Two Image-Template Operations for Binary Image Processing. Hongchi Shi. Department of Computer Engineering and Computer Science
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1 Two Image-Template Operations for Binary Image Processing Hongchi Shi Department of Computer Engineering and Computer Science Engineering Building West, Room 331 University of Missouri - Columbia Columbia, MO Phone: (573) Fax: (573) shi@ece.missouri.edu 1
2 Two Image-Template Operations for Binary Image Processing HONGCHI SHI Department of Computer Engineering and Computer Science, University of Missouri - Columbia, Columbia, MO Abstract This paper presents two new image algebra image-template operations match and mismatch derived from the general image-template product. These image algebra operations extend the binary morphological erosion and dilation operations and can be used to express elegantly most of binary image processing algorithms in a more natural way than binary morphological operations from the image processing viewpoint. In addition, the match and mismatch operations are easy to implement eciently on SIMD bit-serial parallel computers. Keywords image algebra, mathematical morphology, binary image processing 1. Introduction Binary image processing involves manipulating binary images consisting of only 1-pixels and 0-pixels. Binary image processing techniques are employed by binary machine vision systems in the detection and recognition of objects or object defects [3]. Binary morphology is an algebraic system concerning with analysis of shapes. It plays an important role in machine vision, since shape is a prime carrier of information in machine vision [3]. Binary morphology can be used to enhance binary images by removing noise, decompose objects, extract object features, and identify objects in binary images [1, 10, 4]. Binary mathematical morphology is based on the set theory. The set of all the 1-pixels in a binary image gives a complete description of the binary image. The primary morphological operations are dilation and erosion. All other morphological operations such as opening, closing, and hit-and-miss operations are based on these two operations. Since binary mathematical morphology primarily deals with sets, one has to consider binary images as sets, perform operations on sets, and convert sets back to images, when applying mor- 2
3 phological operations on images. Although this process is not complicated, it involves three steps conceptually. Besides, it can not consider both 1-pixels and 0-pixels at the same time. For example, the hit-and-miss operation needs two structuring elements, one for the 1-pixels and the other for the 0-pixels of the object. Another mathematical theory concerning image processing and analysis, image algebra, denes images as its primary operands and deals with images directly [9, 6]. Image algebra extends mathematical morphology and it can manipulate image transformations between dierent domains and transformations between dierent value sets which are ignored in morphology-based image algebras. This paper introduces two image algebra operations and demonstrates how they are used to express binary image processing algorithms in a more natural way for binary image processing. 2. Match and Mismatch Operations Image algebra is a heterogeneous algebra concerned with image processing and analysis. In image algebra, basic objects in image processing such as images, point sets, and templates are formally dened as its operands. Most of image processing and analysis algorithms can be concisely expressed using image algebra [8]. A complete description of image algebra can be found in Ritter's book [7]. Images in image algebra are dened in terms of two other types of elementary operands: value sets and point sets. Given a point set X and a value set F, an F-valued image a on X is a function a : X! F, which is usually expressed in terms of the graph of a by a = f(x; a(x)) : x 2 Xg; where a(x) 2 F. An element (x; a(x)) of a is a pixel. The set of all F-valued images on X is denoted by F X. In image algebra, the denition of a template unies and generalizes the usual concepts of templates, masks, windows, structuring elements, and neighborhood functions into one general mathematical entity. Templates are special types of images. An F-valued template from Y to X is an element of (F X ) Y. If t 2 (F X ) Y, then for notational convenience we dene t y t(y) in order to denote the image t y for each y 2 Y. The pixel values t y (x) of the image t y = f(x; t y (x)) : x 2 Xg 3
