Abstract

In this paper we develop an approach to reduce the noise in stationary and quasi-stationary imagery. Temporal filtering consists of two parts; in the first, a motion detector or estimator is applied to determine the displacement fields, and a 1-D filter is applied along the temporal direction. To date, researchers have approached the problem of restoration of time-varying images using FIR filters; however, simpler and more efficient nonlinear filters can be designed. Median filters have been used for image restoration, but they still possess a considerable amount of computational complexity. Therefore we propose the use of max and of min–max ranked temporal filtering, which requires a very small computation effort. In using the temporal max or min–max estimator we consider a model which consists of k consecutive noisy frame observations: [Z_n-k(x,y),…, Z_n(x,y)]. Each observed frame is Z_n(x,y) = S_n(x,y) − N_n(x,y) where S_n(x,y) is the nth signal and N_n(x,y) is the added noise. For stationary images, the noisy observation becomes Z_n(x,y) = S(x,y) + N_n(x,y). Our aim is to estimate the original signal S_n(x,y) from k observations [Z_n-k(x,y),…, Z_n(x,y)]. The max filter is Ŝ_n,k = max(Z_n-k,…, Z_n), which produces a biased estimation of S_n(x,y). However, the bias is a known random variable hence the original signal can be recovered. Similarly the min-max temporal filter is Š_n,k = ½min(Z_n-k,…,Z_n) + ½max(Z_n-k,…,Z_n). The advantage of the min–max over the max filter is that it yields an unbiased estimate of S_n(x,y). We derived the output distribution of the filtered signals and found a performance criteria for selecting the number of consecutive frames needed to produce sufficiently accurate estimates of S_n(x,y). In particular, letting q_n,k = Ŝ_n+1,k − Ŝ_n,k we select k so that q is arbitrarily small with a given probability. This class of filter is suitable for real-time implementation in applications such as electron microscopy and image coding.

© 1985 Optical Society of America