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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. - PowerPoint PPT Presentation
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 2
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 3
Point Distribution Models (PDMs)Statistical Shape Models (SSMs)
Set of Shapes& Corresponding
PointsShape Space
PCA
ModelPDF
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 4
Adding Image Information
Shape Space Shape & Appearance Space
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 5
● Include image information from
whole region
● Correlation between shape & texture
Adding Image Information
Shape Model Shape & Texture Model
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 6
Active Shape & Appearance Models
ASM Search
AAMSearch
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
The Correspondence Problem
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 8
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?
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 9
Correspondence & Statistics
Shape Space Shape Space
Varying correspondence varies the shape statistics
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 10
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 11
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 12
Arc-Length Parameterisation● Equally-space landmarks around each shape
(Baumberg & Hogg)
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 13
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?
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Automatic Groupwise Correspondence
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 15
Automatic Groupwise Correspondence
Desirable features:
● Groupwise:
Depends on whole set of shapes
● Automatic – little or no user intervention
● 2D & 3D
● Runs in reasonable time!
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 16
Automatic Groupwise Correspondence
Optimisation Problem Framework:
● Method of manipulating correspondence:
2D & 3D
● Objective function:
quantifies the ‘quality’ of the correspondence
● Optimization Scheme
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Manipulating Correspondence
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 18
Manipulating Correspondence● Point-to-Point:
Shape 1 Shape 2
Shape Points
Correspondence Points
Varying correspondence varies shape!
Vary correspondence but not shape!
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 19
Manipulating Correspondence● Continuous parameterisation of shape
● Re-parameterising varies correspondence
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 20
● Generalises to 3D
● Map surface to parameter sphere - no folds or tears
● Varying parameterisation on sphere
Manipulating Correspondence
ShapeSphere & Spherical Polar coordinates
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Objective Function
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 22
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
Shape Space Shape Space
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 23
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 24
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 25
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)
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 26
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)
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
MDL Groupwise Image Registration
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 28
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)
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 29
● Spatial Correspondence between images Shape variation
● Warp one to another Difference is texture variation
● Repeat across group => Appearance model of image set
Image Registration
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 30
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 31
Results● 104 2D brain slices
● Appearance
Model
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Model Evaluation Criteria
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 33
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 34
General but not Specific
Specificity and Generalization
Specific but not General
Training Set:
Sample Set from model pdf:
Space of Shapes/Images
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 35
Specificity
Training Set
Sample Set
:distance on image/shape space
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 36
Generalisation Ability
Sample Set
Training Set
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 37
Validation
● Annotated/Registered Data
● Perturb Registration
GeneralisationSpecificity
Size of Perturbation
Objective function
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 38
Evaluating Brain Appearance Models
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 39
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
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Questions?
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 41
Resources & ReferencesAAMs, 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/
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 42
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.
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 43
Resources & ReferencesFourier 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.
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 44
Resources & ReferencesElastic 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 LengthThis 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.
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 45
Resources & ReferencesMDL for Shape CorrespondenceApproximate MDLNote 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 ShapeThis 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.
Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 46
Resources & ReferencesMDL 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.