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Nonlinear Shape Modelling John Ashburner. Wellcome Trust Centre for Neuroimaging, UCL Institute of Neurology, London, UK. john@fil.ion.ucl.ac.uk

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Motivation Imaging Biomarkers may be derived using pattern recognition methods. –Provide examples of scans and outcomes. –Learn to predict clinical outcomes from scans. Work with similarity measures among scans. –Many ways of measuring similarity. –Include prior knowledge and biological plausibility in similarity model.

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A simple multivariate morphometry illustration: Body Mass Index BMI = weight / height 2 log(BMI) = log(weight) – 2 log(height)

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Inter-subject Variability... diverse and dissimilar fish [brains] can be referred as a whole to identical functions of very different coordinate systems... Suggests a generative model, whereby diverse and dissimilar brains are modeled by a common template ( identical functions ) deformed according to various spatial transformations ( very different coordinate systems ). DArcy Thompson (1917). GROWTH AND FORM.

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Encoding Relative Shapes Most shape models involve adding a smooth displacement field to an identity transform. –This is the small deformation approximation. –Assumes that deformations can be added and subtracted linearly – which is not the case. Should use a model that assumes that deformations are composed – not added.

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Small Displacements Small Deformation Model

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Simple Small Deformation Registration Estimate u that minimizes: E = ½ || L u || 2 + ½ ||f – μ(φ)|| 2 where φ = x + u and L may be something like a Laplacian operator Regularization Matching term – sum of squares difference

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Larger Displacements Bigger Small Deformation Model

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Diffeomorphisms A diffeomorphism is a smooth continuous one- to-one mapping. Diffeomorphisms form a mathematical group under composition (warping one deformation by another). –Composition of group members is another group member –The members have inverses –There is an identity member –The composition operations are associative

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Diffeomorphic Model

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Diffeomorphic Registration Estimate v (a series of velocity fields over unit time) that minimizes: E = ½ t=0 ||Lv t || 2 dt + ½ ||f – μ(φ 1 )|| 2 where dφ t /dt= v t (φ t ) φ 0 = Id. Regularization – minimizes a geodesic distance Matching term 1 c.f. Hamiltons principle of stationary action

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In practice Modeled by the composition of a series of small deformations. Each is one-to-one, so their composition should also be one-to-one. E = (||Lv 1 || 2 + ||Lv 1 || 2 +… ||Lv K || 2 )/2K + ½ ||f – μ o (x+v 1 /K) o (x+v 2 /K) o…o (x+v K /K)|| 2 This is the Large Deformation Diffeomorphic Metric Mapping (LDDMM) algorithm of Beg et al., which is optimized using gradient descent.

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An initial value formulation It turns out that the dynamical system is determined from its initial conditions as: (L L)v t = |D φt | (D φt ) T ((L L)v 0 ) o φ t dφ t /dt= v t (φ t ) These EPDiff equations are used for modeling fluid dynamics. This integration procedure is a form of exponential map.

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vertical horizontal Velocity Field (v) Momentum (Av) Convolve with Greens Function of Differential Operator (K = A + ) Apply Differential Operator (A = L L)

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Some 2D Examples

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Deformations

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Inverse Deformations

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Jacobian Determinants

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Residuals

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Reconstructed from Template & Residuals

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Original Examples

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Grey Matter (550 IXI Brains)

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White Matter (550 IXI Brains)

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Template: Grey Matter

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Template: White Matter

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Template: Other

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Aligned IXI Brains

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Diffeomorphic Eigenwarps 550 scans from the IXI dataset, registered to a common average using a diffeomorphic registration model. Eigen-decomposition of initial velocities/momenta. Exaggerated warps generated by a geodesic shooting method. - Preserves one-to-one mapping. 1st 3rd 10th - +

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Predictions Brain shapes can be used to make predictions about individuals. This figure shows predictions of subjects ages made from IXI dataset using Tippings Relevance Vector Regression (a kernel method). More useful predictions may be possible from other data. Cross-validation results for age predictions.

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Caricatured Differences Used multivariate logistic regression to determine the initial velocity that best separates male from female brains. –Similar principles to Gaussian Process Classification, but with kernel matrix related to that of: Wang, L., Beg, M., Ratnanather, J., Ceritoglu, C., Younes, L., Morris, J.C., Csernansky, J.G. & Miller, M.I. Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE Transactions on Medical Imaging 26(4):462 (2007). Diffeomorphic shooting to obtain exaggerated deformations. Average brain deformed using these exaggerated deformations.

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Another example Predicting the logarithm of the brain/CSF ratio. Used a Gaussian Process Regression approach. Deformed average, from a brain with very little CSF, to one with a lot.

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Some References M. F. Beg, M. I. Miller, A.Trouve & L. Younes. "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". International Journal Of Computer Vision (2003). M. I. Miller, A. Trouve, & L. Younes. "Geodesic Shooting for Computational Anatomy". Journal of Mathematical Imaging and Vision (2004) L. Wang, M. F. Beg, J. T. Ratnanather, C. Ceritoglu, L. Younes, J. C. Morris, J. G. Csernansky & M. I. Miller. "Large Deformation Diffeomorphism and Momentum Based Hippocampal Shape Discrimination in Dementia of the Alzheimer Type". IEEE Trans. Med Imaging (2006) L. Younes, F. Arrate & M. I. Miller. "Evolution Equations in Computational Anatomy". Neuroimage (2008)

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Enjoy your retirement Vin.

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Metrics It is useful if distance measures between scans have the properties of metrics. Requirements are: Non-negativity:d(A,B) 0 Triangle Inequality:d(A,C) d(A,B) + d(B,C) Symmetry:d(A,B) = d(B,A) Identity of discernibles:d(A,B) = 0 iff A=B These are not satisfied by the small deformation approximation, but they are using the diffeomorphism framework of Miller, Younes et al.

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