analysis of planar shapes using geodesic paths on shape space e klassen a srivastava w mio s joshi l.
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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

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
Motivation for Shape Analysis
  • Applications: medical imaging, object, recognition, shape morphing, etc
  • Need: a tool to represent, analyze and interpolate among shapes
existing shape models
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
new approach
New approach

Using tangent angle function θ(s) or curvature k(s)

geometry representation of planar shape using s
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
comparing shape geodesic distance on shape space
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.
numerical methods for finding geodesics
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.
application of shape analysis
Application of Shape analysis
  • Interpolation and extrapolation on shape space
applications cont d
Applications (cont’d)
  • Clustering shapes
applications cont d10
Applications (cont’d)
  • Compute mean shape:
slide11
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
references
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.