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Statistics of Anatomic Geometry: Information Theory and Automatic Model Building. Carole Twining Imaging Science and Biomedical Engineering (ISBE) University of Manchester, UK Contributions from: Rhodri Davies, Stephen Marsland, Tim Cootes, Vlad Petrovic, Roy Schestowitz, & Chris Taylor.

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statistics of anatomic geometry information theory and automatic model building

Statistics of Anatomic Geometry:Information Theory and Automatic Model Building

Carole Twining

Imaging Science and Biomedical Engineering (ISBE)

University of Manchester, UK

Contributions from:

Rhodri Davies, Stephen Marsland, Tim Cootes, Vlad Petrovic,

Roy Schestowitz, & Chris Taylor

overview
Overview
    • Recap of Point Distribution/Statistical Shape Models PDMs/SSMs
  • Correspondence Problem:
    • Shape Representation & Correspondence
    • Correspondence & Statistics
    • Methods for establishing correspondence
  • Automatic Methods for Groupwise Shape Correspondence
    • Manipulating Correspondence not Shape
    • Minimum Description Length objective function
    • Optimisation
  • Extension to Images:
    • MDL Groupwise Registration
      • automatic models from unannotated image sets
  • Model Evaluation Criteria

Slide 2

point distribution models pdms statistical shape models ssms

PCA

Model

PDF

Shape Space

Point Distribution Models (PDMs)Statistical Shape Models (SSMs)

Set of Shapes

& Corresponding Points

Slide 3

adding image information

Shape Space

Shape & Appearance Space

Adding Image Information

Slide 4

adding image information1
Adding Image Information
  • Include image information from

whole region

  • Correlation between shape & texture

Shape & Texture Model

Shape Model

Slide 5

active shape appearance models
Active Shape & Appearance Models

AAM

Search

ASM Search

Slide 6

shape representation correspondence
Shape Representation & Correspondence
  • Non-Local Representations
    • Fourier descriptors (e.g., SPHARM)
    • Medial descriptors (e.g., MREPS)
  • Local Representations
    • Point based (e.g., PDMs/SSMs)
  • Common Representation of training set => Correspondence
    • Non-local tends to give implicit correspondence
    • Point based gives explicit correspondence
  • Why does the correspondence matter?

Slide 8

correspondence statistics

Shape Space

Shape Space

Correspondence & Statistics

Varying correspondence varies the shape statistics

Slide 9

establishing correspondence
Establishing Correspondence
  • Manual landmarking
  • Arbitrary parameterisations
    • Kelemen, Hill, Baumberg & Hogg
  • Shape features
    • Wang, Brett
  • Image registration
    • models from deformation field
    • Christensen, Joshi, Lavalle, Reuckert, Twining

Slide 10

manual methods for correspondence
Manual Methods for Correspondence
  • Manual Landmarks
    • Interpolate for dense

correspondence

    • May need to adjust
  • Problems:
    • Time-consuming
    • Subjective
    • Requires expert anatomical knowledge
    • Very difficult in 3D

Slide 11

arc length parameterisation
Arc-Length Parameterisation
  • Equally-space landmarks around each shape

(Baumberg & Hogg)

Slide 12

shape features
Shape Features
  • e.g. Curvature-based methods
  • Intuitive
  • But:
    • What about regions without such features?
    • Not really groupwise, since depends on local properties of each shape
    • Is it really the best correspondence?

Slide 13

automatic groupwise correspondence1
Automatic Groupwise Correspondence

Desirable features:

  • Groupwise:
    • Depends on whole set of shapes
  • Automatic – little or no user intervention
  • 2D & 3D
  • Runs in reasonable time!

Slide 15

automatic groupwise correspondence2
Automatic Groupwise Correspondence

Optimisation Problem Framework:

  • Method of manipulating correspondence:
    • 2D & 3D
  • Objective function:
    • quantifies the ‘quality’ of the correspondence
  • Optimization Scheme

Slide 16

manipulating correspondence1

Shape Points

Correspondence Points

Manipulating Correspondence
  • Point-to-Point:

Shape 1

Shape 2

Varying correspondence varies shape!

