Is speckle noise more challenging to mitigate than additive noise?

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 $\beta$-H\"older 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, $\hat{\nu}_n$ is an estimate of $f$, and $\sigma_n$ is the standard deviation of the additive Gaussian noise, then $\inf_{\hat{\nu}_n} \sup_f \mathbb{E}_f\| \hat{\nu}_n - f \|^2_2$ decays at the rate $(\max(1,\sigma_n^4)/n)^{\frac{2\beta}{2\beta+1}}$. Note that the rate achieved under purely additive noise is $({\sigma_n^2/n})^{\frac{2\beta}{2\beta+1}}$. 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.