An adaptive hierarchical sensing scheme for sparse signals

In this paper, we present Adaptive Hierarchical Sensing (AHS), a novel adaptive hierarchical sensing algorithm for sparse signals. For a given but unknown signal with a sparse representation in an orthogonal basis, the sensing task is to identify its non-zero transform coefficients by performing only few measurements. A measurement is simply the inner product of the signal and a particular measurement vector. During sensing, AHS partially traverses a binary tree and performs one measurement per visited node. AHS is adaptive in the sense that after each measurement a decision is made whether the entire subtree of the current node is either further traversed or omitted depending on the measurement value. In order to acquire an N-dimensional signal that is K-sparse, AHS performs O(K log N / K) measurements. With AHS, the signal is easily reconstructed by a basis transform without the need to solve an optimization problem. When sensing full-size images, AHS can compete with a state-of-the-art compressed sensing approach in terms of reconstruction performance versus number of measurements. Additionally, we simulate the sensing of image patches by AHS and investigate the impact of the choice of the sparse coding basis as well as the impact of the tree composition.