Minimax Analysis of Estimation Problems in Coherent Imaging
Hao Xing, Soham Jana, Arian Maleki
TL;DR
This work analyzes the fundamental limits of estimating an unknown nonnegative image $x_o$ from multilook coherent-imaging measurements under speckle and additive noise. It introduces a structured image class ${\mathcal C}$ characterized by polynomial covering-number complexity, and derives minimax risk bounds for both varying and fixed forward operators. The main results show that the minimax mean squared error scales as ${R_2(\mathcal{C},m,n,\sigma_z) = O\left( \min\left( \frac{\max\{\sigma_z^4, m^2, n^2\} k \log n}{m^2 n L}, 1 \right) \right)}$, with sharper behavior in sparse-structured regimes; a fixed-$A$ variant adds a mild extra term. The proofs blend Fano-type lower bounds, Rao–Blackwellization for sufficient statistics in the fixed-design case, and delta-net based, concentration-inequality–driven upper bounds for the ML estimator, along with novel decoupling and covering-number arguments tailored to the multiplicative speckle noise model. Collectively, the results establish a near-complete picture of how dimension, noise, and forward-model structure govern coherent-imaging estimation accuracy, highlighting the intrinsic hardness relative to linear regression models and the utility of considering general structure beyond sparsity. The findings have implications for design and analysis of multilook coherent-imaging systems (e.g., SAR, OCT, holography) by quantifying when adding looks, increasing measurements, or tightening forward-model diversity yields meaningful risk reductions.
Abstract
Unlike conventional imaging modalities, such as magnetic resonance imaging, which are often well described by a linear regression framework, coherent imaging systems follow a significantly more complex model. In these systems, the task is to estimate the unknown image ${\boldsymbol x}_o \in \mathbb{R}^n$ from observations ${\boldsymbol y}_1, \ldots, {\boldsymbol y}_L \in \mathbb{R}^m$ of the form \[ {\boldsymbol y}_l = A_l X_o {\boldsymbol w}_l + {\boldsymbol z}_l, \quad l = 1, \ldots, L, \] where $X_o = \mathrm{diag}({\boldsymbol x}_o)$ is an $n \times n$ diagonal matrix, ${\boldsymbol w}_1, \ldots, {\boldsymbol w}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,I_n)$ represent speckle noise, and ${\boldsymbol z}_1, \ldots, {\boldsymbol z}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,σ_z^2 I_m)$ denote additive noise. The matrices $A_1, \ldots, A_L$ are known forward operators determined by the imaging system. The fundamental limits of conventional imaging systems have been extensively studied through sparse linear regression models. However, the limits of coherent imaging systems remain largely unexplored. Our goal is to close this gap by characterizing the minimax risk of estimating ${\boldsymbol x}_o$ in high-dimensional settings. Motivated by insights from sparse regression, we observe that the structure of ${\boldsymbol x}_o$ plays a crucial role in determining the estimation error. In this work, we adopt a general notion of structure based on the covering numbers, which is more appropriate for coherent imaging systems. We show that the minimax mean squared error (MSE) scales as \[ \frac{\max\{σ_z^4,\, m^2,\, n^2\}\, k \log n}{m^2 n L}, \] where $k$ is a parameter that quantifies the effective complexity of the class of images.