Is speckle noise more challenging to mitigate than additive noise?
Reihaneh Malekian, Hao Xing, Arian Maleki
TL;DR
This work provides the first minimax analysis of recovering a β-Hölder function from samples corrupted by both speckle (multiplicative) and additive Gaussian noise. It formalizes the problem with the model y_i = f(x_i) ξ_i + τ_i, ξ_i ~ N(0,1), τ_i ~ N(0, σ_n^2), and constrains f to lie in the Hölder class Σ_h(β,L). The authors derive sharp minimax rates for the L2 and L∞ risks under this mixed-noise setting: R_2(Σ_h(β,L), σ_n) = Θ_{β,L,h}( min{1, (max(1, σ_n^4)/n)^{2β/(2β+1)}} ) and R_∞(Σ_h(β,L), σ_n) = Θ_{β,L,η,h}( (max(1, σ_n^4) log n / n)^{β/(2β+1)} ) in appropriate regimes, highlighting how speckle noise can dominate or interact with additive noise differently than pure additive noise. The analysis combines bump-function constructions and local polynomial estimators to obtain matching lower and upper bounds for both norms, and includes a detailed comparison with the classical additive-noise model to illuminate regimes where speckle makes de-speckling more challenging. The results establish fundamental limits for de-speckling and provide guidance on estimator design under mixed-noise conditions, with implications for SAR, ultrasound, and holography applications where speckle is pervasive.
Abstract
We study the problem of estimating a function in the presence of both speckle and additive noises, commonly referred to as the de-speckling problem. Although additive noise has been thoroughly explored in nonparametric estimation, speckle noise, prevalent in applications such as synthetic aperture radar, ultrasound imaging, and digital holography, has not received as much attention. Consequently, there is a lack of theoretical investigations into the fundamental limits of mitigating the speckle noise.This paper is the first step in filling this gap. Our focus is on investigating the minimax estimation error for estimating a $β$-Hölder continuous function and determining the rate of the minimax risk. Specifically, if $n$ represents the number of data points, $f$ denotes the underlying function to be estimated, $\hatν_n$ is an estimate of $f$, and $σ_n$ is the standard deviation of the additive Gaussian noise, then $\inf_{\hatν_n} \sup_f \mathbb{E}_f\| \hatν_n - f \|^2_2$ decays at the rate $(\max(1,σ_n^4)/n)^{\frac{2β}{2β+1}}$. Note that the rate achieved under purely additive noise is $({σ_n^2/n})^{\frac{2β}{2β+1}}$. We will provide a detailed comparison of this rate with the one obtained in the presence of both noise types across different regimes of their relative magnitudes, and discuss the insights that emerge from these comparisons.