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Statistics of Anatomic Geometry: Information Theory and Automatic Model Building

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### Statistics of Anatomic Geometry:Information Theory and Automatic Model Building

PDF

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

- 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

Model

Shape Space

Point Distribution Models (PDMs)Statistical Shape Models (SSMs)Set of Shapes

& Corresponding Points

Slide 3

Adding Image Information

- Include image information from

whole region

- Correlation between shape & texture

Shape & Texture Model

Shape Model

Slide 5

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

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 Landmarks
- Interpolate for dense

correspondence

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

Slide 11

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 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 Correspondence

Optimisation Problem Framework:

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

Slide 16

Correspondence Points

Manipulating Correspondence- Point-to-Point:

Shape 1

Shape 2

Varying correspondence varies shape!

Vary correspondence but not shape!

Slide 18

Manipulating Correspondence

- Continuous parameterisation of shape
- Re-parameterising varies correspondence

Slide 19

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

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

- 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 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

- 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

- 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

- 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

- 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

- 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

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

Sample Set from model pdf:

General but not Specific

Specific but not General

Specificity and GeneralizationSpace of Shapes/Images

Slide 34

Specificity

Generalisation

Validation- Annotated/Registered Data
- Perturb Registration

Size of Perturbation

Slide 37

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

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 & 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 & 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 & 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 & 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 & 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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