Existence and Uniqueness in Shape from Shading

For the image of a smooth surface object fully contained within the field of view and illuminated in and arbitrary direction,this paper discusses the existence and uniqueness o the conditions for solving a shape-from-shading problem under the conditions that the Fourier series expansion of the image intensity contains only zero and first order terms in a polar coordinate system.Three theorems are established,one for the existence and two for the uniqueness of z-axis symmetric shape from shading.     

Ambiguous Shape from Shading with Critical Points

The eikonal equation can have an infinite number of solutions when the image has critical points. We exhibit a family, indexed by a continuous parameter, of non isomorphic surfaces with one critical point, which give the same simple image. Hence, shape from shading can be an ill-posed problem when no additional condition on the shape is imposed, even when the image has critical points. Also, deformations without distortion are possible, i.e., there can exist a continuous deformation of the surface which does not modify its image.     

Symmetric Shape-from-Shading Using Self-ratio Image

In this paper, we present a symmetric shape-from-shading (SFS) approach to recover both shape and albedo for symmetric objects. Lambertian surfaces with unknown varying albedo and orthographic projections are assumed. In our formulation of symmetric SFS, we have two image irradiance equations. One is the standard equation used in SFS, and the other is a self-ratio image irradiance equation. This new image irradiance equation relates the self-ratio image which is defined as the ratio of two-halves of the input image to light source and surface shape. The introduction of the self-ratio image facilitates the direct use of symmetry cue. Based on the self-ratio image, a new model-based symmetric source-from-shading algorithm is also presented. We then propose symmetric SFS algorithms to recover both shape and albedo from a single image and present experimental results.The new symmetric SFS scheme has one important property: the existence of a unique (global) solution which consists of unique (local) solutions at each point simultaneously obtained using the intensity information at that point and the surrounding local region and the assumption of a C ² surface. Proofs for the existence of a unique solution in the cases of unknown constant and non-constant albedos are provided.     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Periodic strong solution for the optimal filtering problem oflinear discrete-time periodic systems

The periodic Riccati difference equation (PRDE) for the optimal filtering problem of linear periodic discrete-time systems is addressed. Specifically, the author provides a number of results on the existence, uniqueness, and stability properties of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation in the case of nonreversible and nonstabilizable periodic systems. The convergence of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation is also analyzed. The results have been established under weaker assumptions and include both necessary and sufficient conditions. The existence and properties of symmetric periodic nonnegative-definite solutions of the PRDE are established directly from the PRDE     

Closed Form Solution of Local Shape from Shading at Critical Points

In the theory of shape from shading, behaviours of the local solution around a critical point of the image play an important role. This paper shows that the second derivatives of the object surface can be locally determined at these image critical points. Closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown. They are derived as follows: By differentiating the image irradiance equation twice at an image critical point, a set of polynomial equations is obtained that contains the second derivatives of the surface, of the image brightness and of the reflectance map. Regarding these equations as simultaneous equations for unknown surface second derivatives, they are algebraically solved and their explicit expressions are derived. Such a derivation is possible only at image critical points and is impossible at any other image point. The applicability of the derived expressions to noisy images is tested using synthetic images.     

Uniqueness in shape from shading

For general images of smooth objects wholly contained in the field of view, and for illumination symmetric around the viewing direction, it is proven that shape is uniquely determined by shading. Thus, shape from shading is a well-posed problem under these illumination conditions; and regularization is unnecessary for surface reconstruction and should be avoided. Generic properties of surfaces and images are established. Questions of existence are also discussed. Under the conditions above, it is argued that most images are effectively impossible, with no corresponding physically reasonable surface, and that any image can be rendered effectively impossible by a small perturbation of its intensities. This is explicitly illustrated for a synthetic image. The proofs are based on ideas of dynamical systems theory and global analysis.     

Linear interval equations

Necessary and sufficient criteria are given for the existence and uniqueness of solutions of linear interval equations. Explicit formulas are given for the solution set when the solution set is convex. Necessary and sufficient conditions are given for the convexity of the solution set.     

具不定二次项的一般代数RICCATI方程解的存在性问题

讨论具有不定二次项的一般代数RICCATI方程(GARE)的实对称镇定解的存在性问 题,利用GARE相应的微分方程解的性质,建立了GARE实对称镇定解存在性的充分条件.     

