Adaptive filters are commonly used in image processing to enhance or restore data by removing noise without significantly blurring the structures in the image. The adaptive filtering literature is vast and cannot adequately be summarized in a short chapter. However, a large part of the literature concerns one-dimensional (1D) signals [1]. Such methods are not directly applicable to image processing and there are no straightforward ways to extend 1D techniques to higher dimensions primarily because there is no unique ordering of data points in dimensions higher than one. Since higher-dimensional medical image data are not uncommon (2D images, 3D volumes, 4D time-volumes), we have chosen to focus this chapter on adaptive filtering techniques that can be generalized to multidimensional signals.

This chapter assumes that the reader is familiar with the fundamentals of 1D signal processing [2]. Section 2 addresses spatial frequency and filtering. 2D spatial signals and their Fourier transforms are shown to illuminate the similarities to signals in one dimension. Unsharp masking is described as an example of simple image enhancement by spatial filtering. Section 3 covers random fields and is intended as a primer for the Wiener filter, which is introduced in Section 3.2. The Wiener formulation gives a lowpass filter with a frequency characteristic adapted to the noise level in the image. The higher the noise level, the more smoothing of the data. In Section 4 adaptive Wiener formulations are presented. By introducing local adaptation of the filters, a solution more suitable for nonstationary signals such as images can be obtained. For example, by using a "visibility function," which is based on local edge strength, the filters can be made locally adaptive to structures in the image so that areas with edges are less blurred. Section 5 is about adaptive anisotropic filtering. By allowing filters to change from circularly/spherically symmetric (isotropic) to shapes that are closely related to the image structure, more noise can be suppressed without severe blurring of lines and edges. A computationally efficient way of implementing shift-variant anisotropic filters based on a non-linear combination of shift-invariant filter responses is described.

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