Anisotropic diffusion in image processing by Joachim Weickert

By Joachim Weickert

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3. One way to achieve unconditional L∞ -stability for MCM is to approximate uξξ by suitable linear combinations of one-dimensional second-order derivatives along grid directions and to apply an implicit finite difference scheme [95, 13, 146]. Schemes of this type, however, renounce consistency with the original equation as well as rotational invariance: round shapes evolve into polygonal structures. A consistent semi-implicit approximation of MCM which discretizes the firstorder spatial derivatives explicitly and the second-order derivatives implicitly has been proposed by Alvarez [10].

53) can be regarded as the steady-state equation of ∂t u = div (g(|∇u|2) ∇u) + β(f −u). 56) This equation can also be obtained directly as the descent method of the functional Ff (u) := Ω β ·(u−f )2 + λ2 ·ln 1+ |∇u| λ2 2 dx. 56) consists of the Perona–Malik process with an additional bias term β ·(f −u). One of Nordstr¨om’s motivations for introducing this term was to free the user from the difficulty of specifying an appropriate stopping time for the Perona–Malik process. However, it is evident that the Nordstr¨om model just shifts the problem of specifying a stopping time T to the problem of determining β.

4 METHODS OF DIFFUSION–REACTION TYPE 27 [307, 308], for segmentation of textures [433, 437] and remotely sensed data [6, 5], and for target tracking in infrared images [65]. Most applications, however, are concerned with the filtering of medical images [26, 28, 29, 155, 244, 248, 264, 270, 308, 321, 355, 386, 393, 431, 434, 437, 444]. Some of these applications will be investigated in more detail in Chapter 5. 4 Methods of diffusion–reaction type This section investigates variational frameworks, in which diffusion–reaction equations or coupled systems of them are interpreted as steepest descent minimizers of suitable energy functionals.

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