Deterministic mathematical morphology for CAD/CAM

Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed, usually with polynomial complexity and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification. In this second part, the concepts developed in part I (see Diaz de Leon and Sossa Azuela, “Mathematical morphology based on linear combined metric spaces on Z^1 (part I): Fast distance transforms,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 2, pp. 137–154, 2000) are used to prove that it is possible to reduce the complexity of the morphological operators to zero complexity (constant time algorithms) for any regular discrete metric space and to eliminate the use of the structural element. In particular, this is done for an infinite family of metric spaces further defined. The use of the distance transformation is proposed for it comprises the information concerning all the discs included in a region to obtain fast morphological operators: erosions, dilations, openings and closings (of zero complexity) for an infinite (countable) family of regular metric spaces. New constant time, in contrast with the polynomial time algorithms, for the computation of these basics operators for any structural element are next derived by using this background. Practical examples showing the efficiency of the proposed algorithms, final comments and present research are also given here.

Morphological openings and closings are basic operators in mathematical morphology. Morphological area openings and area closings have the same functions as standard morphological openings and closings but avoid distorting object boundaries. The conventional implementation of morphological area operators is based on the heap data structure. A new methodology for implementing morphological area operators, which relies on the disjoint set data structure, is proposed in this paper. The computational complexity of our method is O(N) where N is the image size, and not related to the area parameter. Experiments show that the new implementation of grayscale area operators runs faster than the conventional implementations. The computational complexity of binary area operators is also studied in this paper.

A morphological image enhancement method based on the GAN (General Adaptive Neighborhood) structuring elements is proposed for some problems such as low overall contrast, noise and inter-target edge of some remote sensing images. Firstly, a pixel to be processed is selected as a seed point, and an adaptive neighborhood is defined according to the image pixel characteristic and the neighborhood constraint relation to obtain the self-adaptive structuring elements set. A corresponding self-adaptive morphological operation based on the self-adaptive structuring elements set is constructed with reference to the classical morphological operation definition, and the self-adaptive alternative filter is constructed by using the adaptive morphological operation combination so as to realize the enhancement of the remote sensing image. The experimental results show that the method can preserve the accurate positioning of the objects edge while enhancing the image, avoid the loss of information, has better noise removing performance. Compared with the classical morphological filter operator, the single-scale Retinex algorithm and bi-histogram equalization (BHE) algorithm, the contrast ratio and the signal-to-noise ratio of the enhanced image are improved.

As the semiconductor technology is scaling down to nanometer regime, accurate layout analysis requires operations with simulated contours of VLSI layouts which are non-rectilinear.The basic operations on non-rectilinear shapes include union, connected components and boundary. Whatever precision one would choose for the representation of vertices of shapes and the exact representation of, say, A∩B would require an even higher precision. Consequently rounding of intermediate results of a chain of basic operations seems to be inevitable. The presence of rounding errors is unacceptable because the implementation of algorithms for all operations which involve topology, such as connected components, find boundary, etc, becomes impossible or extremely complexWe present a complete solution to the following problem: Let a VLSI design or a simulated through a lithography process image of a VLSI design be given by a number of two-dimensional point sets. Each point set consists of polygons with arbitrary (not necessarily 90°) angles and with vertices on an integer grid. Perform any amount of sequential basic operations so that:• The resulting point sets are mathematically exact, that is no rounding errors are allowed. In particular the connectivity of the point sets remains intact, the boundaries remain undistorted and the statements like = B) ∪ ∩ B), (A B) is disjoint with B and ∩ ⊂ A always hold.• The amount of time and memory per one basic geometric operation on elementary polygons (trapezoids) is constant. For example it would not be an acceptable solution to represent the vertices of the new shapes which appear as a result of the operations of intersection, difference, etc, by unlimited length rational numbers.

This chapter describes the basic operations and applications of mathematical morphology, which is based on lattice theory and random geometry to study geometric structures. It defines the basic operators such as dilation and erosion and structuring elements of binary images and grayscale images. The commonly used image preprocessing strategies are image filtering and image reconstruction. The chapter describes the basic knowledge of morphological reconstruction as well as the basic operations of morphological reconstruction by introducing geodesic erosion, geodesic dilation, and their applications. It also describes the simplest yet effective segmentation algorithm, namely the watershed transform. The chapter introduces the commonly used watershed transforms based on distance, gradient, and control markers. It shows how to extend the mathematical morphology to the color space and then completed the basic operations such as the dilation and erosion in color space.

This paper presents a Genetic Programming (GP) approach to the design of Mathematical Morphology (MM) algorithms for binary images. The algorithms are constructed using logic operators and the basic MM operators, i.e. erosion and dilation, with a variety of structuring elements. GP is used to evolve MM algorithms that convert a binary image into another containing just a particular feature of interest. In the study we have tested three fitness functions, training sets with different numbers of elements, training images of different sizes, and 7 different features in two different kinds of applications. The results obtained show that it is possible to evolve good MM algorithms using GP.

Fast parallel processing of gray-scale images and exact hard-clip thresholding are two important functionalities necessary in optoelectronic implementations of structural processors. The parallel nature of processing stems from optical implementation of local operations with arrays of active smart pixels. We have demonstrated a morphological image processor composed of arrays of bistable optoelectronic transceivers which are connected in differential pairs and work as comparators. The use of differential pairs of optical thyristors fabricated in GaAs technology allows to realize a dual rail architecture for this photonic morphological image processor. The processor consists of a thresholding module and a binary morphological processing module. The thresholding module decomposes gray level images into series of binary slices. In the binary morphological processing module operations are performed within a neighborhood defined by a structuring element implemented as a diffractive fan-out element. In the prototype set-up we demonstrate median filtering, dilation and erosion operations performed for an image of 8x8 pixels and threshold decomposition of 6 gray level images. In principle all rank order as well as morphological filters can be optically calculated in the set-up. Additional functionality of the processor is achieved with use of the electronic layer with digital cellular processors. The electronic layer, designed as an array of simple digital processors, realizes a set of operations on binary images using 4 bit programmable weights. Simulation results for 0.8 µm CMOS technology are presented. We discuss the limitations of the photonic morphological image processing with respect to bandwidth, parallelism and architecture of the processor.

