Principles of Modern Radar - Volume 2 1891121537

182 CHAPTER 5 Radar Applications of Sparse ReconstructionAs an aside, SR can be extended beyond the case of a sparse vector or set of jointlysparse vectors. One intriguing example is the case of reconstructing low-rank matricesfrom sparse sampling of their entries, for example [119], as inspired by the so-calledNetflix problem. 35 In a related paper [120], the authors reconstruct a matrix as a sum ofa low rank and an entry-wise sparse matrix. This decomposition can remove impulsivenoise and was very effective for various image processing applications. In [121], a fastalgorithm is proposed that can handle this decomposition task in the presence of noise andmissing data. As a final example, [122] estimates the covariance matrix of a data set byassuming that the matrix’s eigen-decomposition can be represented as product of a smallnumber of Givens rotations.5.3.7 Matrix Uncertainty and CalibrationPerfect knowledge of the forward operator A cannot reasonably be expected in manyradar applications, where A may include assumptions about calibration, discretization,and other signal modeling issues. While additive noise can account for some of theseeffects, the impact of matrix uncertainty should be given specific attention. Specifically,we are interested in problems of the formy = (A + E) x true + e (5.38)where the multiplicative noise E is unknown. Notice that this additional error term canaccount for a wide range of signal modeling issues, including calibration, grid error, 36autofocus, sensor placement, and manifold errors. Since the system measurement modelis linear, it is perhaps unsurprising that a relatively straight forward extension to the existingRIP theory can offer limited performance guarantees in the presence of multiplicative errorE [126]. A simple argument in this direction can be made by bounding E and absorbingits effect into the additive noise e.However, as observed in [124], basis mismatch and grid errors can lead to significantperformance loss. Thus, algorithms that can compensate for matrix uncertainty directlyhave been developed. In l 2 regularized reconstruction, the approach of total least squaresis often employed to cope with matrix uncertainty. In [125], the authors extend this ideato sparse regularization with an algorithm known as sparsity-cognizant total least squares(STLS). They provide a low-complexity algorithm that can also cope with some forms ofparametric matrix uncertainty. In [127], the authors propose an algorithm closely relatedto the DS that offers promising performance in Monte Carlo trials. Our own work in [128]leverages GAMP to address matrix uncertainty, including a parametric approach withclose ties to the work in [125]. Particularly as CS is applied to a wider range of practical35 Netflix created a competition for the development of algorithms for matrix completion in support ofits efforts to predict customer preferences from a small sample of movie ratings.36 CS uses a discretized linear model to circumvent the model order selection problems in traditionalnonlinear parametric approaches, but the potential for grid error is a consequence of the underlyingdiscretization. Specifically, in parameter estimation, the columns of A represent the system response tovarious values of a parameter sampled on some discrete grid, for example pixels in an image or frequenciesof sinusoids. When the true parameter values lie between these samples, the true signal is not perfectlyrepresented in the dictionary; indeed, the representation may not even be sparse. See [37,123–125] fordiscussions of this topic.