/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

/pdf/geodesic-active-contours-4wqlqxa13k.pdf

Geodesic active contours

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Geodesic Active Contours

Formulation of Large Deviation Theory in Terms of the Laplace Principle. First Example: Sanov's Theorem. Second Example: Mogulskii's Theorem. Representation Formulas for Other Stochastic Processes. Compactness and Limit Properties for the Random Walk Model. Laplace Principle for the Random Walk Model with Continuous Statistics. Laplace Principle for the Random Walk Model with Discontinuous Statistics. Laplace Principle for the Empirical Measures of a Markov Chain. Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain. Laplace Principle for Continuous-Time Markov Processes with Continuous Statistics. Appendices. Bibliography. Indexes.

A weak convergence approach to the theory of large deviations

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.

Dynamical systems and variational inequalities

/pdf/variational-problems-on-flows-of-diffeomorphisms-for-image-223g1q133q.pdf

Variational problems on flows of diffeomorphisms for image matching

A Weak Convergence Approach to the Theory of Large Deviations: Dupuis/A Weak

In this paper we show that the variational representation $$-\log Ee^{-f(W)} = \inf_v E{1/2 \int_0^1 \parallel v_s \parallel^2 ds + f(W + \int_0^{\cdot} v_s ds)}$$ holds, where $W$ is a standard $d$-dimensional Brownian motion, $f$ is any bounded measurable function that maps $C([0, 1]: \mathbb{R}^d)$ into $\mathbb{R}$ and the infimum is over all processes $v$ that are progressively measurable with respect to the augmentation of the filtration generated by $W$. An application is made to a problem concerned with large deviations, and an extension to unbounded functions is given.

/pdf/a-variational-representation-for-certain-functionals-of-41qmhs4w0r.pdf

Paul Dupuis

Papers

A weak convergence approach to the theory of large deviations

Dynamical systems and variational inequalities

Variational problems on flows of diffeomorphisms for image matching

A Weak Convergence Approach to the Theory of Large Deviations: Dupuis/A Weak

A variational representation for certain functionals of Brownian motion