A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

Biermé, Hermine; Moisan, Lionel; Richard, Frédéric

doi:10.1080/10618600.2014.946603

Cited by 14 publications

(35 citation statements)

References 35 publications

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“…We observe that the tangent field Y x0 is no more and no less than an elementary field using the terminology of [13]. This result shall be crucial when simulating this Gaussian model as detailed in the next section.…”

Section: Tangent Fields At Every Pointmentioning

confidence: 67%

“…To define stationary anisotropic models with global orientation α 0 , one can set h ≡ H in (4) and: for some 0 < α π/2. Note that we then recover the elementary fields of [13], which are a particular case of AFBF. When α = π/2, this model corresponds to the usual isotropic FBF of Hurst index H ( Fig.…”

Section: General Anisotropic Self-similar Gaussian Fieldsmentioning

confidence: 99%

“…But their high complexity is a real problem to produce large textures, and the covariance function is not explicitly known in general case. With respect to the AFBF, a recent fast method has been proposed in [13], called the turning band method, and used here to simulate the textures of Fig. 2, with global orientation α 0 = 0.…”

Section: General Anisotropic Self-similar Gaussian Fieldsmentioning

confidence: 99%

“…The simulation of a LAFBF will first require the simulation of a tangent field at every point x 0 . We follow below the methodology of [13] using the turning bands method.…”

Section: Simulation Of Locally Anisotropic Fractional Brownian Fieldmentioning

confidence: 99%

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Texture modeling by Gaussian fields with prescribed local orientation

Polisano

Clausel

Perrier

et al. 2014

2014 IEEE International Conference on Image Processing (ICIP)

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This paper presents a new framework for oriented texture modeling. We introduce a new class of Gaussian fields, called Locally Anisotropic Fractional Brownian Fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation and a turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation.

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Section: Tangent Fields At Every Pointmentioning

confidence: 67%

Section: General Anisotropic Self-similar Gaussian Fieldsmentioning

confidence: 99%

Section: General Anisotropic Self-similar Gaussian Fieldsmentioning

confidence: 99%

“…The simulation of a LAFBF will first require the simulation of a tangent field at every point x 0 . We follow below the methodology of [13] using the turning bands method.…”

Section: Simulation Of Locally Anisotropic Fractional Brownian Fieldmentioning

confidence: 99%

See 2 more Smart Citations

Texture modeling by Gaussian fields with prescribed local orientation

Polisano

Clausel

Perrier

et al. 2014

2014 IEEE International Conference on Image Processing (ICIP)

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“…We conducted some experiments on AFBF (see (2)) simulated using the turningband method developed by Biermé et al (2015). This method was applied with approximately 500 bands to simulate field realizations on a grid of size 100 × 100.…”

Section: Numerical Resultsmentioning

confidence: 99%

Tests of isotropy for rough textures of trended images

Richard¹

2016

STAT SINICA

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Abstract:In this paper, we propose a statistical methodology to test whether the texture of an image is isotropic or not. This methodology is based on the well-known quadratic variations defined as averages of square image increments. Specific to our approach, these variations are computed in different directions using grid-preserving image rotations. We study these variations asymptotically in a framework of intrinsic random fields allowing us to take into account the presence of polynomial trends in images. We establish a convergence result linking variation and scale logarithms through an asymptotic Gaussian linear model. This model involves direction-dependent intercepts which are equal when textures are isotropic. Hence, we test the texture isotropy using Fisher tests that check the validity of the assumption of the intercept equality. These tests are validated using 6000 realizations of anisotropic fractional Brownian fields simulated on a grid of size 100 × 100. Results show that more than 70% of anisotropic cases can be detected with less than 1% of misclassified isotropic cases. Eventually, we apply our methodology to the segmentation of some biological microscopic images.Key words and phrases: texture analysis, image processing, isotropy, anisotropy, intrinsic random field, statistical test, fractional Brownian field, quadratic variations, biological imaging, microscopy.

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Introduction to Random Fields and Scale Invariance

Biermé

2019

Stochastic Geometry

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In medical imaging, several authors have proposed to characterize roughness of observed textures by their fractal dimensions. Fractal analysis of 1D signals is mainly based on the stochastic modeling using the famous fractional Brownian motion for which the fractal dimension is determined by its so-called Hurst parameter. Lots of 2D generalizations of this toy model may be defined according to the scope. This lecture intends to present some of them. After an introduction to random fields, the first part will focus on the construction of Gaussian random fields with prescribed invariance properties such as stationarity, self-similarity, or operator scaling property. Sample paths properties such as modulus of continuity and Hausdorff dimension of graphs will be settled in the second part to understand links with fractal analysis. The third part will concern some methods of simulation and estimation for these random fields in a discrete setting. Some applications in medical imaging will be presented. Finally, the last part will be devoted to geometric constructions involving Marked Poisson Point Processes and shot noise processes. Definition 2. The distribution of (X t) t∈T is given by all its finite dimensional distribution (fdd) ie the distribution of all real random vectors (X t 1 ,. .. , X t k) for k ≥ 1,t 1 ,. .. ,t k ∈ T. Note that joint distributions for random vectors of arbitrary size k are often difficult to compute. However we can infer some statistics of order one and two by considering only couples of variables. Definition 3. The stochastic process (X t) t∈T is a second order process if E(X 2 t) < + ∞, for all t ∈ T. In this case we define • its mean function m X : t ∈ T → E(X t) ∈ R; • its covariance function K X : (t, s) ∈ T × T → Cov(X t , X s) ∈ R. We have therefore a very simple condition on variogram for Gaussian random fields with stationary increments. Corollary 2. Let X be a centered Gaussian field with stationary increments. If for all ε > 0, there exist c 1 , c 2 > 0, such that for all x ∈ [−1, 1] d , c 1 x 2γ+ε ≤ v X (x) = E((X x − X 0) 2) ≤ c 2 x 2γ−ε , then X has critical Hölder exponent on [0, 1] d equal to γ.

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A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

Cited by 14 publications

References 35 publications

Texture modeling by Gaussian fields with prescribed local orientation

Texture modeling by Gaussian fields with prescribed local orientation

Tests of isotropy for rough textures of trended images

Introduction to Random Fields and Scale Invariance

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