The present work investigates the segmentation of textures by formulating it as a strongly convex optimization problem, aiming to favor piecewise constancy of fractal features (local variance and local regularity) widely used to model real-world textures in numerous applications very different in nature. Two objective functions combining these two features are compared, referred to as joint and coupled, promoting either independent or co-localized changes in local variance and regularity. To solve the resulting convex nonsmooth optimization problems, because the processing of large size images and databases are targeted, two categories of proximal algorithms (dual forward-backward and primal-dual), are devised and compared. An in-depth study of the objective functions, notably of their strong convexity, memory and computational costs, permits to propose significantly accelerated algorithms. A class of synthetic models of piecewise fractal texture is constructed and studied. They enable, by means of large-scale Monte-Carlo simulations, to quantify the benefits in texture segmentation of combining local regularity and local variance (as opposed to regularity only) while using strong-convexity accelerated primal-dual algorithms. Achieved results also permit to discuss the gains/costs in imposing co-localizations of changes in local regularity and local variance in the problem formulation. Finally, the potential of the proposed approaches is illustrated on real-world textures taken from a publicly available and documented database.

本工作通过将纹理分割公式化为一个强凸优化问题来进行研究，旨在支持分形恒定的分形恒定性（局部方差和局部规则性），该分形特征被广泛用于对本质上差异很大的许多应用程序中的真实世界纹理进行建模。比较了结合这两个特征的两个目标函数，称为联合和耦合，促进局部方差和规律性的独立变化或局部变化。为了解决由此产生的凸非平滑优化问题，因为针对大尺寸图像和数据库的处理是有针对性的，所以设计并比较了两类近端算法（双向向前和向后双向）。对目标函数的深入研究，尤其是对它们的强凸性，内存和计算成本的研究，可以提出显着加速的算法。建立并研究了一类分段分形纹理的综合模型。通过使用大型蒙特卡洛模拟，它们可以量化结合局部规则性和局部方差（仅与规则性相反）的纹理分割的好处，同时使用强凸加速的原始对偶算法。取得的成果还允许讨论在问题制定中将局部规律性和局部方差的变化置于共局部化中的收益/成本。最后，在从可公开获取并记录在案的数据库中获取的真实世界纹理上，说明了所提出方法的潜力。