A novel model of a normal vector sphere is proposed in this paper. Two 3D object morphological addition can be calculated by merging two corresponding normal vector spheres. The important property of convex object morphological addition is proved based on integral geometry, which can le popularized to a concave set, then the two objects' morphological operator can be calculated using the two point sets' Minkowsky addition which have the same normal vector. A unified model of graphics morphological operators is developed; this model unifies the morphological addition, subtraction of 2D and 3D objects in algorithm theory, which lays a solid foundation for morphological operator in computer graphics.

A fast and exact Euclidean distance transformation using decomposed grayscale morphological operators is presented. Applied on a binary image, a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels. It is shown that the large structuring element required for the Euclidean distance transformation can be easily decomposed into 3/spl times/3 windows. This is possible because the square of the Euclidean distance matrix changes uniformly both in the vertical and horizontal directions. A simple extension for a 3D Euclidean distance transformation is discussed. A fast distance transform for serial computers is also presented. Acting like thinning algorithms, the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects, significantly reducing execution time. Nonsquare pixels can also be used in this algorithm. An example application, shape filtering using arbitrary sized circular dilation and erosion, is discussed. Rotation-invariant basic morphological operations can be done using this example application. >

Fast and accurate fault detection and localization are important topics in voltage source converter-based high-voltage direct current transmission (VSC-HVDC) grids in order to isolate the fault after the first milliseconds of occurrence. Common signal processing methods used for detecting fault traveling wave (TW) peaks like Fourier transform, S-transform and wavelet-based techniques transform the signal into the frequency domain, requiring complicated computations. In this paper, the application of mathematical morphology (MM)-based filters for detecting and locating faults in VSC-HVDC links is studied. MM-based methods analyze the signal in the time domain and detect the wave peaks accurately. Multiple MM-based filters resulted from basic MM operators are presented and used for TW-based fault study in VSC-HVDC grids. Several fault cases are applied to the CIGRE VSC-HVDC model in PSCAD, and the MM-based scripts are written in MATLAB. The impact of different sizes and types of the structuring elements on the accuracy of peak detection is analyzed. The results show the accuracy of MM filters for detecting and locating fault transient in VSC-HVDC links. The proposed method gives accurate results for both low-impedance and high-impedance faults.

On the basis of Canny operator on image processing, The edge extraction of the fuzzy droplet images is carried out by combination of removing a small target and morphological open operation .First, Use edge detection operator to detect the image edge preliminary, and then call remove a small target which belongs to MATLAB internal function to filtering the results of the edge detection, and finally with morphological operations for final processing.The results showed that: the combination of Canny operator, open operation and remove a small target works much better than the classical Canny operator.

Mathematical morphology is a powerful tool for image processing and analysis of binary, greyscale and colour images. In the paper, an efficient new fuzzy model for morphological colour image processing is introduced. A new vector-ordering scheme that uses fuzzy if–then rules is proposed, and then the basic morphological operations of erosion and dilation are defined. The morphological operators presented, which are vector preserving, are less sensitive to image distortion and perform significantly better in noise removal problems than other reported morphological operators. Experimental results demonstrate these advantageous characteristics on real images. Additionally, illustrative examples that exhibit the applicability of the proposed framework to edge-detection problems are given.

When considering ways to automate the generation of image processing algorithms for object recognition tasks, one critical element is the availability of measures to assess the potential and actual ability of individual operations for making a set of desired discriminations. This paper discusses the analysis and evaluation of grey-level image processing operators or algorithms from the perspective of trying to search automatically through a large space of them for one which satisfactorily performs a given recognition or discrimination task. Performance of an operator may be expressed in terms of accuracy, consistency, and cost, over an entire set of training images.The major issues of evaluating and choosing between operators in this context are discussed, and examples are given of measures which can be used to evaluate classes of operators for applicability, or to evaluate individual operators or parameter settings for actual performance. The paper first describes the form of the analysis for binary morphological operators, and then shows how it may be directly extended to grey-level morphological operators. Several examples are provided to show how grey-level pixel sets may be discriminated on the basis of various combinations of grey level and spatial criteria, as calculated by the basic morphological operators of erosion, dilation, opening, and closing.

Mathematical morphological operations are commonly used as a tool in image processing for extracting image components that are useful in the representation and description of region shape. In this paper, six basic morphological operations are investigated to remove noise and enhance the appearance of binary images. Dilation, erosion, opening, closing, fill and majority operations are tested on twenty-five images and subjectively evaluated based on perceived quality of the enhanced images. Results of the experiments showed that noise can be effectively removed from binary images using combinations of erode-dilate operations. Also, the binary images are significantly enhanced using combinations of majority-close operations.

ABSTACT Image segmentation has been an important and challenging issue in the field of computer vision over decades. It plays a critical role for most image analysis tasks, such as object detection and recognition. The aim of this paper is to obtain segmented image of Printed Circuit Board (PCB)’s track using mathematical morphological operation. Morphological image analysis is based on the idea that images represent a collection of spatial patterns that can be analyzed based on some predefined patterns called structuring elements. Most basic morphological operation, dilation and erosion have been employed in this research. Detail elaboration on this method will be discussed and experimental results are presented.