Mathematical Morphology

Transcription

1 Mathematical Morphology Erosions and Dilations Luc Brun (d après le cours de M. Coster) 25 janvier / 61

2 Plan (1/2) Structuring element Definition, Example, Transpose Set erosion and dilatation Hit Miss Transformation Erosion Example of Erosion Bi colored transformation Erosion and Minkowski s substraction Dilatation Dilatation : Examples Dilatation and Minkowski s addition Properties of set erosion and dilatation Duality, Extensiveness Growth Composition Union, Intersection Composition Above continuity of the erosion Distance computations Distance of a point to a set Distance and slices Distance by erosion : Algorithm Examples External Distance 2 / 61

3 Plan (2/2) Erosion and dilatation of functions Type of structuring element Example of structuring element Erosion of a function Erosion with a volumic structuring element Examples of erosion Function dilatation Dilatation with a volumic structuring element Example of dilatation Morphological Residues Morphological Gradients Morphological Gradients between sets Functional internal Gradient Functional external Gradient Functional symmetric Gradient Morphological Laplacian Morphological Laplacian : Example 3 / 61

4 Basic idea of mathematical morphology Recall : The basic idea of mathematical morphology is to compare the set to be analysed with a set with a known geometry called structuring element. 4 / 61

5 Structuring Element : Definition A struturing element Bis a set with the following characteristics : It has a known geometry A size λ, This element is located by its origin o. The origin o usually belongs to the structuring element but this is not mandatory. Examples : Square Circle Segment Pair of points 5 / 61

6 Transpose of a structuring element Definition : The transpose of a struturing element B (denoted B or B) is the structuring element mirrored in the origin o. 6 / 61

7 Set Erosion and Dilatation Struncturing element Définition, Example, Transpose colorred Set erosions and dilatations Hit Miss Transformation Erosion Example of Erosion Bi colored transformation Erosion and Minkowski s substraction Dilatation Dilatation : Examples Dilatation and Minkowski s addition Properties of set erosion and dilatation Duality, Extensiveness Growth Composition Union, Intersection Composition Above continuity of the erosion Distance computations Distance of a point to a set Distance and slices Distance by erosion : Algorithm Examples External Distance 7 / 61

8 Binary transform A binary transformation of X by B in E,is defined by moving B on the whole set of points x E. For each position, we ask a question relative to the union, the intersection or the inclusion of B with X. The set of points corresponding to a positive answer defines the transformed image. Simplest binary transformations are : The erosion which is a transformation relative to the inclusion. The dilation which is relative to the intersection. It exists more complex binary transformations like the hit or miss using two structuring elements. 8 / 61

9 Erosion Definition : The structuring element B, located by its origin is moved so as to occupy successively all positions of the space E. For each position we wonder does B is completely included in X? The set of positive answers provide the eroded set. E B (X ) = {x E, B x X } 9 / 61

10 Erosion : Example (1) B = (radius 3) X X ( + ) et E B (X )( ) E B (X ) Qualitative properties The size of objects decreases An object with holes or concavities may split into several connected components Small objects and details disappear 10 / 61

11 Erosion : Example (2) B = (radius 5) X X ( + ) et E B (X )( ) E B (X ) 11 / 61

12 Erosion with structuring elements of increasing size X E 5B (X ) E 10B (X ) E 15B (X ) 12 / 61

13 Erosion and structuring elements Quid of E B (X )? B Erosion with different structuring elements 13 / 61

14 Hit-and-Miss transformation The structuring element is composed of two subsets corresponding to two labels : B = B 0 B 1. We say that x belongs to the Hit and Miss transform of X by B iff : B 0 x C E (X ) and B 1 x X An Hit and Miss transform may be written as the intersection of two erosions on X and C E (X ). η B (X ) = η B 0,B 1(X ) = E B 1(X ) E B 2(C E (X )) 14 / 61

15 Illustration B = E B1 (X ) E B2 (C E (X )) η B (X ) 15 / 61

16 Examples of application Corner detection bottom left bottom right top right top left 16 / 61

17 Erosion and Minkowski s subtraction Set erosion is equivalent to Minkowski subtraction by the transposed element E B (X ) = X B = b B X b Demonstration : Let z b B X b...(to do) 17 / 61

