A recent approach to compressed sensing using deterministic sensing matrices formed from discrete frequency-modulated chirps or from Reed–Muller codes is extended to support efficient deterministic reconstruction of signals that are much less sparse than envisioned in the original work. In particular, this allows the application of this approach in imaging. The reconstruction algorithm developed for images incorporates several new elements to improve computational complexity and reconstruction fidelity in this application regime.

An image reconstruction algorithm using compressed sensing (CS) with deterministic matrices of second-order Reed-Muller (RM) sequences is introduced. The 1D algorithm of Howard et al. using CS with RM sequences suffers significant loss in speed and accuracy when the degree of sparsity is not high, making it inviable for 2D signals. This paper describes an efficient 2D CS algorithm using RM sequences, provides medical image reconstruction examples, and compares it with the original 2DCS using noiselets. This algorithm entails several innovations that enhance its suitability for images: initial best approximation, a greedy algorithm for the nonzero locations, and a new approach in the least-squares step. These enhancements improve fidelity, execution time, and stability in the context of image reconstruction.

We propose and analyze a nonparametric region-based active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes pixel intensity is independently identically distributed. Our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region-based image term that uses histograms of pixel intensity to distinguish different regions. More specifically, the region data encourages segmentation so that local histograms within each region are approximately homogeneous. An advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem. We use Wasserstein distance with exponent 1 to determine the dissimilarity between two histograms. The Wasserstein distance is a metric and is able to faithfully measure the distance between two histograms, compared to many pointwise distances. Moreover, it is insensitive to oscillations, and therefore our model is robust to noise. A fast global minimization method based on (Chan et al. in SIAM J. Appl. Math. 66(5):1632–1648, 2006 ; Bresson et al. in J. Math. Imaging Vis. 28(2):151–167, 2007 ) is employed to solve the proposed model. The advantages of using this method are two-fold. First, the computational time is less than that of the method by gradient descent of the associated Euler-Lagrange equation (Chan et al. in Proc. of SSVM, pp. 697–708, 2007 ). Second, it is able to find a global minimizer. Finally, we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes.

This paper examines the issue of scale in modeling texture for the purpose of segmentation. We propose a scale descriptor for texture and an energy minimization model to find the scale of a given texture at each location. For each pixel, we use the intensity distribution in a local patch around that pixel to determine the smallest size of the domain that can be used to generate neighboring patches. The energy functional we propose to minimize is comprised of three terms: The first is the dissimilarity measure using the Wasserstein distance or Kullback-Leibler divergence between neighboring patch distributions; the second maximizes the entropy of the local patch, and the third penalizes larger size at equal fidelity. Our experiments show the proposed scale model successfully captures the intrinsic scale of texture at each location. We also apply our scale descriptor for improving texture segmentation based on histogram matching [15].