Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradient-domain image processing and shape-preserving geometric computation. We propose Poisson coordinates, a novel transfinite interpolation scheme based on the Poisson integral formula, as a rapid way to estimate a harmonic function on a certain domain with desired boundary values. Poisson coordinates are an extension of the Mean Value coordinates (MVCs) which inherit their linear precision, smoothness, and kernel positivity. We give explicit formulae for Poisson coordinates in both continuous and 2D discrete forms. Superior to MVCs, Poisson coordinates are proved to be pseudoharmonic (i.e., they reproduce harmonic functions on n-dimensional balls). Our experimental results show that Poisson coordinates have lower Dirichlet energies than MVCs on a number of typical 2D domains (particularly convex domains). As well as presenting a formula, our approach provides useful insights for further studies on coordinates-based interpolation and fast estimation of harmonic functions.

Image matching is a fundamental problem in computer vision. One of the well-known techniques is SIFT (scale-invariant feature transform). SIFT searches for and extracts robust features in hierarchical image scale spaces for object identification. However it often lacks efficiency as it identifies many insignificant features such as tree leaves and grass tips in a natural building image. We introduce a content adaptive image matching approach by preprocessing the image with a color-entropy based segmentation and harmonic inpainting. Natural structures such as tree leaves have both high entropy and distinguished color, and so such a combined measure can be both discriminative and robust. The harmonic inpainting smoothly interpolates the image functions over the tree regions and so blurrs and reduces the features and their unnecessary matching there. Numerical experiments on building images show

Shape registration is fundamental to 3D object acquisition; it is used to fuse scans from multiple views. Existing algorithms mainly utilize geometric information to determine alignment, but this typically results in noticeable misalignment of textures (i.e. surface colors) when using RGB-depth cameras. We address this problem using a novel approach to color-aware registration, which takes both color and geometry into consideration simultaneously. Color information is exploited throughout the pipeline to provide more effective sampling, correspondence and alignment, in particular for surfaces with detailed textures. Our method can furthermore tackle both rigid and non-rigid registration problems (arising, for example, due to small changes in the object during scanning, or camera distortions). We demonstrate that our approach produces significantly better results than previous methods.

Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings.

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