UPEM Master 2 Informatique SIS. Digital Geometry. Topic 2: Digital topology: object boundaries and curves/surfaces. Yukiko Kenmochi.

Transcription

1 UPEM Master 2 Informatique SIS Digital Geometry Topic 2: Digital topology: object boundaries and curves/surfaces Yukiko Kenmochi October 5, 2016 Digital Geometry : Topic 2 1/34

2 Opening Representations of discrete shapes grid point set graph complex Digital Geometry : Topic 2 2/34

3 Opening Representations of discrete shapes grid point set graph (grid points + adjacent relations) complex Digital Geometry : Topic 2 2/34

4 Opening Representations of discrete shapes grid point set graph (grid points + adjacent relations) complex (grid cells + neighboring relations) Digital Geometry : Topic 2 2/34

5 Point set representation Object boundary in the Euclidean space For A R n, the set of interior points is defined by Int(A) = {x A : r R +, U r (x) A} where U r (x) = {y R n : x y < r}. Digital Geometry : Topic 2 3/34

6 Point set representation Object boundary in the Euclidean space For A R n, the set of interior points is defined by Int(A) = {x A : r R +, U r (x) A} where U r (x) = {y R n : x y < r}. The set of border points is: Br(A) = A \ Int(A). Digital Geometry : Topic 2 3/34

7 Point set representation Object boundary in the Euclidean space For A R n, the set of interior points is defined by Int(A) = {x A : r R +, U r (x) A} where U r (x) = {y R n : x y < r}. The set of border points is: Br(A) = A \ Int(A). Then we obtain the set of boundary points such that Fr(A) = Br(A) Br(A) = Fr(A). Digital Geometry : Topic 2 3/34

8 Point set representation Object boundary in the nd discrete space For A Z n, the set of m-interior points is defined by Int m (A) = {x A : N m (x) A} where N m (x) = {y Z n : d m (x, y) 1} where m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3. A Digital Geometry : Topic 2 4/34

9 Point set representation Object boundary in the nd discrete space For A Z n, the set of m-interior points is defined by where Int m (A) = {x A : N m (x) A} N m (x) = {y Z n : d m (x, y) 1} where m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3. The set of m-boundary points is: where Fr(A) = Br m (A) Br m (A) Br m (A) = A \ Int m (A) Br m (A) = A \ Int m (A) m-interior border, m-exterior border. A Br 4 (A) Br 4 (A) Digital Geometry : Topic 2 4/34

10 Point set representation Neighborhoods in the 2D discrete space Definition (m-neighborhood) The m-neighborhood of a grid point x Z 2 is defined by: N m (x) = {y Z 2 : d m (x, y) 1} where d m (x, y) = x y p for m = 4, 8 if p = 1, respectively. ( ) 1 d p Norm on a d-dimensional vector space: x p = x i p i=1 (Manhattan norm for p = 1, Euclidean norm for p = 2, Maximum norm for p = ) Digital Geometry : Topic 2 5/34

11 Point set representation Neighborhoods in the 3D discrete space Definition (m-neighborhood) The m-neighborhood of a grid point x Z 3 is defined by: for m = 6, 18, 26 where N m (x) = {y Z 3 : d m (x, y) 1} d 6(x, y) = x y 1, d 26(x, y) = x y, { } d6(x, y) d 18(x, y) = max d 26(x, y),. 2 Digital Geometry : Topic 2 6/34

12 Point set representation Object boundary in the nd discrete space The set of m-boundary points is: Fr(A) = Br m (A) Br m (A) where Br m (A) = A \ Int m (A) m-interior border, Br m (A) = A \ Int m (A) m-exterior border. In the discrete space, a set A and its complement A do not have the common boundary. The boundary of A consists of elements in A, and that of A consists of elements in A. (Clifford, 1956) A Br 4 (A) Br 4 (A) Digital Geometry : Topic 2 7/34

13 Point set representation Object boundary in the nd discrete space The set of m-boundary points is: Fr(A) = Br m (A) Br m (A) where Br m (A) = A \ Int m (A) m-interior border, Br m (A) = A \ Int m (A) m-exterior border. In the discrete space, a set A and its complement A do not have the common boundary. The boundary of A consists of elements in A, and that of A consists of elements in A. (Clifford, 1956) Alternative definition of m-border points: Br m (A) = {x A : N m (x) A }. A Br 4 (A) Br 4 (A) Digital Geometry : Topic 2 7/34

