Advances in Digital Imaging and Computer Vision

Transcription

1 Advances in Digital Imaging and Computer Vision Lecture and Lab 8 th lecture Κώστας Μαριάς Αναπληρωτής Καθηγητής Επεξεργασίας Εικόνας 1

2 Τοπολογία Εικόνας Image Topology 2

3 Basic Βασικές σχέσεις ανάμεσα σε pixels Neighbors of pixel p: 4-neighbors Ν 4 (p) The set of the vertical/horizontal neighbors: (x + 1, y), (x - 1, y), (x, y + 1), (x, y - 1) (x - 1, y) (x, y - 1) (x,y) (x, y + 1) (x + 1, y) columns rows 3

4 Basic Βασικές σχέσεις ανάμεσα σε pixels Neighbors of pixel p: 4- diagonal neighbors Ν D (p) It is the set of the four diagonals: (x + 1, y + 1), (x + 1, y - 1), (x - 1, y + 1), (x - 1, y - 1) (x - 1, y - 1) (x - 1, y + 1) (x,y) columns (x + 1, y - 1) (x + 1, y + 1) rows 4

5 Basic Βασικές σχέσεις ανάμεσα σε pixels Neighbors of pixel p: 8-neighbours N 8 (p)=n 4 (p)+ Ν D (p) columns (x - 1, y - 1) (x - 1, y) (x - 1, y + 1) (x, y - 1) (x,y) (x, y + 1) (x + 1, y - 1) (x + 1, y) (x + 1, y + 1) rows In all cases if (x,y) is in the edge of the image the neighbors might fall outside the image grid!!! 5

6 Αποστάσεις For pixels p, q, and z, with coordinates(x, y), (s, t), and (v, w), D is a distance metric if: D(p, q) 0, (D(p, q) = 0 if p = q), D(p, q) = D(q, p) D(p, z) D(p, q) + D(q, z). 6

7 Euclidian distance: Αποστάσεις De(p, q) = [(x - s) 2 + (y - t) 2 ] 1/2 D 4 distance: D 4 (p, q) = Ix s I + I y t I p 1 2 γραμμές στήλες p:(x, y) stable/center q: (s, t) moving D 4 =1: 4-neighbours of p 7

8 D 8 distance: Αποστάσεις D 8 (p, q) = max{ Ix s I, I y t I } στήλες p p:(x, y) fixed q: (s, t) moving D 8 =1: 8-neighbours of p γραμμές 8

9 Adjacency, Connectivity, Regions, and Boundaries Adjacency, Connectivity, Regions, and Boundaries Γειτνίαση, Συνδεσιμότητα, Περιοχές, και Όρια V is the set of image intensities values which we consider for defining neighborhood In a binary image V = {1} οπότε μιλάμε για γειτνίαση pixels που έχουν τιμή 1. In an 8bit image V ={0..255}. Basic 9

10 Adjacency, Connectivity, Regions, and Boundaries Neighborhood (a) 4-neighbourhood. 2 pixels p and q with values from V are 4-neighbors if q belongs to the set Ν 4 (p) (b) 8-neighbourhood. 2 pixels p and q with values from V are 8-neighbors if q belongs to the set Ν 8 (p) Basic 10

11 Adjacency, Connectivity, Regions, and Boundaries A digital path from pixel p (x,y) to q(s,t) is defined as the sequence (x 0, y 0 ), (x 1, y 1 ),, (x n, y n ) if: (x 0, y 0 ) = (x, y) and (x n, y n ) = (s, t) Each pixel pair(x i, y i ) και (x i-1, y i-1 ) are neighbors for i=1..n. The length of the path is n If (x 0, y 0 ) = (x n, y n ) the digital path is closed We can define 4-,8-, or m-paths according to how we define neighborhood. Basic 11

12 Adjacency, Connectivity, Regions, and Boundaries We can define 4-,8-, or m-paths according to how we define neighborhood. Basic 8-path m-path 12

13 Adjacency, Connectivity, Regions, and Boundaries Let R be a subset of the image pixels. R is an image region if it is a connected set Two regions are connected if their union forms a connected set. Else they are disjoint. All the above can be defined for 4-,8- region neighborhoods. Basic 13

14 Adjacency, Connectivity, Regions, and Boundaries In the following example the image regions are disjoint considering 4-neighbourhood. However if we use 8-neibourhood (right) the regions are connected since their union forms a connected set. 14

15 Adjacency, Connectivity, Regions, and Boundaries Boundary is also defined on a 4-,8-, m- neighbourhood basis. Usually 8-neighbourhood is used This pixel doesn t belong to the boundary if 4-neighbourhood is used as foreground! 15

16 Adjacency, Connectivity, Regions, and Boundaries The previous algorithm defines the inner boundary but with reverse hypothesis we can also compute the outer boundary. This is important for algorithms estimating the outer boundary since it is always a closed path! The inner boundary of the image region with 1s is THE SAME region but it isn t a closed path! The outer boundary is indeed a closed path!!! 16

