Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices (0908.2676v2)

Abstract: In this paper we establish the connection between the Orthogonal Optical Codes (OOC) and binary compressed sensing matrices. We also introduce deterministic bipolar $m\times n$ RIP fulfilling $\pm 1$ matrices of order $k$ such that $m\leq\mathcal{O}\big(k (\log_2 n)^{\frac{\log_2 k}{\ln \log_2 k}}\big)$. The columns of these matrices are binary BCH code vectors where the zeros are replaced by -1. Since the RIP is established by means of coherence, the simple greedy algorithms such as Matching Pursuit are able to recover the sparse solution from the noiseless samples. Due to the cyclic property of the BCH codes, we show that the FFT algorithm can be employed in the reconstruction methods to considerably reduce the computational complexity. In addition, we combine the binary and bipolar matrices to form ternary sensing matrices (${0,1,-1}$ elements) that satisfy the RIP condition.

• The paper introduces deterministic constructions for binary, bipolar, and ternary compressed sensing matrices using coding theory, leveraging Orthogonal Optical Codes and BCH codes.
• The construction of bipolar matrices using BCH codes allows for efficient signal reconstruction using FFT due to the cyclic nature of the codes.
• These deterministic designs offer efficient alternatives to random matrices, providing practical advantages for signal processing applications needing guaranteed performance.

Deterministic Construction of Compressed Sensing Matrices

The paper, co-authored by Arash Amini and Farokh Marvasti, presents significant insights into the construction of deterministic compressed sensing matrices, addressing the crucial challenge of sampling sparse signals efficiently. The research extends compressed sensing methodology by employing deterministic matrices over the conventionally preferred random matrices due to their practical advantages in implementation. A notable contribution of this work is the establishment of a relationship between Orthogonal Optical Codes (OOC) and binary compressed sensing matrices, which is complemented by the introduction of novel deterministic RIP-fulfilling matrices.

Overview of Contributions

1. Binary Sampling Matrices via OOC: The authors utilize OOC, traditionally used in optical communications, to demonstrate a method for constructing binary matrices that fulfill the Restricted Isometry Property (RIP). Leveraging the inherent properties of OOC vectors, the research provides an upper bound for the number of columns these binary matrices can possess without compromising the RIP, comparing them to Devore's matrices which claim near-optimality.
2. Bipolar Matrices via BCH Codes: A pivotal advancement in this paper is the development of deterministic bipolar matrices using BCH codes. These matrices contain elements from the set {±1}, improving computational and sampling efficiency. The matrix columns, derived from BCH code vectors with zeros replaced by -1s, facilitate signal reconstruction using greedy algorithms like Matching Pursuit. Due to the cyclic nature of BCH codes, the implementation of FFT in the reconstruction process significantly reduces complexity, making this approach highly efficient.
3. Ternary Matrices Combination: By combining the binary and bipolar matrices, the research further extends the deterministic designs to ternary matrices, comprising elements {0,1,-1}. This novel construction supports a greater number of columns, enhancing the capacity of deterministic designs to accommodate larger dimensions without losing RIP compliance.

Implications and Future Directions

The implications of this research span both theoretical and practical realms. Theoretically, it reinforces the viability of deterministic approaches for compressed sensing matrix design, transcending the need for randomness which lacks performance guarantees in specific realization cases. Practically, deterministic constructions, especially those leveraging BCH cyclic properties for optimized computation, propose efficient alternatives for real-world signal processing applications where computational resources and deterministic performance are critical.

Looking toward future developments, the deterministic construction of RIP matrices could explore further efficiencies in various dimensions, such as reducing matrix size while maintaining RIP compliance. Additionally, the investigation of new coding theory methods to improve minimum distance in binary codes could enhance RIP orders further. Concurrent advancements in computational architecture may also unlock greater efficiencies in employing cyclic properties to reduce complexity, which remains a continued focus for the advancement of compressed sensing technology in real-world applications.

In summary, this paper provides a comprehensive framework for the deterministic construction of RIP matrices that solidly integrate coding theory with compressed sensing. The methodologies described herein pave the way for deeper exploration of deterministic approaches, potentially leading to breakthroughs in efficient signal sampling and processing.