The Weibull manifold in low-level image processing: An application to automatic image focusing

Abstract

In this paper, we introduce a novel framework for low-level image processing and analysis. First, we process images with very simple, differencebased filter functions. Second, we fit the 2-parameter Weibull distribution to the filtered output. This maps each image to the 2D Weibull manifold. Third, we exploit the information geometry of this manifold and solve lowlevel image processing tasks as minimisation problems on point sets. For a proof-of-concept example, we examine the image autofocusing task. We propose appropriate cost functions together with a simple implicitly-constrained manifold optimisation algorithm and show that our framework compares very favourably against common autofocus methods from literature. In particular, our approach exhibits the best overall performance in terms of combined speed and accuracy.

Generalized Pareto Distributions—Application to Autofocus in Automated Microscopy

“…Another advantage of parametric models is the fact that they provide information about the analytical form of the sharpness function. This can be used in the construction of faster optimization methods where the sequence of an already measured set of parameters controls the step-length of the microscopes focus mechanism before the next image is collected [1]. We also want to point out that the autofocus method can also be used for other imaging devices since only the visual properties of the images enter the algorithm.…”

“…The statistical distribution of such edge-type filter systems has previously been investigated in the framework of the Weibull-or more generally in the framework of the generalized extreme value distributions (GEV) (see, for example [1], [9]- [13]). From the construction of the filter functions follows that the filter results follow a mixture distribution consisting of near-zero filter results and…”

“…The magnitude of such a two-dimensional vector is invariant under the group operations while the relation between the two filter results encode the group operation. For many image sources (including microscopy images) the combined magnitude values can be characterized with the help of the generalized extreme value distributions (GEV), of which the more commonly known Weibull distribution is one example (see for example [1] or [2] ). Using the GEVs to analyze the filter results requires that the contribution of flat regions with near-zero filter results has to be excluded in the GEV-fitting.…”

“…Using a parametrized statistical model instead of methods based on empirical measurements, like means and variances, has the advantage that one can use the manifold structure of the distributions to construct more efficient search methods to find the focal position using a reduced number of image acquisitions. Methods from the theory of optimization on manifolds can be used to speed up the autofocus process as described in [1] but this is not included in the experiments described here. The main focus of this study is the investigation of the role of the GPDs in low-level signal processing.…”

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

“…For dynamic focus sequences the change of the probability distributions between consecutive frames can be measured in the Fisher geometry and combining the static sharpness function with the characterization of the dynamic change improves the autofocus results. This approach is similar to the framework developed in [13] where it is shown how geometry-based optimization methods can be used to improve the efficiency of auto-focus control.The study of efficient implementations or the comparison to existing auto-focus methods is outside the scope of this study. The fact that these images can be described by…”