Signal processing issues in diffraction and holographic 3DTV

IEEE Trans. Signal Process.

Arıkan

2010

Self Cite

Abstract-We are concerned with the problem of optimally measuring an accessible signal under a total cost constraint, in order to estimate a signal which is not directly accessible. An important aspect of our formulation is the inclusion of a measurement device model where each device has a cost depending on the number of amplitude levels that the device can reliably distinguish. We also assume that there is a cost budget so that it is not possible to make a high amplitude resolution measurement at every point. We investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error. This problem differs from standard estimation problems in that we are allowed to "design" the number and noise levels of the measurement devices subject to the cost constraint. Our main results are presented in the form of tradeoff curves between the estimation error and the cost budget. Although our primary motivation and numerical examples come from wave propagation problems, our formulation is also valid for other measurement problems with similar budget limitations where the observed variables are related to the unknown variables through a linear relation. We discuss the effects of signal-to-noise ratio, distance of propagation, and the degree of coherence (correlation) of the waves on these tradeoffs and the optimum cost allocation. Our conclusions not only yield practical strategies for designing optimal measurement systems under cost constraints, but also provide insights into measurement aspects of certain inverse problems.

Section: Special Cases 1) Repeated Measurements Of the Field At Amentioning

confidence: 99%

Signal Recovery With Cost-Constrained Measurements

Özçelikkale

IEEE Trans. Signal Process.

Arıkan

2010

Self Cite

“…Two fundamental signal processing problems in holographic 3DTV are what we will refer to as the forward and inverse problems [64].…”

Section: Fundamental Problems In Holographic 3dtvmentioning

confidence: 99%

A Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television

Onural

Gotchev

IEEE Trans. Circuits Syst. Video Technol.

et al. 2007

Self Cite

Abstract-Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory.

“…More precisely, the measured signal is of the form g(x) = L{f (x)} + n(x), (1) where x e Rk, f: IRk --> R is the unknown input random process, n2: Rk --> R is the random process denoting the inherent system noise which is independent of the input f, and g: Rk --> R is the output of the linear system. The dimension k is typically 1 or 2.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Its solutions in free space can be expressed in many forms. One of these is to express the field over one plane in terms of that on another, through a diffraction integral, a convenient approximation of which is well-known as the Fresnel diffraction integral [1]. Generalizations of such so-called quadratic-phase integrals allow one to characterize a broad class of optical systems involving arbitrary concatenations of lenses and sections of free space [2].…”

Section: Introductionmentioning

confidence: 99%

Optimal Measurement under Cost Constraints for Estimation of Propagating Wave Fields

Özçelikkale

2007 IEEE International Symposium on Information Theory

Arıkan

2007

Self Cite

Abstract-We give a precise mathematical formulation of some measurement problems arising in optics, which is also applicable in a wide variety of other contexts. In essence the measurement problem is an estimation problem in which data collected by a number of noisy measurement probes are combined to reconstruct an unknown realization of a random process f (x) indexed by a spatial variable x C Rk for some k > 1. We wish to optimally choose and position the probes given the statistical characterization of the process f(x) and of the measurement noise processes. We use a model in which we define a cost function for measurement probes depending on their resolving power. The estimation problem is then set up as an optimization problem in which we wish to minimize the mean-square estimation error summed over the entire domain of f subject to a total cost constraint for the probes. The decision variables are the number of probes, their positions and qualities. We are unable to offer a solution to this problem in such generality; however, for the metrical problem in which the number and locations of the probes are fixed, we give complete solutions for some special cases and an efficient numerical algorithm for computing the best trade-off between measurement cost and mean-square estimation error. A novel aspect of our formulation is its close connection with information theory; as we argue in the paper, the mutual information function is the natural cost function for a measurement device. The use of information as a cost measure for noisy measurements opens up several direct analogies between the measurement problem and classical problems of information theory, which are pointed out in the paper.