4 are the weights of the template t at the target point y. Let X R n. A template t 2 (F X ) X is a translation invariant template if and only if for x; y 2 X with x + z; y + z 2 X, where z 2 R n, we have t y (x) = t y+z (x + z). Translation invariant templates can be dened pictorially. Figure 1 shows a translation invariant template. Image algebra denes many operations between images and between images and templates. In terms of image processing, image-template operations that combine images and templates are the most powerful tools of image algebra. They combine images and templates by using appropriate binary operations. They may be used for transformations between dierent domains and transformations between dierent value sets. Suppose that F 1, F 2, and F are value sets. The operation : F 1 F 2! F is a binary operation and the operation : F F! F is an associative and commutative binary operation on F. If a 2 F X 1 and t 2 (F X 2 )Y, then the generalized product of a with t is the binary operation : F X (F X 1 2 )Y! F Y dened by a t = f(y; b(y)) : b(y) =? x2xa(x) t y (x); y 2 Yg: We may derive dierent image-template operations by substituting appropriate operations for and in the denition of the generalized image-template product. Ritter has dened the most commonly used image-template operations in image processing such as +, _2, ^2, _, and ^ [7]. Here, we derive two image-template operations mismatch and match for binary image processing. First, we dene two logic operations ~+ and ~+ 0. The operation ~+ is dened as the exclusive-or operation, i.e., a ~+b = 1 if and only if a and b are dierent. The operation ~+ 0 is dened as the complement of the exclusive-or operation, i.e., a ~+ 0 b = a ~+b. The mismatch operation ~_2 is obtained by substituting the logic or _ and the operation ~+ for and. Specically, a ~_2 t = f(y; b(y)) : b(y) = _ x2x a(x) ~+t y (x); y 2 Yg: The match operation ~^2 is obtained by substituting the logic and ^ and the operation ~+ 0 for and. Specically, a ~^2 t = f(y; b(y)) : b(y) = ^ x2x a(x) ~+ 0 t y (x); y 2 Yg: 4
5 Most of binary image processing algorithms require only translation invariant templates. Thus, a template for binary image processing usually can be dened by a conguration of the neighborhood of each target pixel. For example, the template t in Figure 1 species the conguration of the 3 3 neighborhood of each target pixel. Figure 1 goes here. For the mismatch operation, each target pixel in the result image obtains 1 if and only if the conguration of its neighborhood does not match the conguration given by the template. For the match operation, each target pixel gets 1 if and only if the conguration of its neighborhood matches the conguration given by the template. The match and mismatch operations are bit-serial operations. They usually use information from a small neighborhood of each pixel to check if it matches/mismatches the conguration specied by the template. They are perfectly suited for implementation on bit-serial SIMD meshconnected computers such as Lockheed-Martin's CISP computer [11]. 3. Binary Image Processing Using Match and Mismatch Haralick and Shapiro present many applications of binary morphological operations such as erosion, dilation, and hit-and-miss to binary image processing [3]. The two basic binary morphological operations erosion and dilation can be expressed using match and mismatch, respectively. The operations given by match and mismatch are more natural to image processing researchers. The useful morphological operation hit-and-miss can be expressed using match with a single template specifying the foreground (1-pixels) and background (0-pixels) of the object to be matched Erosion and Dilation In morphology, the erosion of a set A E N by another set B E N, denoted by A B, is dened by A B = fy 2 E N : y + x B 2 A; 8x B 2 Bg: 5
6 This can also be expressed as follows: A B = fy 2 E N : B y Ag; where B y is the set of elements of B translated by y. The dilation of a set A E N by another set B E N, denoted by A B, is dened by A B = fz 2 E N : y = x A + x B ; 9x A 2 A and 9x B 2 Bg: This can also be represented as follows: A B = fy 2 E N : x A 2 By ; 9x A 2 Ag; where B = fx :?x 2 Bg. For binary image processing, the set A corresponds to a source binary image a with a(x) = 1 if and only if x 2 A, while the set B corresponds to a template t. To achieve the erosion eect, the template is dened by t y (x) = 1 if x 2 B y. To achieve the dilation eect, the template t is dened by t y (x) = 0 if x 2 B y. For example, if the structuring element B is as shown in Figure 2(a), the corresponding erosion template is t 1 shown in Figure 2(b) and the dilation template is t 2 shown in Figure 2(c). Figure 2 goes here. The match operation implements exactly the morphological erosion operation. Let c = a ~^2 t. By the denition of the match operation, c(y) = V x2by a(x) ~+ 0 t y (x), which means c(y) = 1 if and only if all the elements in B y are in A. The mismatch operation implements exactly the morphological dilation operation. Let c = a ~_2 t. By the denition of the mismatch operation, c(y) = W x2 By a(x) ~+t y (x), which means c(y) = 1 if and only if some element of A is in B y. When we use the match and mismatch operations for binary image processing, we work on images directly and design templates specifying what kind of conguration of the neighborhood to match or to avoid, which is more natural. 6
