Generalized extreme value distributions, information geometry and sharpness functions for microscopy images

Abstract

We introduce the generalized extreme value distributions as descriptors of edge-related visual appearance properties. Theoretically these distributions are characterized by their limiting and stability properties which gives them a role similar to that of the normal distributions. Empirically we will show that these distributions provide a good fit for images from a large database of microscopy images with two visually very different types of images. The generalized extreme value distributions are transformed exponential distributions for which analytical expressions for the Fisher matrix are available. We will show how the determinant of the Fisher matrix and the gradient of the determinant of the Fisher matrix can be used as sharpness functions and a combination of the determinant and the gradient information can be used to improve the quality of the focus estimation.Index Terms-generalized extreme value distribution, information geometry, edge statistics, auto-focus, imagebased screening OVERVIEW AND BACKGROUNDEdge detection is one of the first and most important operations in all technical and biological vision systems. It is therefore important to understand the relation between the statistical properties of the space of input images and the statistical properties of the resulting edge detector values. In this paper we mainly consider images from an automated microscope taking focus series of cells with two different types of staining. We have thus two sets of images with two visually different properties, the optical properties of the system in the form of the focus plane are systematically varied and the optimal focus setting is established with the help of an independent measuring process. Statistical properties of edge detectors have been studied earlier, mainly in the context of natural image statistics (see [1,2,3,4,5,6,7] for some examples). The main observation is that the distributions of these filter results This work was supported by the Swedish Foundation for Strategic Research through grant IIS11-0081 usually have heavy tails, with the (two-parameter) Weibull distribution as the most popular choice. In this paper we will use the generalized extreme value (GEV) distributions which come in three different types: the Fréchet-, (reversed) Weibull-and Gumbel-distributions. We will compare the fitting properties of the GEV distributions with those of the standard Weibull distribution and show that the fitting results for the GEV are comparable or slightly better than those for the Weibull distributions.The second topic we investigate is the usage of concepts from information geometry [8,9] where probability distributions are points on a manifold and methods from differential geometry are used to study the relation between different distributions. Typical concepts used are the distance between distributions or the shortest path (the geodesic) between them. Both, Weibull-and GEV-distributions, can be derived by transformations from the exponential distribution and closed form expressions for the metric ...

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