analysing images of curved surfaces a reflectance map makes the relationship between image intensity and surface orientation explicit trade offs between image intensity and surface orientation emerge which cannot be resolved locally in a single view existing methods for determining surface orientation from a single view embody assumptions about surface curvature the hessian matrix is introduced to represent surface curvature properties of surface curvature are expressed as properties of the hessian matrix for several classes of surface image analysis simplifies this result has already been established for planar surfaces forming trihedral corners similar simplification is demonstrated for developable surfaces and for the subclass of surfaces known as generalized cones these studies help to delineate shape information that can be determined from geometric measurements at object boundaries and shape information that can be determined from intensity measurements over sections of smoothly curved surface a novel technique called photometric stereo is discussed the idea of stereo is to obtain multiple images in order to determine the underlying scene precisely in photometric stereo the viewing direction is constant multiple images are obtained by varying the incident illumination it is shown that this provides sufficient information to determine surface orientation at each image point

finite element methods for active contour models and balloons for 2 d and 3 d images the use of energy minimizing curves known as snakes to extract features of interest in images has been introduced by kass witkin and terzopoulos 1987 a balloon model was introduced by cohen 1991 as a way to generalize and solve some of the problems encountered with the original method a 3 d generalization of the balloon model as a 3 d deformable surface which evolves in 3 d images is presented it is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential we also show properties of energy minimizing surfaces concerning their relationship with 3 d edge points to solve the minimization problem for a surface two simplified approaches are shown first defining a 3 d surface as a series of 2 d planar curves then after comparing finite element method and finite difference method in the 2 d problem we solve the 3 d model using the finite element method yielding greater stability and faster convergence this model is applied for segmenting magnetic resonance images

elastically adaptive deformable models we present a technique for the automatic adaptation of a deformable model s elastic parameters within a kalman filter framework for shape estimation applications the novelty of the technique is that the model s elastic parameters are not constant but spatio temporally varying the variation of the elastic parameters depends on the distance of the model from the data and the rate of change of this distance each pass of the algorithm uses physics based modeling techniques to iteratively adjust both the geometric and the elastic degrees of freedom of the model in response to forces that are computed from the discrepancy between the model and the data by augmenting the state equations of an extended kalman filter to incorporate these additional variables we are able to significantly improve the quality of the shape estimation therefore the model s elastic parameters are always initialized to the same value and they are subsequently modified depending on the data and the noise distribution we present results demonstrating the effectiveness of our method for both two dimensional and three dimensional data

pattern recognition by graph matching using the potts mft neural networks this paper is concerned with programming of the potts mean field theory neural networks for pattern recognition by homomorphic mapping of the attributed relational graphs arg in order to generate the homomorphic mapping from the scene relational graph to the model graph we make use of the recently introduced suganthan technical report nanyang technical university 1994 compatibility functions in relation to the hopfield network an efficient pose clustering algorithm is used to separate and localize different occurrences of any particular object model in the scene the pose clustering algorithm also eliminates spurious hypotheses generated by the network and resolves ambiguities in the final interpretation the performance of the proposed approach to pattern recognition by homomorphism is demonstrated using a number of line patterns silhouette images and circle patterns

third order variable mesh cubic spline methods for singularly perturbed boundary value problems abstract third order variable mesh methods based on cubic spline approximation for nonlinear singularly perturbed boundary value problems of the form ey f x y y a y b are presented the convergence analysis is given and the method is shown to have third order convergence several tet examples are solved to demonstrate the efficiency of the method

high accuracy variable mesh method for nonlinear two point boundary value problems in divergence form in this article using three grid points we discuss variable mesh method of order three for the numerical solution of nonlinear two point boundary value problems p x y f x y y 0 a y 1 b we first establish an identity from which general three point finite difference approximation of various order can be obtained we obtain a family of third order discretization using variable mesh for the differential equations we select the free parameter available in this discretization which leads to the simplest third order method numerical results are provided to illustrate the proposed method and their convergence

fast surface interpolation using hierarchical basis functions an alternative to multigrid relaxation that is much easier to implement and more generally applicable is presented conjugate gradient descent is used in conjunction with a hierarchical multiresolution set of basis functions the resultant algorithm uses a pyramid to smooth the residual vector before the direction is computed simulation results showing the speed of convergence and its dependence on the choice of interpolator the number of smoothing levels and other factors are presented the relationship of this approach to other multiresolution relaxation and representation schemes is also discussed

