some performances of local bivariate quadratic c 1 quasi interpolating splines on nonuniform type 2 triangulations we investigate spline quasi interpolants defined by c1 bivariate quadratic b splines on nonuniform type 2 triangulations and by discrete linear functionals based on a fixed number of triangular mesh points either in the support or close to the support of such b splines we show they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds we also present some numerical and graphical results

conformal geometry and its applications on 3d shape matching recognition and stitching three dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration 3d object recognition and classification however shape matching with noise occlusion and clutter is a challenging problem in this paper we analyze a family of quasi conformal maps including harmonic maps conformal maps and least squares conformal maps with regards to 3d shape matching as a result we propose a novel and computationally efficient shape matching framework by using least squares conformal maps according to conformal geometry theory each 3d surface with disk topology can be mapped to a 2d domain through a global optimization and the resulting map is a diffeomorphism i e one to one and onto this allows us to simplify the 3d shape matching problem to a 2d image matching problem by comparing the resulting 2d parametric maps which are stable insensitive to resolution changes and robust to occlusion and noise therefore highly accurate and efficient 3d shape matching algorithms can be achieved by using the above three parametric maps finally the robustness of least squares conformal maps is evaluated and analyzed comprehensively in 3d shape matching with occlusion noise and resolution variation in order to further demonstrate the performance of our proposed method we also conduct a series of experiments on two computer vision applications i e 3d face recognition and 3d nonrigid surface alignment and stitching

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

reconstructing 3d face model with associated expression deformation from a single face image via constructing a low dimensional expression deformation manifold facial expression modeling is central to facial expression recognition and expression synthesis for facial animation in this work we propose a manifold based 3d face reconstruction approach to estimating the 3d face model and the associated expression deformation from a single face image with the proposed robust weighted feature map rwf we can obtain the dense correspondences between 3d face models and build a nonlinear 3d expression manifold from a large set of 3d facial expression models then a gaussian mixture model in this manifold is learned to represent the distribution of expression deformation by combining the merits of morphable neutral face model and the low dimensional expression manifold a novel algorithm is developed to reconstruct the 3d face geometry as well as the facial deformation from a single face image in an energy minimization framework experimental results on simulated and real images are shown to validate the effectiveness and accuracy of the proposed algorithm

algorithms for computing shape preserving spline approximations to data we treat the problem of approximating data that are sampled with error from a function known to be convex and increasing the approximating function is a polynomial spline with knots at the data points this paper presents results analogous to those in 7 and 9 that describe some approximation properties of polynomial splines and algorithms for determining the existence of a shape preserving approximant for given data

a nonlinear analysis of instability of a pre stressed incompressible elastic plate for an isotropic incompressible elastic plate which is pre stressed into a state of hydrostatic pressure linear stability analysis predicts that when the value of the hydrostatic pressure p is increased above p say or decreased below p say the plate will become unstable in the sense that certain forms of travelling waves will become standing waves whose amplitude grows exponentially in time the upper branch neutral value p is a function of the product of the wave number and plate thickness while the lower branch neutral value p is a constant those waves which occur when the pressure deviates from the neutral values p or p by a small amount are termed near neutral waves in this paper the nonlinear evolutionary behaviour of upper branch near neutral waves are investigated it is shown that the amplitude a of such near neutral waves satisfies an evolution equation of the form d 2 a d 2 v 0 a v 1 a a 2 where v 0 and v 1 are constants and is a slow time scale the general properties of the solution of this equation are studied it is found that when v 1 0 nonlinear effects are stabilizing in the sense that any exponential growth will be suppressed by nonlinearity and turned into an oscillation and when v 1 v 1 are calculated for a selection of the plate thicknesses and wavenumbers using two material models we find that the sign of v 1 is not definite and thus nonlinear effects are stabilizing only over a certain wavenumber regime while destabilizing over the remaining wavenumber regime all the relevant factors which affect the stability of the plate are fully discussed

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

shape preserving piecewise rational interpolation with higher order continuity abstract a united form of the classical hermite interpolation and shape preserving interpolation is presented in this paper the presented interpolation method provides higher order continuous shape preserving interpolation splines the given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations by setting parameter values the interpolation curve can be generated by choosing the classical piecewise hermite interpolation polynomials or the presented piecewise rational expressions for monotonicity preserving and convexity preserving interpolation the appropriate values of a parameter are given on each subinterval numerical examples indicate that the given method produces visually pleasing curves

convexity preserving approximation by univariate cubic splines a convexity preserving approximation method is presented similar to the cubic spline interpolation the given approximation function is univariate cubic spline with c 2 continuity a very simple computing method is described the approximation order and polynomial reproducing property of this convexity preserving approximant are discussed the computing method for interpolating a small quantity of data is given some numerical examples are given to show the effectivity

