Null controllability of an infinite dimensional SDE with state- and control-dependent noise

We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H. We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.     

Learning shape from shading by a multilayer network

The multilayer feedforward network has often been used for learning a nonlinear mapping based on a set of examples of the input-output data. In this paper, we present a novel use of the network, in which the example data are not explicitly given. We consider the problem of shape from shading in computer vision, where the input (image coordinates) and the output (surface depth) satisfy only a known differential equation. We use the feedforward network as a parametric representation of the object surface and reformulate the shape from shading problem as the minimization of an error function over the network weights. The stochastic gradient and conjugate gradient methods are used for the minimization. Boundary conditions for either surface depth or surface normal (or both) can be imposed by adjusting the same network at different levels. It is further shown that the light source direction can be estimated, based on an initial guess, by integrating the source estimation with the surface estimation. Extensions of the method to a wider class of problems are discussed. The efficiency of the method is verified by examples of both synthetic and real images.     

3D Curves Reconstruction Based on Deformable Models

We present a new method, based on curve evolution, for the reconstruction of a 3D curve from two different projections. It is based on the minimization of an energy functional. Following the work on geodesic active contours by Caselles et al. (in Int. Conf. on Pattern Recognition, 1996, Vol. 43, pp. 693–737), we then transform the problem of minimizing the functional into a problem of geodesic computation in a Riemann space. The Euler-Lagrange equation of this new functional is derived and its associated PDE is solved using the level set formulation, giving the existence and uniqueness results. We apply the model to the reconstruction of a vessel from a biplane angiography.     

Approximate conditional least squares estimation of a nonlinear state-space model via an unscented Kalman filter

The problem of estimating a nonlinear state-space model whose state process is driven by an ordinary differential equation (ODE) or a stochastic differential equation (SDE), with discrete-time data is studied. A new estimation method is proposed based on minimizing the conditional least squares (CLS) with the conditional mean function computed approximately via the unscented Kalman filter (UKF). Conditions are derived for the UKF–CLS estimator to preserve the limiting properties of the exact CLS estimator, namely, consistency and asymptotic normality, under the framework of infill asymptotics, i.e. sampling is increasingly dense over a fixed domain. The efficacy of the proposed method is demonstrated by simulation and a real application.     

Backward Stochastic H2/H∞ Control with Random Jumps

This paper is concerned with H₂/H_∞ control of a new class of stochastic systems. The most distinguishing feature, compared with the existing literature, is that the systems are described by backward stochastic differential equations (BSDEs) with Brownian motion and random jumps. It is shown that the backward stochastic H₂/H_∞ control under consideration is associated with the of the corresponding uncontrolled backward stochastic perturbed system. A necessary and sufficient condition for the existence of a unique solution to the control problem under consideration is derived. The resulting solution is characterized by the solution of an uncontrolled forward backward stochastic differential equation (FBSDE) with Brownian motion and random jumps. When the coefficients are all deterministic, the equivalent linear feedback solution involves a pair of Riccati‐type equations and an uncontrolled BSDE. In addition an uncontrolled forward stochastic differential equation (SDE) is given.     

Stability of regime-switching stochastic differential equations

The main result is reduction of the asymptotic stability problem for a stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the analogous wellstudied problem for the ??averaged?? SDE without switching. Applications to the switching stabilization problem and to ordinary differential equations (ODE) with switching are also considered.     

Gibbs Field Approaches in Image Processing Problems

In this paper, we address the problem of image denoising using a stochastic differential equation approach. Proposed stochastic dynamics schemes are based on the property of diffusion dynamics to converge to a distribution on global minima of the energy function of the model, under a special cooling schedule (the annealing procedure). To derive algorithms for computer simulations, we consider discrete-time approximations of the stochastic differential equation. We study convergence of the corresponding Markov chains to the diffusion process. We give conditions for the ergodicity of the Euler approximation scheme. In the conclusion, we compare results of computer simulations using the diffusion dynamics algorithms and the standard Metropolis–Hasting algorithm. Results are shown on synthetic and real data.     

Optimal Algorithm for Shape from Shading and Path Planning

An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian's fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.     

A simulation‐based algorithm for optimal pricing policy under demand uncertainty

We propose a simulation‐based algorithm for computing the optimal pricing policy for a product under uncertain demand dynamics. We consider a parameterized stochastic differential equation (SDE) model for the uncertain demand dynamics of the product over the planning horizon. In particular, we consider a dynamic model that is an extension of the Bass model. The performance of our algorithm is compared to that of a myopic pricing policy and is shown to give better results. Two significant advantages with our algorithm are as follows: (a) it does not require information on the system model parameters if the SDE system state is known via either a simulation device or real data, and (b) as it works efficiently even for high‐dimensional parameters, it uses the efficient smoothed functional gradient estimator.     

