Autosimilarité et anisotropie : applications en imagerie médicale

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1 1 Autosimilarité et anisotropie : applications en imagerie médicale Hermine Biermé journées MAS, Bordeaux, 03/09/2010 ANR-09-BLAN mataim [F. Richard (MAP5)] ( Coll: INSERM U 658 [Pr. C.L. Benhamou (CHRO)], INSERM ERI-20 [Pr. F. Clavel (IGR)] hermine.bierme@mi.parisdescartes.fr

2 OUTLINES 2 Outlines 1 Fractal analysis of medical images Examples Variogram method Fractional Brownian motion modeling Stochastic modeling for images Restriction on a line Fractional Brownian field Anisotropic Gaussian field Validation for bone radiographs

3 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 3 1 Fractal analysis of medical images ROI medical images = textures Goal: use texture analysis to extract diagnostically meaningful information One tool: fractal analysis to characterize texture via statistical self-similarity or scale invariance through a fractal index H (0, 1) Numerous methods and studies! [Lopes and Betrouni, 2009]

4 1 FRACTAL ANALYSIS OF MEDICAL IMAGES Examples Mammography dense breast tissue fatty breast tissue Validation of self-similarity using power spectrum method H [0.33, 0.42] [Heine et al, 2002]. Discrimination of dense H [0.55, 0.75] and fatty H [0.2, 0.35] breast tissue using WTMM method [Kestener et al, 2001]

5 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 5 Trabecular bone microarchitecture Dataset of 211 high-resolution digital X-ray images of calcaneum (a heel bone) with standardized acquisition procedure [Lespessailles et al., 2007]: ROI location control case osteoporotic case Validation of self-similarity using variogram and power spectrum methods on calcaneous bone [Benhamou et al, 94], on cancellous bone [Caldwell et al, 94] Discrimination of osteoporotic cases H mean = ± (osteoporotic cases)/ H mean = ± (control cases) [Benhamou et al, 2001]

6 1 FRACTAL ANALYSIS OF MEDICAL IMAGES Variogram method Let us consider the discrete image I(k, l), 0 k n 1, 0 l n 1. Extract a line of the image I θ (k), 0 k n θ 1 for θ a direction Compute v θ (u) = 1 n θ u n θ 1 u k=0 (I θ (k + u) I θ (k)) 2, 1 u n θ 1.

7 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 7 Average along a set of lines with the same direction v θ (u) and plot log v θ (u) versus log(u). Fractal index H θ = slope(θ)/2 Example on bone radiographs (211 cases): 20 average /+ std deviation 20 average /+ std deviation 20 average /+ std deviation log(v u ) 17 log(v u ) 17 log(v u ) log(u) log(u) log(u) θ = (1, 0) (horizontal) θ = (0, 1) (vertical) θ = (1, 1)/ 2 (diagonal) H θ = 0.51 ± 0.08 H θ = 0.56 ± 0.06 H θ = 0.51 ± 0.08

8 1 FRACTAL ANALYSIS OF MEDICAL IMAGES Fractional Brownian motion modeling Let (Ω, A, P) be a probability space. For H (0, 1), the fractional Brownian motion [Kolmogorov, 1940], [Mandelbrot and Van Ness, 1968] B H = {B H (t) ; t R} is a zero mean Gaussian random process such that ( t R, B H (t) : Ω R real random variable N 0, t 2H). and Cov (B H (t), B H (s)) = 1 2 ( t 2H + s 2H t s 2H). Properties : B H (0) = 0 a.s. Stationary increments: t 0 R, B H (. + t 0 ) B H (t 0 ) fdd = B H (.) Self-similarity of order H: λ > 0, B H (λ.) fdd = λ H B H (.).

9 1 FRACTAL ANALYSIS OF MEDICAL IMAGES H = 0.3 H = 0.5 H = 0.7 H= Self-similarity/Hölder regularity/fractal dimension : 2 H Numerous estimators of H. Law characterized by the variogram v H (t) = Var (B H (t)) = t 2H. ( Spectral representation B H (t) = c H R e itξ 1 ) ξ H 1/2 dw(ξ) v H (t) = c 2 H e itξ 1 2 ξ 2H 1 dξ R spectral density = c 2 H ξ 2H 1 [1/f process].