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

automated allocation of mesh points for stiff two point boundary value problems abstract the numerical solution of a stiff two point boundary value problem is considered where a general procedure to automate the allocation of mesh points is obtained this procedure requires negligible prerequisites to the problem the automation is achieved by constructing a diffeomorphism acting on the domain of the independent variable this smooths out the sharp corners in the solution while it leaves the domain boundary and the boundary conditions unchanged the diffeomorphism is defined by differential equations which are coupled with the original equations to form an augmented system

viewpoint invariant recovery of visual surfaces from sparse data an algorithm for the reconstruction of visual surfaces from sparse data is proposed an important aspect of this algorithm is that the surface estimated from the sparse data is approximately invariant with respect to rigid transformation of the surface in 3d space the algorithm is based on casting the problem as an ill posed inverse problem that must be stabilized using a priori information related to the image and constraint formation to form a surface estimate that is approximately invariant with respect to viewpoint the stabilizing information is based on invariant surface characteristics with appropriate approximations this results in a convex functional to minimize which is then solved using finite element analysis the relationship of this algorithm to several previously proposed reconstruction algorithms is discussed and several examples that demonstrate its effectiveness in reconstructing viewpoint invariant surface estimates are given

dynamic 3d models with local and global deformations deformable superquadrics the authors present a physically based approach to fitting complex three dimensional shapes using a novel class of dynamic models that can deform both locally and globally they formulate the deformable superquadrics which incorporate the global shape parameters of a conventional superellipsoid with the local degrees of freedom of a spline the model s six global deformational degrees of freedom capture gross shape features from visual data and provide salient part descriptors for efficient indexing into a database of stored models the local deformation parameters reconstruct the details of complex shapes that the global abstraction misses the equations of motion which govern the behavior of deformable superquadrics make them responsive to externally applied forces the authors fit models to visual data by transforming the data into forces and simulating the equations of motion through time to adjust the translational rotational and deformational degrees of freedom of the models model fitting experiments involving 2d monocular image data and 3d range data are presented

the computation of visible surface representations a computational theory of visible surface representations is developed the visible surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of 1 integrating multiscale constraints on surface depth and orientation from multiple visual sources 2 interpolating dense piecewise smooth surfaces from these constraints 3 detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation and 4 structuring large scale distributed surface representations to achieve computational efficiency visible surface reconstruction is an inverse problem a well posed variational formulation results from the use of a controlled continuity surface model discontinuity detection amounts to the identification of this generic model s distributed parameters from the data finite element shape primitives yield a local discretization of the variational principle the result is an efficient algorithm for visible surface reconstruction

a linearly implicit predictor corrector method for reaction diffusion equations abstract a novel linearly implicit predictor corrector scheme is developed for the numerical solution of reaction diffusion equations iterative processes are avoided by treating the nonlinear reaction terms explicitly while maintaining superior accuracy and stability properties compared to the well known methods and linearly implicit runge kutta methods the proposed method allows the opportunity of solving large systems of reaction diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system numerical results confirm the enhanced stability accuracy and efficiency of the method when applied to reaction diffusion equations arising in biochemistry and population ecology

a fast scalable algorithm for discontinuous optical flow estimation multiple moving objects partially occluded objects or even a single object moving against the background gives rise to discontinuities in the optical flow field in corresponding image sequences while uniform global regularization based moderately fast techniques cannot provide accurate estimates of the discontinuous flow field statistical optimization based accurate techniques suffer from excessive solution time a weighted anisotropic smoothness based numerically robust algorithm is proposed that can generate discontinuous optical flow field with high speed and linear computational complexity weighted sum of the first order spatial derivatives of the flow field is used for regularization less regularization is performed where strong gradient information is available the flow field at any point is interpolated more from those at neighboring points along the weaker intensity gradient component such intensity gradient weighted regularization leads to euler lagrange equations with strong anisotropies coupled with discontinuities in their coefficients a robust multilevel iterative technique that recursively generates coarse level problems based on intensity gradient weighted smoothing weights is employed to estimate discontinuous optical flow field experimental results are presented to demonstrate the efficacy of the proposed technique