7 3.2. Hit-and-Miss The structuring elements in morphological operations can only specify one kind of pixels either 1-pixels or 0-pixels. The match and mismatch operations extend the morphological erosion and dilation operations by allowing the neighborhood to contain 1-pixels and 0-pixels, making some image processing algorithms more natural and concise. The hit-and-miss transformation is a useful operation for selecting pixels that satisfy certain geometric properties [3]. It is represented in morphology using two structuring elements, one for the object to be hit and the other for the object to be missed. Let J and K be two structuring elements that satisfy T J K = ;. The hit-and-miss transformation of set A by (J; K), denoted by A (J; K), is dened by \ A (J; K) = (A J) (A c K); where A c is the complement of A. When applied to binary image processing, the hit-and-miss operation gives a pixel value 1 if and only if the pixel's neighborhood of 1's specied by J matches J and its neighborhood of 0's specied by K matches K. Let a be the binary image corresponding to set A. Dene a template t by t y (x) = 1 if x 2 J y and t y (x) = 0 if x 2 K y. The match operation a ~^2 t performs the hit-and-miss operation with one template specifying the conguration of the neighborhood to be matched. We give three examples to show the match operation gives more natural expression of some binary image processing algorithms. Finding 8-isolated pixels An 8-isolated pixel is a 1-pixel with its eight 8-neighbors all 0 as shown in Figure 3. Figure 3 goes here. To identify all 8-isolated pixels of an image using the hit-and-miss transform, we need to design two structuring elements J = f(0; 0)g shown in Figure 4(a) and K = f(0; 1); (0;?1); (1; 0); (?1; 0)g shown in Figure 4(b). Using the match operation, we just need to dene a template as shown in Figure 4(c). The conguration given by the template directly reects the denition of 8-isolated 7
8 pixels. Figure 4 goes here. Identifying upper right-hand corner pixels An upper right-hand corner pixel can be dened as a 1-pixel whose south and west neighbors are 1-pixels and whose north, northeast, and east neighbors are 0-pixels as shown in Figure 5. Figure 5 goes here. To identify upper right-hand corner pixels using hit-and-miss, we have to dene two structuring elements J and K as shown in Figure 6. The structuring element J species the object to be hit and K species the object that has to be missed. Using the match operation, we need to derive only one template directly from the denition of upper right-hand corner pixels. The template is dened as shown in Figure 6(c). Figure 6 goes here. Pattern matching In binary image processing, a pattern to be matched in a binary image can be dened as a smaller binary image. Figure 7 gives a pattern of a diamond with side length 3. A pixel at x in the source image is said to match the pattern exactly if the pattern image translated by x matches the subimage at x in the source image. 8
9 Figure 7 goes here. Using the morphological hit-and-miss operation for pattern matching, we have to design two structuring elements J and K, one for the 1-pixels and the other for the 0-pixels of the pattern. To perform pattern matching using the match operation, we only need a template t which can be derived directly from the pattern to be matched. The structuring elements and the template for matching the pattern dened in Figure 7 are shown in Figure 8. Figure 8 goes here. Pattern matching with tolerance can be done by leaving some 1's out in the template conguration. For example, to match diamond-like patterns with side lengths from 2 to 3 using the match operation, we can dene a template as shown in Figure 9. Figure 9 goes here Computing Binary Image Topological Properties We further demonstrate the applicability of the match and mismatch operations by describing two more binary image processing algorithms using these operations. The algorithms given here compute some topological properties of binary images. We only consider 8-connected components. That is, we use 8-connectivity for 1-pixels and 4-connectivity for 0-pixels. Shrinking and Counting Components Levialdi gives an algorithm that counts the components in a binary image by shrinking the components into isolated pixels [5]. The basic idea is to shrink each component to an isolated pixel 9
10 at one corner of its bounding rectangle and then count the isolated pixels. The bounding rectangle of a component is the smallest upright rectangle containing all the 1-pixels of the component. The shrinking process preserves the component connectivity. Let a 2 f0; 1g X denote the source image, where X is an n n point set. The shrink operator ' that shrinks components toward the upper left corners of their bounding rectangles can be dened in terms of the congurations of the 2 2 neighborhood as shown in Figure 10. Figure 10 goes here. The operator ' assigns a value to each pixel as follows: If the pixel's neighborhood has the conguration shown in Figure 11(a), it is assigned 0; If its neighborhood has the conguration shown in Figure 11(b), it is assigned 1; Otherwise, it keeps its old value. Figure 11 goes here. Dening two templates t 1 and t 2 shown in Figure 12, the operator ' can be expressed using the match and mismatch operations as follows. Let a = '(a), the image resulted from applying ' to a. Then, a = a ^ (a ~_2 t 1 ) _ (a ~^2 t 2 ): To count components in a binary image, we apply the shrink operator repeatedly. After each iteration, we use a match operation with the template t dened in Figure 4. Let a 0 be the source binary image and c 0 = P a 0 ~^2 t, the number of isolated pixels in the source image. Dene a k+1 = a k ^ (a k ~_2 t 1 ) _ (a k ~^2 t 2 ) 10
