Representing regions in 2 ways:

Transcription

1 Representing regions in 2 ways: Based on their external characteristics (its boundary): Shape characteristics Based on their internal characteristics (its region): Both Regional properties: color, texture, and 1

2 Describes the region based on a selected representation: Representation boundary or textural features Description length, orientation, the number of concavities in the boundary, statistical measures of region. 2

3 Invariant Description: Size (Scaling) Translation Rotation 3

4 Boundary (Border) Following: We need the boundary as a ordered sequence of points. 4

5 Moore Boundary Tracking Algorithm: 1. Let the starting point, b 0, be the uppermost, leftmost point in the image that is labeled 1. Denote by c 0 the west neighbor of b 0. Clearly c 0 always will be a background point. Then, examine the 8 neighbors of b 0, starting at c 0 and proceeding in clockwise direction. Let b 1 denote the first pixel encountered whose value is 1, and let c 1 be the points immediately preceding b 1 in the sequence. Store the location of b 0 and b Let b=b 1 and c=c Let the 8 neighbors of b, starting at c and proceeding in a clockwise direction be denoted by n 1,n 2,..., n k. Find the first n k whose value is Let b=n k and c=n k Repeat steps 3 and 4 until b=b 0 and the next boundary point found is b 1. 5

6 Example: 6

7 Freeman Chain Code: Code the 4 or 8 connectivity 7

8 Chain Code Problems: Long Code Low noise Robustness Solution: Resampling Starting Point: Solution: Rotary/Circular shift until forms a minimum integer Angle normalization: First difference (Counterclockwise) or (transition between last and first) Useful for integer multiple of used chain code (45 or 90 ) 8

9 Example: Resampling 4 and 8 chain codes 9

10 Example: 10

11 Polygon Approximation: Minimum-Perimeter Polygons Read Pages

12 Example (Cont.): 12

13 Example: 13

14 Polygon Approximation: Merging: Start from a seed point Continue on a line based on local average error (e.g. linear regression) Stop if error exceeds a threshold Continue from the last point... No guarantee for corner detection 14

15 Polygon Approximation: Splitting: Segments to two parts based on a criteria (e.g. maximum internal distance) Check each segment for splitting based on another criteria (e.g. linearity error). 15

16 Signatures: A 1-D functional representation of a boundary Distance vs. Angle (in the polar representation): Invariant to translation Non-Invariant to rotation (may be achieved by start point selection)» Farthest point from centroid» The point on eigen axis» Use chain code solution for the start point Line tangent angle Histogram of tangent angle 16

17 Example: 17

18 Example: 18

19 Boundary Segments: Convex Hull: Smallest convex set containing the S Convex Deficiency: The set difference D=H-S Problem of noise: Smoothing (K-neighbour average) Polygon approximation Convex Hull of Polygon 19

20 Boundary Segments: Skeletonization / Thinning: Medial Axis Transform (MAT) Point p is belong to medial (Region R and Border B): Has more than one closest neighbor in B Chapter 9 and Page

21 Example of Thinning: 21

22 Boundary Descriptor: Simple one: Diameter: Diam B max D p, q Lead to Major axis and minor axis (Perpendicular to major) Basic Rectangle Curvature i j i, j k 1 xy xy R x y 22

23 Shape Number Smallest integers of first difference circular chain code. 23

24 Example: 24

25 Fourier Descriptors: s k x k jy k K 1 uk a u s k exp j2 k 0 K K 1 1 uk s k a uexp j2 K u0 K P1 1 uk sˆ k a uexp j2 P u0 K 25

26 Example: 26

27 Problem of invariance: Rotation K 1 1 j uk ar u sk e exp j2 aue K k0 K Normalization will solve! s 0 s i i as u as u as u aˆ s u,,,... a u a u a i s u 2 i j 27

28 Problem of invariance: Translation t j xy x x st k s k xy x k x j y k y a u u xy Translation to centroid will solve! 28

29 Problem of invariance: Scaling ss k s k as u a u Normalization will solve s 0 s i i as u as u as u aˆ s u,,,... a u a u a i s u 2 i 29

30 Problem of invariance: Starting Point, sp k s k k x k k y k k j2 k0u ap u a uexp K Normalization will solve p 0 p i i ap u ap u ap u aˆ p u,,,... a u a u a i p u 2 i 30

31 Properties in a single table: 31

32 Statistical Moments: Boundary Representation A1 i0 r, v g r v as random variable m i A1 v p v i n v v m p v n i i i0 A1 i0 r, v g r A1 normalized gr ( ) as histogram m i r g r i n v r m g r n i i i0 32

33 Regional Descriptor: The simple one: Area (Number of pixels) Perimeter (Length of boundary) Compactness (Perimeter 2 /Area) Circularity: Ratio of the area to the area of a circle with same perimeter Mean, median, max, min, ratio pixels above/below from intensity data. 33

34 Example: Normalized area: Ratio of light pixels to total light pixels 34

35 Topology Descriptor: Number of holes (H): Invariants to several operators. 35

36 Topology Descriptor: Number of connected components (C) : Invariants to several operators. 36

37 Topology Descriptor: Euler number (E=C-H) : Invariants to several operators

38 Topology Descriptor: Polygonal net: V: # of vertices (7) Q: # of edges (11) F: # of faces (2) E=C-H=V-Q+F C=1, H=3 E=-2 38

