Multidimensional sampling

Abstract: The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments. It is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors. The most efficient lattice (i.e., requiring minimum sampling points per unit hypervolume) is not in general rectangular, nor is a unique reconstruction function associated with a given sampling lattice. The above results also apply to homogeneous wave-number-limited stochastic processes in the sense of a vanishing mean-square error. It is also found that, given a particular sampling lattice, the optimum (mean-square) presampling filter for nonwave-number-limited processes effects an ideal wave-number cutoff appropriate to the specified sampling lattice. Particular attention is paid to isotropic processes: minimum sampling lattices are specified up to eight-dimensional spaces, and a number of typical reconstruction functions are calculated.

Abstract: Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized concept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jittered sampling and N-rooks sampling at the same time. The faster convergence and improved anti-aliasing are demonstrated by numerical experiments. Categories and Subject Descriptors (according to ACM CCS): G.3 [Probability and Statistics]: Probabilistic Algorithms (including Monte Carlo); I.3.2 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism.

Abstract: We present an accurate and efficient method to generate samples based on a Poisson-disk distribution. This type of distribution, because of its blue noise spectral properties, is useful for image sampling. It is also useful for multidimensional Monte Carlo integration and as part of a procedural object placement function. Our method extends trivially from 2D to 3D or to any higher dimensional space. We demonstrate results for up to four dimensions, which are likely to be the most useful for computer graphics applications. The method is accurate because it generates distributions with the same statistical properties of those generated with the brute-force dart-throwing algorithm, the archetype against which all other Poisson-disk sampling methods are compared. The method is efficient because it employs a spatial subdivision data structure that signals the regions of space where the insertion of new samples is allowed. The method has O(N log N) time and space complexity relative to the total number of samples. The method generates maximal distributions in which no further samples can be inserted at the completion of the algorithm. The method is only limited in the number of samples it can generate and the number of dimensions over which it can work by the available physical memory.

Abstract: In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.