Discrete Monogenic Functions in Image processing

Transcription

1 Discrete Function Teory Discrete Monogenic Functions in Image processing U. Käler CIDMA and Departamento de Matemática Universidade de Aveiro MOIMA Scloss Herrenausen Hannover, June 20-24, /34 U. Käler Discrete Monogenic Functions

2 Analytic signal Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae Applying to a given signal f (x) te Hilbert transform Hf (x) = 1 + f (y) dy, y R. 2π y x Consider te complex-valued function Z (x) = f (x) + ihf (x) = a(x)e iθ(x) Z (x) is te analytic signal, a(x) te amplitude and θ(x) te pase ω(x) = θ (x) is called instantaneous frequency Te pair (a, θ) is called canonical modulation pair of f (x) 2/34 U. Käler Discrete Monogenic Functions

3 Monogenic signal Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae Given signal f take Riesz transforms R i f Monogenic signal: f + e i R i f Amplitude: a = f 2 + (R i f ) 2 Local pase direction: u = Local pase angle: tan α = Local pase: Φ = αu ei R i f e i R i f (Ri f ) 2 f 3/34 U. Käler Discrete Monogenic Functions

4 Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae From Continuous to Discrete in iger dimensions Several variables approac: Bobenko/Mercat/Suris, M. Desbrun (discrete olomorpic functions on bricks) Several works in different directions: P. Leopardi, A. Stern (Dirac operators based on discrete exterior derivative) Simplicial Complexes Finite Element Exterior Calculus Going from simplicial complexes to more general meses Script Geometry 4/34 U. Käler Discrete Monogenic Functions

5 Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae From Continuous to Discrete - Clifford Analysis First approac in Clifford analysis: Gürlebeck/Sprößig, operator teory/potential teory; Based on Kircoff laws, one as te concept of a potential function satisfying to te discrete star Laplacian. ] f # (x) := 1 2 [ n i=1 (f # (x + e i ) + f # (x e i )) 2n f # (x). Recently: Cerejeiras, de Ridder, Faustino, Käler, Ku, Sommen (Dirac operator on Z d, upper alf lattice), etc. 5/34 U. Käler Discrete Monogenic Functions

6 Discrete Function Teory Setting Notations - 1 Motivation Discrete Upper/Lower Caucy Formulae Assumptions 1 witout loss of generality, we restrict ourselves d = 3; Tus, we consider te grid Z 3 wit ortonormal basis e k, k = 1,..., 3; 2 forward and backward differences ±j are given by +j f (m) = 1 (f (m + e j ) f (m)), j f (m) = 1 (f (m) f (m e j )) wit lattice constant > 0 and m = m 1 e 1 + m 2 e 2 + m 3 e 3 Z 3. 3 splitting basis element e k (k = 1, 2, 3) into two basis elements e k = e + k + e k corresponding to te forward and backward directions. 6/34 U. Käler Discrete Monogenic Functions

7 Discrete Function Teory Setting Notations - 2 Motivation Discrete Upper/Lower Caucy Formulae Elements e + k, e k, (k = 1, 2, 3) satisfy: e j e k + e k e j = 0, e + j e + k + e+ k e+ j = 0, e + j e k + e k e+ j = δ jk, wit δ jk te Kronecker symbol. Ten e + k, e k, (k = 1, 2, 3) form a Witt basis for te complexified Clifford algebra C 3. 4 Discrete Dirac operator D + = 3 factorizes te star-laplacian as j=1 e+ j +j + e j j wic =: m j=1 +j j = (D + ) 2. 5 Adjoint Dirac operator D + = 3 j=1 e+ j j + e j +j 7/34 U. Käler Discrete Monogenic Functions

