Geometric Filters, Diffusion Flows, and Kernels in Image Processing

Diffusion flows are processes applied to digital images in order to enhance or simplify them. These flows are usually implemented by appropriate discretizations of partial differential equations (PDEs). Iteratively applying these discretizations, called also numerical schemes , to an image results in a series of images with decreasing detail (Fig. 7.1). Using a suitable flow, one can enhance important image features such as edges and objects while filtering the image from undesired noise. This can be done not only to gray-level and color images but also to textures, movies, volumetric medical images, and so on. Diffusion flows are important members of the family of methods for image processing, computer vision, and computer graphics based on the numerical solution of PDEs. Other members of the family include active contours/surfaces for image segmentation, reconstruction of three-dimensional scenes from their shading or stereo images, graphic visualization of natural phenomena, and many others. This family of methods has many advantages, among them theoretical origin due to derivation from a minimization of (usually geometric) cost functions, efficiency, and robustness.

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