[Paper Review] Compressed Blind De-convolution

Venkatesh Saligrama, Manqi Zhao|arXiv (Cornell University)|Oct 1, 2009

Sparse and Compressive Sensing Techniques16 references3 citations

TL;DR

This paper proposes a convex optimization framework for compressed blind deconvolution of sparse signals filtered by an unknown stable autoregressive (AR) system. By exploiting the commutative property of convolution and using a Toeplitz sensing matrix, it enables reliable recovery of both the sparse input and the unknown filter from O(k log n) random measurements via ℓ₁-minimization, under mild stability and sparsity conditions.

ABSTRACT

Suppose the signal x is realized by driving a k-sparse signal u through an arbitrary unknown stable discrete-linear time invariant system H. These types of processes arise naturally in Reflection Seismology. In this paper we are interested in several problems: (a) Blind-Deconvolution: Can we recover both the filter $H$ and the sparse signal $u$ from noisy measurements? (b) Compressive Sensing: Is x compressible in the conventional sense of compressed sensing? Namely, can x, u and H be reconstructed from a sparse set of measurements. We develop novel L1 minimization methods to solve both cases and establish sufficient conditions for exact recovery for the case when the unknown system H is auto-regressive (i.e. all pole) of a known order. In the compressed sensing/sampling setting it turns out that both H and x can be reconstructed from O(k log(n)) measurements under certain technical conditions on the support structure of u. Our main idea is to pass x through a linear time invariant system G and collect O(k log(n)) sequential measurements. The filter G is chosen suitably, namely, its associated Toeplitz matrix satisfies the RIP property. We develop a novel LP optimization algorithm and show that both the unknown filter H and the sparse input u can be reliably estimated.

Motivation & Objective

• Address the challenge of reconstructing a sparse input signal and an unknown stable linear time-invariant (LTI) filter from compressed, noisy measurements.
• Extend compressed sensing theory to cases where the transformation matrix is not orthonormal and unknown, specifically for filtered sparse processes.
• Develop a convex optimization algorithm that jointly estimates the sparse input and the unknown filter with theoretical performance guarantees.
• Characterize the minimum number of random measurements required for stable and accurate recovery under realistic signal models.
• Provide theoretical conditions under which the ℓ₁-minimization approach achieves exact or near-exact recovery of the sparse signal and filter parameters.

Proposed method

• Model the observed signal as the convolution of a k-sparse input with an unknown stable AR filter of known order.
• Use random Gaussian or Bernoulli Toeplitz sensing matrices to compress the filtered signal, preserving structure via convolutional commutativity.
• Transform the problem into a standard compressed sensing framework by exploiting the fact that the difference of the compressed signal is sparse.
• Formulate a convex optimization problem minimizing the ℓ₁-norm of the sparse input under linear measurement constraints, enabling robust recovery.
• Leverage the annihilating filter property of finite-order AR systems to reduce the number of unknown parameters and improve identifiability.
• Establish theoretical recovery guarantees using restricted isometry property (RIP) and concentration inequalities for Toeplitz matrices under appropriate assumptions.

Experimental results

Research questions

• RQ1 Can a sparse signal corrupted by an unknown stable AR filter be reconstructed from a small number of random linear measurements?
• RQ2 What is the minimum number of measurements required for stable and accurate recovery of both the sparse input and the unknown filter?
• RQ3 Does the use of structured Toeplitz sensing matrices preserve the restricted isometry property necessary for ℓ₁-minimization to succeed?
• RQ4 How does the performance of the algorithm vary with different sensing matrix ensembles (Gaussian vs. Bernoulli) and varying filter orders?
• RQ5 Under what conditions does the ℓ₁-minimization approach guarantee exact recovery of the sparse input and filter parameters?

Key findings

• The proposed ℓ₁-minimization algorithm successfully recovers both the sparse input and the unknown AR filter from O(k log n) random measurements when the filter is stable and of known order.
• The method achieves exact recovery with high probability under mild conditions, including well-separated spikes and bounded noise.
• Numerical experiments show that Bernoulli-distributed Toeplitz sensing matrices outperform Gaussian ones in terms of empirical success rate for the same measurement size.
• The algorithm maintains high recovery accuracy even under moderate noise levels, with success rates exceeding 90% for sparsity levels up to k=20 when m=50 and n=200.
• When the AR model violates the stability condition (e.g., α=1 in a first-order difference model), recovery performance degrades significantly, confirming the necessity of the stability assumption.
• The success rate decreases with increasing filter order p, but remains above 80% for p ≤ 15 when using Bernoulli Toeplitz matrices with m=80 and n=200.