By Joachim Weickert, Hans Hagen

Matrix-valued information units – so-called moment order tensor fields – have received major significance in medical visualization and photograph processing because of fresh advancements reminiscent of diffusion tensor imaging. This ebook is the 1st edited quantity that offers the state-of-the-art within the visualization and processing of tensor fields. It comprises a few longer chapters devoted to surveys and tutorials of particular issues, in addition to loads of unique paintings via major specialists that has no longer been released earlier than. It serves as an outline for the inquiring scientist, as a uncomplicated origin for builders and practitioners, and as as a textbook for specialised periods and seminars for graduate and doctoral scholars.

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3 – Sect. 5 the structure tensor is applied to optic flow estimation, texture analysis, and corner detection. Some experiments show the superiority of adaptive structure tensors in comparison to the classic structure tensor and differences between the methods. The chapter is concluded by a brief summary in Sect. 6. 2 Data-adaptive Structure Tensors An early approach to data-adaptive structure tensors is the gray value local structure tensor of Nagel and Gehrke [26], which has been designed for its use in spatio-temporal optic flow estimation.

Such an extension has been provided in [35]: N ∂t uij = div g k,l=1 |∇ukl |2 ∇uij i, j = 1, . . , N . 12) Details can also be found in Chap. 25 by Weickert et al. When setting the initial condition to uij (t = 0) = J0,ij (cf. 2)), this PDE provides the nonlinear structure tensor Jt for some diffusion time t. Here, N is the Fig. 3. Illustration of how the local neighborhood is adapted by an increasing amount of nonlinear diffusion 26 T. Brox et al. e. N = 2 for the spatial structure tensor and N = 3 for its spatio-temporal version.

In Fig. 16 it can be observed that this kind of smoothing leads to the best performance. Fig. 14. Left: Detail of a test image with ideal corner position (50, 50). Right: Larger eigenvalue of the unsmoothed structure tensor J0 2 Adaptive Structure Tensors and their Applications 43 Fig. 15. Cornerness measured by the smaller eigenvalue of a smoothed structure tensor J, and the detected corner. Top: Linear smoothing. Center: Isotropic nonlinear diffusion with TV diffusivity. Bottom: Anisotropic nonlinear diffusion It is also very closely related to the data-adaptive structure tensor proposed by Kothe ¨ [17].

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