18 Set dilatation Dilatation is a binary transformation based on the intersection. Definition : The structuring element B, located by its origin is moved on all positions of the space E. For each position we check does B intersects X? The set of positive answers forms the dilated set. δ B (X ) = {x E, B x X } reponse negative reponse positive X 18 / 61

19 Dilatation : Example (1) B = (radius 3) X δ B (X )( + ) etx ( ) δ B (X ) Qualitative properties The size of objects increases Holes and concavities may be filled Close objects may become connected Small details disappear 19 / 61

20 Dilatation : Example (2) B = (radius 5) X δ B (X )( + ) etx ( ) δ B (X ) 20 / 61

21 Dilatation with structuring elements of growing size B = X δ 5B (X ) δ 10B (X ) δ 15B (X ) 21 / 61

22 Dilatation and Minkowski s addition The dilatation is equivalent to the set addition by the transposed element. δ B (X ) = X B = b B X b Demonstration : Let z b B X b...(to do...) 22 / 61

23 Set erosion and dilatation s properties Structuring element p. 5 Definition, Example, Transpose Set Erosions and dilatations Binary transformation Erosion Erosion : examples Hit and Miss transformation Erosion and Minkowski s subtraction Dilatation Dilatation : Examples Dilatation and Minkowski s addition Properties of the set erosion and dilatation Duality, Extensibility Growth, Composition Union, Intersection Composition above-continuity of erosion Distance computations Distance of a point to a set Distance and slices Distance by erosion : Algorithm Examples External Distance 23 / 61

24 Duality Erosion and Dilatation are related. We obtain the same result by eroding X or by dilating its complementary and taking the complementary of the resulting set. We say that Erosion and Dilatation are two dual operations according to the complementation. { EB (X )) = C E (δ B (C E (X ))) δ B (X )) = C E (E B (C E (X ))) X δ 5B (X ) E 5B (C E (X )) 24 / 61

25 Extensibility Dilatation is an extensive transformation while erosion is counter-extensive. E B (X ) X δ B (X ) EB (X ) = X = + δ B (X ) = / 61

26 Growth Dilatation and erosion are increasing operators { X Y δb (X ) δ B (Y ) X Y E B (X ) E B (Y ) Erosion is decreasing according to the size of the structuring element. B B E B (X ) E B (X ) δ B ( ) δ B ( ) 26 / 61

27 Composition Dilatation with a structuring element of size n is equal to n dilatations with a struturing element of size 1. (idem for erosion) δ nb (X ) = δ 1B δ }{{ 1B (X ) } nfois nb scaling of B by a factor n. Useful for an hardware implementation when the size of the struturing element is bounded by hardware constraints : A larger structuring element may be obtained by using a cascade of operators. 27 / 61

28 Union, Intersection Dilatation commutes with union : δ B (X Y ) = δ B (X ) δ B (Y ) Erosion commutes with intersection : Moreover : E B (X Y ) = E B (X ) E B (Y ) { δb1 B 2 (X ) = δ B1 (X ) δ B2 (X ) E B1 B 2 (X ) = δ B1 (X ) E B2 (X ) But : δ B1 B 2 (X ) δ B1 (X ) δ B2 (X ) E B1 B 2 (X ) E B1 (X ) E B2 (X ) E B (X Y ) E B (X ) E B (X ) 28 / 61

29 Composition Let B 1 and B 2 be two symmetric structuring elements according to their origins. We have : { δδb1 (B 2 ) (X ) = δ B 1 (δ B2 (X )) E EB1 (B 2 ) (X ) = E B 1 (E B2 (X )) Demonstration (for dilatation) : δ δb1 (B 2 ) (X ) = X (B 1 B 2 ) = X (B 1 B 2 ) = X (B 1 B 2 ) = (X B 1 ) B 2 : key point = (X B 1 ) B 2 = δ B1 (δ B2 (X )) 29 / 61

30 Composition : Illustration B 1 : B 2 : δ B1 (B 2 ) : We pass from a O(n 2 ) complexity to a O(2n) one. 30 / 61

31 Continuity Erosion is semi-continuous superiorly B Demonstration : (to do) lim E B(X n ) E B (lim n + X n ) n + Dilatation is a continuous operator. 31 / 61

