Raul Serapioni

On the structure of finite perimeter sets in step 2 Carnot groups

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure.

On the Motion of Non-homogeneous Fluids in the Presence of Diffusion

In this paper we study the motion of a non-viscous fluid consisting of two components, both incompressible, say, saturated salt water and water, taking into consideration the diffusion among these components. The equations of the model are obtained, for example, in Frank and Kamenetskii [6], and the viscous case is studied in Ignat’ev and Kuznetsov [8] and Kazhikhov and Smagulov [9, lo]. For a presentation of this kind of problems see also Kuznetsov [ 111. Observe that in [S-lo] the problem is solved only for fluids whose viscosity ,D is sufficiently large; i.e.,

Pointwise estimates for degenerate parabolic equations

In this paper we establish an invariant Harnack type inequality for positive solutions of linear degenerate parabolic equations. Consequently we prove Holder continuity of weak solutions.The cylinders on which this invariant Harnack inequality holds are not anymore the usual parabolic cylinders. On the contrary their shape is different from point to point depending on the degeneracy of the operator

A New Region-based Active Contour Model for Object Segmentation

We present a novel region-based active contour model that segments one or more image regions that are visually similar to an object of interest, said prior. The region evolution equation of our model is defined by a simple heuristic rule and it is not derived by minimizing an energy functional, as in the classic variational approaches. The prior and the evolving region are described by the probability density function (pdf) of a photometric feature, as color or intensity. The heuristic rule deforms an initial region of the image in order to equalize pointwise the pdfs of the prior and of the region. Such heuristic rule can be modeled by many mathematical monotonic decreasing functions, each defining an evolution equation for the initial image region. The choice of a particular function is remitted to the user, that in this way can even integrate a priori knowledge possibly useful to break down the computational charge of the method and to increase the detection accuracy. Here we propose two different evolution equations for the general purpose of prior detection without a priori information and we discuss empirically the performances of our model on real-world and synthetic datasets. These experiments show that our model is a valid alternative to the classic models.

On edge-aware path-based color spatial sampling for Retinex: from Termite Retinex to Light Energy-driven Termite Retinex

Abstract. Retinex theory estimates the human color sensation at any observed point by correcting its color based on the spatial arrangement of the colors in proximate regions. We revise two recent path-based, edge-aware Retinex implementations: Termite Retinex (TR) and Energy-driven Termite Retinex (ETR). As the original Retinex implementation, TR and ETR scan the neighborhood of any image pixel by paths and rescale their chromatic intensities by intensity levels computed by reworking the colors of the pixels on the paths. Our interest in TR and ETR is due to their unique, content-based scanning scheme, which uses the image edges to define the paths and exploits a swarm intelligence model for guiding the spatial exploration of the image. The exploration scheme of ETR has been showed to be particularly effective: its paths are local minima of an energy functional, designed to favor the sampling of image pixels highly relevant to color sensation. Nevertheless, since its computational complexity makes ETR poorly practicable, here we present a light version of it, named Light Energy-driven TR, and obtained from ETR by implementing a modified, optimized minimization procedure and by exploiting parallel computing.

Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem

Abstract A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meant to stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensional intrinsic Lipschitz graphs are sets with locally finite G-perimeter. From this a Rademacher’s type theorem for one codimensional graphs in a general class of groups is proved.

GRASS: A Gradient-Based Random Sampling Scheme for Milano Retinex

Retinex is an early and famous theory attempting to estimate the human color sensation derived from an observed scene. When applied to a digital image, the original implementation of retinex estimates the color sensation by modifying the pixels channel intensities with respect to a local reference white, selected from a set of random paths. The spatial search of the local reference white influences the final estimation. The recent algorithm energy-driven termite retinex (ETR), as well as its predecessor termite retinex, has introduced a new path-based image aware sampling scheme, where the paths depend on local visual properties of the input image. Precisely, the ETR paths transit over pixels with high gradient magnitude that have been proved to be important for the formation of color sensation. Such a sampling method enables the visit of image portions effectively relevant to the estimation of the color sensation, while it reduces the analysis of pixels with less essential and/or redundant data, i.e., the flat image regions. While the ETR sampling scheme is very efficacious in detecting image pixels salient for the color sensation, its computational complexity can be a limit. In this paper, we present a novel Gradient-based RAndom Sampling Scheme that inherits from ETR the image aware sampling principles, but has a lower computational complexity, while similar performance. Moreover, the new sampling scheme can be interpreted both as a path-based scanning and a 2D sampling.

Remarks on Lipschitz domains in Carnot groups

In this Note we present the basic features of the theory of Lipschitz maps within Carnot groups as it is developed in [8], and we prove that intrinsic Lipschitz domains in Carnot groups are uniform domains.

Intrinsic Difference Quotients

An alternative characterizations of intrinsic Lipschitz functions within Carnot groups through the boundedness of appropriately defined difference quotients is provided. It is also shown how intrinsic difference quotients along horizontal directions are naturally related with the intrinsic derivatives, introduced e.g. in Franchi et al. (Comm Anal Geom 11(5):909–944, 2003) and Ambrosio et al. (J Geom Anal 16:187–232, 2006) and used to characterize intrinsic real valued C1 functions inside Heisenberg groups. Finally the question of the equivalence of the two conditions: (1) boundedness of horizontal intrinsic difference quotients and (2) intrinsic Lipschitz continuity is addressed in a few cases.