Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support

arXiv:0903.5066v5 [cs.IT] 27 Jul 2010 1 Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support Namrata Vaswani and Wei Lu Abstract—We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known” part of the support, denoted T , may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known” part. The idea of our proposed solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T . We obtain sufficient conditions for exact reconstruction using modified-CS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called Regularized Modified-CS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation compar- isons for both sparse and compressible signals are shown. I. INTRODUCTION In this work, we study the sparse reconstruction problem from noiseless measurements when a part of the support is known, although the known part may contain some errors. The “known” part of the support may be available from prior knowledge. For example, consider MR image reconstruction using the 2D discrete wavelet transform (DWT) as the sparsi- fying basis. If it is known that an image has no (or very little) black background, all (or most) approximation coefficients will be nonzero. In this case, the “known support” is the set of indices of the approximation coefficients. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known support”. This latter problem occurs in various practical applications such as real-time dynamic MRI reconstruction, real-time single-pixel camera video imaging or video compression/decompression. There are also numerous other potential applications where sparse reconstruction for time sequences of signals/images may be needed, e.g. see [3], [4]. Sparse reconstruction has been well studied for a while, e.g. see [5], [6]. Recent work on Compressed Sensing (CS) gives conditions for its exact reconstruction [7], [8], [9] and bounds the error when this is not possible [10], [11]. Our recent work on Least Squares CS-residual (LS-CS) [12], [13] can be interpreted as a solution to the problem of sparse reconstruction with partly known support. LS-CS N. Vaswani and W. Lu are with the ECE dept. at Iowa State University (email: {namrata,luwei}@iastate.edu). A part of this work appeared in [1], [2]. This research was supported by NSF grants ECCS-0725849 and CCF- 0917015. Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. Original sequence (a) Top: larynx image sequence, Bottom: cardiac sequence 5 10 15 20 0 0.01 0.02 0.03 Time → |Nt\Nt−1| |Nt|

Cardiac, 99% Larynx, 99% 5 10 15 20 0 0.01 0.02 0.03 Time → |Nt−1\Nt| |Nt|

Cardiac, 99% Larynx, 99% (b) Slow support change plots. Left: additions, Right: removals Fig. 1. In Fig. 1(a), we show two medical image sequences. In Fig. 1(b), Nt refers to the 99% energy support of the two- level Daubechies-4 2D discrete wavelet transform (DWT) of these sequences. |Nt| varied between 4121-4183 (≈0.07m) for larynx and between 1108-1127 (≈0.06m) for cardiac. We plot the number of additions (left) and the number of removals (right) as a fraction of |Nt|. Notice that all changes are less than 2% of the support size. replaces CS on the observation by CS on the LS observation residual, computed using the “known” part of the support. Since the observation residual measures the signal residual which has much fewer large nonzero components, LS-CS greatly improves reconstruction error when fewer measure- ments are available. But the exact sparsity size (total number of nonzero components) of the signal residual is equal to or larger than that of the signal. Since the number of measurements required for exact reconstruction is governed by the exact sparsity size, LS-CS is not able to achieve exact reconstruction using fewer noiseless measurements than those needed by CS. Exact reconstruction using fewer noiseless measurements than those needed for CS is the focus of the current work. Denote the “known” part of the support by T . Our proposed solution (modified-CS) solves an ℓ1 relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T . We derive sufficient conditions

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