8 Discrete Function Teory Discrete Fourier Transform Motivation Discrete Upper/Lower Caucy Formulae l p spaces (1 p < + ) ( ) 1/p u l p (Ω) iff u lp(ω) := u(m) p 3 < m Ω Discrete Fourier transform F : l p (Z 3 ) L p (R 3 ), F u(ξ) = e i m,ξ u(m) 3, ξ [ π/, π/] 3. m Z 3 wit inverse F 1 = R F were i) R is te restriction to te lattice Z 3 ii) Ff (x) = 1 (2π) 3 [ π/,π/] 3 e i x,ξ f (ξ)dξ, x R 3. 8/34 U. Käler Discrete Monogenic Functions

9 Discrete Function Teory Discrete Fundamental Solutions Motivation Discrete Upper/Lower Caucy Formulae Discrete fundamental solution of D + D + E + (m) = δ (m) = { 3 m = 0 0 m 0, m Z3 were δ is te discrete Dirac function. Remark tat te symbol of te discrete star Laplacian is known and tat F ( u)(ξ) = j=1 F (D + u)(ξ) := ξ F u(ξ) = wit ξ D ±j = 1 ( 1 e iξ j ). ( ) sin 2 ξj F u(ξ) =: d 2 F u(ξ) 2 3 e + j ξ j D + e j ξ+j D F u(ξ), j=1 9/34 U. Käler Discrete Monogenic Functions

10 Discrete Function Teory Computation as Fourier Integral Motivation Discrete Upper/Lower Caucy Formulae Tese leads to E + wit lim 0 ξ d 2 Lemma = R F ( ) ξ d 2 = = iξ ξ 2. 3 e + j R F j=1 ( ξ D j d 2 ) + e j R F ( ) ξ D +j d 2 For eac fix point m Z 3 of te lattice, tere exists a constant C > 0 suc tat E + (m) E(m) C m 3, m 0, were E is te fundamental solution of te Dirac operator in R 3. 10/34 U. Käler Discrete Monogenic Functions

11 Discrete Function Teory Discrete Borel-Pompeiu Formulae Motivation Discrete Upper/Lower Caucy Formulae From te discrete Stokes formula and translates of te fundamental solution we obtain Lemma (Upper Discrete Borel-Pompeiu Formula) For f l p (Z 2 Z + ; C 3 ), 1 p < +, we ave n Z 2 + [ E + (n m, m 3 )e f (n, 1) + E (n m, 1 m 3 )e 3 f (n, 0)] 3 n Z 2 Z + E + were m = (m, m 3 ). (n m) [ D + f (n) ] { 3 = 0, if m / Z 2 Z +, f (m), if m Z 2 Z +. Similar lemma olds for te lower alf Lattice Z 2 Z. 11/34 U. Käler Discrete Monogenic Functions

12 Discrete Function Teory Discrete Caucy Formulae Motivation Discrete Upper/Lower Caucy Formulae Lemma (Upper/Lower Discrete Caucy Formulae) 1 For f l p (Z 2 Z + ; C 3 ), 1 p < +, suc tat D + f = 0 we ave [ E + (n m, m 3 )e + 3 f (n, 1) + E + (n m, 1 m 3 )e 3 f (n, 0)] 3 n Z 2 { = 0, if m / Z 2 Z +, f (m), if m Z 2 Z +. 2 For f l p (Z 2 Z ; C 3 ), 1 p < +, suc tat D + f = 0 we ave [ E + (n m, 1 m 3 )e + 3 f (n, 0) + E + (n m, m 3 )e 3 f (n, 1)] 3 n Z 2 = { 0, if m / Z 2 Z, f (m), if m Z 2 Z. 12/34 U. Käler Discrete Monogenic Functions