39 Example: Original Threshold Largest Connected Region Skeleton 39

40 Texture: No formal definition Smooth Coarse Regular 40

41 Statistical Approaches 1 st order grey level statistics From normalised histogram One pixel gray level repeat n times 2 nd order grey level statistics From GLCM (Grey Level Co-ocurrence Matrix) Repeatation of two pixels in a pre-defined neighbourhood Needs: A Positioning Operator, P. GLCM(i,j): # of times that points with gray level Z i occure relative to points with gray level Z j 41

42 Texture feature from 1 st order statistics: n z z m P z, m z P z z z L1 L1 n i i i i i0 i0 R z : Gray Level Contrast (Normalized) 1 L1 2 z :Skewness : Kurtosis, Flatness 2 U z p z i0 L1 log i0 i i : Uniformity e z p z p z i : Entropy 42

43 Example: Smooth Coarse Regular 43

44 Problem with 1 st order histogram Lack of spatial information 44

45 Gray Level Co-Occurence Matrix (GLCM): , z z1 0 z2 1 z3 2, : one pixel to right one below G = P

46 Gray Level Co-Occurence Matrix (GLCM): 46

47 Gray Level Co-Occurence Matrix (GLCM): N g : # of gray levels N N g g n g p i1 j1 ij N N N N g g g g m i p, m j p r ij c ij i1 j1 j1 i1 ij g ij n Ng Ng Ng Ng r i mr pij, c j mc pij i1 j1 j1 i1 47

48 Texture feature from GLCM Max( p ) : Maximum probability (G1) i, j ij i m j m i j r c i i i i j i 2 ( i j) pij : Contrast (G3) p 2 ij r c : Uniformity p ij 1 i : Homogeneity j : Correlation (G2) log ( p i j p ij 2 ij ) : Entropy 48

49 Example: One pixel immediately to he right 49

50 Example: 50

51 Effect of Horizontal offset: Correlation index Horizontal distance between neighbors Noisy Sin Circuit board 51

52 Spectral approaches: Peaks and Frequencies related to periodicity: S(r,θ), S r (θ),s θ (r): 0 S r S R S r S r S r1 S r 52

53 Spectral approaches: 53

54 Spectral approach: 54

55 Moments of Image as a 2D pdf: Moments of order of (p+q) p q m x y f x, y dxdy Central moments of order of (p+q) pq pq x ; p q, x x y y f x y dxdy m m y m m

56 Digital Data: Moments of order of (p+q) m pq pq x y p q x y f ( x, y) p q m ( x x) ( y y) f ( x, y), x ; y m x y m x y m ( x x) ( y y) f ( x, y) m m 0 f ( x, y) x y m00 m ( x x) ( y y) f ( x, y) m m x y m00 m m 56

57 Moments: 2nd ordens centrale momenter pq 2,, rd ordens centrale momenter pq 3,,, Normalized central momens pq pq p q and A set of invariant moments (7 by Hu) 57

58 Hu Moments:

59 Example: 59

60 Results of invariant moments: 60

61 Complex Zernike Moments: m 1 Amn f x, yvmn x, ydxdy 0 x 2 y 2 1 m, n, m n even, n m Where V mn are Zernike polynomials: jn V r, R r e, r, : over unit disc mn mn m n 2 s 1,,, s0 F m, n, s, r mn R r F m n s r m s! r m n m n s! s! s! 2 2 m2s 61

62 Some Zernike Polynomials: R R R R R m, n m, n 0,0 1,1 2,0 2,2 2 2r R3,1 3r 2r R 3,3 R 1 r r r 3 62

63 How to Compute Zernike Polynomials: Scale and Translation invariance transform: Shift to center of gravity (m 10 = m 01 = 0) Fix m 00 at a predetermined value (= b): x y g x, y f x, y, Map Image to the unit disc using polar coordinates. The center of the image is the origin of the unit disc Those pixels falling outside the unit disc are not used in the computation. r x y b m 00 y x 2 2 1, tan 63

64 How to Compute Zernike Polynomials: m 1 A g x y V x y,, m, n m, n x y 64

65 Legendre Moments: 2m12n P x P y dxdy, m, n m, n m n k 1 d Pk r x k k 2 k! dx k 65

66 Fourier-Mellin Moments: Fourier-Mellin Transform: mn, 2 m in F r f ( r, ) e rdrd, m, n 0 0 Fourier-Mellin Moments: in m, n f ( r, ) Qm re rdrd, m, n 2 a m 0 0 Q r : orthogonal polynomial of degree m over the raneg 0 r 1 m Q r Q r rdr a m k m l m 66

67 Principal Component Analysis: Consider a multi (value/channel/spectral) images: x 1 2 K 1 m x Ex x K k 1, K : # of observations 1 C x m x m x x m m C C K T T T x E x x k k x x K k 1 x x x x : Real Positive Definiter Matrix v v : 0, v v, i j x i i j T n T k T A v 1 v2 vn, 1 2 n A A 1 67

68 Karhunen Loève or Hotelling Transform: Analysis: T y = A x - mx m y = 0, C y = ACxA diag 1 2 n Complete Synthesis: Optimal Synthesis: A k 2 T x = A y m Form Using k eigenvectors correspondinf to k-largest eigenvalues T x ˆ = A y m k X e E x xˆ rms j j j j1 j1 jk 1 PCA Analysis n k n X 68

69 Example: 69

70 Data Preparation: 70

71 Eigenvalues Analysis: 71

72 PCA Analysis: Eigneimages 72

73 Example: Using 2 components 73

74 Example: Difference 74

75 PCA in Boundary description: 75

76 PCA in Boundary description: 76