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ROBOTICS Handbook of Computer Vision Algorithms in Image Algebra Part 9

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Tham khảo tài liệu 'robotics handbook of computer vision algorithms in image algebra part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7.3. Opening and Closing Dilations and erosions are usually employed in pairs a dilation of an image is usually followed by an erosion of the dilated result or vice versa. In either case the result of successively applied dilations and erosions results in the elimination of specific image detail smaller than the structuring element without the global geometric distortion of unsuppressed features. An opening of an image is obtained by first eroding the image with a structuring element and then dilating the result using the same structuring element. The closing of an image is obtained by first dilating the image with a structuring element and then eroding the result using the same structuring element. The next section shows that opening and closing provide a particularly simple mechanism for shape filtering. The operations of opening and closing are idempotent their reapplication effects no further changes to the previously transformed results. In this sense openings and closings are to morphology what orthogonal projections are to linear algebra. An orthogonal projection operator is idempotent and selects the part of a vector that lies in a given subspace. Similarly opening and closing provide the means by which given subshapes or supershapes of a complex geometric shape can be selected. The opening of A by B is denoted by A B and defined as AoB A B X B. The closing of A by B is denoted by A B and defined as AiB Ax B B. Image Algebra Formulation Let a 0 1 X denote a source image and B the desired structuring element containing the origin. Define N X 2X is defined by N y x X x yeB . The image algebra formulation of the opening of the image a by the structuring element B is given by b aĐ.r 0A The image algebra equivalent of the closing of a by the structuring element B is given by b a JV 0Ar . Comments and Observations It follows from the basic theorems that govern the algebra of erosions and dilations that A- A 11 L . Í.Í. A A b and A B B A B. This shows the analogy