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A time domain blind source separation algorithm of convolutive sound mixtures is studied based on a compact partial inversion formula in closed form. An L1-constrained minimization problem is formulated to find demixing filter coefficients for source separation while capturing scaling invariance and sparseness of solutions. The minimization aims to reduce (lagged) cross correlations of the mixture signals, which are modeled stochastically. The problem is non-convex, however it is put in a nonlinear least squares form where the robust and convergent Levenberg-Marquardt iterative method is applicable to compute local minimizers. Efficiency is achieved in recovering lower dimensional demixing filter solutions than the physical ones. Computations on recorded and synthetic mixtures show satisfactory performance, and are compared with other iterative 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.

The medial axis transform (MAT) is an important shape representation for shape approximation, shape recognition, and shape retrieval. Despite years of research, there is still a lack of effective methods for efficient, robust and accurate computation of the MAT. We present an efficient method, called {\em Q-MAT}, that uses quadratic error minimization to compute a structurally simple, geometrically accurate, and compact representation of the MAT. We introduce a new error metric for approximation and a new quantitative characterization of unstable branches of the MAT, and integrate them in an extension of the well-known quadric error metric (QEM) framework for mesh decimation. Q-MAT is fast, removes insignificant unstable branches effectively, and produces a simple and accurate piecewise linear approximation of the MAT. The method is thoroughly validated and compared with existing methods for MAT computation.

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.