Collective Sensing: a Fixed-Point 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 Hilbert-space formulation, we argue it is overly-structured to account for the complexity of signals arising from the real-world. Inspired by the works on fractal image decoding and nonlocal image processing, we propose to view an image as the fixed-point of some nonexpansive mapping in the metric space in this paper. Recently proposed BM3D-based 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 clustering-based 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 deterministic-annealing 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 state-of-the-art (SPIHT) coding scheme has been achieved.