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

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 theknown" 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 comparisons for both sparse and compressible signals are shown.

• The paper introduces Modified-CS, which leverages partially known support to solve sparse signal reconstruction via a convex l1 minimization outside the known support.
• It establishes theoretical recovery conditions with relaxed RIP requirements and error bounds compared to traditional compressive sensing methods.
• Empirical results demonstrate that the method achieves accurate reconstruction with fewer measurements, benefiting applications like dynamic MRI and real-time imaging.

Insights on Modified Compressive Sensing for Partially Known Support

The paper by Vaswani and Lu addresses a refined approach to Compressive Sensing (CS), termed Modified Compressive Sensing (modified-CS), targeting scenarios where a portion of the signal's support is known, albeit potentially with some errors. The primary problem is reconstructing a sparse signal from limited linear projections with partial prior support information.

Problem Context

Conventionally, CS facilitates signal reconstruction with fewer measurements than traditional methods require, by leveraging the signal's sparsity. This paper specifically focuses on optimizing this process when part of the signal's support is accessible through prior estimates or knowledge, significant for applications like real-time dynamic MRI or single-pixel camera imaging, where signals evolve over time.

Proposed Approach

The modified-CS method involves solving a convex optimization problem designed to exploit known support while enforcing sparsity outside this support, formulated as:

$$\min_{\beta} \|\beta_{T^c}\|_1 \quad \text{s.t.} \quad y = A\beta$$

This relaxation from the traditional CS problem is pivotal when the support knowledge is approximate. The authors also extend the concept to the Regularized Modified-CS (RegModCS), incorporating prior signal estimate knowledge, thus enhancing reconstruction error handling when exact reconstruction is unattainable.

Theoretical Contributions

The paper provides substantial theoretical insights, offering conditions under which modified-CS achieves exact reconstruction. These conditions are notably weaker compared to standard CS, contingent upon the size of the unknown support and the errors being minimal relative to the entire support size. This is mathematically framed as requirements on the Restricted Isometry Property (RIP) constants and support sizes that govern the feasibility of sparse reconstruction.

Empirical Validation

Extensive simulations substantiate the theoretical claims, demonstrating superior performance of modified-CS and RegModCS over conventional CS, particularly for sparse and compressible signals. In empirical comparisons, scenarios with low measurement availability reveal modified-CS achieves exact reconstruction with fewer measurements, validating the practical efficacy of integrating known support constraints.

Practical Implications and Future Work

The implications for real-time applications are profound. Modifying CS to incorporate prior support knowledge directly translates to enhanced reconstruction capabilities in dynamic contexts. This advance holds potential for various fields, including medical imaging and video compression.

The paper suggests avenues for future exploration, notably robust error bounds for noisy measurements and compressible signals, and adapting the approach for time-varying environments. Ongoing work is likely to examine stability concerns and extend the framework's applicability to scenarios utilizing the Total Variation (TV) norm, a frequent choice in imaging applications.

Conclusion

Vaswani and Lu's work represents a significant methodological step towards more efficient sparse signal reconstruction in constrained environments, promising deeper exploration into adaptive and real-time signal processing domains.