Vary correspondence but not shape!

Slide 18

manipulating correspondence2
Manipulating Correspondence
  • Continuous parameterisation of shape
  • Re-parameterising varies correspondence

Slide 19

manipulating correspondence3

Sphere & Spherical Polar coordinates

Shape

Manipulating Correspondence
  • Generalises to 3D
  • Map surface to parameter sphere - no folds or tears
  • Varying parameterisation on sphere

Slide 20

objective function1

Shape Space

Shape Space

Objective Function
  • Varying Correspondence = Varying Statistics
  • Objective function based on model probability density function
    • number of model modes
    • compactness
    • quality of fit to training data
    • number of model parameters

Slide 22

mdl objective function

Shape Space

MDL Objective Function
  • Transmit training set as encoded binary message
  • Shannon:
    • Set of possible events {i} with probabilities {pi}
    • Optimal codeword length for event i: -log pi
  • Encode whole training set of shapes:
    • Encoded Model: mean shape, model modes etc
      • Reconstruct shape space and model pdf
    • Each training shape: pi from model pdf
      • Reconstruct all training shapes
  • MDL Objective function = total length of message

Slide 23

mdl objective function1
MDL Objective Function
  • Fit between model pdf and training data:
    • Probabilities for training points => better the fit, shorter the message
  • Too complex a model:
    • model parameter term large
  • Too few modes:
    • Bad fit to data & large residual
  • Badly chosen modes:
    • Bad fit to data

Slide 24

optimisation
Optimisation
  • Genetic algorithm search (Davies et al, 2002)
    • All parameters optimised simultaneously
    • Slow, scales badly with no of examples
  • More recent, multi-scale, multi-resolution approaches:
    • better convergence
    • fast enough for routine use
    • scales approximately linearly with no of examples

(Davies et al, IPMI 2003)

Slide 25

results
Results
  • Quantitatively better results compared to SPHARM
  • Differences tend to be subtle
  • Comparing techniques, have to go beyond visual inspection

(see section on Model Evaluation Criteria)

Slide 26

image shape correspondence
Image & Shape Correspondence
  • Groups of Shapes:

groupwise dense correspondence

    • statistical models of shape variability
      • analysis of variation across & between populations
      • assist in analysing unseen examples (ASM & AAM)
  • Groups of Images:

groupwise dense correspondence = groupwise registration

    • statistical models of shape & appearance
      • as above
  • MDL technique for correspondence can be applied to both

(Twining et al 2005)

Slide 28

image registration
Image Registration
  • Spatial Correspondence between images
    • Shape variation
  • Warp one to another
    • Difference is texture variation
  • Repeat across group => Appearance model of image set

Slide 29

groupwise image registration
Groupwise Image Registration
  • MDL Objective Function
    • Combined shape & texture model
  • Define dense correspondence
    • triangulated points on each image & interpolate
  • Manipulate Correspondence
  • Increase resolution of mesh & repeat

Slide 30

results1
Results
  • 104 2D brain slices
  • Appearance

Model

Slide 31

model evaluation criteria1
Model Evaluation Criteria
  • Need to go beyond visual inspection, subtle differences
  • Generalisability:
    • the ability to represent unseen shapes/images which belong to the same class as those in the training set
  • Specificity:
    • the ability to only represent images similar to those seen in the training set
  • Quantitative comparison of models

Slide 33

specificity and generalization

Training Set:

Sample Set from model pdf:

General but not Specific

Specific but not General

Specificity and Generalization

Space of Shapes/Images

Slide 34

specificity

:distance on image/shape space

Specificity

Training Set

Sample Set

Slide 35

validation

Objective function

Specificity

Generalisation

Validation
  • Annotated/Registered Data
  • Perturb Registration

Size of Perturbation

Slide 37

summary
Summary
  • Manipulating Correspondence
    • Shown to produce quantitatively better models
    • Large-scale Optimisation problem - so far, only linear models
    • Extension to other shape representation methods (e.g. MREPS)
    • Topology – manipulate parameter space:
      • simple, fixed topology
    • Multi-part objects
    • Differences tend to be subtle - go beyond visual inspection of results
      • Model evaluation criteria
    • Extension to groupwise image registration

Slide 39

resources references
Resources & References

AAMs, ASMs

  • [1] T. F. Cootes, G. J. Edwards, and C. J. Taylor,

Active appearance models,

IEEE Trans. Pattern Anal. Machine Intell., vol. 23, no. 6, pp. 681-685, 2001.

