1,548 research outputs found
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
Compressed Sensing aims to capture attributes of -sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all -sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of -sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin
Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
On block coherence of frames
Block coherence of matrices plays an important role in analyzing the
performance of block compressed sensing recovery algorithms (Bajwa and Mixon,
2012). In this paper, we characterize two block coherence metrics: worst-case
and average block coherence. First, we present lower bounds on worst-case block
coherence, in both the general case and also when the matrix is constrained to
be a union of orthobases. We then present deterministic matrix constructions
based upon Kronecker products which obtain these lower bounds. We also
characterize the worst-case block coherence of random subspaces. Finally, we
present a flipping algorithm that can improve the average block coherence of a
matrix, while maintaining the worst-case block coherence of the original
matrix. We provide numerical examples which demonstrate that our proposed
deterministic matrix construction performs well in block compressed sensing
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