Publications

We study a gen­er­al­iza­tion of the man­i­fold-­val­ued Rud­in-Osh­er-­Fatemi (ROF) mod­el, which involves an ini­tial datum f map­ping from a curved com­pact sur­face with smooth bound­ary to a com­plete, con­nected and smooth n-di­men­sional Rie­mann­ian man­i­fold. We prove the exis­tence and unique­ness of min­i­miz­ers under cur­va­ture restric­tions on the tar­get and topo­log­i­cal ones on the range of f. We obtain a series of reg­u­lar­ity results on the asso­ci­ated PDE sys­tem of a relaxed func­tional with Neu­mann bound­ary con­di­tion. We apply these results to the ROF model to obtain Lip­schitz reg­u­lar­ity of min­i­miz­ers with­out fur­ther require­ments on the con­vex­ity of the bound­ary. Addi­tion­al­ly, we pro­vide vari­ants of the reg­u­lar­ity state­ment of inde­pen­dent inter­est: for 1-di­men­sional domains (re­lated to sig­nal denois­ing), local Lip­schitz reg­u­lar­ity (mean­ing­ful for image pro­cess­ing) and Lip­schitz reg­u­lar­ity for a ver­sion of the Mosolov prob­lem com­ing from fluid mechanics.

We con­sider man­i­fold con­strained weak solu­tions of qua­si­lin­ear uni­formly ellip­tic sys­tems of diver­gence type with a source term that grows at most qua­drat­i­cally with respect to the gra­di­ent of the solu­tion. As we impose that the solu­tion lies on a Rie­mann­ian man­i­fold, the clas­si­cal small­ness con­di­tion for reg­u­lar­ity can be relaxed to an inequal­ity relat­ing strict con­vex­ity of the squared dis­tance and growth of the lead­ing order term in the tan­gent com­po­nent of the source. As a key tool for the proof of a par­tial reg­u­lar­ity result, we derive a fully intrin­sic Cac­ciop­poli inequal­ity which may be of inde­pen­dent inter­est. Finally we show how the sys­tems under con­sideration have a vari­a­tional nature and arise in the con­text of F- or V-har­monic maps.

We con­sider an abstract con­cept of perime­ter mea­sure space as a very gen­eral frame­work in which one can prop­erly con­sider two of the most well-s­tud­ied vari­a­tional mod­els in image pro­cess­ing: the Rud­in–Osh­er–­Fatemi model for image denois­ing (ROF) and the Mum­ford–Shah model for image seg­men­ta­tion (MS). We show the link­age between the ROF model and the two phases piece­wise con­stant case of MS in perime­ter mea­sure spaces. We show appli­ca­tions of our results to non­lo­cal image seg­men­ta­tion, via dis­crete weighted graphs, and to mul­ti­class clas­si­fi­ca­tion on high dimen­sional spaces.

In this paper we study a vari­ant to Chan–Vese (CV) seg­men­ta­tion model with rec­ti­lin­ear anisotropy. We show exis­tence of min­i­miz­ers in the 2-phases case and how they are related to the (anisotrop­ic) Rud­in–Osh­er–­Fatemi (ROF) denois­ing mod­el. Our analy­sis shows that in the nat­ural case of a piece­wise con­stant on rec­tan­gles image (PCR func­tion in short), there exists a min­i­mizer of the CV func­tional which is also piece­wise con­stant on rec­tan­gles over the same grid that the one defined by the orig­i­nal image. In the mul­ti­phase case, we show that min­i­miz­ers of the CV mul­ti­phase func­tional also share this prop­erty in the case that the ini­tial image is a PCR func­tion. We also inves­ti­gate a mul­ti­phase and anisotropic ver­sion of the Trun­cated ROF algo­rithm, and we com­pare the solu­tions given by this algo­rithm with min­i­miz­ers of the mul­ti­phase anisotropic CV func­tional.

In this paper, we pro­vide a new insight to the two-phase sig­nal seg­men­ta­tion prob­lem. We pro­pose an aug­mented Lagrangian vari­a­tional model based on Chan–Vese’s orig­i­nal one. Using both energy meth­ods and PDE meth­ods, we show, in the one-di­men­sional case, that the set of min­i­miz­ers to the pro­posed func­tional con­tains only binary func­tions and it coin­cides with the set of min­i­miz­ers to Chan–Vese’s one. This fact allows us to obtain two impor­tant fea­tures of the min­i­miz­ers as a by-prod­uct of our analy­sis. First of all, for a piece­wise con­stant ini­tial sig­nal, the jump set of any min­i­mizer is a sub­set of the jump set of the given sig­nal. Sec­ond, all of the jump points of the min­i­mizer belong to the same level set of the sig­nal, in a mul­ti­val­ued sense. This last prop­erty per­mits to design a triv­ial algo­rithm for com­put­ing the min­i­miz­ers.