13 Discrete Function Teory Discrete Caucy Transforms Motivation Discrete Upper/Lower Caucy Formulae Definition (Upper/Lower Caucy Transforms) 1 For a discrete l p-function f, 1 p < +, defined on te boundary layers (n, 0), (n, 1), n Z 2, we define te upper Caucy transform for m Z 3 Z + as C + [f ](m) = [ n Z E + 2 (n m, m 3 )e + 3 f (n, 1) +E + (n m, 1 m 3 )e 3 f (n, 0)] 3 ; 2 For a discrete l p-function f, 1 p < +, defined on te boundary layers (n, 1), (n, 0), unn Z 2, we define te lower Caucy transform for m Z 3 Z as C [f ](m) = [ n Z E + 2 (n m, 1 m 3 )e + 3 f (n, 0)+ +E + (n m, m 3 )e 3 f (n, 1)] 3. 13/34 U. Käler Discrete Monogenic Functions

14 Properties Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae 1 C + [f ] l p (Z 3 +, C 3 ); (resp. C [f ] l p (Z 3, C 3 )); 2 For all m Z 3 suc tat m 3 > 1 we ave D + C + [f ](m) = 0; 3 For all m Z 3 suc tat m 3 < 1 we ave D + C [f ](m) = 0; 4 From te discrete Caucy formulae we obtain a discrete equivalent of boundary beaviour of monogenic functions, tat is, for a discrete monogenic in te upper alf plane it olds [ E + (n m, 1)e + 3 f (n, 1) + E + (n m, 0)e 3 f (n, 0)] 3 = f (m, 1), n Z 2 wile for a discrete monogenic in te lower alf plane [ E + (n m, 0)e + 3 f (n, 0) + E + (n m, 1)e 3 f (n, 1)] 3 = f (m, 1). n Z 2 14/34 U. Käler Discrete Monogenic Functions

15 Boundary Data Discrete Function Teory Motivation Discrete Upper/Lower Caucy Formulae Visualization of te upper / lower boundary data. 15/34 U. Käler Discrete Monogenic Functions

16 Discrete Function Teory Symbols & Fourier Domain Space vs. Fourier domain - ( 1)- and 1-layers Te boundary values equation (at te layer m 3 = 1 (resp. m 3 = 1) of a function wic is discrete monogenic in te upper (resp., lower) alf plane [ n Z E + 2 (n m, 1)e f (n, 1) + E (n m, 0)e 3 f (n, 0)] 3 = f (m, 1) [ n Z E + 2 (n m, 0)e f (n, 0) + E (n m, 1)e 3 f (n, 1)] 3 = f (m, 1) becomes F E + (ξ, 1)e + 3 F f + (ξ, 1) + F E + (ξ, 0)e 3 F f + (ξ, 0) = F f + (ξ, 1) F E + (ξ, 1)e 3 F f (ξ, 1) + F E + (ξ, 0)e + 3 F f (ξ, 0) = F f (ξ, 1) 16/34 U. Käler Discrete Monogenic Functions

17 Discrete Function Teory Symbols & Fourier Domain Fourier symbols on te layers 1, 0, 1 Lemma Te Fourier symbols of te fundamental solution E + on te layers 1, 0, 1 are given by F E + (ξ, 0) = B + (e + 3 e 3 )C, ] [ ξ F E + (ξ, 0) = (A + B) + e + 3 C + e 3 d (A + 2B) 1, 2 ] [ ξ F E + (ξ, 1) = (A + B) e + 3 d (A + 2B) 1 e 3 2 C were A = ξ d ( 2 d d 2 d 2 ), B = ξ d 1 4+, C = d 2 d d /34 U. Käler Discrete Monogenic Functions

18 Discrete Function Teory Symbols & Fourier Domain Caracterization of b.v. of discrete monogenic functions From te last system we obtain Teorem Let f l p (Z 2, C 3 ), wit f = f 1 + e + 3 f 2 + e 3 f 3 + e + 3 e 3 f 4, f i : Z 2 C 2. Ten f is te boundary value of a discrete monogenic function in te discrete upper alf plane iif its 2D-Fourier transform F = F f satisfies d 4+ 2 d 2 2 F 1 + ξ d F 2 = 0, d 4+ 2 d 2 2 F 3 + ξ d (F 1 F 4 ) = 0. (1) 18/34 U. Käler Discrete Monogenic Functions