17 Σήμανση συνδεδεμένων περιοχών Region Labeling Labeling connected components / regions is an algorithmic application of graph theory, where the subsets of linked elements are uniquely labeled with a label. Region Labelling should not be confused with image segmentation. By No machine-readable author provided. Wereon assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., GFDL,

18 Connected components Labelling Labelling is used in artificial sight to detect linked areas in binary digital images, although it can also be done in color images and higher dimensional data. Over the last two decades many new approaches have been proposed, but virtually none of these algorithms have been compared to others on the same basis. Recently, YACCLAB is an example of a C ++ open source framework that collects, runs, and tests labelling algorithms. 18

19 Αλγόριθμος για Σήμανση συνδεδεμένων περιοχών We will present an algorithm for 4- component labeling in binary images. In the algorithm, we define 4- neighboring only 90 up (u) and left (k) for each pixel (p), as shown in the adjacent diagram. Pixels with ( ) are the foreground (= 1 in binary images) of the image and the spaces (= 0 in binary images), the background. k u p 19

20 Αλγόριθμος για Σήμανση συνδεδεμένων περιοχών 1. Scan the image from the upper left pixel (1,1). Suppose we have reached p: If it is in the background, we move on to the next one. Otherwise we look at u, k pixel values. If both are in the background, we put a new label on p. If only one of u, k belongs to the foreground then p 'inherits' its label. If both u, k are in the foreground and have the same label, then p inherits this label. If both u, k are in the foreground and have different labels then p 'inherits' one of the two and the algorithm notes that the labels of u, k are equivalent as they are common 4-neighbors of p. 2. At the end of scanning, all pixels in the background are sorted, but several labels are labeled as equivalent. We group the equivalent labels and assign new labels to each group. 3. Make a second pass in the image by changing the foreground pixel tag to that assigned to its equivalent label group in the previous step. k u p 20

21 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών We will present an algorithm for labeling 4-component binary images. We begin to explain the algorithm with an example in the adjacent image. Pixels ( ) are the foreground of the image and spaces ->background. 21

22 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών 1. Scan the image from the pixel (1,1), that is to the top left. 2. If it is in the background (blank) we proceed to the next pixel (1,2). The 4-neighbors (top-left) are either background or non-existent, so we assign tag 1. We move on to the second line and for the same reason (2,1) we get the label 2. The pixel (2,2) has two foreground 4- neighbors with different labels. We label it 1 and note that labels 1, 2 are equivalent

23 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών The next foreground pixel on the second line is (2, 4) which has no (upstream) 4-neighbors -> we assign a new class 3. On the third line, the first foreground pixel is (3,3), which for the same reason acquires a new label at 4. The adjacent pixel (3, 4) has two foreground 4- neighbors with different labels. We give him tag 3 and note that labels 3, 4 are equivalent

24 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών The next foreground pixel on the fourth line is (4,2) which does not have (upstream) 4 neighbors -> we assign a new class 5. The adjacent pixel (4,3) has two foreground 4- neighbors with different labels. We give him the label 4 and note that the labels 4, 5 are equivalent. By completing the first step, we have 2 equivalent sets of tags: {1,2} and {3,4,5} We give the new label '1' to the first tag group {1,2} and '2' in the second {3,4,5}

25 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών The last step is to go over all the pixels tagged with 1 or 2 (first set of tags {1,2}) and assign them a new tag '1' and tagged with 3 or 4 or 5 (second group {3, 4,5}), the new label '2' This way the algorithm comes to an end. 25

26 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών Extending the algorithm for labeling 8- component binary images is easy. In the algorithm we will define 8- neighboring only a) 90 up and left (k, u) and b) diagonal top left and diagonal top right (d, e) of the pixel (p) as shown in the adjacent diagram. d k u p e Pixels ( ) are the foreground (= 1 in binary images) of the image and the spaces (= 0 in binary images) represent the background. 26

27 Αλγόριθμος για Χαρακτηρισμό συνδεδεμένων περιοχών 1. Scan the image from the upper left pixel (1,1). Suppose we have reached p: If it is in the background, we move on to the next one. Otherwise we look at what are u, k, d, e. If everything is in the background we put a new category / tag on p. d u e If only one of them is in the foreground, then p 'inherits' his label. k p If two or more are in the foreground, then p is inheriting one of them, and the algorithm notes that the labels of u, k, d, e are equivalent since they are common 8-neighbors of p. 2. At the end of scanning, all pixels in the background are sorted, but several labels are labeled as equivalent. We group the equivalent labels and assign new labels to each group. 3. We make a second pass in the image by changing the foreground pixel tag to that assigned to its equivalent label group in the previous step. 27

28 Homework Implement a Matlab/Octave Algorithm for 4 and 8- neighbor labelling (as defined in the lesson). Your input will have to be a binary image. 28

29 References An Introduction to Digital Image Processing with MATLAB by Alasdair McAndrew Digital Image Processing, Rafael C. Gonzalez & Richard E. Woods, Addison-Wesley,

30 Thank you for your attention! 30