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

cooperative robust estimation using layers of support we present an approach to the problem of representing images that contain multiple objects or surfaces rather than using an edge based approach to represent the segmentation of a scene we propose a multilayer estimation framework which uses support maps to represent the segmentation of the image into homogeneous chunks this support based approach can represent objects that are split into disjoint regions or have surfaces that are transparently interleaved our framework is based on an extension of robust estimation methods that provide a theoretical basis for support based estimation we use a selection criteria derived from the minimum description length principle to decide how many support maps to use in describing an image our method has been applied to a number of different domains including the decomposition of range images into constituent objects the segmentation of image sequences into homogeneous higher order motion fields and the separation of tracked motion features into distinct rigid body motions

a two stage algorithm for discontinuity preserving surface reconstruction a two stage algorithm for visual surface reconstruction from scattered data while preserving discontinuities is presented the first stage consists of a robust local approximation algorithm the moving least median of squares mlms of error to clean the data and create a grid from the original scattered data points this process is discontinuity preserving the second stage introduces a weighted bicubic spline wbs as a surface descriptor the wbs has a factor in the regularizing term that adapts the behavior of the spline across discontinuities the weighted bicubic approximating spline can approximate data with step discontinuities with no discernible distortion in the approximating surface the combination of robust surface fitting and wbss removes outliers and reduces gaussian noise either stage by itself would not effectively remove both kinds of noise experimental results with the two stage algorithm are presented

predictor corrector methods for parabolic partial differential equations in this paper we extend predictor corrector methods commonly used for the numerical solution of ordinary differential equations o d e s to parabolic partial differential equations p d e s typically of the form ut auxx u ux x t we describe linear multistep methods for p d e s the nonlinear algebraic equations arising from implicit formulae being solved using a corrector analogous to those used for o d e s a sufficient condition for convergence of the iteration is then derived and is found in most cases to be far less restrictive than that obtained from the usual method of lines approach numerical results are presented to investigate the necessity of this condition they also indicate that we can accelerate convergence by reducing the time increment this allows us to achieve convergence within a prescribed number of iterations and so to construct pcm methods corresponding to p ec m methods for o d e s numerical results are also given to test the absolute stability of the crank nicolson corrector for various predictors p and iterations m

ranging through gabor logons a consistent hierarchical approach in this work the correspondence problem in stereo vision is handled by matching two sets of dense feature vectors inspired by biological evidence these feature vectors are generated by a correlation between a bank of gabor sensors and the intensity image the sensors consist of two dimensional gabor filters at various scales spatial frequencies and orientations which bear close resemblance to the receptive field profiles of simple v1 cells in visual cortex a hierarchical stochastic relaxation method is then used to obtain the dense stereo disparities unlike traditional hierarchical methods for stereo feature based hierarchical processing yields consistent disparities to avoid false matchings due to static occlusion a dual matching based on the imaging geometry is used

an efficient computational method for second order boundary value problems of nonlinear differential equations we will discuss a class of boundary value problems bvps of nonlinear differential equations in the reproducing kernel space the existence of the solution and a iterative method are established for the kind of problems some examples are displayed to demonstrate the computation efficiency of the method

an implicit high accuracy variable mesh scheme for 1 d non linear singular parabolic partial differential equations abstract in this paper we propose a new two level implicit difference scheme of o k 2 h l 1 kh l h l 3 for the solution of non linear parabolic equation eu xx x t u u x u t 0 x t 0 subject to appropriate initial and dirichlet boundary conditions prescribed where k 0 h l 0 are mesh sizes in t and x coordinates respectively and e 0 is a small parameter in addition we also discuss a new explicit variable mesh difference scheme of o kh l h l 3 for the estimates of u x in all cases we require only three spatial variable grid points the proposed schemes require less algebra and three evaluations of function a special technique is required to solve singular parabolic equations the proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all h l 0 and k 0 computational results are provided to support our derived schemes and analysis

