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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

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

- Recap of Point Distribution/Statistical Shape Models PDMs/SSMs

- Shape Representation & Correspondence
- Correspondence & Statistics
- Methods for establishing correspondence

- Manipulating Correspondence not Shape
- Minimum Description Length objective function
- Optimisation

- MDL Groupwise Registration
- automatic models from unannotated image sets

Slide 2

PCA

Model

Shape Space

Set of Shapes

& Corresponding Points

Slide 3

Shape Space

Shape & Appearance Space

Slide 4

- Include image information from
whole region

- Correlation between shape & texture

Shape & Texture Model

Shape Model

Slide 5

AAM

Search

ASM Search

Slide 6

The Correspondence Problem

- 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

Shape Space

Shape Space

Varying correspondence varies the shape statistics

Slide 9

- 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 Landmarks
- Interpolate for dense
correspondence

- May need to adjust

- Interpolate for dense
- Problems:
- Time-consuming
- Subjective
- Requires expert anatomical knowledge
- Very difficult in 3D

Slide 11

- Equally-space landmarks around each shape
(Baumberg & Hogg)

Slide 12

- 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

Optimisation Problem Framework:

- Method of manipulating correspondence:
- 2D & 3D

- Objective function:
- quantifies the ‘quality’ of the correspondence

- Optimization Scheme

Slide 16

Manipulating Correspondence

Shape Points

Correspondence Points

- Point-to-Point:

Shape 1

Shape 2

Varying correspondence varies shape!

Vary correspondence but not shape!

Slide 18

- Continuous parameterisation of shape
- Re-parameterising varies correspondence

Slide 19

Sphere & Spherical Polar coordinates

Shape

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

Slide 20

Objective Function

Shape Space

Shape Space

- 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

Shape Space

- 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

- Encoded Model: mean shape, model modes etc
- MDL Objective function = total length of message

Slide 23

- 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

- 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

- 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

MDL Groupwise Image Registration

- Groups of Shapes:
groupwise dense correspondence

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

- statistical models of shape variability
- Groups of Images:
groupwise dense correspondence = groupwise registration

- statistical models of shape & appearance
- as above

- statistical models of shape & appearance
- MDL technique for correspondence can be applied to both
(Twining et al 2005)

Slide 28

- Spatial Correspondence between images
- Shape variation

- Warp one to another
- Difference is texture variation

- Repeat across group => Appearance model of image set

Slide 29

- 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

- 104 2D brain slices
- Appearance
Model

Slide 31

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

Training Set:

Sample Set from model pdf:

General but not Specific

Specific but not General

Space of Shapes/Images

Slide 34

:distance on image/shape space

Training Set

Sample Set

Slide 35

Training Set

Sample Set

Slide 36

Objective function

Specificity

Generalisation

- Annotated/Registered Data
- Perturb Registration

Size of Perturbation

Slide 37

Slide 38

- 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

Questions?

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

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

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

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

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

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