analysis
  • Find the transformations Ti which minimise
  • Where
  • Under the constraint that

12
Aligning Shapes Algorithm
  • Normalise all so CoG at origin, size1
  • Let
  • Align each shape with m
  • Re-calculate
  • Normalise m to default size, orientation
  • Repeat until convergence

13
Aligned Shapes
  • Need to model the aligned shapes

14
Statistical Shape Models
  • For shape synthesis
  • Parameterised model preferable
  • For image matching we can get away with only
    knowing p(x)
  • Usually more efficient to reduce dimensionality
    where possible

15
Dimensionality Reduction
  • Co-ords often correllated
  • Nearby points move together

16
Principal Component Analysis
  • Compute eigenvectors of covariance,S
  • Eigenvectors main directions
  • Eigenvalue variance along eigenvector

17
Dimensionality Reduction
  • Data lies in subspace of reduced dim.
  • However, for some t,

18
Building Shape Models
  • Given aligned shapes,
  • Apply PCA
  • P First t eigenvectors of covar. matrix
  • b Shape model parameters

19
Hand shape model
  • 72 points placed around boundary of hand
  • 18 hand outlines obtained by thresholding images
    of hand on a white background
  • Primary landmarks chosen at tips of fingers and
    joint between fingers
  • Other points placed equally between

20
Hand Shape Model
21
Face Shape Model
22
Brain structure shape model
23
Example Hip Radiograph
24
Spine Model
25
Distribution of Parameters
  • Learn p(b) from training set
  • If x multivariate gaussian, then
  • b gaussian with diagonal covariance
  • Can use mixture model for p(b)

26
Conclusion
  • We can build statistical models of shape change
  • Require correspondences across training set
  • Get compact model (few parameters)
  • Next Matching models to images
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