Structure Adaptive Methods in Image Denoising

The nonparametric regression originated in mathematical statistics offers an original approach to signal processing problems (e.g., [1], [2]). It basically results in linear Þltering with the linear Þlters designed using some moving window local approximations. In many applications like speech recognition or image denoising, nonlinear or locally adaptive methods have been shown to be more efficient than the linear ones. The typical examples are given by non-linear wavelet thresholding, [3], and pointwise adaptive kernel smoothing, [4], [5]. The Þrst local pointwise (varying window size) adaptive nonparametric regression statistical procedure was suggested by Lepski [6] (see also [4], [5], and [7]). This approach has been further developed in application to various signal and image processing problems [8]—[12]. Particularly, [12] offered another view on the problem of local adaptive estimation based on the link between adaptive estimation and multiple testing. This allows to treat in a uniÞed way different types of images, including Gaussian and Poissonian. Another important feature is that the problem of choosing the tuning parameters of the procedure is carefully addressed, leading to an efficient automatic procedure. The presentation extends these ideas to a general approach to spatially adaptive local parametric estimation. Suppose we have independent observations {Zi}i=1 of the form Zi = (Xi, Yi). Here Xi denotes a vector of “features” or explanatory variables which determines the distribution of the random “observation ” Yi. The d-dimensional vector Xi ∈ R can be viewed as a location in time or space and Yi as the “observation at Xi”. Our model assumes that the distribution of each Yi is determined by a parameter fi which may depend on the location Xi, fi = f(Xi). In many cases the natural parametrization is chosen which provides the relation fi = E{Yi}. The estimation problem is to reconstruct f(x) from the data {Zi}i=1,...,n. This set-up includes Gaussian images Yi = f(Xi) + εi with a regression function f(·) and i.i.d. Gaussian errors εi ∼ N (0, σ); Poisson images with P (Yi = k|Xi = x) = f(x) exp(−f(x))/k!; Bernoulli (binary response) images with P (Yi = 1) = f(Xi), P (Yi = 0) = 1 − f(Xi). The joint distribution of the samples Y1, . . . , Yn is given by the log-likelihood L = !n i=1 log p(Yi, f(Xi)). In the parametric setup, the whole function f(·) is determined by a parameter vector θ: f(·) = f(·,θ). This reduces the problem of estimating f to the problem of estimating θ ∈ Θ ∈ R. The maximum likelihood approach yields the estimates