14 Graph representation Adjacency graph Definition (m-adjancency) If a grid point x is m-neighboring from another distinct grid point y, x and y are m-adjacent, denoted by x A m (y) and y A m (x). Digital Geometry : Topic 2 8/34

15 Graph representation Adjacency graph Definition (m-adjancency) If a grid point x is m-neighboring from another distinct grid point y, x and y are m-adjacent, denoted by x A m (y) and y A m (x). Definition (Adjacency graph (Rosenfeld 1970)) For a given grid point set X Z n, the adjacency graph is defined by G = (X, E m ) where E m = {(x, y) X X : y A m (x)}. Digital Geometry : Topic 2 8/34

16 Graph representation Path Definition (m-path) Let X be a set of grid points. An m-path in X joining two points p and q of X is a sequence π = (p 0,..., p n ) of points in X such that p 0 = p, p n = q and p i A m (p i 1 ) for i = 1,..., n. (Klette, Rosenfeld, 2003) Digital Geometry : Topic 2 9/34

17 Graph representation Path Definition (m-path) Let X be a set of grid points. An m-path in X joining two points p and q of X is a sequence π = (p 0,..., p n ) of points in X such that p 0 = p, p n = q and p i A m (p i 1 ) for i = 1,..., n. (Klette, Rosenfeld, 2003) In general, m = 4, 8 for 2D and m = 6, 18, 26 for 3D. Digital Geometry : Topic 2 9/34

18 Graph representation Discrete object (connected component) Definition (m-object) A set X of grid points is an m-object if there exists an m-path in X for every pair p and q of X. Digital Geometry : Topic 2 10/34

19 Graph representation Discrete object (connected component) Definition (m-object) A set X of grid points is an m-object if there exists an m-path in X for every pair p and q of X. In other words, an m-object is a connected component of a graph G = (X, E m ). Digital Geometry : Topic 2 10/34

20 Graph representation Connected component labeling (of a graph) Algorithm (Connected components) Input: Graph G, starting vertex s Put s in the queue (or stack) L. while L do pull a vertex t from L. Label all the neighbors of t that are not labelled and put them in L. (Hopcropft and Tarjan, 1973) Digital Geometry : Topic 2 11/34

21 Graph representation Connected component labeling (of a graph) Algorithm (Connected components) Input: Graph G, starting vertex s Put s in the queue (or stack) L. while L do pull a vertex t from L. Label all the neighbors of t that are not labelled and put them in L. It allows to calculate the connected components of a graph in linear time. (Hopcropft and Tarjan, 1973) Digital Geometry : Topic 2 11/34

22 Graph representation Connected component labeling (of a graph) Algorithm (Connected components) Input: Graph G, starting vertex s Put s in the queue (or stack) L. while L do pull a vertex t from L. Label all the neighbors of t that are not labelled and put them in L. It allows to calculate the connected components of a graph in linear time. breadth-first search (Hopcropft and Tarjan, 1973) Digital Geometry : Topic 2 11/34

23 Graph representation Connected component labeling (of a graph) Algorithm (Connected components) Input: Graph G, starting vertex s Put s in the queue (or stack) L. while L do pull a vertex t from L. Label all the neighbors of t that are not labelled and put them in L. It allows to calculate the connected components of a graph in linear time. breadth-first search depth-first search (Hopcropft and Tarjan, 1973) Digital Geometry : Topic 2 11/34

24 Graph representation Discrete curve An m-path π is also called an m-curve. Digital Geometry : Topic 2 12/34

25 Graph representation Discrete curve An m-path π is also called an m-curve. Definition (m-curve) An m-path π is an m-curve if for all the elements p i of π, i = 1,..., n, p i has exactly two m-adjacent points in π, except for p 0 and p n that have only one. Digital Geometry : Topic 2 12/34

26 Graph representation Discrete curve An m-path π is also called an m-curve. Definition (m-curve) An m-path π is an m-curve if for all the elements p i of π, i = 1,..., n, p i has exactly two m-adjacent points in π, except for p 0 and p n that have only one. Definition (closed m-curve) An m-curve π = (p 0,..., p n ) is a closed m-curve if p 0 = p n. An m-curve π that is not closed is also called an m-arc. Digital Geometry : Topic 2 12/34