11 and c k+1 = c k + X a k+1 ~^2 t: The number of components in a 0 is given by c K, where K is the smallest integer for which a K = 0. Figure 12 goes here. Computing Binary Image Euler Number Euler number is a topological descriptor for binary images. It is useful in image processing because the topological counts in an image are of intrinsic interest in characterizing and recognizing the image. The Euler number of a binary image is dened to be the number of connected components minus the number of holes inside the connected components. Gray studied computation of Euler numbers using local information [2]. Given a direction, let X denote a -facing convexity and V denote a -facing concavity as shown in Figure 13. Gray proved that E = #(X ) + #(V ); where #(X ) is the number of -facing convexities and #(V ) is the number of -facing concavities. Choosing to point to the northwest direction, we have an X only if we have the conguration shown in Figure 14(a). A V can only arise from the conguration shown in Figure 14(b). Figure 13 goes here. Figure 14 goes here. 11
12 Thus, we can design two templates t 1 and t 2 as shown in Figure 15 and compute the Euler number of a binary image a as follows: E = X a ~^2 t 1? X a ~^2 t 2 : Figure 15 goes here. 4. Conclusion We have derived two image algebra image-template operations match and mismatch. They extend the binary morphological erosion and dilation operations. We have demonstrated how to design templates directly from binary image processing problems. When applied to binary image processing, the match and mismatch operations give more natural algorithms. Acknowledgements The author wishes to thank Dr. Gerhard Ritter and Dr. Joseph Wilson for their advice on this work. The author also wishes to thank Dr. Patrick Coeld of Wright Laboratory, Eglin AFB, for his continued support of this research. References [1] T. R. Crimmins and W. R. Brown. Image algebra and automatic shape recognition. IEEE Transactions on Aerospace and Electronic Systems, 21:60{69, [2] S. B. Gray. Local properties of binary images in two dimensions. IEEE Transactions on Computers, 20(5):551{561, May
13 [3] R. M. Haralick and L. G. Shapiro. Computer and Robot Vision, volume 1. Addison-Wesley, Reading, MA, [4] R. M. Haralick, S. R. Sternberg, and X. Zhuang. Image analysis using mathematical morphology. IEEE Transactions on Pattern Analysis and Machine Intelligence, 9:523{550, [5] S. Levialdi. On shrinking binary picture patterns. Communications of the ACM, 15:7{10, [6] G. X. Ritter. Recent developments in image algebra. In P. Hawkes, editor, Advances in Electronics and Electron Physics, 80, pages 243{308. Academic Press, New York, NY, [7] G. X. Ritter. Image algebra, in preparation. [8] G. X. Ritter and J. N. Wilson. Handbook of Computer Vision Algorithms in Image Algebra. CRC Press, [9] G. X. Ritter, J. N. Wilson, and J. L. Davidson. Image algebra: An overview. Computer Vision, Graphics, and Image Processing, 49(3):297{331, March [10] J. Serra. An introduction to mathematical morphology. Computer Vision, Graphics, and Image Processing, 35:283{305, [11] M. S. Tomassi and R. D. Jackson. An evolving SIMD architecture approach for a changing image processing environment. DSP & Multimedia Technology, pages 1{7, October
14 t = Figure 1: Template t specifying a 3 3 neighborhood conguration 14
15 t = t 2 = (a) (b) (c) Figure 2: A structuring element and its corresponding templates 15
16 Figure 3: An 8-isolated pixel in the center 16
17 J K t = (a) (b) (c) Figure 4: Structuring elements and template for nding 8-isolated pixels 17
18 Figure 5: An upper right-hand corner pixel in the center 18
19 J K 0 0 t = (a) (b) (c) Figure 6: Structuring elements and template for identifying upper right-hand corner pixels 19
20 Figure 7: A pattern image 20
21 J K (a) (b) t = (c) Figure 8: Structuring elements and template for pattern matching 21
22 t = Figure 9: Template for pattern matching with tolerance 22
23 Figure 10: Neighborhood for the Levialdi shrinking operator ' 23
24 (a) (b) Figure 11: Neighborhood congurations for the Levialdi shrinking operator ' 24
25 t 1 = 0 t 2 = Figure 12: Templates for the Levialdi shrinking operator ' 25
26 V θ X V θ θ V θ X θ X θ θ Figure 13: -facing convexities and concavities 26
27 (a) * (b) Figure 14: Congurations for convexity and concavity, where is 0 or 1 27
28 t 1 = 1 0 t 2 = Figure 15: Templates derived from the congurations for convexity and concavity 28
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