discontinuity preserving and viewpoint invariant reconstruction of visible surfaces using a first order regularization this paper describes the application of a first order regularization technique to the reconstruction of visible surfaces our approach is a computationally efficient first order method that simultaneously achieves approximate invariance and preservation of discontinuities it is also robust with respect to the smoothing parameter spl lambda the robustness property to spl lambda allows a free choice of spl lambda without struggling to determine an optimal spl lambda that provides the best reconstruction a new approximately invariant first order stabilizing function for surface reconstruction is obtained by employing a first order taylor expansion of a nonconvex invariant stabilizing function that is expanded at the estimated value of the squared gradient instead of at zero the data compatibility measure is the squared perpendicular distance between the reconstructed surface and the constraint surface this combination of stabilizing function and data compatibility measure is necessary to achieve invariance with respect to rotations and translations sharp preservation of discontinuities is achieved by a weighted sum of adjacent pixels the results indicate that the proposed methods perform well on sparse noisy range data in addition the volume between two surfaces normalized by the surface area interpreted as average distance between two surfaces is proposed as an invariant measure for the comparison of reconstruction results

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

surface interpolation networks orthogonal wavelets can be used as models for receptive fields in the human visual system they may also be used to solve spatial interpolation problems formulated either as regularization or 2 d kalman filtering the solutions take the form of simple feedback networks and only a few iterations are required for convergence

a fourth order finite difference method for a class of singular two point boundary value problems a method is described based on a non uniform mesh for the class of singular two point boundary value problems x ay f x y 0

high order variable mesh off step discretization for the solution of 1 d non linear hyperbolic equation abstract in this paper we propose a new high order three level implicit method based on off step discretization on a non uniform mesh for the solution of 1 d non linear hyperbolic partial differential equation of the form u tt u xx g x t u u x u t subject to appropriate initial and dirichlet boundary conditions we use only three evaluations of the function g and three grid points at each time level in a compact cell our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation the method is convergent for uniform mesh numerical results are provided to justify the usefulness of the proposed method

variable mesh methods for the numerical solution of two point singular perturbation problems abstract in this paper variable mesh difference methods of third order are derived for the solution of the two point second order singular perturbation problems y f x y y e these methods are applied to a number of examples and their superiority over the existing methods is shown

photometric stereo via expectation maximization this paper presents a robust and automatic approach to photometric stereo where the two main components namely surface normals and visible surfaces are respectively optimized by expectation maximization em a dense set of input images is conveniently captured using a digital video camera while a handheld spotlight is being moved around the target object and a small mirror sphere in our approach the inherently complex optimization problem is simplified into a two step optimization where em is employed in each step 1 using the dense input the weight or importance of each observation is alternately optimized with the normal and albedo at each pixel and 2 using the optimized normals and employing the markov random fields mrfs surface integrabilities and discontinuities are alternately optimized in visible surface reconstruction our mathematical derivation gives simple updating rules for the em algorithms leading to a stable practical and parameter free implementation that is very robust even in the presence of complex geometry shadows highlight and transparency we present high quality results on normal and visible surface reconstruction where fine geometric details are automatically recovered by our method

regularization of inverse visual problems involving discontinuities inverse problems such as the reconstruction problems that arise in early vision tend to be mathematically ill posed through regularization they may be reformulated as well posed variational principles whose solutions are computable standard regularization theory employs quadratic stabilizing functionals that impose global smoothness constraints on possible solutions discontinuities present serious difficulties to standard regularization however since their reconstruction requires a precise spatial control over the smoothing properties of stabilizers this paper proposes a general class of controlled continuity stabilizers which provide the necessary control over smoothness these nonquadratic stabilizing functionals comprise multiple generalized spline kernels combined with noncontinuous continuity control functions in the context of computational vision they may be thought of as controlled continuity constraints these generic constraints are applicable to visual reconstruction problems that involve both continuous regions and discontinuities for which global smoothness constraints fail