27 Graph representation Discrete curve An m-path π is also called an m-curve. Definition (m-curve) An m-path π is an m-curve if for all the elements p i of π, i = 1,..., n, p i has exactly two m-adjacent points in π, except for p 0 and p n that have only one. Definition (closed m-curve) An m-curve π = (p 0,..., p n ) is a closed m-curve if p 0 = p n. Definition (simple closed m-curve) An m-curve π is a simple closed m-curve if every element of π is m-adjacent to exactly two other points of π. Digital Geometry : Topic 2 12/34

28 Graph representation Jordan curve theorem Theorem (Jordan curve theorem (Jordan, 1887)) Let C be a simple closed curve in the plane R 2, called a Jordan curve. Then, its complement R 2 \ C consists of exactly two components, the interior and exterior, and C is their boundary. Digital Geometry : Topic 2 13/34

29 Graph representation Jordan curve theorem Theorem (Jordan curve theorem (Jordan, 1887)) Let C be a simple closed curve in the plane R 2, called a Jordan curve. Then, its complement R 2 \ C consists of exactly two components, the interior and exterior, and C is their boundary. Problem The discrete version of Jordan theorem does not hold for simple closed m-curve. If the curve is connected, it does not disconnect its interior from its exterior (8-connectedness); if it is totally disconnected it does disconnect them (4-connectedness). (Rosenfeld, Pflatz, 1966) Digital Geometry : Topic 2 13/34

30 Graph representation Good adjacency pairs for 2D binary images Theorem (Separation theorem (Duda, Hart, Munson, 1967)) A simple closed m-curve C m -separates all pixels inside C from all pixels outside C, for (m, m ) = (4, 8), (8, 4). (Klette, Rosenfeld, 2003) Digital Geometry : Topic 2 14/34

31 Graph representation Good adjacency pairs for 2D binary images Theorem (Separation theorem (Duda, Hart, Munson, 1967)) A simple closed m-curve C m -separates all pixels inside C from all pixels outside C, for (m, m ) = (4, 8), (8, 4). (Klette, Rosenfeld, 2003) Definition (Generarisation: good adjacency pairs (Kong, 2001)) (α, β) is called a good pair iff, for (m, m ) {(α, β), (β, α)}, any simple closed m-curve m -separates its (at least one) m -holes from the background and any totally m-disconnected set cannot m -separate any m -hole from the background. Digital Geometry : Topic 2 14/34

32 Graph representation Exercise: shape border Question Consider the set A of discrete points (black points) in the figure. 1 Illustrate the 4-interior border Br 4 (A). 2 Illustrate the 8-interior border Br 8 (A). 3 Does each Br m (A) for m = 4, 8 form a simple closed curve? If the answer is positive, what adjacency pair (a, a ) should be used for the curve and its complement so that the discrete version of Jordan separation theorem holds? Digital Geometry : Topic 2 15/34

33 Graph representation Exercise: shape boundary (answers) 1-2 Digital Geometry : Topic 2 16/34

34 Graph representation Exercise: shape boundary (answers) Br 4 (A) does not form a simple closed curve; there is a point that has more than two adjacent points. On the other hand, Br 8 (A) forms a simple closed 4-curve if the adjacency pair (4, 8) is used. (Similarly, the following answer is also accepted: Br 4 (A) forms a simple closed m-curve if one of the adjacency pairs (m, m ) = (4, 4), (4, 8) is used.) Digital Geometry : Topic 2 16/34

35 Graph representation Inter-pixel boundary of a discrete shape Let us consider a discrete space as a pair (V, W ) where V is a countable set and W is a symmetric relation on V V. For example: (V, W ) = (Z 2, E 4), (Z 2, E 8). Digital Geometry : Topic 2 17/34

36 Graph representation Inter-pixel boundary of a discrete shape Let us consider a discrete space as a pair (V, W ) where V is a countable set and W is a symmetric relation on V V. For example: (V, W ) = (Z 2, E 4), (Z 2, E 8). Definition (Inter-pixel boundary) Let (V, W ) be a discrete space, and X be a subset of V. The boundary of X and its complement X is defined by (X, X) = {(u, v) W : u X v X}. Note that every element of (X, X) is directed. Digital Geometry : Topic 2 17/34