10 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 10 Estimation Let us observe { I θ (k) = c θ B H ( k n) ;0 k n 1 }. V 1,u (B H ) := c2 θ n u E(V 1,u (B H )) = n 1 u k=0 c2 θ n u ( ( ) k + u B H n n 1 u k=0 v H ( u n ) B H ( k n )) 2 = c 2 θn 2H u 2H. Ergodic Theorem: V 1,u (B H ) /E(V 1,u (B H )) 1 a.s. n + Ĥ = 1 ( ) 2 log V1,u (B H ) ( u ) / log V 1,v (B H ) v Remark: ( v u ) H n =variogram at small scales = Hölder regularity spectral density at high frequencies

11 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 11 Rate of convergence: asymptotic normality? P K (x) = (1 x) K = K a l x l filter of order K 1 l=0 K ( ) t + lu t R, P K(B u H)(t) = a l B H n l=0 = c H e i tξ n P R uξ i (e n stationary ergodic Gaussian process Generalized quadratic variations [Istas and Lang, 1997] ) K ξ H 1 2 dw(ξ) V K,u (B H ) = 1 n Ku n Ku+1 k=0 (P K u (B H)(k)) 2 with, E(V K,u (B H )) = c K,H n 2H u 2H. Ergodic Theorem: V K,u (B H ) /E(V K,u (B H )) n + 1 a.s.

12 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 12 ( ) VK,u (B H ) n Ku E ( V K,u (B H ) ) 1 fdd = n Ku 1 1 n Ku k=0 H 2 (X u ) (k), with H 2 (x) = x 2 1 = 2th-Hermite polynomial and X u is a centered stationary Gaussian time series which admits for spectral density 2π ( P F Xu (ξ) = e iuξ) 2K ξ + 2kπ 2H 1, c K,H u 2H ie Cov(X u (k + k ), X u (k )) = 1 e ikξ F Xu (ξ)dx. 2π π Theorem:[Breuer-Major, 1983] If ( ) R P e iuξ 4K ξ 4H 2 dξ < +, then, F Xu L 2 ([ π, π]) and, as n +, ( ) VK,u (B H ) n Ku E ( V K,u (B H ) ) 1 N(0, σu) 2 with σ 2 u = 2F 2 X u (0). k Z +π

13 1 FRACTAL ANALYSIS OF MEDICAL IMAGES 13 New proof by [Nualart, Ortiz-Latorre, 2008], [Nourdin, Peccatti, 2009] + vectorial CLT from [Peccatti, Tudor, 2004] Theorem:[B., Bonami and León, 2010] Let Y (t) = ( R e itξ 1 ) f(ξ) 1/2 dw(ξ), with ( c f(ξ) = ξ + O 2H+1 ξ + 1 ξ 2H+1+γ ), Then, Ĥ = 1 ( ) 2 log VK,u (Y ) ( u ) /log V K,v (Y ) v H a.s. n + + asymptotic normality for K > H + 1/4 and γ > 1/2.

14 2 STOCHASTIC MODELING FOR IMAGES 14 2 Stochastic modeling for images {X(x); x R 2 } a zero mean Gaussian random field Spectral representation: [Bonami and Estrade, 2003] ( ) X(x) = e ix ξ 1 f(ξ) 1/2 dw(ξ), x R 2 R 2 well defined for f 0 even and R 2 min(1, ξ 2 )f(ξ)dξ < +. Properties : X(0) = 0 a.s. Stationary increments: x 0 R 2, X(. + x 0 ) X(x 0 ) fdd = X(.) Cov (X(x), X(y)) = 1 (v 2 X(x) + v X (y) v X (x y)) with v X (x) = e ix ξ 1 2 f(ξ)dξ R 2 f spectral density

15 2 STOCHASTIC MODELING FOR IMAGES Restriction on a line For all direction θ S 1, {X(tθ); t R} is a zero mean Gaussian random process with spectral density given by R θ f(τ) = f(τθ + γθ )dγ a.e. τ R Self-similarity: R θ f(τ) = c θ τ 2H 1 f(ξ) = c(arg(ξ)) ξ 2H 2, a.e. ξ R 2 R λ > 0, X(λ ) fdd = λ H X( ). Isotropy: R θ f = R θ f for all θ S 1 f is radial M O 2 (R), X(M ) fdd = X( ).