  • [2] T. F. Cootes, C. J. Taylor, D. H. Cooper and J. Graham,

Active shape models – their training and application,

Computer Vision and Image Understanding, 61(1), 38-59, 1995

  • [3] T. F. Cootes, A. Hill, C. J. Taylor, and J. Haslam,

The use of active shape models for locating structures in medical images,

Image and Vision Computing, vol. 12, no. 6, pp. 276-285, July 1994.

  • [4] B. van Ginneken, A.F.Frangi, J.J.Stall, and B. ter Haar Romeny,

Active shape model segmentation with optimal features,

IEEE Trans. Med. Imag., vol. 21, pp. 924-933, 2002.

  • [5] P. Smyth, C. Taylor, and J. Adams,

Vertebral shape: Automatic measurement with active shape models,

Radiology, vol. 211, no. 2, pp. 571-578, 1999.

  • [6] N. Duta and M. Sonka,

Segmentation and interpretation of MR brain images: An improved active shape model,

IEEE Trans. Med. Imag., vol. 17, pp. 1049-1067, 1998.

Further references, as well as notes on the historical meanderings in the development of these techniques

can be found on Tim Cootes’ website:

http://www.isbe.man.ac.uk/~bim/

Slide 41

resources references1
Resources & References

MREPS

  • [7] S. M. Pizer, D. Eberly, D. S. Fritsch, and B. S. Morse,

Zoom-invariant vision of figural shape: The mathematics of cores,

Computer Vision and Image Understanding, vol. 69, no. 1, pp. 055-071, 1998.

Fourier descriptors, spherical harmonics & SPHARM

  • [8] C. Brechb¨uhler, G. Gerig, and O. Kubler,

Parameterisation of closed surfaces for 3D shape description,

Computer Vision, Graphics and Image Processing, vol. 61, pp. 154-170, 1995.

  • [9] A. Kelemen, G. Szekely, and G. Gerig,

Elastic model-based segmentation of 3D neurological data sets,

IEEE Trans. Med. Imag., vol. 18, no. 10, pp. 828-839, Oct. 1999.

  • [10] C. Brechb¨uhler, G. Gerig, and O. K uhler,

Parametrization of closed surfaces for 3D shape description,

Computer Vision and Image Understanding, vol. 61, no. 2, pp. 154-170, 1995.

  • [11] G. Szekely, A. Kelemen, C. Brechbuhler, and G. Gerig,

Segmentation of 2-D and 3-D objects from MRI volume data using constrained elastic deformations

of flexible fourier contour and surface models,

Medical Image Analysis, vol. 1, pp. 19-34, 1996.

Slide 42

resources references2
Resources & References

Fourier descriptors, spherical harmonics & SPHARM

  • [12] D. Meier and E. Fisher,

Parameter space warping: Shape-based correspondence between morphologically different objects,

IEEE Trans. Med. Imag., vol. 21, no. 1, pp. 31-47, Jan. 2002.

  • [13] M. Styner, J. Liberman, and G. Gerig,

Boundary and medial shape analysis of the hippocampus in schizophrenia,

in Proc. International Conference on Medical Image Computing and Computer Aided Intervention

(MICCAI), 2003, pp. 464-471.

Feature-Based Shape correspondence

  • [14] A. D. Brett, A. Hill, and C. J. Taylor,

A method of automatic landmark generation for automated 3D PDM construction,

Image and Vision Computing, vol. 18, pp. 739-748, 2000.

  • [15] Y. Wang, B. S. Peterson, and L. H. Staib,

Shape-based 3D surface correspondence using geodesics and local geometry,

in Proc. IEEE conference on Computer Vision and Pattern Recognition (CVPR), 2000, pp. 644-651.

