Structured Compressed Sensing: From Theory to Applications (1106.6224v2)

Abstract: Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discrete-to-discrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuous-time signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.

• The paper introduces structured compressed sensing by integrating domain-specific measurement models that improve signal recovery under practical hardware constraints.
• It classifies various structured matrices, such as subsampled incoherent bases, circulant, and separable matrices, to support efficient applications in imaging and communications.
• It details algorithmic advances like model-based CoSaMP and structured basis pursuit, offering enhanced recovery guarantees and reduced sampling rates in real-world systems.

Structured Compressed Sensing: From Theory to Applications

Overview

The paper "Structured Compressed Sensing: From Theory to Applications" by Duarte and Eldar marks a significant advancement in the domain of Compressed Sensing (CS), focusing particularly on the structured forms of CS that extend beyond the traditional methods. This comprehensive review navigates the theoretical underpinnings, algorithmic developments, and practical applications of structured CS, providing valuable insights into various configurations and implications.

Structured Compressed Sensing

CS traditionally relies on sparse signal representations and random measurement matrices to enable signal reconstruction from fewer samples than dictated by the Nyquist rate. The authors highlight the necessity to move beyond random matrices to structured acquisition models, encapsulating the unique constraints posed by real-world signal acquisition systems.

The primary innovation discussed is the shift from employing completely random matrices to structured matrices that integrate domain-specific knowledge into the sensing framework, which aligns better with practical hardware constraints and signal structures.

Measurement Matrices and Sparsity Models

The review classifies structured CS into several categories:

1. Subsampled Incoherent Bases: These matrices, derived from subsampling orthonormal bases incoherent with the signal's sparsity basis, allow practical implementation in systems such as Magnetic Resonance Imaging (MRI) and single pixel cameras.
2. Structurally Subsampled Matrices: This includes Vandermonde matrices and those inheriting structural properties from the hardware, applicable to analog signal acquisition such as in the random demodulator.
3. Subsampled Circulant Matrices: Employed in scenarios where convolution operations are inherent, such matrices can be efficiently implemented using Fast Fourier Transforms (FFT), significantly aiding in applications like wideband channel estimation and optical imaging.
4. Separable Matrices: These utilize Kronecker products of lower-dimensional matrices, thus extending CS to multidimensional signals. This approach is particularly efficacious in high-dimensional data scenarios such as hyperspectral imaging.

Advanced Signal Models

The paper explores more intricate signal models beyond simple sparsity:

1. Multiple Measurement Vectors (MMV): This model leverages joint sparsity across multiple signal vectors sharing a common support, enhancing robustness and reconstruction efficiency. MMV is crucial in applications like EEG/MEG source localization and multichannel communications.
2. Unions of Subspaces: Structured sparsity within signals is captured via unions of subspaces models. Signals are assumed to lie within a structured subspace union, which is instrumental in applications like image recovery from wavelet coefficients where the support of large coefficients conforms to a tree structure.
3. Finite Rate of Innovation (FRI): For continuous-time signals, FRI models capture bandlimited signals with few degrees of freedom, enabling sampling at below-Nyquist rates. Applications include super-resolution imaging and ultrasound imaging, wherein the structured nature of signal innovations is exploited for efficient recovery.

Practical Implementations and Applications

The review also details practical implementations and applications:

1. Multiband Signal Sampling: employing structured analog filters and subsampling strategies such as the Modulated Wideband Converter (MWC), achieving significant reductions in sampling rates while retaining the ability to reconstruct wideband signals.
2. Radar and Ultrasound: Utilizing structured sampling to achieve super-resolution imaging and target detection, the authors demonstrate improved resolution and reduced sampling rates in practical radar and ultrasound applications.

Algorithmic Developments

Key algorithmic developments discussed include:

1. Greedy Algorithms and Thresholding: These are adapted to structured models, such as model-based CoSaMP and Iterative Hard Thresholding (IHT), which integrate signal model structure into the iterative reconstruction process.
2. Optimization Approaches: Adaptations of Basis Pursuit (BP) and related algorithms to structured sparsity models, using mixed norms and other regularization techniques to promote the desired signal structure.

Implications and Future Directions

From a theoretical perspective, structured CS provides stronger recovery guarantees under realistic signal and hardware constraints. Practically, these advances enable the design of more efficient sensing systems, paving the way for high-performance, low-power applications in areas ranging from medical imaging to wireless communications.

Future research will likely explore deeper integration of CS into various domains, enhancing the theoretical framework for new signal structures and developing more sophisticated algorithms that push the boundaries of current capabilities.

Conclusion

By offering a detailed exploration of structured CS, the authors present a roadmap for researchers and practitioners to leverage advanced signal models and structured measurement matrices. This shift not only aligns theoretical developments with practical needs but also opens up new avenues for innovation in signal processing technology.