Parametric shape-from-shading by radial basis functions

We present a new method of shape from shading by using radial basis functions to parameterize the object depth. The radial basis functions are deformed by adjusting their centers, widths, and weights such that the intensity errors are minimized. The initial centers and widths are arranged hierarchically to speed up convergence and to stabilize the solution. Although the smoothness constraint is used, it can be eventually dropped out without causing instabilities in the solution. An important feature of our parametric shape-from-shading method is that it offers a unified framework for integration of multiple sensory information. We show that knowledge about surface depth and/or surface normals anywhere in the image can be easily incorporated into the shape from shading process. It is further demonstrated that even qualitative knowledge can be used in shape from shading to improve 3D reconstruction. Experimental comparisons of our method with several existing ones are made by using both synthetic and real images. Results show that our solution is more accurate than the others     

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

We give an improved qualitative method to solve the osmosis K(2, 2) equation. This method combines several characteristics of other methods. Using this method, the existence of symmetric and non-symmetric wave solutions of the osmosis K(2, 2) equation is studied. Besides abundant symmetric forms such as smooth wave solutions, peaked waves, cusped waves, looped waves, stumpons and fractal-like waves, this equation also admits non-symmetric ones including breaking kink wave solutions, breaking anti-kink wave solutions and rampons. As regards this equation most of those solutions, either symmetric or non-symmetric solutions, have not appeared in the literature. We also study the limiting behavior of all periodic solutions as the parameters tend to some special values.     

基于表面局部最高点估计的明暗恢复形状算法

基于数据逼近强约束的针图恢复算法是近年来提出的一种较为成功的从明暗恢复形状（shape from shading）的算法,但由于该算法在非垂直光线下得到的初始化针图的误差较大,并且不能保证法向量有解或有唯一解,为了解决SFS算法存在的问题,提出了一种改进的SFS算法。该改进算法从分析非垂直光线下图像梯度图与针图之间的关系入手,首先检测图像局部最亮点位置;然后根据照度方程估计表面局部最高点的位置,同时对梯度方向进行调整,并建立方程组;最后针对方程组解的不同情况,提出了相应的处理方法。改进后的算法,对于垂直光线和非垂直光线下的情况同样有效,从而扩大了基于数据逼近强约束的SFS算法的适用范围。从合成图像和实际图像的实验结果可以看出,采用改进的算法可以得到比基于数据逼近强约束的算法更接近真实表面的初始化针图和初始化高度。     

Existence and uniqueness of open-loop Nash equilibria in linear-quadratic discrete time games

We present existence and uniqueness results for an equilibrium in an M-person Nash game with quadratic performance criteria and a linear difference equation as constraint, describing the system dynamics under an open-loop information pattern. The approach used is the construction of a value function which leads to existence assertions in terms of solvability of certain symmetric and nonsymmetric Riccati difference equations.     

Asymptotic behavior of certain Riccati difference equations

Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.     

Positive solutions of operator equations on ordered Banach spaces and applications

This paper is concerned with an operator equation on ordered Banach spaces. The existence and uniqueness of its’ positive solutions is obtained by using the properties of cones and monotone iterative technique. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for two classes of integral equations.     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients

In this paper, a class of fuzzy random differential equations with non-Lipschitz coefficients is studied. The existence and uniqueness of solutions for fuzzy random differential equations with non-Lipschitz coefficients is first proved. Then the dependence of fuzzy random differential equations on initial values is discussed. Finally the non-confluence property of the solution for fuzzy random differential equation is investigated. Our main tool is the Gronwall lemma.     

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 shape-from-shading method of polyhedral objects using prior information

We propose a new method for recovering the 3D shape of a polyhedral object from its single 2D image using the shading information contained in the image and the prior information on the object. In a strict sense, we cannot recover the shape of a polyhedron from an incorrect line drawing, even if it is practically almost correct. In order to overcome this problem, we propose a flexible face positioning method that can permit inconsistencies in the recovered shape that arise from vertex-position errors contained in incorrect line drawings. Also, we propose to use prior information about the horizontality and verticality of special faces and the convex and concave properties of the edges in order to attain good solutions and present a method of formulating such prior information as physical constraints. The shape-from-shading method is formulated as a minimization problem of a nonlinear cost function with the nonlinear constraints and its solution is searched by a global optimization algorithm. In the experiments with a synthetic image and three kinds of real images, shapes that are similar to those of the actual objects were recovered in all cases. As a result, the proposed method has proven to be effective in the shape recovery of simple-shape polyhedral objects.     

Mathematical Modelling Of Wreck Events Originated By Dam Collapse

Based on Saint-Venant (shallow water) equations, in this paper the mathematical model of wreck events produced by dam collapse is constructed. A two-layer difference scheme with non-linear regularisation is used for the numerical solution of the aforementioned model. The convergence of this difference scheme in Eulerian variables with non-linear regularisation to the smooth solutions of one-dimensional Saint-Venant equations are considered for a Cauchy problem with periodic (in spatial variables) solutions. The proof of difference scheme convergence is conducted using the energetic method. The existence and uniqueness of the difference scheme solution is proved. That the difference scheme converges in mesh norm $L_2$ with speed $O\lpar h^2\rpar$ in the class of sufficiently smooth solutions of the difference scheme is also proved.