CHARACTERIZATION OF LINEAR AND MORPHOLOGICAL OPERATORS. Gerald Jean Francis Banon

Transcription

1 OF LINAR AND MORPHOLOGICAL OPRATORS Gerald Jean Francis Banon Instituto Nacional de Pesquisas spaciais INP Divisão de Processamento de Imagens DPI CP , São José dos Campos, SP, Brazil São José dos Campos, October, G. Banon

2 CONTNT Introduction Image processing objects Image operations Image decomposition Morphism of commutative monoids Separable measures Translation invariant operators Window operators tiw operator characterization tiw morphism characterization Linear operator characterization Morphological operator characterization rosion example References 2 G. Banon

3 INTRODUCTION (1/1) Linear and morphological operators are important pieces in image and signal processing. Linear operators are models for many optical sensors and they can be used as filters for sensor simulation and image restoration. Combinations of morphological operators can be used as filters for image segmentation. Operator characterization shows where are the differences and the similarities between the linear and morphological operators. Operator characterization helps to understand how these operators are built. 3 G. Banon

4 IMAG PROCSSING OBJCTS (1/4) f : K (image) K { } f : K l : (gray scale transform or lut) K { } l : K K K K 4 G. Banon

5 IMAG PROCSSING OBJCTS (2/4) m : 2 1 (domain transform) 1 2 m : 2 1 : (measure) Read Pixel Measure f x (f) f (x) x x K x K { } 5 G. Banon

6 IMAG PROCSSING OBJCTS (3/4) f x : K K (impulse image creator) s f x, s (u) e K { } K x e f x x s if u x e otherwise. : 1 2 (operator) f Radiometric Operator l (f) l f l K K 6 G. Banon

7 IMAG PROCSSING OBJCTS (4/4) (,, o) Geometric Operator m x : o m x (y) y x m x x f x (f) f m x K K W (,, o) Geometric Operator W o m x : W m x (y) y x m x x f x (f) f m x m x (W) K K W 7 G. Banon

8 IMAG OPRATIONS (1/3) Operations on gray scale (K,,e) is a commutative monoid iff x y y x y x y x commutativity x x y z (x y) z y z x (y z) associativity e x e x e x null element 8 G. Banon

9 IMAG OPRATIONS (2/3) Let K R (set of real numbers) then (K,,,) is a linear vector space and (K,,) is a commutative monoid. Let K [,k] Z and let s t max{s,t} and s t min{s,t}, then (K,,) is a lattice and (K,,) and (K,,k) are two commutative monoids. 9 G. Banon

10 IMAG OPRATIONS (3/3) Operation extension to images f g f g (1 5) (2 2) e e (5 6) (4 3) (K,,e) is a commutative monoid. Let K R, then (K,,o) is a commutative monoid. Let K [,k] Z, then (K,,o) and (K,,i) are commutative monoids. 1 G. Banon

11 IMAG DCOMPOSITION (1/1) Image decomposition (consequence of (K,,e) being a commutative monoid) Let {x 1,x 2,,x n } and let K [,k]. xample: n 4 and k x1 f x x x 1 x 1 x 2 x 3 x 4 x n f x n x n x n 11 G. Banon

12 MORPHISM OF COMMUTATIV MONOIDS (1/1) Let (A, A,e A ) and (B, B,e B ) be two commutative monoids. A mapping from A to B is a morphism iff A B e A e B 12 G. Banon

13 SPARABL MASURS (1/1) Let {x 1,x 2,,x n }. A measure from K 1 to is separable if there exists a family {l 1,l 2,,l n } of luts from to such that f(x 1 ) x1 l 1 f x 1 f(x n ) (f) x n l n x n Proposition (separability morphism) The measures which are morphisms of commutative monoids are separable and their characteristic luts are morphisms too. (This is a consequence of the image decomposition.) 13 G. Banon

14 TRANSLATION INVARIANT OPRATORS (1/1) Let (,, o) be an Abelian group. is translation invariant (ti operator) iff, for any x in x x x x 14 G. Banon

15 WINDOW OPRATORS (1/1) Let B be a mapping from 2 to ( 1 ). is a window operator (w operator) iff, for any y in 2, fb(y) gb(y) (f)(y) (g)(y). Let be w operator, then there exists a family ( ) of measures such that (f)(y) (y)(fb(y)) B(y) y f (f) 1 2 fb(y) (y) (y)(fb(y)) 15 G. Banon

16 tiw OPRATOR (1/1) Let {x 1,x 2,,x n }. Let be tiw operator, then there exists a measure such that. e x1 f x1 f x 1 W x 1 (f) e x n f x n W x n x n 16 G. Banon

17 tiw MORPHISM (1/1) Let W {u 1,u 2,,u n }. Let be tiw morphism of commutative monoids, then there exists a family {l 1,l 2,,l n } of luts which are morphisms of commutative monoids such that u1 f u 1 l 1 (f) u n u n l n 17 G. Banon

