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Weighted tensorized fractional Brownian textures

Céline Esser, Claire Launay, Laurent Loosveldt, Béatrice Vedel

TL;DR

The paper addresses realistic texture modeling by relaxing the tensor-product structure of fractional Brownian sheets through the weighted tensorized fractional Brownian field (WTFBF), parameterized by $\alpha$ and $H$. The WTFBF is defined as a Gaussian field with a spectral-kernel $\phi_{\alpha,H}$ that interpolates between the fractional Brownian sheet ($\alpha=0$) and fields closer to LFBM ($\alpha=1$), and it retains self-similarity with exponent $2H$ and stationary rectangular increments. A variance bound for rectangular increments is established, together with a nearly rectangular Hölder regularity result, and an anisotropic extension using $\beta_1,\beta_2$ leads to operator-scaling Gaussian fields with $D=\mathrm{diag}(\beta_1,\beta_2)$. Simulations via a spectral representation on a grid illustrate the transition across $\alpha$ and the anisotropic case, confirming the model’s flexibility for texture-like patterns and potential applications in medical and urban imagery. The work provides a bridge between classical fBs and fractional Brownian sheets, highlighting practical tools for texture synthesis and analysis of anisotropic regularity properties.

Abstract

This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. Statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties are obtained. An operator scaling extension is defined and we provide simulations of the fields using their spectral representation.

Weighted tensorized fractional Brownian textures

TL;DR

The paper addresses realistic texture modeling by relaxing the tensor-product structure of fractional Brownian sheets through the weighted tensorized fractional Brownian field (WTFBF), parameterized by and . The WTFBF is defined as a Gaussian field with a spectral-kernel that interpolates between the fractional Brownian sheet () and fields closer to LFBM (), and it retains self-similarity with exponent and stationary rectangular increments. A variance bound for rectangular increments is established, together with a nearly rectangular Hölder regularity result, and an anisotropic extension using leads to operator-scaling Gaussian fields with . Simulations via a spectral representation on a grid illustrate the transition across and the anisotropic case, confirming the model’s flexibility for texture-like patterns and potential applications in medical and urban imagery. The work provides a bridge between classical fBs and fractional Brownian sheets, highlighting practical tools for texture synthesis and analysis of anisotropic regularity properties.

Abstract

This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. Statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties are obtained. An operator scaling extension is defined and we provide simulations of the fields using their spectral representation.
Paper Structure (6 sections, 3 theorems, 26 equations, 2 figures, 1 table)

This paper contains 6 sections, 3 theorems, 26 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definition of the fields
  3. Basic properties
  4. Variance of rectangular increments
  5. Anisotropic extension
  6. Simulation

Key Result

Proposition 3.1

For all $\alpha \in [0,1]$ and $H \in (0,1)$, the process $X^{\alpha,H}$ is self-similar: for all $a>0$, $\{X^{\alpha,H}_{(a x_1, a x_2)}\}_{(x_1,x_2)\in \mathbb{R}^2}\stackrel{(d)}{=}\{a^{2H} X^{\alpha,H}_{(x_1, x_2)}\}_{(x_1,x_2)\in \mathbb{R}^2}$.

Figures (2)

  • Figure 1: Weighted tensorized fractional Brownian fields simulated using a spectral representation approximation method, with parameters (a-c) $H~=~0.3$ or (d-f) $H~=~0.7$ and (a,d) $\alpha~=~0$, (b,e) $\alpha~=~0.5$ or (c,f) $\alpha~=~1$.
  • Figure 2: Anisotropic weighted tensorized fractional Brownian fields simulated using a spectral representation approximation method, with parameters (a-c) $H=0.4$, $\beta_1=0.7$, $\beta_2=1.3$ and $\alpha = 0$, $\alpha = 0.5$ or $\alpha = 1$, and (d-f) $H=0.6$, $\beta_1=0.85$, $\beta_2=1.15$ and $\alpha = 0$, $\alpha = 0.5$ or $\alpha = 1$.

Theorems & Definitions (7)

  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof