We recently introduced the polyharmonic local sine transform (PHLST), a new $u+v$ model for hierarchical image decomposition. We first split the original domain into a quadtree of rectangular blocks. Then in each block, the $u$ component is obtained by solving the Laplace or biharmonic equations with appropriate boundary conditions. The $v$ component can be represented by the quickly decaying Fourier sine series.
In this talk, we discuss two generalizations of this idea after briefly reviewing the original PHLST: one is its extension to 3D data processing; and the other is image analysis of an object whose boundary is a smooth closed curve. We try to decouple the geometry and internal information of the object by solving the elliptic boundary value problem using Fast Multipole Method on the domain where the object is supported. This allows us to analyze the internal} information of the object, such as textures, without being disturbed by the object boundary, and to approximate and compress the object information efficiently.
This is a joint work with Jean-Francois Remy of France Telecom, Katsu Yamatani of Shizuoka Univ., Japan, and Noel Smith, Jim Zhao of UC Davis.
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