## On Generalized Entropies and Scale-Space (1997)

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### BibTeX

@MISC{Sporring97ongeneralized,

author = {Jon Sporring and Joachim Weickert and Universitetsparken E},

title = {On Generalized Entropies and Scale-Space},

year = {1997}

}

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### Abstract

this paper we show that the generalized entropies are such functionals. It should be noted that this behavior is not seen for the number of critical points: Although critical points most often disappear when scale is increased, creation of critical points with increasing scale is a generic event [16, 14, 7]. Secondly, generalized entropy is the basis for the theory of Multi-Fractal [11, 18] and it is known that there are very strong algebraic similarities to the fundamental equations of Statistical Mechanics. These are thus well known functions, and while images are not physical systems in classical thermodynamic sense, Linear Scale-Space is governed by the Linear Heat Diffusion Equation, and one could thus without great difficulty extend the view of images to be a classical thermodynamical system for which the Linear Heat Diffusion is valid. Such a system is an ideal gas. These interpretations of images will be discussed in detail in this chapter. Finally, as will be demonstrated the generalized entropies offer practical, mathematical well founded functions to study scaling behaviors of images for scale-selection and texture analysis. Related to this work is Vehel et al. [29], where images are studied in the multi-fractal setting, focusing on certain dimensions, and Brink & Pendock [6], and Brink [5] have used the entropy and the closely related Kullback measure to do local thresholding of images. This article is organized as follows. First, in Section 2 will be given a brief introduction to Linear ScaleSpace and linear entropy. Then, in Section 3 will we discuss the generalized entropies, what the difference is to linear entropy, and what their properties are in Scale-Space. Following this, in Section 4 we will discuss a physical interpretation of images both from the...