Principles of Modern Radar - Volume 2 1891121537

5.3 SR Algorithms 181only a subset of vectors that are well separated is used for communication. This separationallows the symbols to be accurately distinguished by the receiver. CS techniques thatleverage structured sparsity operate on a similar principle.Before delving into more elaborate approaches, we mention that the simplest priorknowledge that we might exploit would be differing prior probabilities on which coefficientsof x true are nonzero. This information could easily be available in various radarscenarios and other problems. For example, in a surveillance application looking for vehicleson a road network, it might make sense to drastically reduce the probability oftarget presence at large distances from roadways. This approach can be implemented byreplacing λ with a diagonal weighting matrix when solving QP λ . This modification canbe incorporated into any CS algorithm; see problem 9. Notice that this scheme is closelyrelated to the iterative reweighting reconstruction algorithms. Indeed, the authors in [86]suggest incorporating prior information into the reconstruction weights. As another example,in [110] the authors determine optimal weighting schemes for a signal with twogroups of coefficients that share different prior probabilities.However, we can also incorporate information about the structure of the sparsitypattern rather than simply assigning weights to specific elements. An excellent intuitiveexample of this idea is found in [111]. The authors seek to detect changes between images.Naturally, these change detection images will be sparse. However, they will also beclumpy, in that the nonzero coefficients will tend to occur in tightly clustered groups. Theauthors use a Markov random field as a prior on x true to promote clustering of nonzerocoefficients. Motivated by the Markov model, they propose an intuitive greedy algorithmknown as LaMP. The algorithm is reminiscent of CoSaMP. The primary difference is thatrather than simply pruning to enforce the desired sparsity after computing the error term,LaMP uses the known structure of the sparsity pattern to determine the most likely supportset for the nonzero coefficients. LaMP performs significantly better than CoSaMP onsignals that satisfy this structural assumption.Baraniuk et al. [112] extend the RIP framework to deal with structured sparse signalsand provide RIP-like performance guarantees for modified versions of CoSaMP and IHT.Like the LaMP algorithm, the key to the improved performance is the exploitation ofadditional signal structure. For some classes of structured sparsity, the algorithm is able toreconstruct sparse signals from order s measurements, as opposed to the order s log(N/s)required by algorithms that focus on simple sparsity. In particular, their approach is developedfor signals characterized by wavelet trees and block sparsity. See [113] for additionaldiscussion of these ideas with connections to graphical models. Block sparsity is also exploitedin [114]. One can also consider the joint sparsity of multiple signals as in [115].In [116], the authors address extensions of the RIP condition to address block-sparsesignals and develop recovery algorithms for them.On a final note, [117] combines belief-propagation-based CS algorithms with ideasfrom turbo equalization [118] to develop an innovative framework for incorporating structuredsparsity information. The algorithm alternates between two decoding blocks. Thefirst block exploits the structure of the measurements, while the second block leveragesthe known structure of the sparsity patterns. By exchanging soft decisions, that is probabilitiesthat individual elements of x true are nonzero, these two blocks cooperativelyidentify the support of the true signal. The paper demonstrates excellent results for simulationsusing data derived from a simple Markov chain model. This approach appearspromising for incorporating structural and prior information into SR and related inferencetasks.