Shih_Image_Processing_and_Mathematical_Morpholo.pdf

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Decomposition of Morphological Structuring Elements 283 3 7 6 4 8 5 6 5 9 8 5 4 10 7 12 5 9 7 11 {{ = max 4 3 10 9 7 × 4 × × 5 × 8 × 10 9 × × × × × 4 6 × 12 11 × × 10 9 7 4 –3 1 0 ⊕ 6 5 8 7 7 –3 –2 –3 –8 × 4 × × 5 {{ = max 1 0 –3 ⊕ 6 5 8 7 7 –2 × 0 ⊕ 4 6 × 12 11 × × × 8 × × × × × × × × × × × –2 , –4 , × × 10 9 × –3 –8 = max{ { 1 0 ⊕ 6 –3 –2 0 4 × ⊕ × 5 4 8 × 7 × –2 7 × 0 , –4 × × 5 × ⊕ ⊕ 4 × 6 × × 10 12 11 , 9 × × × × , 0 FIGURE 9.24 The combined row-and-column decomposition of a two-dimensional structuring element. 9.3.4 Complexity Analysis In this section, we discuss the computational complexity of the structuring element decomposition. The worst case in decomposing the one-dimensional arbitrary grayscale structuring element is when the size is n, it is decomposed into (n - 1)/2 smaller structuring components of size 3 as shown in Figure 9.25. Supposing that an image is performed several dilations by l SEs of size N, S1 N, S2 N, º , Sl N, for each pixel it needs l · N processing time and l · N comparisons (maximum operation) in a general sequential architecture as shown in Figure 9.26a. If all the SEs are decomposable without utilizing the maximum selection operation as shown in Figure 9.26b, it needs 3 · [(N - 1)/2 + (l - 1)] processing time and 3 · l · (N - 1)/2 comparisons in a pipeline architecture with l stages. However, in the worst case if all the SEs can only be decomposed into the format as shown in Figure 9.25, it needs 3 · [(N - 1)/2 + (l - 1)] processing time and 3 · l · (N - 1)/2 + l · N comparisons in a pipelined architecture with l stages. Table 9.4 shows the computational complexity of decomposing one-dimensional SEs in different conditions. ,
284 Image Processing and Mathematical Morphology SN = [ a1 , a2 , a3 , . . ., aN ] = max{[a1 , a2 , a3 , . . ., aN–2 ] ⊕ [0, ×, × ], [×, aN–1 , aN ] N–3 } [a1 , a2 , a3 , . . ., aN–2 ] ⊕ [0, ×, × ] ⊕ [0, ×, × ], = max [0, ×, × ] ⊕ [×, aN–3 , aN–2 ] N–5 , [×, aN–1 , aN ] N–3 N–3 terms { 2 [ a1 , a2 , a3 ]⊕[0,×,×]⊕[0, ×, × ]⊕ · · · ⊕[0, ×, × ] [×, a4 , a5 ] 2 ⊕[0,×,×]⊕ · · · ⊕[0,×,×] N–5 N–1 = max terms 2 terms 2 [×, aN–3 , aN–2 ] N–5 , ⊕ [0,×,×] [×,aN–1 , aN ] N–3 {{ = max{[a 1 , a 2 , a 3 ],[×,a 4 , a 5 ] 2 , . . .,[ ×,a N–3 ,a N–2 ] N–5 ,[×,a N–1 , a N ] N–3 } FIGURE 9.25 The worst case of one-dimensional structuring element decomposition. For the two-dimensional arbitrary grayscale structuring element decomposition, the computational complexity of row- or column-based algorithm is based on the one-dimensional structuring element decomposition. The worst case in the combined row-and-column algorithm is when a two- dimensional structuring element of size n ¥ n is decomposed into n pairs of dilations of row and column vectors. Figure 9.27 shows a row-major example of decomposing a 5 ¥ 5 structuring element. In the fi rst iteration, the fi rst row is selected for the decomposition. Note that the fi rst combined pair of row and column can determine only some pixels as shown in Figure 9.27b. The symbol • indicates that the value cannot be determined in the current iteration and will be determined in the next iteration, and the symbol ¥ indicates the don’t care pixel in the current iteration. Figures 9.27c through f show the determined pixels of the SEs in the 2nd–5th pairs of the combined row and column decompositions, respectively. Sussner and Ritter [1997, 2000] indicated that the best case occurs when the rank of a matrix is 1, and the decomposition of two-dimensional SEs only needs one pair of combined row and (a) (b) N Pipeline (l stages) l structuring elements N • • • 3 3 • • • 3 3 3 • • • 3 3 • • • 3 (N – 1) / 2 FIGURE 9.26 The sequential and pipelined architectures. • • • • • • 3 N { {{ l – 1
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To my loving wife and children, to
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viii Contents 3.3 Grayscale Opening
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x Contents 7.2 Corner Detection by
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xii Contents 10.1.2.3 Additional Si
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Preface Image processing and mathem
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Preface xvii morphological fi lters
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Acknowledgments Portions of the boo
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xxii Author Dr. Shih has contribute
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1 Introduction to Mathematical Morp
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Introduction to Mathematical Morpho
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12 Image Processing and Mathematica
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4 Basic Morphological Algorithms In
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Basic Morphological Algorithms 39 N
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Basic Morphological Algorithms 41 0
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Basic Morphological Algorithms 45 i
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Basic Morphological Algorithms 49 F
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Basic Morphological Algorithms 51 s
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Basic Morphological Algorithms 53 S
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100 Image Processing and Mathematic
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7 Feature Extraction Feature extrac
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Feature Extraction 185 morphology f
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Feature Extraction 187 If the cente
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Feature Extraction 189 FIGURE 7.4 A
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Feature Extraction 191 FIGURE 7.8 U
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Feature Extraction 193 FIGURE 7.11
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Feature Extraction 195 (a) (b) (c)
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Feature Extraction 197 The original
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Feature Extraction 205 4. The outpu
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Feature Extraction 207 A question a
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Feature Extraction 209 We present a
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Feature Extraction 211 In additiona
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8 Object Representation Prior to ob
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Object Representation 215 a single
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Object Representation 217 dimension
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Object Representation 219 FIGURE 8.
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Object Representation 221 convergen
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Object Representation 223 - - - - -
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Object Representation 225 than or e
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Object Representation 229 Proof of
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400 Index Base point defi nitions,
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402 Index Decomposition (Continued)
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404 Index Erosion (Continued) sweep
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406 Index H Heijmans, Henk, 5 Hemie
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408 Index Mathematical morphology a
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410 Index O Object representation,
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412 Index Rotational erosion operat
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414 Index Sweep morphology (Continu
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