32 Superior continuity of erosion : Example Let (X n ) n IN be a sequence of 2 squares separated by a distance of 1 n. B = X n X = lim X n n + lim n + E B (X n ) E B (X ) 32 / 61

33 Summary Dilatation removes small holes, small isthmus and makes objects bigger. Erosion removes small objects, small isthmus, and shrinks objects. Dilatation and erosion are not topological transformations. If X is connected δ B (X ) is connected Dilatation and erosion are non reversible operations Dilatation and erosion are dual but not inverse operations : dilatation X δ B (X ) complémentation C E (X ) E B (C E (X )) 33 / 61

34 Calculs de Distances Structuring element p. 5 Definition, Example, Transpose Set Erosions and dilatations Binary transformation Erosion Erosion : examples Hit and Miss transformation Erosion and Minkowski s subtraction Dilatation Dilatation : Examples Dilatation and Minkowski s addition Properties of the set erosion and dilatation Duality, Extensibility Growth, Composition Union, Intersection Composition above-continuity of erosion Distance computations Distance of a point to a set Distance and slices Distance by erosion : Algorithm Examples External Distance 34 / 61

35 Distance of a point to the boundary of its set Let x X d(x, X ) = d(x, C E (X )) = inf d(x, y) = inf d(x, y) y C E (X ) y X X x d(x, X ) 35 / 61

36 Distance and slices : dist=0 : dist=1 : dist=2 : dist=3 : dist= / 61

37 Distance by erosion : Algorithm bin1=initial image Initialization of an empty image grey0 copy of bin1 in grey0 (f (x) = 1 if x X, 0 otherwise) While bin1 not empty do bin1 E 1B (bin1) grey0 grey0 + bin1 End While grey0 = distance function / 61

38 distance Functions : Examples Shapes Erosion based distance Euclidean distance Distance by erosion is only an approximation of the Euclidean distance. 38 / 61

39 External distance Distance to a set obtained by dilatation 39 / 61

40 Extension to the distance between two sets d(x, Y ) = inf d(x, y) (x,y) X Y bin 0 = binary image containing X extraction of the connected component X i which is copied into the image bin 1 bin 2 = bin 0 /bin 1 (we put in bin 2 all others connected components) Creation of an empty image bin3 d = 0 (initialisation of the value of the distance between X i and ( X /X i ) While bin 3 is empty do bin1 δ 1B (bin 1 ) d = d + 1 bin 3 = bin 1 bin 2 End while 40 / 61

41 Erosion and dilatation of functions Structuring element Set erosion and dilatation Properties of set erosion and dilatation Distance computations Erosion and dilatation of functions Type of structuring element Example of structuring element Erosion of a function Erosion with a volumic structuring element Examples of erosion Function dilatation Dilatation with a volumic structuring element Example of dilatation Morphological Residues Morphological Gradients Morphological Gradients between sets Functional internal Gradient Functional external Gradient Functional symmetric Gradient Morphological Laplacian Morphological Laplacian : Example 41 / 61

42 Erosion and dilatation of functions Pb : Erosion and dilatation have been defined only in the framework of set theory. What to do with f : IR 2 IR? Idea : Consider a function as a set Graph of a function : Shadow of a function : G(f (x)) = {(x, t) t = f (x)} U(f (x)) = {(x, t) t f (x)} G(f(x)) U(f(x)) 42 / 61

43 Type of structuring element U(f (x)) is composed of couples (x, t) with t = f (x). The structuring element is composed of couples (x, b(x)) with x belonging to a bounded support. B = {(x, t) x B and t b(x)} b fulfils the following conditions : { x B b(x) ± b(x) = x B b(x) = We distinguish the flat structuring element defined by : b(x) = { x B b(x) = 0 x B b(x) = 43 / 61

44 Examples of structuring elements G(f(x)) U(f(x)) Non flat structuring elements are called volumic structuring elements. 44 / 61

45 Erosion of a function The erosion of a function is defined as the set-erosion of its shadow. E B (U(f (x))) = {(x, t) B (x,t) U(f (x))} Question : Given x what is the maximal value of t such that B (x,t) U(f (x))? Answer : Using a flat structuring element : the smallest value of f on its support B. E B (f )(x) = inf f (x u) u B U(f(x)) 45 / 61