Invariant surface segmentation through energy minimization with discontinuities

The computational problems in segmenting range data into surface patches based on the invariant surface properties, i.e., mean curvature H and Gaussian curvature K, are investigated. The goal is to obtain reliable HK surface maps. Two commonly encountered problems are: firstly the noise effect in computing derivative estimates, and secondly the smoothing across discontinuities. Here, the segmentation is formulated as finding minimization solutions of energy functionals involving discontinuities. A two-stage approach to the goal is presented: stage (1) from a range image to curvature images and stage (2) from the curvature images to the HK maps. In both stages, solutions are found through minimizing energy functionals that measure the degree of bias of a solution from two constraints: the closeness of the solution to the data, and the smoothness of the solution controlled by predetermined discontinuities. Propagation across discontinuities is prevented during minimization, which preserves the original surface shapes. Experimental results are given for a variety of test images.This work was supported by a TC Scholarship from the British Council.     

Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.     

Stochastic Optimal Control of Finite Ensembles of Nanomagnets

We control ferromagnetic N-spin dynamics in the presence of thermal fluctuations by minimizing a quadratic functional subject to the stochastic Landau–Lifshitz–Gilbert equation. Existence of a weak solution of the stochastic optimal control problem is shown. The related first order optimality conditions consist of a coupled forward–backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the new stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise.     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

带噪声散乱数据的光滑曲面重构——变分水平集方法

带噪声散乱数据点的光滑曲面重构应用广泛,基于变分水平集方法提出一种求解该问题的新的能量模型,并由此能量得到一新的微分方程,该微分方程演化后得到的极限曲面即为要重构的光滑曲面.给出了一种快速建立初始曲面的方法,节约了重构时间;然后对该微分方程的初值问题运用水平集方法求解,其中的空间方向离散化采用本质无震荡或加权本质无震荡技术,时间方向采用具有高精度的TVD Runge-Kutta技术.提出一种变步长的TVDRunge-Kutta方法来重新初始化符号距离函数,保证了Runge-Kutta方法中每一欧拉步都满足迎风设计要求.实验结果表明,该方法高效且能产生良好的重建效果.     

图像引导的二阶总广义变分稀疏深度图的稠密重构

利用图像颜色信息进行深度图重构,可以恢复对象边界处的深度不连续性,但无法保证对象内部的深度均匀性。为解决该问题,提出图像引导下总广义变分正则化的深度图重构模型。该模型利用扩散张量将图像提供的边缘信息引入二阶总广义变分正则项,使得重构深度在保持对象边缘的同时逼近分段仿射平面,从而保证恢复深度既保持对象边界处的不连续性,又具有对象内部的均匀性。通过Legendre-Fenchel变换将模型转换成等效的凸凹鞍点问题,从而得到高效的一阶原始对偶求解算法。实验结果表明,该方法能够恢复尖锐的对象边缘,同时保持对象内部的深度均匀性。与现有算法相比,所提方法具有更高的峰值信噪比、归一化互协方差和更低的平均绝对误差。     

Robust and efficient surface reconstruction from contours

We propose a new approach for surface recovery from planar sectional contours. The surface is reconstructed based on the so-called “equal importance criterion,” which suggests that every point in the region contributes equally to the reconstruction process. The problem is then formulated in terms of a partial differential equation, and the solution is efficiently calculated from distance transformation. To make the algorithm valid for different application purposes, both the isosurface and the primitive representations of the object surface are derived. The isosurface is constructed by means of a partial differential equation, which can be solved iteratively. The traditional distance interpolating method, which was used by several researchers for surface reconstruction, is an approximate solution of the equation. The primitive representations are approximated by Voronoi diagram transformation of the surface space. Isosurfaces have the advantage that subsequent geometric analysis of the object can be easily carried out while primitive representation is easy to visualize. The proposed technique allows for surface recovery at any desired resolution, thus avoiding the inherent problems of correspondence, tiling, and branching.     

Representation of the bilinear system output by multiple stochastic integrals

Consideration was given to the scalar stochastic differential Ito equation whose drift and diffusion coefficients are affine functions of the phase coordinate. Its solution was represented in terms of the stochastic exponent which plays the part of the equation resolvent. Expansion of the stochastic exponent in series in the Hermit polynomials induces expansion of the solution of the bilinear equation in series in multiple stochastic integrals. Obtained were the moment characteristics of the stochastic integrals solving the problem of statistical analysis of the approximate solutions of the bilinear stochastic systems. A model of the financial (B, S)-market and its optimization in terms of a nonsquare criterion were considered by way of example of a bilinear system.     

