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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
The Correspondence Problem
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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?
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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
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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
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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
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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)
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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?
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Automatic Groupwise Correspondence
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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!
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Manipulating Correspondence
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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!
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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
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Objective Function
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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
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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
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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
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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)
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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)
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
MDL Groupwise Image Registration
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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)
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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
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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
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Model Evaluation Criteria
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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
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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
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 36
Generalisation Ability
Sample Set
Training Set
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Slide 38
Evaluating Brain Appearance Models
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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
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Combining the strengths of UMIST andThe Victoria University of Manchester MICCAI 2005: Statistics of Anatomic Geometry
Questions?
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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/
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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.
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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.
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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.
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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.
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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,
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