46 Erosion with a volumic structuring element Heights b(u) must be in U(f (x)) E B (f )(x) = inf f (x u) b(u) u B f (x u) f (x u) b(u) inf f (x u) b( b(u) b(u ) B 46 / 61

47 Example of erosion (1) Original Eroded (disc of radius 3) The eroded image is darker, its pics shrink or disapear. 47 / 61

48 Examples of erosion (2) I Erosion with a flat structuring element having a circular support Original E2B (I ) E3B (I ) E5B (I ) E7B (I ) E9B (I ) 48 / 61

49 Dilatation of a function Defined the same way using the function s shadow δ B (U(f (x))) = {(x, t) B (x,t) U(f (x)) } Question : given x what is the maximal value of t such that B (x,t) U(f (x))? Answer : In the case of a flat structuring element : The maximal value of f on its support B : δ B (f )(x) = sup u B f (x u) 49 / 61

50 Dilatation with a volumic structuring element De façon symétrique à l érosion : f (x u) + b(u) δ B (f )(x) = sup u B f (x u) + b(u) x sup f (x u) + b(u) f (x u) b(u) b(u ) B 50 / 61

51 Example of dilatation (1) Original dilated (disc of radius 3) The dilated image is more bright. Narrow valleys disappear. 51 / 61

52 Example of dilatation (2) I Dilatation with a circular, flat structuring element Original δ2b (I ) δ3b (I ) δ5b (I ) δ7b (I ) δ9b (I ) 52 / 61

53 Morphological Residues Within set morphology a residue corresponds to : The symmetric difference between the initial set X and its transformation ψ(x ). The symmetric difference between two transformations ψ1 (X ) et ψ 2 (X ) of a set. Within functional morphology : Arithmetic difference between f (x) and ψ(f (x)), Arithmetic difference between ψ1 (f (x)) and ψ 2 (f (x)). When transformations ψ, ψ i are defined as erosion or dilatation we speak of morphological gradients. X E B (X ) =, X = + X E B (X ) 53 / 61

54 Morphological gradients Structuring element Set erosion and dilatation Properties of set erosion and dilatation Distance computations Erosion and dilatation of functions Type of structuring element Example of structuring element Erosion of a function Erosion with a volumic structuring element Examples of erosion Function dilatation Dilatation with a volumic structuring element Example of dilatation Morphological Residues Morphological Gradients Morphological Gradients between sets Functional internal Gradient Functional external Gradient Functional symmetric Gradient Morphological Laplacian Morphological Laplacian : Example 54 / 61

55 Morphological Gradients Three types of gradients : The symmetric morphological gradient (from Beucher) : B (X ) = δ B (X ) E B (X ) and B (f (x)) = δ B (f (x)) E B (f (x)) Internal (or by erosion) morphological gradient : B (X ) = X E B(X ) and B (f (x)) = f (x) E B(f (x)) External (or by dilatation) morphological gradient : + B (X ) = δ B(X ) X and + B (f (x)) = δ B(f (x)) f (x) 55 / 61

56 Morphological set gradients Sets : E B (X ) = X = + δb (X ) = + + Gradients : B (X )= + + B (X )= (X )= B 56 / 61

57 Internal morphological gradient I f (x) : I EB (f (x)) : I B (f (x)) : I EB (I ) B (I ) 57 / 61

58 External morphological functional gradient I f (x) : I δb (f (x)) : I + B (f (x)) : I B (f (x)) : I δb (I ) B (I ) 58 / 61

59 Symmetric morphological functional gradient I f (x) : I δb (f (x)) : I EB (f (x)) : I B (f (x)) : I B (f (x)) : I + B (f (x)) : I δb (I ) EB (I ) B (I ) 59 / 61

60 Morphological Laplacian The morphological Laplacian is defined as the residue of the external and internal gradients. f (x) : L B (f ) = + B (f ) B (f ) δb (f (x)) : E B (f (x)) : L B (f (x)) : B (f (x)) : + (f (x)) : 60 / 61

61 Morphological Laplacian : Example I B (I ) + B (I ) LB (I ) 61 / 61