167 research outputs found
Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach
Total variation regularization has proven to be a valuable tool in the
context of optimal control of differential equations. This is particularly
attributed to the observation that TV-penalties often favor piecewise constant
minimizers with well-behaved jumpsets. On the downside, their intricate
properties significantly complicate every aspect of their analysis, from the
derivation of first-order optimality conditions to their discrete approximation
and the choice of a suitable solution algorithm. In this paper, we investigate
a general class of minimization problems with TV-regularization, comprising
both continuous and discretized control spaces, from a convex geometry
perspective. This leads to a variety of novel theoretical insights on
minimization problems with total variation regularization as well as tools for
their practical realization. First, by studying the extremal points of the
respective total variation unit balls, we enable their efficient solution by
geometry exploiting algorithms, e.g. fully-corrective generalized conditional
gradient methods. We give a detailed account on the practical realization of
such a method for piecewise constant finite element approximations of the
control on triangulations of the spatial domain. Second, in the same setting
and for suitable sequences of uniformly refined meshes, it is shown that
minimizers to discretized PDE-constrained optimal control problems approximate
solutions to a continuous limit problem involving an anisotropic total
variation reflecting the fine-scale geometry of the mesh.Comment: 47 pages, 12 figure
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
SΓ£o Carlos, Brasil. 3-7 february 2014
Morphological bilateral filtering
International audienceA current challenging topic in mathematical morphology is the construction of locally adaptive operators; i.e., structuring functions that are dependent on the input image itself at each position. Development of spatially-variant filtering is well established in the theory and practice of Gaussian filtering. The aim of the first part of the paper is to study how to generalize these convolution-based approaches in order to introduce adaptive nonlinear filters that asymptotically correspond to spatially-variant morphological dilation and erosion. In particular, starting from the bilateral filtering framework and using the notion of counter-harmonic mean, our goal is to propose a new low complexity approach to define spatially-variant bilateral structuring functions. Then, in the second part of the paper, an original formulation of spatially-variant flat morphological filters is proposed, where the adaptive structuring elements are obtained by thresholding the bilateral structuring functions. The methodological results of the paper are illustrated with various comparative examples
ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ Π½Π΅Π²ΡΠ·ΠΊΠΈ Π² Π·Π°Π΄Π°ΡΠ΅ ΡΠ΅ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠΈ Π½Π΅ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ Ρ Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊ ΡΠΈΡΡΠ΅ΠΌΡ Π²ΡΠΎΡΠΎΠ³ΠΎ ΠΏΠΎΡΡΠ΄ΠΊΠ°
Π ΡΡΠ°ΡΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡΡΡ Π΄Π²Π΅ Π·Π°Π΄Π°ΡΠΈ Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠΈ Π½Π΅ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊ ΡΠΈΡΡΠ΅ΠΌΡ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ, ΠΎΠΏΠΈΡΡΠ²Π°ΡΡΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΡ Π΄ΠΈΡΡΡΠ·ΠΈΠΈ ΠΈΠ½Π½ΠΎΠ²Π°ΡΠΈΠΉ, ΠΏΠΎ Π½Π΅ΡΠΎΡΠ½ΡΠΌ ΠΈΠ·ΠΌΠ΅ΡΠ΅Π½ΠΈΡΠΌ ΡΠ°Π·ΠΎΠ²ΡΡ
