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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

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Statistics of anatomic geometry information theory and automatic model building l.jpg

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 l.jpg

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 l.jpg

    PCA

    Model

    PDF

    Shape Space

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

    Set of Shapes

    & Corresponding Points

    Slide 3


    Adding image information l.jpg

    Shape Space

    Shape & Appearance Space

    Adding Image Information

    Slide 4


    Adding image information5 l.jpg

    Adding Image Information

    • Include image information from

      whole region

    • Correlation between shape & texture

    Shape & Texture Model

    Shape Model

    Slide 5


    Active shape appearance models l.jpg

    Active Shape & Appearance Models

    AAM

    Search

    ASM Search

    Slide 6


    The correspondence problem l.jpg

    The Correspondence Problem


    Shape representation correspondence l.jpg

    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 l.jpg

    Shape Space

    Shape Space

    Correspondence & Statistics

    Varying correspondence varies the shape statistics

    Slide 9


    Establishing correspondence l.jpg

    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 l.jpg

    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 l.jpg

    Arc-Length Parameterisation

    • Equally-space landmarks around each shape

      (Baumberg & Hogg)

    Slide 12


    Shape features l.jpg

    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 l.jpg

    Automatic Groupwise Correspondence


    Automatic groupwise correspondence15 l.jpg

    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 correspondence16 l.jpg

    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 correspondence l.jpg

    Manipulating Correspondence


    Manipulating correspondence18 l.jpg

    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 correspondence19 l.jpg

    Manipulating Correspondence

    • Continuous parameterisation of shape

    • Re-parameterising varies correspondence

    Slide 19


    Manipulating correspondence20 l.jpg

    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 function l.jpg

    Objective Function


    Objective function22 l.jpg

    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 l.jpg

    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 function24 l.jpg

    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 l.jpg

    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 l.jpg

    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


    Mdl groupwise image registration l.jpg

    MDL Groupwise Image Registration


    Image shape correspondence l.jpg

    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 l.jpg

    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 l.jpg

    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


    Results31 l.jpg

    Results

    • 104 2D brain slices

    • Appearance

      Model

    Slide 31


    Model evaluation criteria l.jpg

    Model Evaluation Criteria


    Model evaluation criteria33 l.jpg

    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 l.jpg

    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 l.jpg

    :distance on image/shape space

    Specificity

    Training Set

    Sample Set

    Slide 35


    Generalisation ability l.jpg

    Training Set

    Generalisation Ability

    Sample Set

    Slide 36


    Validation l.jpg

    Objective function

    Specificity

    Generalisation

    Validation

    • Annotated/Registered Data

    • Perturb Registration

    Size of Perturbation

    Slide 37


    Evaluating brain appearance models l.jpg

    Evaluating Brain Appearance Models

    Slide 38


    Summary l.jpg

    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


    Questions l.jpg

    Questions?


    Resources references l.jpg

    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 references42 l.jpg

    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 references43 l.jpg

    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 references44 l.jpg

    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 references45 l.jpg

    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 references46 l.jpg

    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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