A Formalism for Analizing Degraded Edges Using Modified Heaviside Functions

We present a mathemathical formalism to analyze saturated edge imaging using various modified Heaviside functions. Introduction. For the past few decades an enormous amount of work has been done in the field of the theory of edge detection. This area of optical, hybrid and digital image processing interrelates different aspects of image formation, visual optics and robotic vision. It is well established (e.g., Marr and Hildreth) that edge detection is a primary mechanism in visual object recognition. These have been studied as a major application for pattern recognition. Studying the established models and formalisms one is concerned with the specific phenomena related to blurred edge detection. From the literature, we find that only approximate calculations have been done in the sense of explaining the different contributions to the total image intensity distribution of a blurred or degraded edge. Our main goal in our current studies is to establish well-defined blurred (or degraded) edge functions to interpret different pattern and object recognition in a two dimensional scene. The idea is illustrated in the scheme shown in Fig.1. These edge functions depend on a variable parameter which accounts for the degree of degradation or edge blurring. We are interested in defining a unique edge generator as a general function to be applied to represent different features of edge image processing. The calculation of a first derivative defines a line object. From these two functions one can either represent Edge Spread Function(ESF), Line Spread Function (LSF) and the associated Modulation Transfer Function (MTF), through the well established reciprocity for perfect optical systems. On the other hand, one could obtain a larger class of degraded Basic and Clinical Applications of Visual Science, The Professor Jay M. Enoch Festschrift Volume, (Ed. V. Lakshinarayanan), Kluwer Academic Publishers, DORDRECHT, 77-81, 1997 (or blurred) edges and the corresponding class of thick lines. Calculating second derivatives from the first function, one is able to define a class of differential operators to be applied for edge detection mechanism, namely, zero-crossing location. This aspect has been also largely discussed and our main interest is to achieve a generalization of this procedure. Fig. 1: Scheme of the proposed processes As for resolution criterion applied to blurred edges very little information is available in the literature, in part because it is not easy to design a simple experiment from which one could infer objective criteria, and partially also because there is an inadequate formalism. Recently, Marshall et al. have shown that identification of two identical images except for the nature of boundaries between different regions of the picture (sharp edge and blurred edge) is related to the so-called occlusion edge blur on the relative depth seen on a scene, which is related to depth perception. To interpret their experiment, they define an approximate model in which the total intensity (stimuli) is given as a sum of two contributions: 1) A convolution of a Gaussian blur kernel and the intensity image of the blurred object. 2) A vignetting function defined as the convolution of the Gaussian kernel and a perfect edge, multiplied by the intensity image of the sharp object. We propose a more rigorous formalism as an alternative approach to those in that the Gaussian kernel is substituted by the exact LSF of an incoherent and diffraction-limited system. This formalism is also consistent with the definition of resolution criteria and a large class of second order differential operators to be applied to various blurred edge detection processes. Proposed Formalism: LSF Definition and Blurred Edge Representation. Fig.2 displays various representations for different analytical formulations of the LSF available in the literature. The LSF in terms of the Struve function can be represented as a polynomial expansion or represented in terms of a Bessel functions series(dashed line), as proposed earlier by Steel. Here we have found a major error. Approximate formulations are also given by Barakat-Houston (dots) and Gaskill-Hufnagel (solid line). We have applied the exact representation obtained by us as a Bessel functions expansion, showing the appropriate high speed of convergence of the series. 1.5 1 0.5 0 0.5 1 1.5 3 2 1 0 1 2 3 1.00943 -1.39971 LSF1( ) x LSF2( ) x LSF3( ) x LSF5( ) x 2.995 -3 x Fig. 2: Comparison of different definitions for the LSF. The representation of blurred edges has been defined as in the earlier work of Shanmugen et al. in terms of a sigmoid where the blurring parameter gives the degree of blur. We have introduced an alternative representation called saturated degraded edges (Fig.3) in terms of the so-called degradation parameter.