on three dimensional surface reconstruction methods a survey is presented of some of the surface reconstruction methods that can be found in the literature the focus is on a small recent and important subset of the published reconstruction techniques the techniques are classified based on the surface representation used implicit versus explicit functions a study is made of the important aspects of the surface reconstruction techniques one aspect is the viewpoint invariance of the methods this is an important property if object recognition is the ultimate objective the robustness of the various methods is examined it is determined whether the parameter estimates are biased and the sensitivity to obscuration is addressed the latter two aspects are particularly important for fitting functions in the implicit form a detailed description is given of a parametric reconstruction method for three dimensional object surfaces that involves numeric grid generation techniques and variational principle formulations this technique is invariant to rigid motion in dimensional space

robust parameter estimation in computer vision estimation techniques in computer vision applications must estimate accurate model parameters despite small scale noise in the data occasional large scale measurement errors outliers and measurements from multiple populations in the same data set increasingly robust estimation techniques some borrowed from the statistics literature and others described in the computer vision literature have been used in solving these parameter estimation problems ideally these techniques should effectively ignore the outliers and measurements from other populations treating them as outliers when estimating the parameters of a single population two frequently used techniques are least median of squares lms p j rousseeuw j amer statist assoc 79 1984 pp 871 880 and m estimators robust statistics the approach based on influence functions f r hampel et al john wiley 1986 robust statistics p j huber john wiley 1981 lms handles large fractions of outliers up to the theoretical limit of 50 for estimators invariant to affine changes to the data but has low statistical efficiency m estimators have higher statistical efficiency but tolerate much lower percentages of outliers unless properly initialized while robust estimators have been used in a variety of computer vision applications three are considered here in analysis of range images images containing depth or x y z measurements at each pixel instead of intensity measurements robust estimators have been used successfully to estimate surface model parameters in small image regions in stereo and motion analysis they have been used to estimate parameters of what is called the fundamental matrix which characterizes the relative imaging geometry of two cameras imaging the same scene recently robust estimators have been applied to estimating a quadratic image to image transformation model necessary to create a composite mosaic image from a series of images of the human retina in each case a straightforward application of standard robust estimators is insufficient and carefully developed extensions are used to solve the problem

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

a class of non uniform mesh three point arithmetic average discretization for y f x y y and the estimates of y abstract we propose two new non uniform mesh three point arithmetic average discretization strategy of order two and three to solve non linear ordinary differential equation y f x y y a x b and the estimates of first order derivative y where y y x subject to the essential boundary conditions y a a y b b both methods are compact and directly applicable to singular problems there is no need to discuss any special scheme for the singular problems error analysis of a proposed method is discussed numerical experiments are performed to study the convergence behaviors and efficiency of the proposed methods

robust b spline image modeling with application to image processing we present a new approach to two dimensional 2 d robust spline image smoothing based on the m estimator algorithm unlike in other m estimator based image processing algorithms the new algorithm takes into consideration the spatial relations between picture elements the contribution of the sample to the model depends not only on the current residual of that sample but also on the neighboring residuals a smoothing parameter is estimated separately for each processing window and it adapts to the local structure of the image the proposed algorithm is applied to image filtering the resulting filter preserves details and suppresses additive gaussian and impulsive noise efficiently

spline function approximation for differential equations abstract a spline function which depends on a parameter p 0 has been constructed the spline function is of trigonometric and polynomial order one for p 0 it reduces to ordinary cubic spline the application of the parametric spline to the solution of the system of ordinary and partial differential equations is discussed the numerical results are given

orientation matching minimization for image denoising and inpainting in this paper we propose an orientation matching functional minimization for image denoising and image inpainting following the two step tv stokes algorithm rahman et al in scale space and variational methods in computer vision pp 473 482 springer heidelberg 2007 tai et al in image processing based on partial differential equations pp 3 22 springer heidelberg 2006 bertalmio et al in proc conf comp vision pattern rec pp 355 362 2001 a regularized tangential vector field with zero divergence condition is first obtained then a novel approach to reconstruct the image is proposed instead of finding an image that fits the regularized normal direction from the first step we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction this functional yields a new nonlinear partial differential equation pde for reconstructing denoised and inpainted images the equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field providing reconstructed images which have sharp edges and smooth regions the additive operator splitting aos scheme is used for discretizing euler lagrange equations we present the results of various numerical experiments that illustrate the improvements obtained with the new functional

