Unit 5. Mathematical Morphology. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 85

Transkript

1 Unit 5 Mathematical Morphology Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 85

2 Introduction to Mathematical Morphology Use of algebraic analysis for detecting shapes in (image) data Binary morphology works with b/w, i.e. 0/1, data. Fuzzy morphology works with gray value data (interval [0, 1]). Central concept: structuring element, which corresponds to the shape under consideration Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 86

3 Binary Dilation Let X be the n-dimensional Euclidean space. The dilation of a pixel set A with respect to the structuring element B is defined as the Minkowski addition as follows: A B = {x X x = a + b for some a A and b B} Characterization of dilation: A B = a A B a, where A t = {x X x = a + t for some a A}. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 87

4 Properties of the Dilation/Minkowski Addition Commutativity: A B = B A Associativity: A (B C) = (A B) C Neutral element: A {0} = A Non-decreasingness: if A B then A C B C Extensivity: if 0 B then A B A Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 88

5 Binary Erosion Let X be the n-dimensional Euclidean space. The erosion of a pixel set A with respect to the structuring element B is defined as follows: A B = {x X x + b A for every b B} Characterization of erosion: A B = {x X B x A} = b B A b Note that this is not exactly the same as the Minkowski subtraction! Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 89

6 Properties of the Erosion Commutativity is not fulfilled Associativity is not fulfilled Neutral element: A {0} = A Non-decreasingness in first argument: if A B then A C B C Non-increasingness in second argument: if B C then A B A C Anti-extensivity: if 0 B then A B A Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 90

7 Relationships Between Erosion and Dilation Theorem (Erosion Dilation Duality). (A B) = ( A) ( B) (A B) = ( A) ( B) Theorem (Chaining Rule for Erosion). A (B C) = (A B) C Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 91

8 Some Other Properties of Erosion and Dilation (A B) C = (A C) (B C) (A B) C (A C) (B C) A (B C) = (A B) (A C) A (B C) (A B) (A C) (A B) C (A C) (B C) (A B) C = (A C) (B C) A (B C) = (A B) (A C) A (B C) (A B) (A C) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 92

9 The Hit-and-Miss Transform Given a pair of two structuring elements (B, C), the Hitand-Miss Transform of a pixel set A is defined as A (B, C) = (A B) ( A C). Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 93

10 Thickening and Thinning Given a pair of two structuring elements (B, C), thickening and thinning operators with respect to the pair (B, C) are defined as follows: A (B, C) = A A (B, C) A (B, C) = A\A (B, C) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 94

11 Template Matching Assume we are given an image A, a template T, and a window of interest W T in which the template should be localized. Then the set of match points is given by the following hit-and-miss transform: A (T, W \T ) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 95

12 Openings and Closures Opening: dilation of erosion Closure: erosion of dilation A B = (A B) B A B = (A B) B Note we call a set A B-open if A B = A and B-closed if A B = A, respectively. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 96

13 Properties of Openings A B A A B implies A C B C A is B-open if and only if there exists a C such that A = C B A B is the largest B-open subset of A (A B) B = A B Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 97

14 Properties of Closures A B A A B implies A C B C A is B-closed if and only if there exists a C such that A = C B A B is the smallest B-closed superset of A (A B) B = A B Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 98

15 Example 1: Separating a Disc From a Beam (cf. [Haralick & Shapiro, 1992]) Left: original image ( ) Middle: opening with respect to a large disc-shaped structuring element (radius 80) Right: residual (left middle) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 99

16 Example 2: Separating Horizontal and Vertical Parts (cf. [Haralick & Shapiro, 1992]) Left: original image ( ) Middle: opening with respect to a (100 5) structuring element Right: opening with respect to a (20 200) structuring element (100 5) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 100

17 Example 3: Extracting Pinheads (cf. [Haralick & Shapiro, 1992]) Left: original image ( ) Right: opening with respect to a disc-shaped structuring element (radius 5) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 101

18 An Example from Industrial Practice I (cf. [Haralick & Shapiro, 1992]) Goal: extraction of pin areas of integrated circuit (B s refer to binary images, G s refer to grayscale images) (a) original image G 0 (b) thresholded image B 1 computed from G 0 using a threshold value of 200 (c) B 2 = B 1 box(5, 5) (d) B 3 = B 2 box(80, 100) (e) B 4 = B 3 box(160, 90) (f) G 5 = min(g 0, B 4 ) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 102

19 An Example from Industrial Practice II (cf. [Haralick & Shapiro, 1992]) (g) G 6 : remove noise from G 5 (h) thresholded image B 7 computed from G 6 using a threshold value of 174 (i) B 8 = B 7 box(8, 2) (j) B 9 = B 8 box(20, 3) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 103

20 Gray-Scale Morphology: Overview Threshold/level set approach Umbra approach Fuzzy morphology Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 104

