Compressed Shattering

TL;DR

Compressed Shattering is a novel method that adapts compressed sensing to signals with unknown but bounded sparsity, reducing measurements by creating fixed sparsity shattered signals and using a simple sensing matrix.

Contribution

It introduces a new technique to handle sparsity range in compressed sensing by spectrum permutation and filtering, enabling fewer measurements.

Findings

Requires significantly fewer measurements than traditional CS for given sparsity range.

Creates fixed sparsity shattered signals to simplify sensing.

Achieves low measurement count with a simple deterministic sensing matrix.

Abstract

The central idea of compressed sensing is to exploit the fact that most signals of interest are sparse in some domain and use this to reduce the number of measurements to encode. However, if the sparsity of the input signal is not precisely known, but known to lie within a specified range, compressed sensing as such cannot exploit this fact and would need to use the same number of measurements even for a very sparse signal. In this paper, we propose a novel method called Compressed Shattering to adapt compressed sensing to the specified sparsity range, without changing the sensing matrix by creating shattered signals which have fixed sparsity. This is accomplished by first suitably permuting the input spectrum and then using a filter bank to create fixed sparsity shattered signals. By ensuring that all the shattered signals are utmost 1-sparse, we make use of a simple but efficient deterministic sensing matrix to yield very low number of measurements. For a discrete-time signal of length 1000, with a sparsity range of $5 - 25$, traditional compressed sensing requires $175$ measurements, whereas Compressed Shattering would only need $20 - 100$ measurements.