high order differentiation filters that work reliable derivatives of digital images have always been hard to obtain especially but not only at high orders we analyze the sources of errors in traditional filters such as derivatives of the gaussian that are used for differentiation we then study a class of filters which is much more suitable for our purpose namely filters that preserve polynomials up to a given order we show that the errors in differentiation can be corrected using these filters we derive a condition for the validity domain of these filters involving some characteristics of the filter and of the shape our experiments show a very good performance for smooth functions

constructing intrinsic parameters with active models for invariant surface reconstruction a technique for constructing a canonical surface parameterization in terms of lines of curvature is presented two methods of computing the canonical invariant representation are also presented in the first method a static instance of the controlled continuity spline is used for the stabilizer ways to modify it to reflect a change of parameters to the lines of curvature are described in the second method the dynamic instance of the controlled continuity spline called the deformable model is used a force field defined in terms of the principal vectors is synthesized and applied to the parameter curves of the deformable model to coerce them along the lines of curvature in essence any transformation of parameters requires a modification of the stabilizer in the first method whereas in the second method it is tantamount to synthesizing a new force field experimental results with real and synthetic range data are included

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

a uniform mesh finite difference method for a class of singular two point boundary value problems abstract a method is described based on a uniform mesh for the class of singular two point boundary value problems x y f x y 0 x 1 y 0 a y 1 b it is shown to be order h 4 convergent for all 0 1

high accuracy implicit variable mesh methods for numerical study of special types of fourth order non linear parabolic equations in this paper we propose a new implicit high order two level finite difference method of o k 2 k h l h l 3 for the solution of special type of 1d fourth order non linear parabolic partial differential equation of the form u x x x x 2 u x x t u t t f x t u u x u x x u t u x x x u x t a 0 and hl 0 are the mesh sizes in time and space coordinates respectively the third order variable mesh methods proposed are based on only three spatial grid points xl xl 1 meaning that no fictitious points are required for incorporating the boundary conditions we discuss how our formulation is able to handle linear singular problem the proposed two level finite difference method is shown to be unconditionally stable for a model linear problem we applied the proposed methods to find the numerical solution of more complex fourth order non linear equations like second order benjamin ono equation and the good boussinesq equation it is evident from the numerical experiments that the numerical results agree well with the exact solutions hence demonstrating the efficiency of the methods derived

the local structure of image discontinuities in one dimension the detailed structure of intensities in the local neighborhood of an edge can often indicate the nature of the physical event givinig rise to that edge we argue that the limit as we approach arbitrarily close to either side of an edge of such image parameters as type of texture texture gradient color appropriate directional derivatives of intensity etc is a key aspect of this structure however the general problem of capturing this local structure is surprisingly complex thus we restrict ourselves in this paper to a relatively simple domain one dimensional cuts through idealized images modeled by piecewise smooth c1 functions corrupted by gaussian noise within this domain we define local structure to be the limit of the uncorrupted intensity and of its derivatives as we approach arbitrarily close to either side of a discontinuity we develop a technique that captures this local structure while simultaneously locating the discontinuities and demonstrate that these tasks are in fact inseparable the technique is an extension using estimation theory of the classical definition of discontinuity it handles in a consistent fashion both jump discontinuities in the function and jump discontinuities in its first derivative so called step edges are a special case of the former and roof edges of the latter it also integrates again in a consistent fashion information derived from a number of different neighborhood sizes

