Psychophysical experiments show that the perceived colour of an object is relatively independent of the spectrum of the incident illumination and depends only on the surface reflectance. We demonstrate a possible solution to this undetermined problem by expanding the illumination and surface reflectance in terms of a finite number of basis functions. This yields a number of nonlinear equations for each colour patch. We show that given a sufficient number of surface patches with the same illumination it is possible to solve these equations up to an overall scaling factor. Generalizations to the spatial dependent situation are discussed. We define a method for detecting material changes and illustrate a way of detecting the colour of a material at its boundaries and propagating it inwards.

No description available.

A Markov Decision Process (MDP) is a model employed to describe problems in which a decision must be made at each one of several stages, while receiving feedback from the environment. This type of model has been extensively studied in the operations research community and fundamental algorithms have been developed to solve associated problems. However, these algorithms are quite inefficient for very large problems, leading to a need for alternatives; since MDP problems are provably hard on compressed representations, one becomes content even with algorithms which may perform well at least on specific classes of problems. The class of problems we deal with in this thesis allows succinct representations for the MDP as a dynamic Bayes network, and for its solution as a weighted combination of basis functions. We develop novel algorithms for producing, improving, and calculating the error of approximate solutions for MDPs using a compressed representation. Specifically, we develop an efficient branch-and-bound algorithm for computing the Bellman error of the compact approximate solution regardless of its provenance. We introduce an efficient direct linear programming algorithm which, using incremental constraints generation, achieves run times significantly smaller than existing approximate algorithms without much loss of accuracy. We also show a novel direct linear programming algorithm which, instead of employing constraints generation, transforms the exponentially many constraints into a compact form more amenable for tractable solutions. In spite of its perceived importance, the efficient optimization of the Bellman error towards an approximate MDP solution has eluded current algorithms; to this end we propose a novel branch-and-bound approximate policy iteration algorithm which makes direct use of our branch-and-bound method for computing the Bellman error. We further investigate another procedure for obtaining an approximate solution based on the dual of the direct, approximate linear programming formulation for solving MDPs. To address both the loss of accuracy resulting from the direct, approximate linear program solution and the question of where basis functions come from we also develop a principled system able not only to produce the initial set of basis functions, but also able to augment it with new basis functions automatically generated such that the approximation error decreases according to the user's requirements and time limitations.

Digital multimedia can be distributed via the internet efficiently with superior compression technologies. The chance of distributing digital intellectual properties, such as image, music, films, and software, being large-scale unauthorized copied and distributed are much increasing one possible and practical solution for the copyright protection is information hiding technology. Information hiding technology embeds a special data into multimedia data for copyright protection. However, the embedded data may be damaged by malicious attacks or common signal processing. In this thesis, an information hiding technique based on Fractal Orthonormal Basis is proposed. First, the original image is divided into NxN Range blocks, each range block is substituted by several Domain blocks (Fractal Orthonormal Basis), then the watermark information is embedded into the coefficients of the fractal orthonormal basis. Besides, our technique will be compare with the other two watermarking algorithm (using DCT and DWT). After the attacks of cropping, down-scaling, median filter, smoothing, noise, JPEG, SPIHT and EZW compression, the Fractal Orthonormal Basis watermarking technique shows better result of capacity, transparency and robustness. In addition, we only store parts of compression fractal codes and the permutation seed, and these can be the secret key for the security.

RBFs are useful in scientific computing. In this thesis, we are interested in the positions of collocation points and RBF centers which causes the matrix for RBF interpolation singular and ill-conditioned. We explore the best bases by minimizing error function in supremum norm and root mean squares. We also use radial basis function to interpolate shifted data and find the best basis in certain sense. In the second part, we solve ill-posed problems by radial basis collocation method with different radial basis functions and various number of bases. If the solution is not unique, then the numerical solutions are different for different bases. To construct all the solutions, we can choose one approximation solution and add the linear combinations of the difference functions for various bases. If the solution does not exist, we show the numerical solution always fail to satisfy the origin equation.

Includes bibliographical references.

This paper covers basic theory of Grobner Bases and an algebraic analysis of the linear constant coefficient partial differential operators, specifically the Cauchy-Fueter operator. We will review examples and theory of regular and biregular functions in several quaternionic variables. / text

We show the existence of a basis for a vector space over a number field with two key properties. First, the n-th basis vector has a small twisted height which is bounded above by a quantity involving the n-th successive minima associated with the twisted height. Second, at each place v of the number field, the images of the basis vectors under the automorphism associated with the twisted height satisfy near-orthogonality conditions analagous to those introduced by Korkin and Zolotarev in the classical Geometry of Numbers. Using this basis, we bound the Mahler product associated with the twisted height. This is the product of a successive minimum of a twisted height with the corresponding successive minimum of its dual twisted height. Previous work by Roy and Thunder in [12] showed that the Mahler product was bounded above by a quantity which grows exponentially as the dimension of the vector space increases. In this work, we demonstrate an upper bound that exhibits polynomial growth as the dimension of the vector space increases. / text

We consider different ways of describing a matroid to a Turing machine by listing the members of various families of subsets, and we construct an order on these different methods of description. We show that, under this scheme, several natural matroid problems are complete in classes thought not to be equal to P. We list various results linking parameters of basis graphs to parameters of their associated matroids. For small values of k we determine which matroids have the clique number, chromatic number, or maximum degree of their basis graphs bounded above by k. If P is a class of graphs that is closed under isomorphism and induced subgraphs, then the set of matroids whose basis graphs belong to P is closed under minors. We characterise the minor-closed classes that arise in this way, and exhibit several examples. One way of choosing a basis of a matroid at random is to select a total ordering of the ground set uniformly at random and use the greedy algorithm. We consider the class of matroids having the property that this procedure chooses a basis uniformly at random. Finally we consider a problem mentioned by Oxley. He asked if, for every two elements and n - 2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that meets every cocircuit. We show that a slightly stronger property holds for regular matroids.

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Fulton B. Gonzalez. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 72-73). Access restricted to members of the Tufts University community. Also available via the World Wide Web;