A Subspace Learning of Dynamics on a Shape Manifold: A Generative Modeling Approach

Auteurs : Sheng YI
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A Subspace Learning of Dynamics on a Shape Manifold: A Generative Modeling  Approach


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10/15/13 1 A Subspace Learning of Dynamics on a Shape Manifold: A Generative Modeling Approach Sheng Yi* and H. Krim VISSTA, ECE Dept., NCSU Raleigh NC 27695 *GE Research Center, NY Thanks to AFOSR Outline •  Motivation •  Statement of the problem •  Highlight key issues and brief review •  Proposed model and solution •  Experiments 10/15/13 2 Problem Statement X(t) Z(t) Looking for a subspace that preserve geometrical properties of data in the original space Related Work •  Point-wise subspace learning –  PCA, MDS, LLE, ISOMAP, Hessian LLE, Laplacian Mapping, Diffusion Map, LTSA [T. Wittman, "Manifold Learning Techniques: So Which is the Best?“, UCLA ] •  Curve-wise subspace learning –  Whitney embedding [D. Aouada, and H. K., IEEE Trans. IP, 2010] •  Shape manifold –  Kendall’s shape space •  Based on landmarks –  Klassen et al. shape space •  Functional representation •  Concise description of tangent space & Fast Implementation –  Michor&Mumford’s shape space •  Focus on parameterization •  Complex description of tangent space& Heavy computation –  Trouve and Younes Diff. Hom Approach 10/15/13 3 Contribution Summary •  Proposed subspace learning is Invertible Original seq. Reconstructed seq. Subspace seq.
 Contribution Summary •  The parallel transport of representative frames defined by a metric on the shape manifold preserves curvatures in the subspace •  Ability to apply an ambient space calculus instead of relying essentially on manifold calculus 10/15/13 4 Shape Representation •  From curve to shape [Klassen et al.] α(s)= x(s),y(s)( )∈R2 ⇒ ∂ ∂s ∂α ∂s = cosθ(s),sinθ(s)( ) (simpleandclosedθ(s))\Sim(n) Closed: cosθ(s)ds 0 2π ∫ = 0 sinθ(s)ds 0 2π ∫ = 0 Rotation: 1 2π θ(s)ds 0 2π ∫ = π Dynamic Modeling on a Manifold •  The Core idea27 ( ) ti t XV X T M∈ dXt Process on M Driving Process on Rn dXt = Vi (Xt )dZi (t) i=1 dim( M ) ∑ ∈TXt M dim( M ) dZii (t) Zii (t) 10/15/13 5 Parallel Transport span Tangent along curve Tangent along curve Parallel Transport X0 X1 M [ Yi et al. IEEE IP, 2012] The core idea •  Adaptively select frame to represent in a lower dimensional space ( ) ti t XV X T M∈ dXt Process on M Driving Process on Rn dXt PCA on vectors parallel transported to a tangent space dZi (t) Rdim( M ) 10/15/13 6 Formulation of Curves on Shape Manifold A shape at the shape manifold Vectors span the tangent space of the shape manifold[ Yi et al. IEEE IP, 2012] A Euclidean driving process Vectors span a subspace A driving process in a subspace In original space: In a subspace: Core Idea •  Restrict the selection of V to be parallel frames on the manifold •  Advantage of using parallel moving frame: •  Angles between tangent vectors are preserved. With the same sampling scheme, curvatures are preserved as well. •  Given the initial frame and initial location on manifold, the original curve could be reconstructed 10/15/13 7 Core Idea •  Find an L2 optimal V Euclidean distance is used here because it is within a tangent space of the manifold Core Idea Given a parallel transport on shape manifold, with some mild assumption we can obtain a solution as a PCA 10/15/13 8 Parallel Transport Flow Chart By definition of parallel transport Discrete approx. of derivation Tangent space of shape manifold is normal to b1,b2,b3 Tangent space of shape manifold is normal to b1,b2,b3 A linear system Experiments •  Data Ø Moshe Blank et al., ICCV 2005 http://www.wisdom.weizmann.ac.il/~vision/SpaceTimeActions.html Ø Kimia Shape Database Sharvit, D. et al.,. Content-Based Access of Image and Video Libraries,1998 •  Walk •  Run •  Jump •  Gallop sideways •  Bend •  One-hand wave •  Two-hands wave •  Jump in place •  Jumping Jack •  Skip 10/15/13 9 Reconstruction Experiment PCA in Euclidean space The proposed method More Reconstructions and Embeddings 10/15/13 10 Other Embedding Result Experiment on curvature preservation 10/15/13 11 Experiment on curvature preservation 10/15/13 12 Generative Reconstruction 10/15/13 13 Conclusions •  A low dimensional embedding of a parallelly transported shape flow proposed •  A learning-based inference framework achieved •  A generative model for various shape- based activities is obtained