adaptive solution of bvps in singularly perturbed second order odes by the extended numerov method combined with an iterative local grid h refinement the extended numerov scheme of chawla is one of high order compact finite difference discretisations applicable to bvps in second order odes and to pdes the use of such discretisations in connection with adaptive grids remains largely unexplored in order to make up this deficiency we study the combination of the extended numerov discretisation with the iterative local adaptive grid h refinement previously tested by the present author at the assumption of the conventional finite difference discretisation a detailed formalism enabling a posteriori error analysis of the extended numerov scheme is developed together with several alternative choices for error estimators grid refinement indicators and regridding strategies the adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed odes including equations relevant for electrochemistry the most satisfactory algorithm involves deferred approach to error estimation based on virtual grid coarsening for truncation error evaluation local discretisation errors as refinement indicators and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement the extended numerov scheme proves more efficient than the conventional discretisation when used together with the iterative h refinement

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

bootstrap technology and applications bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis although these methods depend on the availability of fast inexpensive computing they offer the potential for highly accurate methods of inference moreover they can even eliminate the need to impose a convenient statistical model that does not have a strong scientific basis in this article we review some bootstrap methods emphasizing applications through illustrations with some real data special attention is given to regression problems with dependent data and choosing tuning parameters for optimal performance

third order accurate variable mesh discretization and application of tage iterative method for the non linear two point boundary value problems with homogeneous functions in integral form in this paper we discuss the application of two parameter alternating group explicit tage iterative method to an efficient third order variable mesh method for the solution of non linear differential equations with integral homogeneous functions subject to natural boundary conditions the proposed method is applicable only when the internal grid points of the solution space are odd in number the proposed iterative method is also applicable to the integro differential equations with singular coefficients comparative numerical results are given to demonstrate the usefulness of the proposed method

scale invariant image recognition based on higher order autocorrelation features we propose a framework and a complete implementation of a translation and scale invariant image recognition system for natural indoor scenes the system employs higher order autocorrelation features of scale space data which permit linear classification an optimal linear classification method is presented which is able to cope with a large number of classes represented by many as well as very few samples in the course of the analysis of our system we examine which numerical methods for feature transformation and classification show sufficient stability to fulfill these demands the implementation has been extensively tested we present the results of our own application and several classification benchmarks

a new stable variable mesh method for 1 d non linear parabolic partial differential equations we propose a new stable variable mesh implicit difference method for the solution of non linear parabolic equation u x email protected x t u u x u t 00 subject to appropriate initial and dirichlet boundary conditions prescribed we require only 3 3 spatial grid points and two evaluations of the function f the proposed method is directly applicable to solve parabolic equation having a singularity at x 0 the proposed method when applied to a linear diffusion equation is shown to be unconditionally stable the numerical tests are performed to demonstrate the convergence of the proposed new method

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

partial smoothing splines for noisy boundaries with corners investigates the estimation of 2 d boundary functions from sampled data sets where both noise and corners are present the approach is based on the partial smoothing spline in which the estimated boundary function consists of an ordinary smoothing spline and a parametric function that describes the discontinuities i e corners of the boundary prior knowledge about the boundary such as the number of corners their locations noise levels and the amount of smoothness is not required for the boundary estimate the smoothing parameter and the corner locations of the spline which are parts of the estimate are determined by the generalized cross validation method whereby statistical properties are gathered from the input sampled data rather than specified a priori this approach enables the smoothing of a noisy boundary while retaining an accurate description of the boundary corners extensive experiments were conducted to verify its ability to smooth noise while retaining a good representation of boundary corners and do not rely on any prior information

a variable mesh c splage method of accuracy o k2hl 1 khl hl3 for 1d nonlinear parabolic equations a new high accuracy two level implicit cubic spline formulation for the solution of nonlinear parabolic equation u x email protected x t u u x u t is presented the strategy of the cubic spline alternating group explicit c splage method is developed the proposed c splage method requires only 3 spatial grid points and applicable to singular problems the convergence analysis of the proposed c splage method is given the obtained c splage method is compared to the corresponding successive over relaxation sor method both in terms of stability and performance

variable mesh difference schemes for solving a nonlinear schrodinger equation with a linear damping term abstract this paper describes moving variable mesh finite difference schemes to numerically solve the nonlinear schrodinger equation including the effects of damping and nonhomogeneity in the propagation media these schemes have accurately predicted the location of the peak of soliton compared to the uniform mesh for the case in which the exact solution is known numerical results are presented when damping and nonhomogeneous effects are included and in the absence of these effects the results were verified with the available exact solution