37 Graph representation Inter-pixel boundary of a discrete shape Let us consider a discrete space as a pair (V, W ) where V is a countable set and W is a symmetric relation on V V. For example: (V, W ) = (Z 2, E 4), (Z 2, E 8). Definition (Inter-pixel boundary) Let (V, W ) be a discrete space, and X be a subset of V. The boundary of X and its complement X is defined by (X, X) = {(u, v) W : u X v X}. Note that every element of (X, X) is directed. (Klette, Rosenfeld, 2003) Digital Geometry : Topic 2 17/34

38 Cell complex approach Cell complex Definition (Cell complex) A cell complex is a set C of cells such that the empty cell is included in C, all the faces of every cell of C also belong to C, the intersection of two cells is one of their common faces. The r-cell is an r-dimensional convex polyhedron. Simplicial complex Cubical complex Digital Geometry : Topic 2 18/34

39 Cell complex approach Face of complex Definition (Face) A face of an r-cell σ is an s-cell that is included in the boundary of σ with s < r. 3-cell its faces Digital Geometry : Topic 2 19/34

40 Cell complex approach Digital image and complex representations Grid point set (digital image) Adjacency graph Kovalevsky topology (cubical complex) Digital Geometry : Topic 2 20/34

41 Cell complex approach Kovalevsky topology For a digital image, we define cells as Definition Kovalevsky topology is defined by C = (A, B) such that A is a set of cells and B A A is a set of their orders. Digital Geometry : Topic 2 21/34

42 Cell complex approach Order of cells cell order If an r-cell σ is a face of s-cell τ, then σ < τ. Note: r < s. Digital Geometry : Topic 2 22/34

43 Cell complex approach nd discrete object and boundary (Kovalevsky topology) Definition (nd discrete object (n-complex)) Given an m-object A Z n (m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3) we obtain the set K n of n-cells whose centroid are the points of A. The nd discrete object is then represented by the n-complex given by K = K n {τ < σ : σ K n }. Digital Geometry : Topic 2 23/34

44 Cell complex approach nd discrete object and boundary (Kovalevsky topology) Definition (nd discrete object (n-complex)) Given an m-object A Z n (m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3) we obtain the set K n of n-cells whose centroid are the points of A. The nd discrete object is then represented by the n-complex given by K = K n {τ < σ : σ K n }. Definition (Boundary ((n-1)-cell set)) The boundary of a nd discrete object in Kovalevky topology is the set of (n 1)-cells located between the interior border and the exterior border. Digital Geometry : Topic 2 23/34

45 Cell complex approach nd discrete object and boundary (Kovalevsky topology) Definition (nd discrete object (n-complex)) Given an m-object A Z n (m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3) we obtain the set K n of n-cells whose centroid are the points of A. The nd discrete object is then represented by the n-complex given by K = K n {τ < σ : σ K n }. Definition (Boundary ((n-1)-cell set)) The boundary of a nd discrete object in Kovalevky topology is the set of (n 1)-cells located between the interior border and the exterior border. The same boundary as the inter-pixel/voxel boundary in the dual sense. Digital Geometry : Topic 2 23/34

46 Cell complex approach Inter-pixel boundary following Algorithm: Inter-pixel boundary following (Aztzy et al., 1981) Input: Binary image in a discrete space (Z 2, m), starting 1-cell s Output: Set F of 1-cells that form the boundary containing s Put s in a list F and in a queue Q. while Q do Pull f from Q. for each successor neighbor g of f do if g is in F, put g in F and in Q. 1-cells which form the boundary are also called bel. Digital Geometry : Topic 2 24/34

47 Cell complex approach Inter-pixel boundary following Algorithm: Inter-pixel boundary following (Aztzy et al., 1981) Input: Binary image in a discrete space (Z 2, m), starting 1-cell s Output: Set F of 1-cells that form the boundary containing s Put s in a list F and in a queue Q. while Q do Pull f from Q. for each successor neighbor g of f do if g is in F, put g in F and in Q. 1-cells which form the boundary are also called bel. The graph structure and the similar idea to the graph traversal are used. Digital Geometry : Topic 2 24/34

