Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics


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Besides extending the main concepts such as dilation, erosion, etc. In the s and s, MM gained a wider recognition, as research centers in several countries began to adopt and investigate the method. MM started to be applied to a large number of imaging problems and applications.


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In , Serra further generalized MM, this time to a theoretical framework based on complete lattices. This generalization brought flexibility to the theory, enabling its application to a much larger number of structures, including color images, video, graphs , meshes , etc. At the same time, Matheron and Serra also formulated a theory for morphological filtering , based on the new lattice framework. The s and s also saw further theoretical advancements, including the concepts of connections and levelings. The basic idea in binary morphology is to probe an image with a simple, pre-defined shape, drawing conclusions on how this shape fits or misses the shapes in the image.

This simple "probe" is called the structuring element , and is itself a binary image i. The basic operations are shift-invariant translation invariant operators strongly related to Minkowski addition. The erosion of the binary image A by the structuring element B is defined by:. When the structuring element B has a center e. For example, the erosion of a square of side 10, centered at the origin, by a disc of radius 2, also centered at the origin, is a square of side 6 centered at the origin.


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Example application: Assume we have received a fax of a dark photocopy. Everything looks like it was written with a pen that is bleeding. Erosion process will allow thicker lines to get skinny and detect the hole inside the letter "o". The dilation of A by the structuring element B is defined by:. If B has a center on the origin, as before, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A.

In the above example, the dilation of the square of side 10 by the disk of radius 2 is a square of side 14, with rounded corners, centered at the origin. The radius of the rounded corners is 2. Example application: Dilation is the dual operation of the erosion. Figures that are very lightly drawn get thick when "dilated". The opening of A by B is obtained by the erosion of A by B , followed by dilation of the resulting image by B :.

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In the case of the square of side 10, and a disc of radius 2 as the structuring element, the opening is a square of side 10 with rounded corners, where the corner radius is 2. Example application: Let's assume someone has written a note on a non-soaking paper and that the writing looks as if it is growing tiny hairy roots all over. Region-based segmentation algorithms operate iteratively by grouping together neighboring pixels that have similar properties such as gray level, texture, color, shape and splitting groups of pixels that are dissimilar in value [ 20 , 21 ]. There are a variety of approaches of region-based segmentation.

These methods can be classified into two categories:. It is also classified as a pixel-based segmentation method since it involves the selection of initial seed points. This approach to segmentation groups of pixels or subregions into larger regions is based on predefined criteria. First, a set of seed points is selected based on some user criterion e. Second, the regions are grown from these seed pixels to neighboring pixels, which are examined to ascertain if they should be added to the region, according to a region membership criterion e.

Third, the second process is iterated on, in the same manner as general data clustering algorithms. Region-growing-based techniques are better than the edge-based techniques in noisy images where edges are difficult to detect. However, it is computationally expensive. Region splitting starts with the whole image as a single region and subdivides it into subsidiary regions recursively while a condition of homogeneity is not satisfied. Region merging is the opposite of region splitting and works as a way of avoiding over-segmentation.

Mathematical Morphology in Computer Graphics, Scientific Visualization and Visual Exploration

It starts with small regions and merges the regions that have similar characteristics such as gray level, variance. There are a great number of various approaches [ 22 — 26 ] excepting for the methods described above, also including the improvements of above methods [ 27 — 37 ]; for example, matching-based segmentation, clustering-based segmentation, fuzzy-inference-based segmentation, generalized PCA principal component analysis -based segmentation.

Each segmentation method has its advantages and disadvantages. A universal algorithm of segmentation does not exist, as each type of image corresponds to a specific approach. Therefore, choice of technique depends on peculiar characteristic of individual problems. The emphasis of this paper lies on an improved method of scene image segmentation based on mathematical morphological operator-toggle operator. This chapter is organized as follows: Section 1 presents an overview of methodologies and algorithms for image segmentation.

A new proposed image segmentation method is then introduced in Section 2. In Section 3, the experimental results are analyzed to prove the validity of the proposed method. Finally, the chapter concludes in Section 4.

Lecture - 37 Object Representation and Description - I

Signs and public notices are ubiquitous indoors and outdoors, and they are often used for route finding, finding public places and other locations. The texts in natural scene images contain important information. Therefore, text detection has attracted wide interest due to its usefulness in a variety of real-world applications, such as robots navigation, assisting visually impaired people, tourist navigation, enhancing safe vehicle driving, and so on [ 38 , 39 ].

To date, a great number of algorithms have been proposed for detecting text on scene images or video [ 40 — 49 ]. However, most approaches proposed in the past research contribute to detect the text regions by analyzing the entire image. The image is segmented into text regions and non-text regions according to their features, respectively. The performance of these methods relies on the text detection algorithm and image complexity. Actually, scene text is usually presented on signboards. Because of the uniform color for the background of signboard, the ideal way for extracting text from scene images is to cut out the signboard regions first, and then detect text from the signboard regions.

Thus, this chapter aims to propose an algorithm for segmenting a natural scene image into homogeneous regions. In our method, we first perform the image segmentation in order to detect homogeneous regions. Signboard regions are then detected with a simple criterion in order to remove the noise, such as trees and other non-signboard areas.

A Mathematical Morphology Approach to the Star/Galaxy Characterization

In the following subsections, the proposed method is described and discussed in detail. A natural scene image, I rgb , is supposed to be a bitmap image based on the RGB red-green-blue color model. The EPSF is applied independently to every pixel using different coefficients as shown in the following convolution mask:. That is,. The filtering of the image is achieved by applying the convolution mask, Eq. Factor p in Eq. The target pixel of the convolution mask is set to zero to remove impulsive noise.

A measure of region homogeneity is variance i. In this section, a mathematical morphological operator, Toggle Mapping TM [ 34 ], is introduced to segment a grayscale image into homogenous regions according to the pixel intensity. This is a simple way to segment a grayscale image into homogeneous regions based on a toggle operator.

Such operator is defined as follows:. In order to meet the needs of the application, Dorini [ 35 ] and Fabrizio [ 36 ] have modified and improved this operator by adding new factors or weight coefficients. In their algorithms, the toggle operator is used one time for segmenting an image, so the values of thresholds and coefficients are fixed in their algorithms.

Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics
Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics Mathematics of Shape Description: A Morphological Approach to Image Processing and Computer Graphics

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