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Analysis of Planar Shapes Using Geodesic Paths on Shape Space E. Klassen, A. Srivastava, W. Mio, S. Joshi

Analysis of Planar Shapes Using Geodesic Paths on Shape Space E. Klassen, A. Srivastava, W. Mio, S. Joshi. Nhon Trinh EN-161 final project Initial presenation Nov 8, 2004. Motivation for Shape Analysis. Applications: medical imaging, object, recognition, shape morphing, etc.

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Analysis of Planar Shapes Using Geodesic Paths on Shape Space E. Klassen, A. Srivastava, W. Mio, S. Joshi

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  1. Analysis of Planar Shapes Using Geodesic Paths on Shape Space E. Klassen, A. Srivastava, W. Mio, S. Joshi Nhon Trinh EN-161 final project Initial presenation Nov 8, 2004

  2. Motivation for Shape Analysis • Applications: medical imaging, object, recognition, shape morphing, etc • Need: a tool to represent, analyze and interpolate among shapes

  3. Existing Shape Models • Most represent shape as finite number of salient points (landmarks) • Drawbacks: • outcome and accuracy of analysis heavily dependent on the choice of landmarks • Difficult to automate the selection of landmarks

  4. New approach Using tangent angle function θ(s) or curvature k(s)

  5. Geometry Representation of Planar Shape Using θ(s) • Each shape is represented as a function θ: [0, 2π)  R2. θ is point the pre-shape manifold C • Constraints: • Invariant to rotation: mean = π • Closure condition: • Let S be the re-parameterization group (change of initial point along the curve). The shape space is C/S

  6. Comparing Shape: Geodesic Distance on Shape Space • Geodesics on a manifold embedded in a Euclidean space is defined to be a constant speed curve on the manifold, whose acceleration is always perpendicular to the manifold. Geodesic is the shortest-distance curve to travel between two points on a manifold.

  7. Numerical Methods for Finding Geodesics • Task: Given two shapes θ1 and θ2, find an geodesic path to go from θ1 to θ2. • Method: among all directions in tangent space Tp of the shape space at θ1, find the direction that leads to θ2. • Difficulty: Tp(S) is infinite-dimensional • Solution: Approximate elements of Tp(S) with finite-dimensional Fourier series.

  8. Application of Shape analysis • Interpolation and extrapolation on shape space

  9. Applications (cont’d) • Clustering shapes

  10. Applications (cont’d) • Compute mean shape:

  11. Plan • Now  Thanksgiving • Implement code to compute geodesic distance between two shapes • Implement shape interpolation • Thanksgiving  Final • Implement shape averaging • Test on LEMS’ shape database

  12. References • Klassen, E., A. Srivastava, W. Mio, S. Joshi. Analysis of Planar Shapes Using Geodesic Paths on Shape Space. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2004. • Lang, S. Fundamentals of Differential Geometry. Springer. 1999. • Marques, J. and A. Abrantes. Shape alignment – optimal initial point and pose estimation. Pattern Recognition Letters. 1997.

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