ΡΠΎΡΡΠΎΡΠ½ΠΈΠΉ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΈΠΉ Π²Π°ΡΠΈΠ°Π½Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΠΈΡ
Π·Π°Π΄Π°Ρ. ΠΡΠ΅Π΄ΠΏΠΎΠ»Π°Π³Π°Π΅ΡΡΡ, ΡΡΠΎ ΡΠΈΡΡΠ΅ΠΌΠ° ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΡΠ΅Ρ Π½Π° Π·Π°Π΄Π°Π½Π½ΠΎΠΌ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΌ Π²ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΌ ΠΈΠ½ΡΠ΅ΡΠ²Π°Π»Π΅. ΠΠ²ΠΎΠ»ΡΡΠΈΡ ΡΠ°Π·ΠΎΠ²ΠΎΠ³ΠΎ ΡΠΎΡΡΠΎΡΠ½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌΡ, ΡΠΎ Π΅ΡΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ, ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΠ΅ΡΡΡ Π½Π΅ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠΌ Π²Ρ
ΠΎΠ΄ΠΎΠΌ. Π’ΠΎΡΠ½ΠΎΠ΅ Π²ΠΎΡΡΡΠ°Π½ΠΎΠ²Π»Π΅Π½ΠΈΠ΅ ΠΈΡΡΠΈΠ½Π½ΠΎΠ³ΠΎ, Π΄Π΅ΠΉΡΡΠ²ΡΡΡΠ΅Π³ΠΎ Π½Π° ΡΠΈΡΡΠ΅ΠΌΡ, Π²Ρ
ΠΎΠ΄Π°, Π²ΠΎΠΎΠ±ΡΠ΅ Π³ΠΎΠ²ΠΎΡΡ, Π½Π΅Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Π² ΡΠΈΠ»Ρ ΠΏΠΎΠ³ΡΠ΅ΡΠ½ΠΎΡΡΠΈ ΠΈΠ·ΠΌΠ΅ΡΠ΅Π½ΠΈΠΉ. ΠΠΎΡΡΠΎΠΌΡ ΠΌΡ ΠΏΡΠ΅Π΄ΠΏΠΎΠ»Π°Π³Π°Π΅ΠΌ ΠΏΠΎΡΡΡΠΎΠΈΡΡ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ Π΅Π³ΠΎ Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΡ. ΠΠΎΡΡΠ΅Π±ΡΠ΅ΠΌ, ΡΡΠΎΠ±Ρ Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΡ Π±ΡΠ»Π° ΡΠΊΠΎΠ»Ρ ΡΠ³ΠΎΠ΄Π½ΠΎ Π±Π»ΠΈΠ·ΠΊΠ° ΠΊ ΠΈΡΡΠΈΠ½Π½ΠΎΠΌΡ Π²Ρ
ΠΎΠ΄Ρ ΠΏΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎΠΉ ΠΌΠ°Π»ΠΎΡΡΠΈ ΠΈΠ·ΠΌΠ΅ΡΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΎΡΠΈΠ±ΠΎΠΊ ΠΈ ΡΠ°ΡΡΡΠΎΡΠ½ΠΈΡ ΠΌΠ΅ΠΆΠ΄Ρ ΠΌΠΎΠΌΠ΅Π½ΡΠ°ΠΌΠΈ ΠΈΠ·ΠΌΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΠ°Π·ΠΎΠ²ΡΡ
ΡΠΎΡΡΠΎΡΠ½ΠΈΠΉ. ΠΠ° ΠΎΡΠ½ΠΎΠ²Π΅ Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ Π²Π°ΡΠΈΠ°Π½ΡΠ° ΠΌΠ΅ΡΠΎΠ΄Π° Π½Π΅Π²ΡΠ·ΠΊΠΈ ΡΠΊΠ°Π·Π°Π½Ρ Π΄Π²Π° Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° ΡΠ΅ΡΠ΅Π½ΠΈΡ Π·Π°Π΄Π°Ρ ΡΠ΅ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠΈ: ΠΏΠ΅ΡΠ²ΡΠΉ ΠΎΡΠΈΠ΅Π½ΡΠΈΡΠΎΠ²Π°Π½ Π½Π° ΡΠ»ΡΡΠ°ΠΉ ΠΈΠ·ΠΌΠ΅ΡΠ΅Π½ΠΈΡ Π²ΡΠ΅Ρ
ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ ΡΠ°Π·ΠΎΠ²ΠΎΠ³ΠΎ Π²Π΅ΠΊΡΠΎΡΠ°, Π²ΡΠΎΡΠΎΠΉ - Π½Π° ΡΠ»ΡΡΠ°ΠΉ ΠΈΠ·ΠΌΠ΅ΡΠ΅Π½ΠΈΡ ΡΠ°ΡΡΠΈ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ. ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΡΠ΅ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΡΠ²Π»ΡΡΡΡΡ ΡΡΡΠΎΠΉΡΠΈΠ²ΡΠΌΠΈ ΠΏΠΎ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ ΠΊ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΎΠ½Π½ΡΠΌ ΠΏΠΎΠΌΠ΅Ρ
Π°ΠΌ ΠΈ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΠΌ ΠΎΡΠΈΠ±ΠΊΠ°ΠΌ ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡΡ ΡΠΎΠ±ΠΎΠΉ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΡΠ΅ ΡΠ΅Π³ΡΠ»ΡΡΠΈΠ·ΠΈΡΡΡΡΠΈΠ΅ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ Π΄Π»Ρ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΈΠ· Π²Π°ΡΠΈΠ°Π½ΡΠΎΠ² ΠΎΠ±ΡΠ°ΡΠ½ΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ Π΄ΠΈΠ½Π°ΠΌΠΈΠΊΠΈ
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes Β· Topp model Β· Reduced planar quartic Toppsystem Β· Singular point Β· Limit cycle Β· Hopf-saddle-node bifurcation Β· Perioddoubling bifurcation Β· Shilnikov homoclinic orbit Β· Chao
Mathematical Methods for the Quantification of Actin-Filaments in Microscopic Images
In cell biology confocal laser scanning microscopic images of the actin filament of human osteoblasts are produced to assess the cell development. This thesis aims at an advanced approach for accurate quantitative measurements about the morphology of the bright-ridge set of these microscopic images and thus about the actin filament. Therefore automatic preprocessing, tagging and quantification interplay to approximate the capabilities of the human observer to intuitively recognize the filaments correctly. Numerical experiments with random models confirm the accuracy of this approach
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