16 2 STOCHASTIC MODELING FOR IMAGES Fractional Brownian field f(ξ) = c H ξ 2H 2 a.e. ξ R 2 and x R 2, v X (x) = Var(X(x)) = x 2H. Exact method of simulation [Stein, 2002] H = 0.3 H = 0.5 H = 0.7

17 2 STOCHASTIC MODELING FOR IMAGES 17 Detectability of spots on mammograms [Grosjean, Moisan, 2009] Simulated spots with identical contrast on a real mammogram Burgess Law [Burgess et al, 2001] Link between size and contrast for human perception of opacities and mammograms

18 2 STOCHASTIC MODELING FOR IMAGES 18 Simulated spots with identical contrast WN H = 0.3 H = 0.7 radius 5 radius 10 radius 50

19 2 STOCHASTIC MODELING FOR IMAGES Anisotropic Gaussian field f(ξ) = c(arg(ξ)) ξ 2H 2 a.e. ξ R 2 x R 2, v X (x) = Var(X(x)) = C(arg(x)) x 2H. C = topothesy function [Davies and Hall, 1999] with θ, C(θ) > 0 Simulations for H = 0.4 [Coll: Moisan, Richard (MAP5)] c(θ) = 1 [π/4,3π/4] ( θ ) c(θ) = sin(θ) c(θ) = cos(θ) Identification of c [Istas, 2007] Anisotropy test for C [Coll: Bonami, León, Richard]

20 2 STOCHASTIC MODELING FOR IMAGES 20 Anisotropic self-similarity? Problem: Let θ 1 = (1, 0) and θ 2 = (0, 1). Find X such that {X(tθ 1 ); t R} =FBM of order H 1 {X(tθ 2 ); t R} =FBM of order H 2 with H 1 < H 2 Remark: It is not possible to get more than 2 different directions due to S.I. Toy example solution: X(x) = X(x 1, x 2 ) = B H1 (x 1 ) + B H2 (x 2 ) with B H1 and B H2 ind. FBMs. Then v X (x) = x 1 2H 1 + x 2 2H 2 no spectral density. Note that λ > 0, v X (λx 1, λ a x 2 ) = λ 2H 1 v X (x 1, x 2 ) with a = H 1 /H 2.

21 2 STOCHASTIC MODELING FOR IMAGES 21 Operator scaling property For a = H 1 /H 2, λ > 0, {X(λx 1, λ a x 2 ); (x 1, x 2 ) R 2 } fdd = λ H 1 {X(x 1, x 2 );(x 1, x 2 ) R 2 } A solution with spectral density f: f(λξ 1, λ a ξ 2 ) = λ β f(ξ 1, ξ 2 ), β = 2H a One example: f(ξ 1, ξ 2 ) = ( ξ ξ 2a 2 ) β/2 Theorem:[B., Meerschaert, Scheffler, 2007]: X is well defined and satisfies the operator scaling property fdd λ > 0, X(λ E ) = λ H 1 X( ), with E = a

22 2 STOCHASTIC MODELING FOR IMAGES 22 Simulations for H 2 = 0.6 H 1 = 0.55 H 1 = 0.5 H 1 = 0.45 What for other directions? If θ = (θ 1, θ 2 ) S 1 with θ 1 0 and θ 2 0, R θ f(p) + c θ p 2H 1 1 with c θ = 1 θ 1 R (s 2a + θ 2 1 ) β/2 ds v θ (t) = Var (X(tθ)) 0 d θ t 2H 1 with d θ > 0.

23 2 STOCHASTIC MODELING FOR IMAGES Validation for bone radiographs [Benhamou, B., Richard, 2009] Implementation issues Black = out of lattice. Precision of red = 1, green = 2 Estimation on oriented lines without interpolation. Precision is not the same in all directions. Accuracy of orientation analysis Precision of the image.

24 2 STOCHASTIC MODELING FOR IMAGES 24 Comparison of the index in different directions H H H H 1 H 1 H 3 H θ3 vs H θ1 H θ4 vs H θ1 H θ4 vs H θ H H H H 1 H 2 H 2 H θ2 vs H θ1 H θ3 vs H θ2 H θ4 vs H θ2 Directions: 1: θ 1 = (1, 0) (horizontal), 2: θ 2 = (0, 1) (vertical), 3: θ 3 = (1, 1)/ 2 (diagonal), 4: θ 4 = ( 1, 1)/ 2 (diagonal).