Principles of Modern Radar - Volume 2 1891121537

5.3 SR Algorithms 177One can argue that this so-called oracle solution is the best that we can hope to dogiven knowledge of Ɣ and no other information, see, for example [1]. Given additionalknowledge about e or x true , we might be able to improve this estimate, but the basic idearemains that knowledge of the support set greatly simplifies our problem.Greedy algorithms attempt to capitalize on this notion by identifying the support set ofx true in an iterative manner. These techniques are far older than CS, tracing back to at leastan iterative deconvolution algorithm known as CLEAN [89], which is equivalent to themore-recent Matching Pursuits (MP) algorithm [90]. Here, we will describe the OrthogonalMatching Pursuits (OMP) algorithm [91] as a typical example of greedy methods. Asalready suggested, OMP computes a series of estimates of the support set denoted by Ɣ k .OMP is based on the idea of viewing A as a dictionary whose columns representpotential basis elements that can be used to represent the measured vector y. At eachiteration, the residual error term y − A ˆx k is backprojected or multiplied by A H to obtainthe signal q k = A H ( y − A ˆx k ). The largest peak in q k is identified and the correspondingatom is added to the index set, that is Ɣ k+1 = Ɣ k ∪ argmax i |qi k |. The new estimate of thesignal is then computed as ˆx k+1 = ( AƔ H ) −1k+1A Ɣk+1 AHƔk+1y.Basically, at each iteration OMP adds to the dictionary the atom that can explain thelargest fraction of the energy still unaccounted for in the reconstruction of ˆx. Figure 5-7provides an example that may clarify this process. The example involves a single radarpulse with 500 MHz of bandwidth used to reconstruct a range profile containing threepoint targets. The initial set Ɣ 0 is empty. The top left pane shows the plot of A H y. Thepeak of this signal is selected as the first estimate of the signal x true , and the correspondingamplitude is computed. The resulting estimate is shown on the top right of the figure.Notice that the estimate of the amplitude is slightly off from the true value shown with across. At the next iteration, the dominant central peak is eliminated from the backprojectederror, and the algorithm correctly identifies the leftmost target. Notice in the middle rightpane that the amplitude estimate for the middle target is now correct. OMP reestimates allof the amplitudes at each iteration, increasing the computational cost but improving theresults compared with MP. The final iteration identifies the third target correctly. Noticethat the third target was smaller than some of the spurious peaks in the top left rangeprofile, but OMP still exactly reconstructs it after only three iterations.For very sparse signals with desirable A matrices, OMP performs well and requires lesscomputation than many other methods. Unfortunately, as discussed in detail in [91,92], theperformance guarantees for this algorithm are not as general as the RIP-based guaranteesfor algorithms like IHT or BPDN. Indeed, the guarantees lack uniformity and only holdin general in a probabilistic setting. Two issues with the OMP algorithm are perhapsresponsible for these limitations. The first is that the algorithm only selects a single atomat each iteration. The second, and perhaps more crucial limitation, is that the algorithmcannot correct mistakes in the support selection. Once an atom is in the set Ɣ, it stays there.Two virtually identical algorithms—compressive sampling matching pursuits(CoSaMP) [39] and subspace pursuit (SP) [93]—were developed to overcome these limitations.They select atoms in a large group at each iteration and include a mechanism forremoving atoms from the dictionary when they are found to be redundant. 33 We can then33 These algorithms are closely related to hard-thresholding approaches. Indeed, many of the SR algorithmsbeing developed blur the lines between the classes of algorithms we have used throughout ourdiscussion.