19 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Discrete Upper Hilbert Transform Teorem Let f l p (Z 2, C 3 ) be a boundary value of a discrete monogenic function in te upper alf space. Ten its 2D-Fourier transform F = F f satisfies ξ e 2 d d 2 3 F + e d d d 2 3 F = F. 2 Definition (Discrete upper Hilbert transform) H +f = F 1 ξ d e 3 2 F + e d d 2 3 d d 2 F f. 2 19/34 U. Käler Discrete Monogenic Functions

20 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Discrete Lower Hilbert Transform In a similar way, H f = F 1 ξ d e e d d 2 3 d 4 2 d 2 F f 2 wic satisfies H 2 f = f. Moreover (H + ) 2 = id = (H ) 2 20/34 U. Käler Discrete Monogenic Functions

21 Function Spaces Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Definition (Discrete Hardy spaces) We define te upper discrete Hardy space p + as te space of discrete functions f l p (Z 2, C 3 ) for wic te discrete 2D-Fourier transform fullfil d 4+ 2 d 2 F ξ F d 2 = 0, d 4+ 2 d 2 F ξ (F d 1 F 4 ) = 0, (2) and te lower discrete Hardy space p as te space of discrete functions f l p (Z 2, C 3 ) for wic te discrete 2D-Fourier transform fullfil d 4+ 2 d 2 F 2 2 ξ F d 1 = 0 (3) d 4+ 2 d 2 (F 2 1 F 4 ) ξ F d 3 = 0. 21/34 U. Käler Discrete Monogenic Functions

22 Discrete Function Teory Plemelj projections Discrete Hilbert Transforms Discrete Hardy spaces Due to te properties of H + and H we can introduce te projectors into te respective Hardy spaces Definition P + = 1 2 (I + H +), Q + = 1 2 (I H +). P = 1 2 (I + H ), Q = 1 2 (I H ). Lemma f + p iff P + f = f ; f p iff P f = f. 22/34 U. Käler Discrete Monogenic Functions

23 Discrete Function Teory Decomposition - Remarks Discrete Hilbert Transforms Discrete Hardy spaces We conclude wit te following remarks 1 we ave convergence of our discrete Caucy transform to te continuous Caucy transform; 2 in te limit case ( 0) te symbols of bot H ± tend to i ξ ξ ; 3 in te limit case ( 0) we obtain P + f + P f = f as in te continuous case. 23/34 U. Käler Discrete Monogenic Functions

24 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Decomposition into Hardy Spaces An arbitrary function f l p (Z 2, C 3 ) can be decomposed into a pair of functions P + f and Q + f were P + f p +, i.e. it can be extended to te zero layer by ξ e 3 F +,0 = A + F +,1 = d d d 2 2 ( ) e 3 F +,1 1 + e 3 e+ 3 F +, /34 U. Käler Discrete Monogenic Functions

25 Discrete Function Teory Upper and lower trace operators Discrete Hilbert Transforms Discrete Hardy spaces Upper and lower trace operator Given f l p (Z 3 ), we define te i) upper trace operator tr + : l p (Z 3 ) l p (Z 2 ) l p (Z 2 ) as tr + [f ] := ( e 3 A +[ e + 3 f 1 ], e + 3 f 1), wit f 1 (m) := f ((m, 1)). ii) lower trace operator tr : l p (Z 3 ) l p (Z 2 ) l p (Z 2 ) as tr [f ] := ( e + 3 A [f 1 ], e 3 f 1), wit f 1 (m) := f ((m, 1)). 25/34 U. Käler Discrete Monogenic Functions