18 LINAR OPRATOR (1/3) Let K R, then (K,,,o) is a linear vector space. A linear operator is a morphism from (K,,,o) to (K,,,o). In other words, for any f, g in K and in K, (f g) (f) (g) and ( f) (f). 18 G. Banon

19 LINAR OPRATOR (2/3) Let l be linear lut, then there exists a real number such that s l l(s) s s R l (, ) R 43 R l (, ) R G. Banon

20 LINAR OPRATOR (3/3) Let W {u 1,u 2,,u n }. Let be linear tiw operator on R, then there exists a family { 1, 2,, n } of real numbers such that u1 f u 1 1 (f) R R u n u n n f f* (h\) * h\ h : u i i 2 G. Banon

21 MORPHOLOGICAL OPRATOR (1/7) Let [,k 1 ] Z, [,k 2 ] Z and let s t max{s,t}, then (K 1,,o) and (K 2,,o) are two commutative monoids. A dilation is a morphism from (K 1,,o) to (K 2,,o). In other words, for any f, g in K 1, (f g) (f) (g) and (o) o. o o 21 G. Banon

22 MORPHOLOGICAL OPRATOR (2/7) Let l be a lut which is a dilation from {,1} to {,1}, then there exists an integer number in {,1} such that s l l(s) s s G. Banon

23 MORPHOLOGICAL OPRATOR (3/7) Let W {u 1,u 2,,u n }. Let be tiw operator which is a dilation on{,1}, then there exists a family { 1, 2,, n } of s and 1 s such that u1 f u 1 1 (f) {, 1} {, 1} u n u n n X X B B B {x i : i 1} 23 G. Banon

24 MORPHOLOGICAL OPRATOR (4/7) Let [,k 1 ] Z, [,k 2 ] Z and let s t min{s,t}, then (K 1,,i), and (K 2,,i) are two commutative monoids. An erosion is a morphism from (K 1,,i) to (K 2,,i). In other words, for any f, g in K 1, (f g) (f) (g) and (i) i. i i 24 G. Banon

25 MORPHOLOGICAL OPRATOR (5/7) Let l be a lut which is an erosion from {,1} to {,1}, then there exists an integer number in {,1} such that s l l(s) s s G. Banon

26 MORPHOLOGICAL OPRATOR (6/7) Let W {u 1,u 2,,u n }. Let be tiw operator which is an erosion on{,1}, then there exists a family { 1, 2,, n } of s and 1 s such that u1 f u 1 1 (f) {, 1} {, 1} u n u n n X X B B B {x i : i 1} 26 G. Banon

27 MORPHOLOGICAL OPRATOR (7/7) Non binary case Let be tiw operator which is a dilation (erosion) from [,k 1 ] to [,k 2 ], then there exists a family {l 1,l 2,,l n } of luts which are dilations (erosions) such that u1 f u 1 l 1 (f) u n u n l n l is a dilation l is increasing and l() l is an erosion l is increasing and l(k 1 ) k 2 xamples of morphological luts (k 1 k 2 7) (see next transparency) 27 G. Banon

28 dilation erosion 7 K Heijmans Heijmans t t 1 1 s s 1 1 K 7 Sinha & Dougherty Heijmans Heijmans t t 2 1 s s 2 1 K 7 Bloch & Maitre Baets et al. Banon t t 3 3 s s 3 3 K 7 Bloch & Maitre Banon Baets et al. t t 4 3 s s K 7 K 28 G. Banon

29 ROSION XAMPL (1/1) Non binary case W f (e x ) (f) e 2 e 1 e e 1 e G. Banon

30 RFRNCS G. J. F. Banon, Characterization of translation invariant elementary operators for gray level morphology, Neural, Morphological, and Stochastic Methods in Image and Signal Processing, ditors:. Dougherty, F. Preteux and S. Shen, SPI vol. 2568, <dpi.inpe.br/banon/1997/ > G. J. F. Banon, Characterization of translation invariant elementary morphological operatores between gray level images, Research Report INP 5616 RPQ/671, Instituto Nacional de Pesquisas spaciais, São José dos Campos, SP, Brazil, <dpi.inpe.br/banon/1995/ > G. J. F. Banon, Formal introduction to digital image processing, Booklet INP 6969 PUD/34, Instituto Nacional de Pesquisas spaciais, São José dos Campos, SP, Brazil, p. <dpi.inpe.br/banon/1998/ > H. J. A. M. Heijmans, Theoretical aspects of gray level morphology, I trans. on PAMI, Vol. PAMI 13, No. 6, pp , June, H. J. A. M. Heijmans, Morphological Image Operators, Advances in letronics and lectron Physics. Supplement 24. Academic Press, Inc., Boston, G. Banon