Hierarchical Reconstruction of Sparse Signals

We consider a family of constrained `p minimizations: min x∈Rn {||x||`p ∣∣ Ax = Ax∗}, where only A, Ax∗ and ||x∗||`0 are known. Such family of problems has been extensively studied in the Compressed Sensing Community and are used to recover sparse signals. We then go through the reasoning on why p = 1 would be the most suitable choice. Due to the possible ill-posedness of the constrained `1 minimization, we introduce the Tikhonov Regularization to arrive at an unconstrained version, min x∈Rn {||x||`1 + λ 2 ||Ax∗ − Ax||2`2}. Through this regularized version, we construct the approximation based on a given scale λ. We also show that the two problems are equivalent when λ→∞. Using the idea of multiscale approximation, we adopt the method of Hierarchical Decomposition from Image Processing to reconstruct sparse signals on layers of dyadic scales. We proceed to show that this Hierarchical Decomposition approach alleviate the dependence on the regularization parameter λ and be can used to de-noise the corrupted signal. We are also able to show that the difference between the approximation from the Hierarchical Reconstruction, call it xHRSS and x∗ is in the Null(A); and various numerical examples support the fact this approach offer a better approximation to the original x∗. 1 Background: A Constrained Minimal `1-norm Problem The ingenious Nyquist-Shannon Sampling Theorem addresses the question about the possibility to recover any signal using finite number of sampling (measurements); however, according to the theorem, the number of sampling to take is two times the highest frequency in a signal. The pursuit of improvement on reducing the sampling rates have been non-stop since 1949; and significant progress has been made especially regarding the sparse signal recovery. In fact, hundreds of papers in the Compressed Sensing community have been published to reduce the sampling rates to a significant level. The theory regarding sparse signal recovery has been ∗[email protected] †[email protected]