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Intuitively, an isotropy means that a signal y(x) has more or less the same behavior in all directions in a neighborhood of x. A constant y(x)=const is an ideal isotropic signal with no variation in the signal at all. A more complex linear model y(x)=∑ d j=1 a j x j might have different derivatives with respect to the directions/variables x j . However, this is again a trivial case because this signal is invariant in the space of the parameters a j . In a nonparametric context, anisotropy mainly denotes unpredictable nonparametric changes in the behavior of a signal. Examples are a univariate y with jumps (Blocks in Fig. 6.3) or the piecewise constant/linear two-variable signals shown in Figs. 2.10 and 2.11. These signals have different values for x on different sides of the jump points in Blocks or change-lines in Figs. 2.10 and 2.11. Points, lines, edges, and textures are present in all images. They are locally defined by position, orientation, size, or scale. Often being of small size, these specific features encode a great proportion of the information contained in images. An image intensity y is a typical example of an anisotropic nonparametric signal. In what follows, we mainly consider anisotropic methods in the context of 2D imaging. The idea that biological visual systems analyze images by orientation and scale dates back to the 1960s, when two points of view became quite popular. The first was that visual systems compute directional derivatives along space and time, and the second was that biological visual systems implement the concept of “matched filtering.” In the computational vision and image-recognition literature, corresponding approaches of analyzing images along multiple orientations appeared at the beginning of the 1970s with works in early vision systems. Respectively, methods have been proposed for differentiation and matching typical visual elements in images.
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