We investigate the problem of reconstructing signals from a subsampled convolution of their modulated versions and a known filter. The problem is studied as applies to a specific imaging architecture that relies on spatial phase modulation by randomly coded “masks.” The diversity induced by the random masks is deemed to improve the conditioning of the deconvolution problem while maintaining sampling efficiency. We analyze a linear model of the imaging system, where the joint effect of the spatial modulation, blurring, and spatial subsampling is represented concisely by a measurement matrix. We provide a bound on the conditioning of this measurement matrix in terms of the number of masks @math , the dimension (i.e., the pixel count) of the scene image @math , and certain characteristics of the blurring kernel and subsampling operator. The derived bound shows that the stable deconvolution is possible with high probability even if the number of masks (i.e., @math ) is as small as @math , meaning that the total number of (scalar) measurements is within a logarithmic factor of the image size. Furthermore, beyond a critical number of masks determined by the extent of blurring and subsampling, use of every additional mask improves the conditioning of the measurement matrix. We also consider a more interesting scenario where the target image is known to be sparse. We show that under mild conditions on the blurring kernel, with high probability the measurement matrix is a restricted isometry when the number of masks is within a logarithmic factor of the sparsity of the scene image. Therefore, the scene image can be reconstructed using any of the well-known sparse recovery algorithms such as the basis pursuit. The bound on the required number of masks grows linearly in sparsity of the scene image but logarithmically in its ambient dimension. The bound provides a quantitative view of the effect of the blurring and subsampling on the required number of masks, which is critical for designing efficient imaging systems.
The article introduces digital image restoration to the reader who is just beginning in this field, and provides a review and analysis for the reader who may already be well-versed in image restoration. The perspective on the topic is one that comes primarily from work done in the field of signal processing. Thus, many of the techniques and works cited relate to classical signal processing approaches to estimation theory, filtering, and numerical analysis. In particular, the emphasis is placed primarily on digital image restoration algorithms that grow out of an area known as "regularized least squares" methods. It should be noted, however, that digital image restoration is a very broad field, as we discuss, and thus contains many other successful approaches that have been developed from different perspectives, such as optics, astronomy, and medical imaging, just to name a few. In the process of reviewing this topic, we address a number of very important issues in this field that are not typically discussed in the technical literature. ▪ Abstract Digital image reconstruction is a robust means by which the underlying images hidden in blurry and noisy data can be revealed. The main challenge is sensitivity to measurement noise in the input data, which can be magnified strongly, resulting in large artifacts in the reconstructed image. The cure is to restrict the permitted images. This review summarizes image reconstruction methods in current use. Progressively more sophisticated image restrictions have been developed, including (a) filtering the input data, (b) regularization by global penalty functions, and (c) spatially adaptive methods that impose a variable degree of restriction across the image. The most reliable reconstruction is the most conservative one, which seeks the simplest underlying image consistent with the input data. Simplicity is context-dependent, but for most imaging applications, the simplest reconstructed image is the smoothest one. Imposing the maximum, spatially adaptive smoothing permitted by the data results in t...