Dimension Reduction on Polyspheres with Application to Skeletal Representations

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14317

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We present a novel method that adaptively deforms a polysphere (a product of spheres) into a single high dimensional sphere which then allows for principal nested spheres (PNS) analysis. Applying our method to skeletal representations of simulated bodies as well as of data from real human hippocampi yields promising results in view of dimension reduction. Specifically in comparison to composite PNS (CPNS), our method of principal nested deformed spheres (PNDS) captures essential modes of variation by lower dimensional representations.

Dimension Reduction on Polyspheres with Application to Skeletal Representations

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application/pdf Dimension Reduction on Polyspheres with Application to Skeletal Representations Benjamin Eltzner, Sungkyu Jung, Stephan Huckemann

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We present a novel method that adaptively deforms a polysphere (a product of spheres) into a single high dimensional sphere which then allows for principal nested spheres (PNS) analysis. Applying our method to skeletal representations of simulated bodies as well as of data from real human hippocampi yields promising results in view of dimension reduction. Specifically in comparison to composite PNS (CPNS), our method of principal nested deformed spheres (PNDS) captures essential modes of variation by lower dimensional representations.

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Introduction Deformation Skeletal Representations Conclusion Dimension Reduction on Polyspheres with Application to Skeletal Representations joint work with Stephan Huckemann and Sungkyu Jung Benjamin Eltzner University of Göttingen conference on Geometric Science of Information, 2015-10-30 Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Dimension Reduction on Manifolds PCA relies on linearity. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Dimension Reduction on Manifolds PCA relies on linearity. Tangent space approaches ignore geometry and periodic topology. Intrinsic approaches rely on manifold geometry. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Dimension Reduction on Manifolds PCA relies on linearity. Tangent space approaches ignore geometry and periodic topology. Intrinsic approaches rely on manifold geometry. Two classes: Forward methods: Submanifold dimension d = 1, 2, 3, . . . Needs “good” geodesics and a construction scheme. Backward methods: d = D − 1, D − 2, D − 3, . . . Needs rich (parametric) set of submanifolds. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Polysphere Dimension Reduction Almost all geodesics of PD = Sd1 r1 × · · · × SdK rK are dense in (S1 )K . Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Polysphere Dimension Reduction Almost all geodesics of PD = Sd1 r1 × · · · × SdK rK are dense in (S1 )K . Low symmetry isom(PD ) = SO(d1 + 1) × · · · × SO(dK + 1), no generic rich set of submanifolds. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Deformation for Unit Spheres Dimension reduction methods exist for spheres: GPCA1 , HPCA2 , PNS3 Recursively deform polysphere to sphere f : PD → SD . Squared line elements of two unit spheres: ds2 1 = d1 k=1   k−1 j=1 sin2 φ1,j   dφ2 1,k, ds2 2 = d2 k=1   k−1 j=1 sin2 φ2,j   dφ2 2,k Deformation: ds2 = ds2 2 + d2 j=1 sin2 φ2,j ds2 1 1 S. Huckemann and H. Ziezold. Advances in Applied Probability 2.38 (2006), pp. 299–319. 2 S. Sommer. Geometric Science of Information. Vol. 8085. Lecture Notes in Computer Science. 2013, pp. 76–83. 3 S. Jung, I. L. Dryden, and J. S. Marron. Biometrika 99.3 (2012), pp. 551–568. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Deformation for Unit Spheres Dimension reduction methods exist for spheres: GPCA1 , HPCA2 , PNS3 Recursively deform polysphere to sphere f : PD → SD . Squared line elements of two unit spheres: ds2 1 = d1 k=1   k−1 j=1 sin2 φ1,j   dφ2 1,k, ds2 2 = d2 k=1   k−1 j=1 sin2 φ2,j   dφ2 2,k Deformation: ds2 = ds2 2 + d2 j=1 sin2 φ2,j ds2 1 Degrees of freedom: Rotation and ordering of spheres. 1 S. Huckemann and H. Ziezold. Advances in Applied Probability 2.38 (2006), pp. 299–319. 2 S. Sommer. Geometric Science of Information. Vol. 8085. Lecture Notes in Computer Science. 2013, pp. 76–83. 3 S. Jung, I. L. Dryden, and J. S. Marron. Biometrika 99.3 (2012), pp. 551–568. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Fixing Degrees of Freedom Rotation: Embed Sdi ri into Rdi+1 . Determine Fréchet mean ˆµi and use rotation along a geodesic to move it to positive xi,di+1-direction (north pole). Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Fixing Degrees of Freedom Rotation: Embed Sdi ri into Rdi+1 . Determine Fréchet mean ˆµi and use rotation along a geodesic to move it to positive xi,di+1-direction (north pole). Ordering: Data spread: si = N n=1 d2 (ψi,n, ˆµi) Choose permutation p such that sp−1(1) is maximal and sp−1(K) is minimal. Minimizes distortion due to factors sin2 φj, i. e. deviation from polysphere geometry. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Mapping Data Points Embedding Sdi 1 ⊂ Rdi+1 we get ∀1 ≤ j ≤ d2 : yj = x2,j, ∀1 ≤ k ≤ d1 + 1 : yd2+k = x2,d1+1x1,j Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Mapping Data Points Embedding Sdi 1 ⊂ Rdi+1 we get ∀1 ≤ j ≤ d2 : yj = x2,j, ∀1 ≤ k ≤ d1 + 1 : yd2+k = x2,d1+1x1,j For different radii, rescale ∀1 ≤ j ≤ d1 + 1 : x1,j → ˜x1,j = R1x1,j, ∀i > 1 ∀1 ≤ j ≤ di : xi,j → ˜xi,j = Rixi,j and use ˜x in definition of y coordinates. This yields an ellipsoid x ∈ Rd2+d1+1 d2 k=1 R−2 2 x2 2,k + d1+1 k=1 R−2 1 (x2,d2+1x1,k)2 = 1 . Normalize all y-vectors to length R := K j=1 Rj 1 K as final step. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Illustration for Different Radii 1. Map from blue polysphere to green ellipsoid. 2. Map to red sphere. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion A Brief Review of Principal Nested Spheres (PNS) PNS determines a sequence SK ⊃ SK−1 ⊃ · · · ⊃ S2 ⊃ S1 ⊃ {µ}. Recursively fit small subsphere Sd−1 ⊂ Sd minimizing sum of squared geodesic projection distances. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion A Brief Review of Principal Nested Spheres (PNS) PNS determines a sequence SK ⊃ SK−1 ⊃ · · · ⊃ S2 ⊃ S1 ⊃ {µ}. Recursively fit small subsphere Sd−1 ⊂ Sd minimizing sum of squared geodesic projection distances. At every projection, save signed projection distance (residuals). Parameter space dimension for Sd−1 ⊂ Sd is p = d + 1, compared to linear PCA where for Rd−1 ⊂ Rd it is p = d. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Skeletal Representation (s-rep) Parameter Space S-rep consists of 1. A two-dimensional mesh of m × n skeletal points. 2. Spokes from mesh points to the surface. Image from: J. Schulz et al. Journal of Computational and Graphical Statistics 24.2 (2015), p. 539 Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Skeletal Representation (s-rep) Parameter Space S-rep consists of 1. A two-dimensional mesh of m × n skeletal points. 2. Spokes from mesh points to the surface. Parameters: Size of centered mesh, spoke lengths, normalized mesh-points, spoke directions: Q = R+ × RK + × S3mn−1 × S2 K Polysphere deformation on S3mn−1 × S2 K yields Q = S5mn+2m+2n−5 Image from: J. Schulz et al. Journal of Computational and Graphical Statistics 24.2 (2015), p. 539 Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Dimension Reduction for Real S-reps PNDS: Deform polysphere to sphere and apply PNS. CPNS: PNS on spheres individually and linear PCA on joint residuals. 0 10 20 30 40 50 Dimension 0 20 40 60 80 100 Variances[%] PNDS CPNS Figure : PNDS vs. CPNS: residual variances for s-reps of 51 hippocampi5 . 5 S. M. Pizer et al. Ed. by M. Breuß, Bruckstein, and Maragos. Springer, Berlin, 2013, pp. 93–115. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Dimension Reduction for Simulated S-reps −100 −50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 component 3 variance = 0.64% −100 −50 0 50 100 −100 −50 0 50 100 components 3 and 2 −100 −50 0 50 100 −100 −50 0 50 100 components 3 and 1 −100 −50 0 50 100 −100 −50 0 50 100 components 2 and 3 −100 −50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 component 2 variance = 5.95% −100 −50 0 50 100 −100 −50 0 50 100 components 2 and 1 −100 −50 0 50 100 −100 −50 0 50 100 components 1 and 3 −100 −50 0 50 100 −100 −50 0 50 100 components 1 and 2 −100 −50 0 50 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 component 1 variance = 92.02% −100 −50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 component 3 variance = 2.17% −100 −50 0 50 100 −100 −50 0 50 100 components 3 and 2 −100 −50 0 50 100 −100 −50 0 50 100 components 3 and 1 −100 −50 0 50 100 −100 −50 0 50 100 components 2 and 3 −100 −50 0 50 100 0.0 0.1 0.2 0.3 0.4 0.5 component 2 variance = 32.10% −100 −50 0 50 100 −100 −50 0 50 100 components 2 and 1 −100 −50 0 50 100 −100 −50 0 50 100 components 1 and 3 −100 −50 0 50 100 −100 −50 0 50 100 components 1 and 2 −100 −50 0 50 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 component 1 variance = 62.73% Figure : PNDS vs. CPNS for simulated twisted ellipsoids: scatter plots of residual signed distances for the first three components. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Reflection on Parameter Space Dimension −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Figure : Simulated twisted ellipsoid data projected to the second component (a small two-sphere) in PNDS with first component (a small circle) inside. Parameter space dimensions: PNS on SD : p = 1 2 D(D + 3) − 1. PCA on RD : p = 1 2 D(D + 1). Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations Introduction Deformation Skeletal Representations Conclusion Conclusion We propose a deformation procedure mapping data on a polysphere to sphere. The construction aims at minimizing geometric distortion. We achieve lower dimensional representations than CPNS. The success of our method is rooted in the higher parameter space dimension. Benjamin Eltzner University of Göttingen Dimension Reduction on Polyspheres with Application to Skeletal Representations