Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (e.g., bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. We also evaluate the performance of the relaxed alternative in various numerical experiments.
Due to the diverse shape geometries and sharp intensity transitions on the boundaries, this class of visual signals, like many other real-world signals, are neither bandlimitted nor belong to a shift invariant subspace. Hence, the classical sampling results do not apply here. A similar scenario happens for the class of 1D signals studied in @cite_24 @cite_15 , known as signals with finite rate of innovation (FRI). It is shown that the discrete samples can lead to perfect signal recovery, although the signals are not necessarily bandlimitted. A generalization to 2D FRI signals is presented in @cite_19 @cite_17 @cite_21 , with the goal of recovering convex polygonal shapes from the gray-scale pixels. A different approach is devised in @cite_2 by considering the boundary curves in a shape image as the zero-level-sets of specific 2D FRI signals. Due to the FRI requirements, exact recovery relies on the PSF satisfying the so-called Strang-Fix condition. Furthermore, the FRI model admits limited shape geometries.
@cite_21 @cite_24 @cite_19 @cite_2 @cite_15 @cite_17
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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. The finite rate of innovation (FRI) principle is developed for sampling a class of non-bandlimited signals that have a finite number of degrees of freedom per unit of time, i.e., signals with FRI. This sampling scheme is later extended to three classes of sampling kernels with compact support and applied to the step edge reconstruction problem by treating the image row by row. In this paper, we regard step edges as 2D FRI signals and reconstruct them block by block. The step edge parameters are obtained from the 2D moments of a given image block. Experimentally, our technique can reconstruct the edge more precisely and track the Cramer-Rao bounds (CRBs) closely with a signal-to-noise ratio (SNR) larger than 4 dB on synthetic step edge images. Experiments on real images show that our proposed method can reconstruct the step edges under practical conditions, i.e., in the presence of various types of noise and using a real sampling kernel. The results on locating the corners of data matrix barcodes using our method also outperform some state-of-the-art barcode decoders. This paper builds theoretical foundations for the recovery of a newly proposed class of smooth graph signals, approximately bandlimited graph signals, under three sampling strategies: uniform sampling, experimentally designed sampling, and active sampling. We then state minimax lower bounds on the maximum risk for the approximately bandlimited class under these three sampling strategies and show that active sampling cannot fundamentally outperform experimentally designed sampling. We propose a recovery strategy to compare uniform sampling with experimentally designed sampling. As the proposed recovery strategy lends itself well to statistical analysis, we derive the exact mean square error for each sampling strategy. To study convergence rates, we introduce two types of graphs and find that 1) the proposed recovery strategy achieves the optimal rates; and 2) the experimentally designed sampling fundamentally outperforms uniform sampling for Type-2 class of graphs. To validate our proposed recovery strategy, we test it on five specific graphs: a ring graph with @math nearest neighbors, an Erdős–Renyi graph, a random geometric graph, a small-world graph, and a power-law graph and find that experimental results match the proposed theory well. This paper also presents a comprehensive explanation for when and why sampling for semi-supervised learning with graphs works. Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose the use of the alternating direction method - a classic approach for optimization problems with separable variables (D. Gabay and B. Mercier, ?A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,? Computer and Mathematics with Applications, vol. 2, pp. 17-40, 1976; R. Glowinski and A. Marrocco, ?Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de Dirichlet nonlineaires,? Rev. Francaise dAut. Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975) - for signal reconstruction from partial Fourier (i.e., incomplete frequency) measurements. Signals are reconstructed as minimizers of the sum of three terms corresponding to total variation, ?1-norm of a certain transform, and least squares data fitting. Our algorithm, called RecPF and published online, runs very fast (typically in a few seconds on a laptop) because it requires a small number of iterations, each involving simple shrinkages and two fast Fourier transforms (or alternatively discrete cosine transforms when measurements are in the corresponding domain). RecPF was compared with two state-of-the-art algorithms on recovering magnetic resonance images, and the results show that it is highly efficient, stable, and robust.