Nonlinear Diffusion Filtering
Omer Demirkaya, Musa H. Asyali, Prasanna K. Sahoo · 2008
Linear filtering techniques (e.g., Gaussian filtering) are perhaps the most frequently used filtering techniques in image processing, but they suffer from two major drawbacks. First, they smooth image features such as edges and unwanted noise indiscriminately. Second, they dislocate image CONTENTS 5.1 Introduction ........................................................................................... 189 5.2 Diffusion as a Physical Phenomenon ................................................. 190 5.3 Application of the Concept of Diffusion to Image Filtering ........... 191 5.4 Perona-Malik Diffusion ....................................................................... 192 5.5 Edge Enhancement ............................................................................... 192 5.6 Linear Versus Nonlinear Diffusion .................................................... 196 5.7 Diffusivity Functions ........................................................................... 197 5.8 Numerical Implementation ................................................................. 199 5.8.1 Discretization in Multidimensions ........................................ 201 5.9 Numerical Stability ............................................................................... 202 5.10 Noise Threshold .................................................................................... 203 5.11 Regularization of Perona-Malik Scheme .......................................... 204 5.11.1 Gaussian Regularization.......................................................... 204 5.11.2 Median Regularization ............................................................ 205 5.11.3 Comparison of the Gaussian and Median Regularization ........................................................................... 205 5.12 Energy Functional Approach to Diffusion Filtering ....................... 209 5.13 Total Variational-Based Diffusion ...................................................... 210 5.14 Mean Curvature Diffusion .................................................................. 213 5.15 Afine Invariant Curvature Motion ......................................................214 5.16 Anisotropic Diffusion of Color Images ............................................. 215 5.16.1 Orthogonal Decomposition of Perona-Malik Diffusion ..... 217 Problems .......................................................................................................... 219 References ........................................................................................................ 219 features (such as edges). To tackle the latter problem, the notion of scalespace was introduced by Witkin [1] and Koenderick [2] (see also Chapter 9). Scale-space representation consists of a set of images filtered with a Gaussian kernel with varying σ to change the resolution from coarse to fine. In scalespace analysis, to find the correct location of a feature, one tracks the feature in the scale-space from the coarse level back to the fine level. To alleviate the previously mentioned disadvantages of linear filtering, nonlinear diffusion filtering methods have been proposed. The lead work in this area is the seminal work of Perona and Malik [3], in which they proposed a nonlinear diffusion filtering method for scale-space representation, enhancement, and segmentation (edge detection) of images. The filters in this category address the two important disadvantages of linear filtering. This particular class of filters has found widespread application in medical image processing [4-17] owing to their superb performance in removing noise while preserving edge sharpness. Noise is a major problem in almost all medical imaging modalities, and these filters have provided a solution to the problem of resolution loss when removing noise with linear filtering methods. In this chapter we will discuss some of these nonlinear diffusion filtering techniques that have been applied in medical and biological image processing and analysis.