48 Cell complex approach Adjacency of 2-cells Definition (2-cell adjacency) Given a complex C, two distinct 2-cells of C are adjacent if they have the common 1-face. (Lachaud, Malgouyres, 2003) Digital Geometry : Topic 2 25/34

49 Cell complex approach Adjacency of 2-cells Definition (2-cell adjacency) Given a complex C, two distinct 2-cells of C are adjacent if they have the common 1-face. Property (Voss, 1993) In the boundary of a 6-object, each 2-cell is adjacent to exactly four neighboring 2-cells such that two of them are its successors and the others are its predecessors. (Lachaud, Malgouyres, 2003) Digital Geometry : Topic 2 25/34

50 Cell complex approach 3D boundary following algorithm Algorithm: 3D boundary following (Aztzy et al., 1981) Input: 6-object, starting 2-cell s Output: Set F of 2-cells that form the boundary Put s in a list F and in a queue Q. while Q do Pull f from Q. for each successor neighbor g of f do if g is not in F, put g in F and in Q. (Lachaud, Malgouyres, 2003) The similar idea to the algorithm for connected component labeling is used. Digital Geometry : Topic 2 26/34

51 Cell complex approach Data structure for Kovalevsky topology Data structure If the size of input 3D image is N N N (3D array size), the number of elements of its Kovalevsky topology is 2N 2N 2N = 8N 3. Digital Geometry : Topic 2 27/34

52 Cell complex approach Data structure for Kovalevsky topology Data structure If the size of input 3D image is N N N (3D array size), the number of elements of its Kovalevsky topology is 2N 2N 2N = 8N 3. The dimension of each cell σ is determined by the 3D array index (i, j, k): if all of the integers i, j, k are even numbers, then σ is a 3-cell; if one of the integers i, j, k is odd, then σ is a 2-cell; if one of the integers i, j, k is even, then σ is a 1-cell; if all of the integers i, j, k are odd numbers, then σ is a 0-cell. Digital Geometry : Topic 2 27/34

53 Cell complex approach Data structure for Kovalevsky topology Data structure If the size of input 3D image is N N N (3D array size), the number of elements of its Kovalevsky topology is 2N 2N 2N = 8N 3. The dimension of each cell σ is determined by the 3D array index (i, j, k): if all of the integers i, j, k are even numbers, then σ is a 3-cell; if one of the integers i, j, k is odd, then σ is a 2-cell; if one of the integers i, j, k is even, then σ is a 1-cell; if all of the integers i, j, k are odd numbers, then σ is a 0-cell. The order between two cells is arithmetically defined depending on their 3D array indices. Digital Geometry : Topic 2 27/34

54 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? Digital Geometry : Topic 2 28/34

55 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? Digital Geometry : Topic 2 28/34

56 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? Digital Geometry : Topic 2 28/34

57 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? Digital Geometry : Topic 2 28/34

58 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? 3 What are the inter-pixel boundary between X and X? Digital Geometry : Topic 2 28/34

59 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? 3 What are the inter-pixel boundary between X and X? Digital Geometry : Topic 2 28/34

60 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? Digital Geometry : Topic 2 28/34

61 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? Digital Geometry : Topic 2 28/34

62 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? Digital Geometry : Topic 2 28/34

63 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? Digital Geometry : Topic 2 28/34

64 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? 3 What are the inter-pixel boundary between X and X? Digital Geometry : Topic 2 28/34

65 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? 3 What are the inter-pixel boundary between X and X? Digital Geometry : Topic 2 28/34

66 Cell complex approach Exercise: connected components and inter-pixel boundary Question Let X be the set of discrete points (black points) in the figure. The boundary does not form a manifold! 1 What are the 8-connected components of X and the 4-connected components of X? 2 What are the 4-connected components of X and the 8-connected components of X? 3 What are the inter-pixel boundary between X and X? Digital Geometry : Topic 2 28/34

67 Well-composed images Adjacency pair Definition (Good adjacency pair (Duda et al., 1967)) An image I is a (4, 8)- (resp. (8, 4)-) image if the 4- (resp. 8-) adjacency is used for the black pixels and the 8- (resp. 4-) adjacency is used for the white pixels. (4, 8)-image Connected components Digital Geometry : Topic 2 29/34

