Collective Sensing: a FixedPoint Approach in the Metric Space Conventional wisdom in signal processing heavily relies on the concept of inner product defined in the Hilbert space. Despite the popularity of Hilbertspace formulation, we argue it is overlystructured to account for the complexity of signals arising from the realworld. Inspired by the works on fractal image decoding and nonlocal image processing, we propose to view an image as the fixedpoint of some nonexpansive mapping in the metric space in this paper. Recently proposed BM3Dbased denoising and nonlocal TV filtering can be viewed as the special cases of nonexpansive mappings while differ on the choice of clustering techniques. The physical interpretation of clusteringbased nonexpansive mappings is that they convey organizational principles of the dynamical system underlying the signals of interest. There is an interesting analogy between phases of matters in statistical physics and types of structures in image processing. From this perspective, image reconstruction can be solved by a deterministicannealing based global optimization approach which collectively exploits the a priori information about unknown image. The potential of this new paradigm, which we call collective sensing, is demonstrated on the lossy compression application where significant gain over current stateoftheart (SPIHT) coding scheme has been achieved.

Variations on the Convolutional Sparse Coding Model Over the past decade, the celebrated sparse representation model has achieved impressive results in various signal and image processing tasks. A convolutional version of this model, termed convolutional sparse coding (CSC), has been recently reintroduced and extensively studied. CSC brings a natural remedy to the limitation of typical sparse enforcing approaches of handling global and highdimensional signals by local, patchbased, processing. While the classic field of sparse representations has been able to cater for the diverse challenges of different signal processing tasks by considering a wide range of problem formulations, almost all available algorithms that deploy the CSC model consider the same   ell _1  ell _2   problem form. As we argue in this paper, this CSC pursuit formulation is also too restrictive as it fails to explicitly exploit some local characteristics of the signal. This work expands the range of formulations for the CSC model by proposing two convex alternatives that merge global norms with local penalties and constraints. The main contribution of this work is the derivation of efficient and provably converging algorithms to solve these new sparse coding formulations.