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

the order of monotone piecewise cubic interpolation fritsch and carlson siam j numer anal 17 1980 pp 238 246 developed an algorithm which produces a monotone c 1 piecewise cubic interpolant to a monotone function we show that their algorithm yields a third order approximation while a modification is fourth order accurate

smoothing under diffeomorphic constraints with homeomorphic splines in this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms this diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time dependent vector field this solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques this method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost in a one dimensional setting incorporating diffeomorphic constraints is equivalent to imposing monotonicity thus as an illustration it is shown that such a monotone spline can be used to make monotone any unconstrained estimator of a regression function and that this monotone smoother inherits the convergence properties of the unconstrained estimator some numerical experiments are proposed to illustrate its finite sample performances and to compare them with another monotone estimator we also provide a two dimensional application on the computation of diffeomorphisms for landmark and image matching

face recognition from a single training image under arbitrary unknown lighting using spherical harmonics in this paper we propose two novel methods for face recognition under arbitrary unknown lighting by using spherical harmonics illumination representation which require only one training image per subject and no 3d shape information our methods are based on the result which demonstrated that the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace we provide two methods to estimate the spherical harmonic basis images spanning this space from just one image our first method builds the statistical model based on a collection of 2d basis images we demonstrate that by using the learned statistics we can estimate the spherical harmonic basis images from just one image taken under arbitrary illumination conditions if there is no pose variation compared to the first method the second method builds the statistical models directly in 3d spaces by combining the spherical harmonic illumination representation and a 3d morphable model of human faces to recover basis images from images across both poses and illuminations after estimating the basis images we use the same recognition scheme for both methods we recognize the face for which there exists a weighted combination of basis images that is the closest to the test face image we provide a series of experiments that achieve high recognition rates under a wide range of illumination conditions including multiple sources of illumination our methods achieve comparable levels of accuracy with methods that have much more onerous training data requirements comparison of the two methods is also provided

quadratic and related exponential splines in shape preserving interpolation abstract for quadratic and related exponential splines necessary and sufficient conditions are given under which the properties of convexity or monotonicity carry over from the data set to the interpolants it turns out that for quadratic splines the problems of convex or monotone interpolation may be not solvable however when using the more general exponential splines the shape is preserved if the parameters occuring now are chosen appropriately furthermore since convex or monotone spline interpolants are in general not uniquely determined a strategy for selecting one of them is proposed

positive quartic monotone quintic c 2 spline interpolation in one and two dimensions abstract this paper is concerned with shape preserving interpolation of discrete data by polynomial splines we show that positivity can be always preserved by quartic c2 splines and monotonicity by quintic c2 splines this is proved for one dimensional interpolation as well as for two dimensional interpolation on rectangular grids

stability and bifurcation of inflation of elastic cylinders a method of obtaining a full two dimensional nonlinear stability analysis of inhomogeneous deformations of arbitrary incompressible hyperelastic materials is presented the analysis that we develop replaces the second variation condition expressed as an integral involving two arbitrary perturbations with an equivalent third order system of ordinary differential equations the positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well behaved function the general theory is illustrated by applying it to the problem of the inflation of axially stretched thick walled tubes the bifurcation theory of such deformations is well known and we compare the bifurcation results with the new stability analysis

monotone and convex approximation by splines error estimates and a curve fitting algorithm we obtain jackson type estimates for the approximation of increasing or convex functions by splines with the same property these estimates are new when the knots are not uniformly spaced the method of proof motivates an algorithm for fitting a curve to data with local convexity preservation the properties of this algorithm are developed and some examples are given

face relighting from a single image under arbitrary unknown lighting conditions in this paper we present a new method to modify the appearance of a face image by manipulating the illumination condition when the face geometry and albedo information is unknown this problem is particularly difficult when there is only a single image of the subject available recent research demonstrates that the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace using a spherical harmonic representation moreover morphable models are statistical ensembles of facial properties such as shape and texture in this paper we integrate spherical harmonics into the morphable model framework by proposing a 3d spherical harmonic basis morphable model shbmm the proposed method can represent a face under arbitrary unknown lighting and pose simply by three low dimensional vectors i e shape parameters spherical harmonic basis parameters and illumination coefficients which are called the shbmm parameters however when the image was taken under an extreme lighting condition the approximation error can be large thus making it difficult to recover albedo information in order to address this problem we propose a subregion based framework that uses a markov random field to model the statistical distribution and spatial coherence of face texture which makes our approach not only robust to extreme lighting conditions but also insensitive to partial occlusions the performance of our framework is demonstrated through various experimental results including the improved rates for face recognition under extreme lighting conditions

on the symmetry of energy minimising deformations in nonlinear elasticity ii compressible materials let a bf x in mathbb r n a 0 be the region occupied by an incompressible nonlinearly hyperelastic material in its reference configuration deformations bf u it a rightarrow mathbb r n are therefore isochoric maps and so satisfy the incompressibility constraint rm det nabla bf u 1 the displacement and the mixed displacement zero traction boundary value problems are considered here a displacement boundary condition of the form bf u bf x sigma bf x is prescribed on one of the boundary components where 0 is a given constant and the displacement on the remaining boundary component is either prescribed in the same form in the case of the pure displacement boundary value problem or left unspecified in the case of the mixed boundary value problem in this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of possibly non symmetric isochoric deformations of the region a

monotonicity preserving surface interpolation c 1 monotone quadratic splines are analyzed this motivates an algorithm for interpolating monotone data given on a rectangular grid with a c 1 monotone quadratic spline surface error estimates an operation count and numerical examples are given