Declipping and the recovery of vectors from saturated measurements

TL;DR

The paper develops a frame-theoretic framework for saturation recovery (declipping) of vectors from clipped measurements, introducing $\lambda$-saturation via $\phi_\lambda$ and $\Phi_\lambda$. It characterizes when saturation recovery is possible on $\alpha B_H$ and connects the optimality of frames to multi-packing problems in real projective space, elucidating the roles of full-spark frames and equiangular tight frames. A non-linear adaptation of the classical frame algorithm is proposed, with convergence guarantees under $\lambda>\lambda_c$ and a Parseval case yielding exponential convergence, alongside comparisons to linear approaches on unsaturated coordinates. Numerical experiments in $\mathbb{R}^{10}$ with random frames demonstrate substantial performance gains (over $9$ dB mean error reduction) of the $\lambda$-saturated frame algorithm over linear methods, highlighting practical impact for declipping and saturation recovery tasks.

Abstract

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ allows for a linear and stable reconstruction of any vector $x\in H$ from the linear measurements $(\langle x,x_j\rangle)_{j\in J}$. However, there are many situations where some information in the frame coefficients is lost. In applications where one is using sensors with a fixed dynamic range, any measurement above that range is registered as the maximum, and any measurement below that range is registered as the minimum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We initiate a frame theoretic approach to saturation recovery in a similar way to what [BCE06] did for phase retrieval. We characterize when saturation recovery is possible, show optimal frames for use with saturation recovery correspond to minimal multi-fold packings in projective space, and prove that the classical frame algorithm may be adapted to this non-linear problem to provide a reconstruction algorithm.

Figures (4)

• Figure 1: Saturation of continuous frame coefficients: (a) Unsaturated frame coefficients (b) Saturated frame coefficients.
• Figure 2: Saturation of discrete frame coefficients: (a) Unsaturated frame coefficients (b) Saturated frame coefficients.
• Figure 3: Mean error $\Vert y_{k}-x\Vert$ for the linear frame algorithm ($\circ$) and non-linear frame algorithm ($\square$) using random frames of 30 vectors for $\mathbb{R}^{10}$: (a) $\lambda=0.16$ (b) $\lambda=0.24$ (c) $\lambda=0.32$ (d) $\lambda=0.4$.
• Figure 4: Error reduction of the $\lambda$-saturated frame algorithm relative to the linear frame algorithm using random frames of 30 vectors for $\mathbb{R}^{10}$: (a) $\lambda=0.16$ (b) $\lambda=0.24$ (c) $\lambda=0.32$ (d) $\lambda=0.4$.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Proposition 3.3
• proof
• Corollary 3.5
• proof
• Corollary 3.6
• proof