68 Well-composed images Adjacency pair Definition (Good adjacency pair (Duda et al., 1967)) An image I is a (4, 8)- (resp. (8, 4)-) image if the 4- (resp. 8-) adjacency is used for the black pixels and the 8- (resp. 4-) adjacency is used for the white pixels. (8, 4)-image Connected components Digital Geometry : Topic 2 29/34

69 Well-composed images Well-composed image Definition (Well-composed image (Latecki, 1998)) An image is well-composed if each 8-connected component for the black (and white) points is also a 4-connected component. Well-composed image 8-connected components Digital Geometry : Topic 2 30/34

70 Well-composed images Well-composed image Definition (Well-composed image (Latecki, 1998)) An image is well-composed if each 8-connected component for the black (and white) points is also a 4-connected component. Well-composed image 4-connected components Digital Geometry : Topic 2 30/34

71 Well-composed images Well-composed image Definition (Well-composed image (Latecki, 1998)) An image is well-composed if each 8-connected component for the black (and white) points is also a 4-connected component. Ill-composed image 8-connected components Digital Geometry : Topic 2 30/34

72 Well-composed images Well-composed image Definition (Well-composed image (Latecki, 1998)) An image is well-composed if each 8-connected component for the black (and white) points is also a 4-connected component. Ill-composed image 4-connected components Digital Geometry : Topic 2 30/34

73 Well-composed images Properties of well-composed images Property (Well-composed image (Latecki, 1998)) Any well-composed image is characterized by 2 2 local configuration, and has no topological paradox related to Jordan Theorem. Ill-composed image 4-connected components Digital Geometry : Topic 2 31/34

74 Well-composed images Properties of well-composed images Property (Well-composed image (Latecki, 1998)) Any well-composed image is characterized by 2 2 local configuration, and has no topological paradox related to Jordan Theorem. Well-composed image 4-connected components Digital Geometry : Topic 2 31/34

75 Well-composed images Repairing images to be well-composed If a given image is not well-composed, it can be repaired. (Latecki, 1998; Rosenfeld et al., 1998) Digital Geometry : Topic 2 32/34

76 Well-composed images Repairing images to be well-composed If a given image is not well-composed, it can be repaired. (Latecki, 1998; Rosenfeld et al., 1998) Iterative method: modify every critical configuration iteratively such that the well-composedness is satisfied. Digital Geometry : Topic 2 32/34

77 Well-composed images Repairing images to be well-composed If a given image is not well-composed, it can be repaired. (Latecki, 1998; Rosenfeld et al., 1998) Iterative method: modify every critical configuration iteratively such that the well-composedness is satisfied. Up-sampling method: increase the resolution and repair each critical configuration locally. Digital Geometry : Topic 2 32/34

78 Summary Summary nd discrete object boundary is expected to be a Jordan curve/surface, a n-manifold. Digital Geometry : Topic 2 33/34

79 Summary Summary nd discrete object boundary is expected to be a Jordan curve/surface, a n-manifold. For this aim, we need to choose appropriate shape representations, such as inter-pixel boundary, with Digital Geometry : Topic 2 33/34

80 Summary Summary nd discrete object boundary is expected to be a Jordan curve/surface, a n-manifold. For this aim, we need to choose appropriate shape representations, such as inter-pixel boundary, with good adjacency pair, or Digital Geometry : Topic 2 33/34

81 Summary Summary nd discrete object boundary is expected to be a Jordan curve/surface, a n-manifold. For this aim, we need to choose appropriate shape representations, such as inter-pixel boundary, with good adjacency pair, or verifying if a given image is well-composed. Digital Geometry : Topic 2 33/34

82 References References Reinhard Klette and Azriel Rosenfeld. Chapters 4 and 7 in Digital geometry: geometric methods for digital picture analysis, San Diego: Morgan Kaufmann, David Coeurjolly, Annick Montanvert, et Jean-Marc Chassery. Eléments de base, Chapitre 1 dans Géométrie discrète et images numériques, Hermès Lavoisier, Jacques-Olivier Lachaud et Rémy Malgouyres. Topologie, courves et surfaces discrètes, Chapitre 3 dans Géométrie discrète et images numériques, Hermès Lavoisier, Gabor T. Herman. Geometry of digital spaces, Birkhäuser, Longin Jan Latecki. Discrete representation of spatial objects in computer vision, Kluwer Academic Publishers, Digital Geometry : Topic 2 34/34