Image and Signal Reconstruction

Problems of reconstructing images or signals, which are affected by random noise, may also be viewed as problems in the field of nonparametric statistics. Then, one may consider a noisy image or signal as an empirical observation, where the true image or signal is the quantity that we are interested in.

Such problems of signal recovery can be classified by the dimensionality of the underlying signal. Conventional images are usually observed on a plane so that the dimension d is two. In this case, we have a bivariate problem. Models in which the dimension of the image is d > 1 are called multivariate problems; spatial problems, that is, d = 3, also have their applications; while, of course, the most elementary – and also useful – situation is the univariate problem d = 1.

In the current section, we assume that the image is observed at a finite number of points only. We classify the following model of the true images and the corresponding observation scheme:

(PGI) Pixel grid image. Such an image consists of finitely many grid points only. To describe the model mathematically, we denote the grid points by ${x}_{1},{x}_{2},\ldots \in {\mathbb{R}}^{d}$. In many applications, they may be assumed to be located in the d-dimensional cube [−1,1]^d. At each grid point, we observe the brightness or the intensity of the image, which may be represented by a real-valued random variable Y _j , j. Apparently, Y _j may be written as the true (but unobserved) intensity y _j ∈ ℝ of the image at the grid point x _j , contaminated by some i.i.d. random variable ε _j , which is centered by the condition Eε _j = 0, that is, ${Y }_{j} = {y}_{j} + {\epsilon }_{j}$. Our goal is to detect the true values y ₁,y ₂,{…} of the image. Pixel grid image occur in electronic imaging and scanning, in particular. Of course, any computer is able to save only finitely many objects.