  • [16] G. Subsol, J. Thirion, and N. Ayache,

A scheme for automatically building three-dimensional morphometric anatomical atlases: application

to a skull atlas,

Medical Image Analysis, vol. 2, no. 1, pp. 37-60, 1998.

Slide 43

resources references3
Resources & References

Elastic and Distortion based methods of shape correspondence

  • [17] M. Kaus, V. Pekar, C. Lorenz, R. Truyen, S. Lobregt, and J. Weese,

Automated 3-D PDM construction from segmented images using deformable models,

IEEE Trans. Med. Imag., vol. 22, no. 8, pp. 1005-1013, Aug. 2003.

  • [18] C. Shelton,

Morphable surface models,

International Journal of Computer Vision, vol. 38, pp. 75-91, 2000.

  • [19] S. Sclaroff and A. P. Pentland,

Modal matching for correspondence and recognition,

IEEE Trans. Pattern Anal. Machine Intell., vol. 17, no. 6, pp. 545-561, 1995.

  • [20] F. L. Bookstein,

Landmark methods for forms without landmarks: morphometrics of group differences in outline shape,

Medical Image Analysis, vol. 1, no. 3, pp. 225-244, 1997.

Minimum Description Length

This is the information theory stuff behind MDL.

  • [21] J. Rissanen, Lectures on Statistical Modeling Theory,

http:\\www.cs.tut.fi\~rissanen\papers\lectures.pdf

  • [22] J. Rissanen,

Stochastic Complexity in Statistical Inquiry,

World Scientific Press, 1989.

Slide 44

resources references4
Resources & References

MDL for Shape Correspondence

Approximate MDL

Note that the freely available code distributed by Thodberg is only approximate MDL, not full state-ofthe-

art MDL as used by other groups. In fact, the objective function used in these papers is equivalent

to what is used to initialise other algorithms. This fact has caused a little confusion in the literature.

  • [23] H. Thodberg,

MDL shape and appearance models,

in Proc. 18th Conference on Information Processing in Medical Imaging (IPMI), 2003, pp. 51-62.

  • [24] H. Thodberg and H. Olafsdottir,

Adding curvature to MDL shape models,

in Proc. 14th British Machine Vision Conference (BMVC), vol. 2, 2003, pp. 251-260.

  • [25] T. Heimann, I. Wolf, T. G. Williams, and H.-P. Meinzer,

3D Active Shape Models Using Gradient Descent Optimization of Description Length ,

IPMI 2005.

MDL for 2D Shape

This uses the initial genetic algorithm search, which was later improved upon.

  • [26] R. H. Davies, C. J. Twining, T. F. Cootes, J. C. Waterton, and C. J. Taylor,

A minimum description length approach to statistical shape modelling,

IEEE Trans. Med. Imag., vol. 21, no. 5, pp. 525-537, May 2002.

  • [27] R. H. Davies, C. J. Twining, P. D. Allen, T. F. Cootes, and C. J. Taylor,

Building optimal 2D statistical shape models,

Image and Vision Computing, vol. 21, pp. 1171-1182, 2003.

Slide 45

resources references5
Resources & References

MDL for 3D Shape

  • [28] R. H. Davies, C. J. Twining, T. F. Cootes, J. C. Waterton, and C. J. Taylor,

3D statistical shape models using direct optimisation of description length,

in Proc. 7th European Conference on Computer Vision (ECCV), 2002, pp. 3-21.

MDL for Image Registration

  • [29] C. J. Twining, T. Cootes, S. Marsland, V. Petrovic, R. Schestowitz, and C. J. Taylor,

A Unified Information-Theoretic Approach to Groupwise Non-Rigid Registration and Model

Building, Presented at IPMI 2005

  • [30] C. J. Twining, S. Marsland, and C. J. Taylor,

Groupwise Non-Rigid Registration: The Minimum Description Length Approach,

In Proceedings of BMVC 2004.

  • [31] C.J. Twining and S. Marsland,

A Unified Information-Theoretic Approach to the Correspondence Problem in Image Registration,

International Conference on Pattern Recognition (ICPR), Cambridge, U.K. 2004.

Slide 46

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