26 Discrete Function Teory Upper and lower trace operators Discrete Hilbert Transforms Discrete Hardy spaces Upper and lower trace operator Given f l p (Z 3 ), we define te i) upper trace operator tr + : l p (Z 3 ) l p (Z 2 ) l p (Z 2 ) as tr + [f ] := ( e 3 A +[ e + 3 f 1 ], e + 3 f 1), wit f 1 (m) := f ((m, 1)). ii) lower trace operator tr : l p (Z 3 ) l p (Z 2 ) l p (Z 2 ) as tr [f ] := ( e + 3 A [f 1 ], e 3 f 1), wit f 1 (m) := f ((m, 1)). 25/34 U. Käler Discrete Monogenic Functions

27 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Upper and lower trace operators-2 Lemma Given f l p (Z 3 ), ten C + tr + [C + tr + [f ]] = C + tr + [f ]; C tr [C tr [f ]] = C tr [f ]. Definition - Boundary data operators Given g l p (Z n 1 ), we define te upper boundary generator G + : l p (Z n 1 ) l p (Z n 1 ) l p (Z n 1 ) as G + [g] := ( e n A + [ e + n g], e + n g ). lower boundary generator G : l p (Z n 1 ) l p (Z n 1 ) l p (Z n 1 ) as G [g] := ( e + n A g, e n g ). 26/34 U. Käler Discrete Monogenic Functions

28 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Discrete Riemann boundary value problem I Problem I Te problem D + f (n) = 0, n Z 3 \ { n 3 = 0 }, f ( n, 1 ) = g(n), n Z 2. wit g + p (Z 2 ) as a unique solution given by te respective Hardy/Plemelj-projections. 27/34 U. Käler Discrete Monogenic Functions

29 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Discrete Riemann boundary value problem II Problem II - Jump problem { D + f (m) = 0, m Z n \ {m n = 0}, e n A + ( f ( n, 1 )) e + n A ( f ( n, 1 )) = g(m), m Z n 1. as a unique solution given by { C f (m) = + [e n + e n g, en A 1 + ( e n g)](m), m n +1, C [en e n g, e n + A 1 (e n g)](m), m n 1. 28/34 U. Käler Discrete Monogenic Functions

30 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Continuous case - Monogenic Signal: f M = trc[f ] = Ae ωu, were A is amplitude, u is pase direction and ω is pase angle Discrete case - Discrete Monogenic Signal: f M = tr + C + [f ] 29/34 U. Käler Discrete Monogenic Functions

31 Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces - Calçadas de Portugal 30/34 U. Käler Discrete Monogenic Functions

32 Discrete Function Teory Calçadas de Portugal Discrete Hilbert Transforms Discrete Hardy spaces First order discrete Riesz transforms of te original image (image pixel, wit computing time: ca. 1, sec) 31/34 U. Käler Discrete Monogenic Functions

33 Discrete Function Teory Calçadas de Portugal Discrete Hilbert Transforms Discrete Hardy spaces Absolute value and pase of te original image 32/34 U. Käler Discrete Monogenic Functions

34 References Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces (1) Cerejeiras, Käler, Ku, Sommen, Discrete Hardy spaces, Journal of Fourier Analysis and Applications 20 (2014), (2) Cerejeiras, Käler, Ku, Discrete Hilbert boundary value problems on alf lattices, Journal of Difference Equations and Applications 21 (2015), (3) Gürlebeck, Hommel, On finite difference Dirac operators and teir fundamental solutions, Adv. Appl. Cliff. Alg. 11(S2) (2001), (4) De Ridder, De Scepper, Käler, Sommen, Discrete function teory based on skew Weyl relations, Proceeding of te American Matematical Society 138(9) (2010), (5) De Ridder, Discrete Clifford Analysis, PD Tesis, Gent University, /34 U. Käler Discrete Monogenic Functions

35 Aknowledgments Discrete Function Teory Discrete Hilbert Transforms Discrete Hardy spaces Tank you for your attention! Tis work is supported by Portuguese funds troug te CIDMA - Center for Researc and Development in Matematics and Applications, and te Portuguese Foundation for Science and Tecnology (FCT - Fundação para a Ciência e a Tecnologia ), witin project UID/MAT/04106/ /34 U. Käler Discrete Monogenic Functions