2. Course 3 Binary Image

3. Binary Images have only two gray levels: “1” and “0”, i.e., black / white. —— save memory —— fast processing —— many features of an intensity image appear in binary form, e.g. edge, contour, … —— human can understand binary line drawings.

4. 1. Neighbors/path/connectivity (1) Neighbors of a pixel a. 4-neighbors: the upper, lower, left and right neighbor pixels of a pixel. b. 8-neighbors: the 4-neighbors pixels plus the diagonal neighbors. 4-neighbor 8-neighbor [i-1,j] [i,j-1][i,j][i,j+1] [i+1,j] [i-1,j-1][i-1,j][i-1,j+1] [i,j-1][i,j][i,j+1] [i+1,j-1][i-1,j][i+1,j+1]

5. (2) Path A path form pixel to the pixel is a sequence of pixel indices: where the pixel is a neighbor of pixel for all. ——4 pathif 4-neighbor is used ——8 pathif 8-neighbor is used

6. (3) Connectivity: A pixel is said to be connected to if there is a path from to consisting of pixels of. Connected component: A set of pixels in which each pixel is connected to all other pixels of the set.                 4-connected 8-connected

7. 2. Geometric attributes of a binary image/component: (1) size (area): for all B[.,.] = 1 pixels. (2) position: the position of a binary image component is defined by its mass center.

8. (3)Orientation: The orientation of binary image component is defined by a straight line that the distance sum from all the pixels to the line is minimized,i.e., where is the distance from pixel [ i, j ]to the line. Let the line in polar coordinates be: Then: So:

9. Set, and We get: where is the center of the component, and where

10. Note: may minimize or maximize. The has two values; the corresponding lines are called maximum axis and minimum axis of the image component. Maximum axis Minimum axis

11. (4) Foreground / Background : Foreground: the set f all “1” pixels in an image. Background: the set of all connected components of that have points on the border of the image. Hole: the connected component in but it does not have border points. Note: foreground and background must use different connectivities. Hole or not? 010 101 010

12. (1) Boundary: boundary of S is a set of pixels in S that has 4-neighbours in, denoted by. (2) Interior (pixels in S but not in ) Surrounds: if any 4-path from any point of S to image border must intersect region T, we say region T surrounds region S.

13. (7) Perimeter Def.1: The total length of lines that separate pixels of S from. Def.2: The number of boundary pixels. (8) Compactness Where P— perimeter, A— area. The smallest value of c is. If c =, S is a circular region. When c is larger, S is elongated when, S is a line segment.

14. 3. Binary Processing: (1) Thresholding: Thresholding converts an intensity image into a binary image, partitioning the original image to regions. Let image has gray level ranged in [0,L], T be threshold, 0<T<L Then, If you are interested in image information of certain range of gray level, you can choose double thresholds

15. Then, Original image Image by threshold double threshold (L=64) T=48 T 1 = 2, T 2 = 48

16. (2) Run-length encode: —— Run-length code is a compact way to represent a binary image. —— Run-length encoding is a row-based operation. Method 1: only encode “1” pixels. Using start position and length of runs of “1” to represent the runs. Method 2: encode “1” runs and “0” runs alternatively. Using only one number to indicate the length of the run. If an image row starts with a “0” run, put a “0” as the header of the code of the row, otherwise the header will not appear.

17. 1110001100011110110111 0000111111111111101111 1110000000000000111111 Method1: (1,3), (7,2), (12,4), (17,2), (20,3) (5,13), (19,4) (1,3), (17,6) Method 2: 3, 3, 2, 3, 4, 1, 2, 1, 3; 0, 4,13,1, 4; 3, 13, 6

18. (3) Boundary-following: Track boundary pixels of a region in a particular order (chain sequence)

19. Algorithm: a) Use systematic scan (from left to right, and from top to bottom) to find the first pixels as a starting pixel. b) Set c=s, and b=left neighbor of c,. c) From pixel b, in clockwise order check the 8-neighbors of c, ; find the first such that. d) Set e) Repeat step c) and d) until c=s.

20. (4) Projections Projecting a binary image onto a line (usually horizontally or vertically) may provide partition information of the image. It is often used in OCR for character separation. Horizontal projection: Vertical projection: For a well alined text binary image, the projection has comb shape, image can be easily partitioned.

22. (5) Clustering Clustering, or component labeling, is a common operation in computer vision to find connected components. Each connected component may represent an object or a part of an object.

23. Algorithm (Sequential) a) Scan the image from left to right, top to bottom. b) If find a pixel of value “1”, then check its left, upper-left, upper and upper-right neighbor pixels: i) If one or more neighbor pixels having the same label, then copy the label. ii) If neighbor pixels have different labels, then copy the label anyone of the neighbor pixel. Add labels in equivalence table as equivalent labels. iii) Otherwise, assign a new label to this pixel and enter this label in equivalence table. c) If there are more pixels to consider, then go to step b).

24. D) Find the lowest label for each equivalent set in the equivalence table. E) Remember the equivalence table to eliminate the gaps of labels. F) Rescan the image; replace each label by the new label of equivalence table. (6) Distance transformation: a) Distance measurements: Euclidean distance: City-block distance:

25. Chessboard distance: a) Distance transformation: Distance transformation computes the minimum distance from a pixel of S to the background, for all pixels in S. Algorithm: Iteratively visit image m times: where N —— 4-neighbors of [ i, j];

26. 11111 11111 11111 11111 11111 11111 12221 12221 12221 11111 11111 12221 12321 12221 11111 1 st iteration 2 nd iteration 3 rd iteration Medial axis: The set of pixels in S with the distance from the pixels to background locally maximized is called medial axis of region S, or skeleton, or symmetric axis.

28. (7) Thinning: —— thinning operation reduces a binary image S to the lines that approximate their central lines —— thinning is an iterative operation. —— thinning operation usually applied to elongated shapes, such as line draws or characters. Thinning operation must satisfy that: —— not break connectivity of a pattern. —— thinned result is 8-connected. —— thinned result approximates the medial lines. —— extraneous spurs keep minimized.

29. (a) Algorithm 1 (iterative method): Definition: connectivity number of a pixel is the number of value changes (1 or 0) countered by clockwise visiting its 8-neighbors n 1, n 2,…,n 8. No=1 No=2 No=3 No=4 No=0 101 111 111 101 111 101 101 010 010 101 010 101 000 010 000

30. The algorithm is a template-based method: i) Find a pixel [i, j] where the pixel in image S matches template M 1 ii) If center pixel is not an endpoint, and has connectivity number = 1, mark this pixel for later detection. x0x x1x x1x xxx 011 xxx x1x x1x x0x xxx 110 xxx M 1 M 2 M 3 M 4

31. iii) Repeat step i) and step ii) for all pixels matching the templates M 1 iv) Repeat steps 1-3 for templates M 2, M 3, M 4. v) If any pixels have marked for detection, change the pixel values to “0” vi) If any pixels were changed value in step 5, repeat entire process from step i); otherwise stop.

33. Algorithm 2 (line following) i) Set a window across the line draw. ii) Adjust the window size and position, such that the distance from window boundaries to the draw equal a preset width (e.g.2 pixel wide). iii) Set the center of the window be a skeleton point. iv) Move window forward. If no branch junction is met, go to step ii). v) If a branch junction is met, use BFS method, move window to one of the branch, go to step ii).

34. vi) If window moves to the end of a line or branch, and the queque of BFS is empty, stop. —— One pass method, fast —— Relink problem at branches.