21 The Threshold Approach The threshold approach is designed for gray-scale images and binary structuring elements The image A : X [0, 1] is quantized into a set of level sets (A α ) α I, where A α = {x X A(x) α and I ]0, 1]. Then all morphological operations discussed so far can be applied to the individual level sets Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 105

22 Morphological Operations in the Threshold Approach For all α I, we carry out dilation and erosion for all level sets: (A B) α = A α B (A B) α = A α B Operations like hit-and-miss transform, opening, closure, etc., can be defined analogously. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 106

23 From Level Sets to a Gray-Scale Image Given level sets (C α ) α I, we can determine a gray-scale image C in the following way (with the convention sup = 0): C (x) = sup{α x C α } Then C α = C α holds for all α I. In general, C is only an approximation of C. If I =]0, 1] of I C(X), then C = C holds. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 107

24 The Umbra Approach The umbra approach is designed generally for grayscale images and gray-scale structuring elements The idea behind the umbra approach is based on considering the areas above and below the graph of the image and its structuring element and applying binary morphology in the corresponding n + 1-dimensional space Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 108

25 Umbras and Top Surfaces Given a function/image f : A R (A X), the Umbra of A is a subset of A R and defined as U[f] = {(x, v) A R v f(x)}. Given a set Y X R, its top surface is an F R function/image defined as T [Y ](x) = sup{v R (x, v) Y }, where F = {x X (x, v) Y for some v R}. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 109

26 Dilation in the Umbra Approach Given an image f : F R and a structuring element k : K R, the dilation of f w.r.t. k is defined as a F K R function in the following way: f k = T [U[f] U[k]] Note that this computation can be represented as (f k)(x) = sup{f(x z) + k(z) z K, x z F }. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 110

27 Erosion in the Umbra Approach Given an image f : F R and a structuring element k : K R, the erosion of f w.r.t. k is defined as a F K R function in the following way: f k = T [U[f] U[k]] Note that this computation can be represented as (f k)(x) = sup{f(x + z) k(z) z K}. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 111

28 Fuzzy Morphology Now let us assume that A and B are fuzzy sets. In order to be able to generalize dilations and erosions to fuzzy sets, we have to generalize the conjunction, the implication, the universal and the existential quantifier. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 112

29 Operations Conjunction: A triangular norm (t-norm) is a binary operation on the unit interval [0, 1] which is commutative, associative, non-decreasing in both components, and has 1 as neutral element. Implication: The residual implication of a left-continuous t- norm is defined as T (x, y) = sup{u [0, 1] T (x, u) y}. Quantifiers: inf can be used as universal and sup can be used as existential quantifier. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 113

30 Important t-norms Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 114

31 Corresponding Implications Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 115

32 Basic Properties Lemma. The following holds for all x, y, z [0, 1]: 1. x y if and only if T (x, y) = 1 2. T (x, y) z if and only if x T (y, z) 3. T ( T (x, y), T (y, z) ) T (x, z) 4. T (1, y) = y 5. T ( x, T (x, y) ) y 6. y T ( x, T (x, y) ) Furthermore, T is non-increasing and left-continuous in the first argument and non-decreasing and right-continuous in the second argument. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 116

33 Fuzzy Dilation and Erosion (A B)(x) = sup y X T (B(x y), A(y)) (A B)(x) = inf y X T (B(y x), A(y)) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 117

34 Properties of the Fuzzy Dilation Commutativity: A B = B A Associativity: A (B C) = (A B) C Neutral element: A {0} = A Non-decreasingness: if A B then A C B C Extensivity: if 0 B then A B A Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 118

35 Properties of the Fuzzy Erosion Commutativity is not fulfilled Associativity is not fulfilled Neutral element: A {0} = A Non-decreasingness in first argument: if A B then A C B C Non-increasingness in second argument: if B C then A B A C Anti-extensivity: if 0 B then A B A Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 119

36 Some Other Properties of Fuzzy Erosion and Dilation (A B) C = (A C) (B C) (A B) C (A C) (B C) A (B C) = (A B) (A C) A (B C) (A B) (A C) (A B) C (A C) (B C) (A B) C = (A C) (B C) A (B C) = (A B) (A C) A (B C) (A B) (A C) Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 120

37 Openings and Closures Opening: dilation of erosion Closure: erosion of dilation A B = (A B) B A B = (A B) B Note we call a set A B-open if A B = A and B-closed if A B = A, respectively. Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 121

38 Properties of Openings A B A A B implies A C B C A is B-open if and only if there exists a C such that A = C B A B is the largest B-open subset of A (A B) B = A B Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 122

39 Properties of Closures A B A A B implies A C B C A is B-closed if and only if there exists a C such that A = C B A B is the smallest B-closed superset of A (A B) B = A B Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 123

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