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 $ -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, $\hat {\nu }_{n}$ 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 $\left ({{\max (1,\sigma _{n}^{4})/n}}\right)^{\frac {2\beta }{2\beta +1}}$ . Comparing this rate with the rate achieved under purely additive noise, namely $\left ({{\sigma _{n}^{2}/n}}\right)^{\frac {2\beta }{2\beta +1}}$ , leads to the following insights: 1) When $ \sigma _{n} = \omega (1) $ , the additive noise appears to be the dominant component in the de-speckling problem. However, the presence of speckle noise significantly complicates the task of mitigating its effects. As a result, the risk increases from the rate $ \left ({{ \sigma _{n}^{2} / n }}\right)^{\frac {2\beta }{2\beta +1}} $ , which characterizes the problem with only additive noise, to $ \left ({{ \sigma _{n}^{4} / n }}\right)^{\frac {2\beta }{2\beta +1}} $ in the presence of both speckle and additive noise. 2) When $ \sigma _{n} = o(1) $ , the variance of the additive noise does not contribute to the risk in the de-speckling problem. This suggests that, in this regime, speckle noise is the primary bottleneck. Interestingly, the resulting risk rate matches the rate for mitigating purely additive noise with $ \sigma _{n} = \Theta (1) $ . 3) When $\sigma _{n} = \Theta (1)$ , the two rates coincide, suggesting that both the speckle noise and additive noise are contributing to the overall error.