Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying MATLAB toolbox called ShearLab (www.ShearLab.org) is provided. And, third, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

In this paper, we present a new framework to compress the geometry of images. This framework generalizes the standard quad partitioning approaches in compression of image geometry (e.g. wedgelet) in two ways. First, we employ an adaptive rectangular partitioning rather than quadratic partitioning. Second, our coder uses an overcomplete collection of (stripe-like) atoms which contains wedgelets as a special case. We present an information-theoretical analysis based on Kolmogorov's e- entropy to show that this collection provides a near-optimal representation of a class of cartoon images with piecewise polynomial boundaries. Furthermore, we develop a provably near-optimal greedy algorithm that significantly reduces the complexity of the exhaustive search method required to achieve the entropy bound. Simulation results for the rate distortion shows a 1.5-2 dB improvement over the standard wedgelets for the "Cameraman" image.

In this paper, some detection-theoretic, estimation-theoretic, and information-theoretic methods are investigated to analyze the problem of determining resolution limits in imaging systems. The canonical problem of interest is formulated based on a model of the blurred image of two closely spaced point sources of unknown brightness. To quantify a measure of resolution in statistical terms, the following question is addressed: "What is the minimum detectable separation between two point sources at a given signal-to-noise ratio (SNR), and for prespecified probabilities of detection and false alarm (Pdand Pf)?". Furthermore, asymptotic performance analysis for the estimation of the unknown parameters is carried out using the Crameacuter-Rao bound. Although similar approaches to this problem (for one-dimensional (1-D) and oversampled signals) have been presented in the past, the analyzes presented in this paper are carried out for the general two-dimensional (2-D) model and general sampling scheme. In particular the case of under-Nyquist (aliased) images is studied. Furthermore, the Kullback-Liebler distance is derived to further confirm the earlier results and to establish a link between the detection-theoretic approach and Fisher information. To study the effects of variation in point spread function (PSF) and model mismatch, a perturbation analysis of the detection problem is presented as well

This paper concerns the problem of resolvability power in the frequency domain. The canonical case of interest is to distinguish whether the received noise-corrupted signal is a single-frequency sinusoid or a two-frequency sinusoid, where the amplitudes, phases and frequencies are unknown to the receiver. Using a model-based hypothesis testing approach, we quantify a measure of attainable resolution between sinusoids with nearby frequencies, in the presence of noise. An explicit relationship is derived for the minimum detectable difference between the frequencies of two tones, for any particular false alarm and detection rate, and at a given SNR. An associated algorithm is proposed that produces significantly better performance compared to the standard subspace-based methods like MUSIC and can be effectively used in practice as a postprocessing step for the existing spectral estimation methods.

The present paper is concerned with the statistical analysis of the resolution limit in a so-called "diffraction-limited" imaging system. The canonical case study is that of incoherent imaging of two closely-spaced sources of possibly unequal brightness. The objective is to study how far beyond the classical Rayleigh limit of resolution one can reach at a given signal to noise ratio. The analysis uses tools from statistical detection and estimation theory. Specifically, we will derive explicit relationships between the minimum detectable distance between two closely-spaced point sources imaged incoherently at a given SNR. For completeness, asymptotic performance analysis for the estimation of the unknown parameters is carried out using the Cramér-Rao bound. To gain maximum intuition, the analysis is carried out in one dimension, but can be well extended to the two-dimensional case and to more practical models.