  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.
