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Beverly Greer
6 years ago
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Outils pour le traitement d image Synthèse de

(014-015) Analyse de textures par TO monogène (016)

(016) Thèse de Kévin Polisano (LJK) Marianne

(Gipsa-Lab) > TEXTURE MODELING BY GAUSSIAN FIELDS

1 Outils pour le traitement d image Synthèse de texture (013 [ ] 015) Super-résolution de lignes ( ) Analyse de textures par TO monogène (016) Super-résolution de sinusoïdes par TO monogène (016) Thèse de Kévin Polisano (LJK) Marianne Clausel, Valérie Perrier (LJK) Laurent Condat (Gipsa-Lab) > TEXTURE MODELING BY GAUSSIAN FIELDS WITH PRESCRIBED LOCAL ORIENTATION ANR ASTRES - ENS Lyon, le 19 novembre 015

The basic component : Fractional Brownian Field

complex Brownian measure H=0. H=0.7 Polisano et al.

2 The basic component : Fractional Brownian Field (FBF) Harmonizable representation B H e ix 1 (x) = R k k H+1 d W c ( ) [Samorodnitsky,Taqqu,1997] roughness indicator complex Brownian measure H=0. H=0.7 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation

3 Elementary field (Bonami-Estrade 003, B H 0, (x) = Biermé-Moisan- Richard 015) (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 =0 = Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 3

4 Elementary field B H 0, (x) = (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.7 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 4

5 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.6 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 5

6 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.5 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 6

7 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.4 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 7

8 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.3 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 8

9 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0. Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 9

10 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.1 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 10

11 B H 0, (x) = Elementary field (e ix 1) 1 [, ](arg 0 ) R k k H+1 d c W ( ) Texture orientation x 0 ~V x0 0 = 3 = 0.05 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 11

12 Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary field B H 0, (x) = (e ix 1) 1 [, ](arg 0 (x)) R k k H+1 dw c ( ) [Polisano et al.,014] Texture orientation x 0 ~V x0 f 1/ (x 0, ) = c 0, (x 0, arg ) k k H+1 The orientation may varies spatially. 0 is now a differentiable function on R Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 1

Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary field B H 0, (x) = (e ix 1) 1 [, ](arg 0

13 Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary field B H 0, (x) = (e ix 1) 1 [, ](arg 0 (x)) R k k H+1 dw c ( ) [Polisano et al.,014] Texture orientation x 0 ~V x0 (a) (b) Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 13

14 Tangent field B H 0, (x) = (e ix 1) 1 [, ](arg 0 (x)) R k k H+1 d c W ( ) Tangent field. For a random field X locally asymptotically self-similar of order H, X (x 0 + h) X (x 0 ) H L!!0 Y x0 Y x0 : tangent field of X at point x 0 R [Benassi,1997] [Falconer,00] Taylor s expansion Deterministic case Tangent field Stochastic case Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 14

15 Tangent field B H 0, (x) = (e ix 1) 1 [, ](arg 0 (x)) R k k H+1 d c W ( ) Theorem. The LAFBF B H 0, admits for tangent field Y x0 : Y x0 (x) = Y x0 (e ix 1) 1 [, ](arg 0 (x 0 )) R k k H+1 d c W ( ) elementary field with global orientation 0 (x 0 ) constant B H 0, (x 0 ) Y x0 (x = x 0 ) Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 15

16 Elementary field : simulation by turning bands Y x0 [n] (x) = (H) 1 nx i=1 p ic 0, (x 0, i )B H i (x u( i )) x 0 Different bands (1D FBM) are implied in the sum x 0 ~V x0 ~V x0 0 = 3 0 = The grayscale repartition completely changes between two close angles = 0.1 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 16

grayscale repartition completely changes between two close

17 LAFBF simulation by tangent fields [T.B] B H 0, (x 0 ) Y x0 (x = x 0 ) Vector field : 0 (x) = + x Orientation curves : 1 ln cos x + y 0 (y (x)=tan (x) ) The texture is properly oriented We observe bands artifacts The grayscale repartition completely changes between two close angles 0 = 67 0 = 55 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 17

18 Elementary field : simulation by Cholesky Variance and covariance of an elementary field. v H, 1, (x) = H 1 (H)C H, 1, (arg x)kxk H Cov (Y H, 1, (x), Y H, 1, (y)) = v H, 1, (x)+v H, 1, (y) v H, 1, (x y) Cholesky method. = LL T N (0, 1) Y L [Drawback : very time and memory consuming] The grayscale repartition is now the quite the same for two close angles use the same random vector for the computation of all tangent field Y 0 = 3 = = Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 18

19 Elementary field simulation Comparison Turning bands VS. Cholesky method The grayscale repartition completely changes between two close angles The grayscale repartition is now the quite the same for two close angles 0 = 3 = = Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 19

20 Elementary field simulation Comparison Turning bands VS. Cholesky method The «transversal» variance varies The «transversal» variance doesn t vary enough 0 = 3 = 0.1 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 0

expected ln 1 cos x + y 0 The grayscale repartition is now the quite the same for two close angles

21 LAFBF simulation by tangent fields [Chol] B H 0, (x 0 ) Y x0 (x = x 0 ) Vector field : 0 (x) = + x Orientation curves : Bands artifacts disappear The grayscales don t fit the orientation as we expected ln 1 cos x + y 0 The grayscale repartition is now the quite the same for two close angles 0 = 67 0 = 55 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 1

22 Why the grayscale curves are like that? Orientation curve Parallel lines of elementary fields rotated C 0 (x) = + x Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation

23 Why the grayscale curves are like that? C Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 3

24 Equations of the grayscale curves? B H 0, (x 0 ) Y x0 (x = x 0 ) Vector field : 0 (x) = + x Orientation curves : ln 1 cos x + y 0 Theoritical grayscale curves : C cos x + x tan(x) We observe the green curve look likes better but does not fit exactly the grayscale variations either Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 4

25 Why does it not fit exactly? C=0.8 C=0 The green lines is varying faster than the gray lines! Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 5

26 LAFBF simulation by tangent fields [Chol] B H 0, (x 0 ) Y x0 (x = x 0 ) Vector field : We observe the accurate orientations but they seem to appear for high frequencies 0 (x) = + y Orientation curves : arcsin(sin(y 0 )e x ) Definition of orientation in high frequencies? Wavelet decomposition? Tangent field formulation is well adapted to the simulation of this oriented model? Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 6

27 LAFBF simulation by tangent fields [Chol] B H 0, (x 0 ) Y x0 (x = x 0 ) Vector field : We observe the accurate orientations but they seem to appear for high frequencies 0 (x) = + y Orientation curves : arcsin(sin(y 0 )e x ) Orientation contenue dans les hautes fréquences (le champ tangent est HF) Définir l orientation locale comme orientation du champ tangent Definition of orientation in high frequencies? Wavelet decomposition? Tangent field formulation is well adapted to the simulation of this oriented model? Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 7

Local orientations estimation Monogenic Wavelet Histogram of

28 Local orientations estimation Monogenic Wavelet Histogram of orientations + Von Mises distribution Working on empirical covariance? Olhede S., Detection directionality in random fields using the monogenic signal, preprint 013. Richard F., Tests of isotropy for rough textures of trended images, preprint 014. Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation 8

Artifacts and Textured Region Detection

Artifacts and Textured Region Detection 1 Vishal Bangard ECE 738 - Spring 2003 I. INTRODUCTION A lot of transformations, when applied to images, lead to the development of various artifacts in them. In