a family of variable mesh methods for the estimates of du dr and solution of non linear two point boundary value problems with singularity in this article using three grid points we discuss variable mesh methods of order two and three for the numerical solution of the nonlinear differential equation u f r u u 0

interpreting line drawings as three dimensional surfaces understanding how line drawings convey tri dimensionality is of fundamental importance in explaining surface perception when photometry is either uninformative or too complex to model analytically we put forward here a computational model for interpreting line drawings as three dimensional surfaces based on constraints on local surface orientation along extremal and discontinuity boundaries specific techniques are described for two key processes recovering the three dimensional conformation of a space curve e g a surface boundary from its two dimensional projection in an image and interpolating smooth surfaces from orientation constraints along extremal boundaries the relevance of the model to a general theory of low level vision is discussed

a fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations the method uses evaluation at two off steps three neighbouring knots and it reduces to simple five terms recurrence relations the formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method the root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method

application of tage iterative algorithms to an efficient third order arithmetic average variable mesh discretization for two point non linear boundary value problems abstract in this article we aim to discuss the application of two parameter alternating group explicit tage iterative method to an efficient third order numerical method for two point non linear boundary value problems on a variable mesh there is no need to use any special technique to handle singular problems for the non linear problems the proposed method leads to a non linear system whose jacobian is tridiagonal we also discuss newton tage algorithm for general non linear differential equations the proposed tage and newton tage iterative methods are suitable for use on parallel computers the convergence theory is discussed in details numerical results are given to justify the utility of the proposed iterative methods

edge preserving image denoising and estimation of discontinuous surfaces in this paper we are interested in the problem of estimating a discontinuous surface from noisy data a novel procedure for this problem is proposed based on local linear kernel smoothing in which local neighborhoods are adapted to the local smoothness of the surface measured by the observed data the procedure can therefore remove noise correctly in continuity regions of the surface and preserve discontinuities at the same time since an image can be regarded as a surface of the image intensity function and such a surface has discontinuities at the outlines of objects this procedure can be applied directly to image denoising numerical studies show that it works well in applications compared to some existing procedures

a new finite difference method for a class of singular two point boundary value problems we present a new three point finite difference method based on uniform mesh for a class of singular two point boundary value problem x ay f x y y 0 a y 1 b 0 a 1 we show that the method based on uniform mesh provides o h 4 convergent approximations this method is illustrated by two numerical examples one is linear and another is non linear

analysis of pseudo profiles in organ pharmacokinetics and toxicokinetics in general the pharmacokinetic model parameters like rate constants area under the curve auc etc are estimated via a two stage procedure where the values obtained from concentration time relationships within one subject experimental unit are considered to be functionally related to the drug concentrations measured in many cases mean estimators and their respective standard errors are calculated afterwards the determination of drug concentrations in organs as well as in the serum of small animals mice rats in dependence of the time after administration often does not permit the establishment of reasonable profiles within one subject suited for conventional pharmacokinetic analyses and tolerability studies frequently only one experimental value per organ or animal is recorded the consequence is that most pharmacokinetic parameters are to be estimated on the basis of the mean concentrations rather than via the mean of individual parameter estimates in all cases of a non linear relationship between a target item and the concentration the mean concentration based estimators and the two stage profile based estimators need not coincide in addition in the former case variance estimators may be either difficult to obtain or not deducible in order to get variance estimators as well as to enable comparisons between different treatment regimens in addition to bioequivalence testing as a step towards human dose finding studies various resampling techniques parametric and non parametric bootstrap were applied to generate pseudo profiles from independent measurements and compared to their more conventional counterparts where applicable simulation studies based on different predefined pharmacokinetic models first order elimination after i v bolus first order elimination after first order absorption simple capacity limited kinetics revealed that even the non parametric pseudo profile stratified bootstrap resampling with replacement per time point performs quite satisfactorily