Stochastic Resonance in Continuous and Spiking Neuron Models With Levy Noise

Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jump-noise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the ItÔ calculus show that white Levy noise will benefit subthreshold neuronal signal detection if the noise process's scaled drift velocity falls inside an interval that depends on the threshold values. These results generalize earlier “forbidden interval” theorems of neuronal “stochastic resonance” (SR) or noise-injection benefits. Global and local Lipschitz conditions imply that additive white Levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation (SDE). Simulation results show that the same noise benefits still occur for some infinite-variance stable Levy noise processes even though the theorems themselves apply only to finite-variance Levy noise. The Appendix proves the two ItÔ-theoretic lemmas that underlie the new Levy noise-benefit theorems.      

A Deterministic Algorithm for Reconstructing Images with Interacting Discontinuities

The most common approach for incorporating discontinuities in visual reconstruction problems makes use of Bayesian techniques, based on Markov random field models, coupled with stochastic relaxation and simulated annealing. Despite their convergence properties and flexibility in exploiting a priori knowledge on physical and geometric features of discontinuities, stochastic relaxation algorithms often present insurmountable computational complexity. Recently, considerable attention has been given to suboptimal deterministic algorithms, which can provide solutions with much lower computational costs. These algorithms consider the discontinuities implicitly rather than explicitly and have been mostly derived when there are no interactions between two or more discontinuities in the image model. In this paper we propose an algorithm that allows for interacting discontinuities, in order to exploit the constraint that discontinuities must be connected and thin. The algorithm, called E-GNC, can be considered an extension of the graduated nonconvexity (GNC), first proposed by Blake and Zisserman for noninteracting discontinuities. When applied to the problem of image reconstruction from sparse and noisy data, the method is shown to give satisfactory results with a low number of iterations.     

A Maximum Likelihood Stereo Algorithm

A stereo algorithm is presented that optimizes a maximum likelihood cost function. The maximum likelihood cost function assumes that corresponding features in the left and right images are normally distributed about a common true value and consists of a weighted squared error term if two features are matched or a (fixed) cost if a feature is determined to be occluded. The stereo algorithm finds the set of correspondences that maximize the cost function subject to ordering and uniqueness constraints. The stereo algorithm is independent of the matching primitives. However, for the experiments described in this paper, matching is performed on the $cf4$individual pixel intensities.$cf3$ Contrary to popular belief, the pixel-based stereo appears to be robust for a variety of images. It also has the advantages of (i) providing adensedisparity map, (ii) requiringnofeature extraction, and (iii)avoidingthe adaptive windowing problem of area-based correlation methods. Because feature extraction and windowing are unnecessary, a very fast implementation is possible. Experimental results reveal that good stereo correspondences can be found using only ordering and uniqueness constraints, i.e., withoutlocalsmoothness constraints. However, it is shown that the original maximum likelihood stereo algorithm exhibits multiple global minima. The dynamic programming algorithm is guaranteed to find one, but not necessarily the same one for each epipolar scanline, causing erroneous correspondences which are visible as small local differences between neighboring scanlines. Traditionally, regularization, which modifies the original cost function, has been applied to the problem of multiple global minima. We developed several variants of the algorithm that avoid classical regularization while imposing several global cohesiveness constraints. We believe this is a novel approach that has the advantage of guaranteeing that solutions minimize the original cost function and preserve discontinuities. The constraints are based on minimizing the total number of horizontal and/or vertical discontinuities along and/or between adjacent epipolar lines, and local smoothing is avoided. Experiments reveal that minimizing the sum of the horizontal and vertical discontinuities provides the most accurate results. A high percentage of correct matches and very little smearing of depth discontinuities are obtained. An alternative to imposing cohesiveness constraints to reduce the correspondence ambiguities is to use more than two cameras. We therefore extend the two camera maximum likelihood toNcameras. TheN-camera stereo algorithm determines the “best” set of correspondences between a given pair of cameras, referred to as the principal cameras. Knowledge of the relative positions of the cameras allows the 3D point hypothesized by an assumed correspondence of two features in the principal pair to be projected onto the image plane of the remainingN− 2 cameras. TheseN− 2 points are then used to verify proposed matches. Not only does the algorithm explicitly model occlusion between features of the principal pair, but the possibility of occlusions in theN− 2 additional views is also modeled. Previous work did not model this occlusion process, the benefits and importance of which are experimentally verified. Like other multiframe stereo algorithms, the computational and memory costs of this approach increase linearly with each additional view. Experimental results are shown for two outdoor scenes. It is clearly demonstrated that the number of correspondence errors is significantly reduced as the number of views/cameras is increased.