By Joachim Weickert

ISBN-10: 3519026066

ISBN-13: 9783519026068

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6 Applications Nonlinear diffusion filters have been applied for postprocessing fluctuating data [269, 415], for visualizing quality-relevant features in computer aided quality control [299, 413, 418], and for enhancing textures such as fingerprints [418]. 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].

22 CHAPTER 1. PARTIAL DIFFERENTIAL EQUATIONS where Ψ is strictly increasing and uniformly bounded in IR, and |Φ′ (s)| = O(Ψ′(s)) as s → ±∞. This regularization was physically motivated by introducing a relaxation time τ into the diffusivity. For the corresponding initial boundary value problem with homogeneous Neumann boundary conditions they proved the existence of a unique generalized solution. They also showed that smooth solutions may become discontinuous within finite time, before they finally converge to a piecewise constant steady-state.

In this case we have to renounce invariance under monotone grey level transformations, but we gain an interesting insight into a process which has very much in common with Gaussian scale-space. A dilation of an image f with a structuring function b : IR2 → IR is defined as (f ⊕ b) (x) := sup {f (x−y) + b(y)}. 65), since one can recover dilation with a structuring element B by considering the flat structuring function b(x) := 0 −∞ (x ∈ B), (x ∈ B). 84) Van den Boomgaard [50, 51] and Jackway [206] proposed to dilate an image f (x) with quadratic structuring functions of type b(x, t) = − |x|2 4t (t > 0).

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Anisotropic diffusion in image processing by Joachim Weickert


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