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PDE methods for Image Segmentation and Shape Analysis: From the Brain to the Prostate and Back presented by John Melonakos – NAMIC Core 1 Workshop – 30/May/2007

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2 Outline Bhattacharyya Segmentation Segmentation Results --------------------------------------- Shape Analysis Shape-Driven Segmentation

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3 Contributors Georgia Tech- Yogesh Rathi, Sam Dambreville, Oleg Michailovich, Jimi Malcolm, Allen Tannenbaum

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4 Publications S. Dambreville, Y. Rathi, and A. Tannenbaum. A framework for Image Segmentation using Shape Models and Kernel Space Shape Priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, (to appear 2007). O. Michailovich, Y. Rathi, and A. Tannenbaum. Image Segmentation using Active Contours Driven by Informaion-Based Criteria. IEEE Transactions on Image Processing, (to appear 2007). Y. Rathi, O. Michailovich, and A. Tannenbaum. Segmenting images on the Tensor Manifold. In CVPR, 2007. Eric Pichon, Allen Tannenbaum, and Ron Kikinis. A statistically based surface evolution method for medical image segmentation: presentation and validation. In International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), volume 2, pages 711-720, 2003. Note: Best student presentation in image segmentation award. Y. Rathi, O. Michailovich, J. Malcolm, and A. Tannenbaum. Seeing the Unseen: Segmenting with Distributions. In Intl. Conf. Signal and Image Processing, 2006. J. Malcolm, Y. Rathi, A. Tannenbaum. Graph cut segmentation with nonlinear shape priors. In Intl. Conf. Signal and Image Processing, 2007.

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5 Segmentation Hierarchy Threshold-basedEdge-basedRegion-based Parametric methods Implicit methods Parameterized representation of the curve (shape) Implicit representation of the curve using level sets

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6 Geometric Active Contours Image Use Calculus of variations

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7 Our Contributions Segmentation by separating intensity based probability distributions (not just intensity moments as in previous works). Novel formulation of the Bhattacharyya distance in the level set framework so as to optimally separate the region inside and outside the evolving contour.

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8 Bhattacharyya Distance The Bhattacharyya distance gives a measure of similarity between two distributions: where z Z is any photometric variable like intensity, color vector or tensors. B can also be thought of as the cosine of the angle between two vectors.

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9 Bhattacharyya Distance Let x R 2 specify the co-ordinates in the image plane and I : R 2 Z be a mapping from image plane to the space of photometric variable Z. Then the pdf is given by: This is the nonparametric density estimate of the pdf of z given the kernel K.

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10 In the Level Set Framework The pdf’s written in terms of the level set function is given by :

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11 The First Variation For segmentation purposes, we would like to minimize the Bhattacharyya distance. This is achieved using calculus of variations, by taking the first variation of B as follows :

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12 The First Variation (cont.) The first variation of P in and P out is given by : where,

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13 Resulting PDE Plugging in all the components, we get the following PDE (partial differential equation) for separating the distributions :

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14 Additional Terms In numerical experiments, an additional regularizing term is added to the resulting PDE that penalizes the length of the contour making it smooth. Thus, the final PDE is given by:

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15 Outline Bhattacharyya Segmentation Segmentation Results --------------------------------------- Shape Analysis Shape-Driven Segmentation

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16 Segmenting the Caudate Nucleus

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17 Caudate Nucleus

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18 Zebra

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19 The Unseen!

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20 The Unseen! Toy example: Region inside and outside was obtained by sampling from a Rayleigh distribution with the same mean and variance. Template ImageGenerated Image

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21 The Unseen! Starting distributionFinal distributionActual distribution

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22 Application to Tensors Intensity is not enough to segment several types of images. Diffusion Tensor MRI images have become common, where at each pixel a tensor is computed from a set of gradients. Color coded Fractional Anisotropy image

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23 Structure Tensors Structure tensor reveals many features like edges, corners or texture of an image. A structure tensor for a scalar valued image I is given by: (K is a Gaussian kernel) Color structure tensor is given by:

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24 The Tensor Manifold The space of n x n positive definite symmetric matrices, is not a vector space, but forms a manifold (a cone). Many past methods by Wang- Vemuri, Lenglet et.al., have however assumed the tensor space to be Euclidean. The active contour based segmentation was performed under this assumption. Structure tensor space for a typical image.

