Mathematical Model for Image Processing Using Graph Theory

Mathematical Model for Image Processing Using Graph Theory

Rajshree Dahal, Ritwika Das Gupta, Debabrata Samanta
DOI: 10.4018/978-1-6684-4580-8.ch003
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Abstract

Image segmentation being an important aspect of computer systems, graph theory provides the most elemental way of representing various parts of an image into mathematical structures. There are many applications of image segmentations including face recognition systems, remote sensing, detecting images sent by satellites, optometry, medical image reading, and many more. Bi-partite graphs are useful in determination of cuts in the segmentation process. These structures are analysed by considering each vertex as pixel, and each weight is some aspect of dissimilarity for two vertices connected by an edge with weights. This makes the problem-solving part very flexible, and their computation becomes easy and fast. The problem is usually parted into small subgraphs that are bound under some continuous forms of graphs like spanning trees, cut vertices or edges, shortest paths graphs, and so on. The cluster formation is proved to be one of most commonly used methods in image segmentation.
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Introduction

Image Segmentation is a traditional and important concept that is studied under the field of computer science. It is mainly concerned with breaking up of an image into many tiny subsets that are disjoint structures and each one of these subsets represent an important aspect of the image. The quality of segmentation obtained is highly valuable to whole process of image processing as it have lot of influence to the total extraction and consecutive steps. A lot of research activities have been taking place in this field in computer science background and still there is lot of scope for its further development. Many famous works in this field have been completed with amazing results in medical background, preconisation, tracking and reconstruction. From the initial stage the basic idea behind image segmentation has been the cluster formations of the data. In 1938 gestalt theory expressed this idea in a distinct manner by identifying a group of principles for example likeness, closeness and continuity that clarifies the idea of man-made awareness system. This theory has been the base of many research work conducted in image segmentation since then in the hope that significant clusters could imitate some the local or universal aspects of an image. Image intensity and colour has been the main focus during the initial research activities like that of Robert edge detector method, Canny edge detector method or the Sobel edge detector method. More over some work has been done due to thresholding process for differentiation of objects and background as many distinct structures has been seen in them. The PDE based methods or partial differentiation-based methods gives segmentation considering the constant change occurring in curves in space minimizing the energy to obtain a required segmentation. Breaking up of regions and assimilation of region is also another popular method which uses iterations to obtain a particular regularity condition. Although rigorous research has been conducted in image and video segmentation and numerous algorithms and methods are established: Thresholding, Edge detection, Region growing methods, clustering methods, and compression-based methods, Watershed transformations, Morphically methods, Graph partitioning methods etc. but in general there exists no segmentation algorithm which can be commonly applied on all the domains. (Chen and L. Pan et al. 2018)

The main step is pre-processing of the image that removes the variety of instabilities of the image including noise by filtration and particular noise removing procedures. Further, the obtained image is partitioned or segmented by the algorithms available for image segmentation. The most problematic stage of image segmentation is the processing stage followed by the structure extraction process. These structures are further used in organisation of images. Some of the application of image processing is in areas like OCR, recognition of face and finger print, biometric, image detection by satellites, image analysis in medical field and so on.

A flowchart of various step included in image processing step is summarized in Figure 1 below:

Figure 1.

Flowchart of image processing method

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The various segmentation techniques and their subdivisions are summarized in Figure 2 below:

Figure 2.

Flowchart of segmentation techniques and sub-techniques

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The following table 1 gives a brief description of the various Image Segmentation methods along with their advantage and disadvantage.

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