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25 Riemannian vs Euclidean Manifold Euclidean distance d(A,B) = d(A,C) = d(C,B) = d 1 Riemannian distance d r (A,B) = d(A,C) + d(C,B) = 2d 1 Thus, under Euclidean manifold assumption, one obtains an erroneous estimate for mean and variance used in many active contour based segmentation algorithms.

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26 Basic Riemannian Geometry For a tensor manifold (cone), T x M is the set of all symmetric matrices and forms a vector space. TxMTxM x Y Y’ Log Map M

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27 Tensor Space A recent method proposed by Lenglet et.al. (2006), incorporates the Riemannian geometry of the tensor space and performs segmentation by assuming a Gaussian distribution of the object and background. By using the Bhattacharyya distance and taking into account the Riemannian structure of the tensor manifold, we propose to extend the above segmentation technique to any arbitrary and non-analytic probability distribution.

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28 Segmentation Framework Compute Mean on the Riemannian Manifold Map all points onto the Tangent Space T M at the mean Euclidean Space Compute Target points or bins Perform Curve Evolution using the PDE described earlier. Accepted for publication in IEEE CVPR 2007

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29 Duck Segmentation using Bhattacharyya flow, but using Riemannian metric Segmentation using Bhattacharyya flow, but assuming Euclidean distance between tensors

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30 Tiger Segmentation using Bhattacharyya flow, but assuming Euclidean distance between tensors Segmentation using Bhattacharyya flow, but using Riemannian metric

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31 Butterfly Segmentation assuming Euclidean metric Segmentation assuming Riemannian metric Color structure tensor:

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32 Segmentation Summary No assumption on the distribution of the object or background. Computationally very fast, since we only need to update the probability distribution instead of having to map each point in the image from Riemannian space to tangent space after each iteration to compute the mean and variance (under a Gaussian assumption).

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33 Outline Bhattacharyya Segmentation Segmentation Results --------------------------------------- Shape Analysis Shape-Driven Segmentation

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34 Contributors Georgia Tech- Delphine Nain, Xavier LeFaucheur, Yi Gao, Allen Tannenbaum

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35 Publications D. Nain, S. Haker, A. Tannenbaum. Multiscale 3D shape representation and segmentation using spherical wavelets. IEEE Trans. Medical Imaging, 26 (2007). pp 598-618. D. Nain, S. Haker, A. Bobick, and A. Tannenbaum. Shape-Driven 3D Segmentation using using Spherical Wavelets. In Proceedings of MICCAI, Copenhagen, 2006. Note: Best Student Paper Award in the category Segmentation and Registration. D. Nain, S. Haker, A. Bobick, and A. Tannenbaum. Multiscale 3D Shape Analysis using Spherical Wavelets. In Proceedings of MICCAI, Palm Springs, 2005.

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36 Overview 3D, Parametric, Data-driven prior Caudate nucleus dataset Prostate dataset

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37 Overview of ASM K shapes in training set, N landmarks

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38 Limitations of ASM Rank of the covariance matrix DD T is at most K-1 Small training set: only first K-1 major variations captured by shape prior E.g. Reconstruction, given new shape s Ground TruthReconstructed with ASM shape prior

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39 Multi-scale prior Hierarchical decomposition: shape is represented at different scales [Davatzikos03] Learn variations at each scale

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40 The Algorithm Step 1: Find Landmarks [Conformal Mapping] Step 2: Multi-scale representation [Spherical Wavelets] Step 3: Find independent bands of variation [Spectral Graph Partitioning]

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41 [1] Shape Registration and Re-meshing Nonlinear area-preserving mapping [Brechbuhler95], Conformal mapping [Haker04]

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42 [2] Spherical Wavelets A function decomposed in wavelet space is uniquely described by a Weighted sum of scaling functions and wavelet functions that are localized in space and scale Spherical scaling and wavelet functions are defined on a multi-resolution grid Scaling, level 0 Wavelet, level 1Wavelet, level 2Wavelet, level 3

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43 [2] Spherical Wavelets In matrix notation: is signal on the sphere Analysis: Synthesis:

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44 [2] Shape Representation After the registration step, all shapes have the required mesh structure Given a shape S k, we find a 3 1D signals: We take the wavelet transform of each signal and represent the shape as: Original Shape Shape representation using a weighted combination of the lowest resolution scaling functions and wavelet functions up to j th resolution j=0j=1j=2j=3

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45 [2] Compression Compression: from 2562 to 649 coefficients, mean error 5.10 -3 At each scale, we can truncate least significant coefficients based on spectrum analysis of population Results in local compression at each scale

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46 [3] Scale-Space Prior Previous approach [Davatzikos 03] We propose a more principled approach where for each scale, we cluster highly correlated coefficients into a band, with the constraint that coefficients across bands have minimum cross-correlation

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47 Band Decomposition Covariance Matrix level 1 1 2 3 4 5 6 123456123456 2 3 6 1 4 5 236145236145 Rearranged Covariance Matrix level 1

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48 Band Decomposition Spectral Graph Partitioning technique [Shi00] Fully connected graph G = (V,E) where nodes V are wavelet coefficients for a particular scale Weight on each edge: w(i, j) is covariance between coefficients i and j Stopping criterion: validating whether the subdivided band correspond to two independent distributions based on KL divergence

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49 Band Decomposition Color is influence of coefficients in a band: red (high), blue (none)

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50 Building the Prior Assuming K shapes in training set, for each band, we obtain (K-1) eigenvectors In total we have B(K-1) eigenvectors, where B is number of bands

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51 Experiments Dataset of N samples randomly into T training samples and [N − T] testing samples, where T = [5, 10, 25] Reconstruction task: Test: Compare to ASM, other wavelet band decomposition Effect of noise Effect of truncation

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52 Results GT Noise WAV rec. from GTWAV rec. from Noise ASM rec. from GTASM rec. from Noise GTNoise WAV rec. from GTWAV rec. from Noise ASM rec. from GTASM rec. from Noise

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53 Results Reconstruction from Ground TruthReconstruction from Noisy Shape

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54 Outline Bhattacharyya Segmentation Segmentation Results --------------------------------------- Shape Analysis Shape-Driven Segmentation

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55 Shape-Driven Segmentation End goal is to derive a parametric surface evolution equation by evolving the wavelet coefficients directly so that we can include the shape prior directly in the flow

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56 Segmentation via Shape Prior Likelihood Evolve , p Shape Representation Shape Prior Constrain , p Segmented Shape Segmentation Probability Distribution Learn shape space (evectors U) Learn bounds within shape space Pose: Rotations, Translations, Scaling shape

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57 Energy Minimization Region-based energy Data term Region inside evolving surface [Rousson05] Points on Evolving surface Points on Evolving surface

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58 Evolution Update equations Run until step size Start with lowest resolution s Run for 1 iteration Constrain Add next resolution s when Coarse to fine evolution

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59 Experiments 1. Evolution with PDM shape prior (Active Shape Model) 2. Evolution with WDM shape prior Two types image input: Using Ground Truth image data (binary): to test convergence Using real image data Quantitative measurements: Compare to Ground truth (manually segmented) Details: caudate nucleus shapes from MRI scans training set of 24 shapes, testing set of 5 shapes 4 subdivision levels, 16 bands in the shape prior Start with mean shape, mean position

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60 Ground Truth Example Image data is binary GT in red; ASM in blueGT in red; Mscale in blue

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61 Caudate Nucleus Example GT in red; ASM in yellowGT in red; Mscale in blue

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62 Caudate Nucleus Example MRI image intensity GT in red; ASM in blue GT in red; Mscale in blue

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63 Conclusions (Speculations) Geodesic tractography (tomorrow) Fast non-rigid registration (tomorrow) Estimation and filtering techniques from tracking?

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64 Questions?

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