stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

three dimensional face shape by local fitting to a single reference model the authors present a simple method to estimate the three dimensional shape of a face from an input image using a single reference model based on least squares between the output of the linear nonlinear ln neuronal model applied to blocks from an intensity image and blocks from a depth reference model the authors present the results obtained by varying the ln model parameters and estimate their best values which provide an acceptable reconstruction of each subject s depth the authors show that increasing the light source angle over the horizontal plane in the input image produces slight increases in reconstruction error but increasing the ambient light proportion produces greater increases in reconstruction error the authors applied the method to predict each subject s unknown depth using different individual reference models and an average reference model which provides the best results as a noise reduction technique the authors perform a point by point weighted averaging with the average reference model with weights equal to the fractions of the squares of the laplacian of a gaussian applied to the prediction and to the reference depth over the sum of both finally the authors present acceptable visual results obtained from external images of faces under arbitrary illumination having performed an illumination estimation previously

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

physically based adaptive preconditioning for early vision several problems in early vision have been formulated in the past in a regularization framework these problems when discretized lead to large sparse linear systems in this paper we present a novel physically based adaptive preconditioning technique which can be used in conjunction with a conjugate gradient algorithm to dramatically improve the speed of convergence for solving the aforementioned linear systems a preconditioner based on the membrane spline or the thin plate spline or a convex combination of the two is termed a physically based preconditioner for obvious reasons the adaptation of the preconditioner to an early vision problem is achieved via the explicit use of the spectral characteristics of the regularization filter in conjunction with the data this spectral function is used to modulate the frequency characteristics of a chosen wavelet basis and these modulated values are then used in the construction of our preconditioner we present the preconditioner construction for three different early vision problems namely the surface reconstruction the shape from shading and the optical flow computation problems performance of the preconditioning scheme is demonstrated via experiments on synthetic and real data sets

evaluating kube and pentland s fractal imaging model the paper assesses the validity of a model proposed by kube and pentland 1988 that relates a rough surface to its image texture simulation was used to assess whether a linear approximation is appropriate and whether the optimal linear filter agrees with the predictions of kube and pentland s model the predictions of the model about the image directionality were also assessed on real images it was found that a linear model is capable of modeling the imaging process for surfaces of moderate roughness and lambertian reflectance and that subject to a small modification kube and pentland s model accurately predicts the relationship between surface and image spectra

a green s function approach to shape from shading we present a new algorithm for shape from shading inspired on the recently introduced disparity based approach to photometric stereo dbps assuming that the single input image will be matched to a second image through a uniform disparity field we construct an irradiance conservation equation and solve it for the matching image via green s function when a linear expansion of the reflectance map is considered this leads to a closed form approximate expression for the surface function whose parameters can be estimated via a structure from motion approach already used for dbps

the eikonal equation some results applicable to computer vision in this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape particular emphasis is given to eikonal equations two results are presented first we prove that a special type of eikonal equation has only one convex and positive c2 solution in some neighborhood of a singular point using this result we show that a restricted form of this equation has exactly two solutions these results have application in scanning electron microscopy

coefficients for studying one step rational schemes for ivps in odes ii abstract in 1 we derived variable order variable stepsize one step nonlinear methods in which global formulation is facilitated by defining some certain coefficients the analysis has a broader application and facilitates the derivation of new formulae the methods are based on a variant self adjusting rational interpolant of luke et al 2 and otunta and ikhile 3 4 in this paper we motivate an extension of this analysis to obtain a more generalized class of the rational methods in 1 for an arbitrarily desired order

learning shape correspondence for n d curves we present a learning method that introduces explicit knowledge into the shape correspondence problem given two input curves to be matched our approach establishes a dense correspondence field between them where the characteristics of the matching field closely resemble those in an a priori learning set we build a shape distance matrix from the values of a shape descriptor computed at every point along the curves this matrix embeds the correspondence problem in a highly expressive and redundant construct and provides the basis for a pattern matching strategy for curve matching we selected the previously introduced observed transport measure as a shape descriptor as its properties make it particularly amenable to the matching problem synthetic and real examples are presented along with discussions of the robustness and applications of the technique

two step runge kutta methods and hyperbolic partial differential equations the purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation jameson recently introduced the use of one step runge kutta methods for the numerical solution of the euler equations improvements in efficiency up to 80 may be achieved by using two step runge kutta methods instead of the classical onestep methods these two step runge kutta methods were first introduced by byrne and lambert in 1966 they are designed to have the same number of function evaluations as the equivalent one step schemes and thus they are potentially more efficient by solving a nonlinear programming problem which is specified by stability requirements optimal two step schemes are designed the optimization technique is applicable for stability regions of any shape

estimating three dimensional motion parameters of a rigid planar patch ii singular value decomposition we show that the three dimensional 3 d motion parameters of a rigid planar patch can be determined by computing the singular value decomposition svd of a 3 3 matrix containing the eight so called pure parameters furthermore aside from a scale factor for the translation parameters the number of solutions is either one or two depending on the multiplicity of the singular values of the matrix

surface topography using shape from shading abstract this paper demonstrates how a recently reported shape from shading scheme can be used to extract topographic information from 2d intensity imagery worthington and hancock ieee trans pattern anal mach intell 21 1999 1250 1267 the shape from shading scheme has two important features which enhance the recovered surface description firstly it uses a geometric update procedure which allows the image irradiance equation to be satisfied as a hard constraint this not only improves the data closeness of the recovered needle map but also removes the necessity for extensive parameter tuning secondly we use curvature information to impose topographic constraints on the recovered needle map the topographic information is captured using the shape index of koenderink and van doorn image vision comput 10 1992 557 565 and consistency is imposed using a robust error function we show that the new shape from shading scheme leads to a meaningful topographic labelling of 3d surface structures moreover the resulting topographic information proves to be useful in a simple histogram based object recognition scheme

direct shape from shading with improved rate of convergence in this paper new direct shape from shading algorithms with improved rate of convergence for inclined and perpendicular light source are presented the algorithms are mainly based on optimal dynamic system control with dynamic programming unlike the previous shading algorithms which first estimate the gradient of surface instead of depth using variational approaches our algorithms provide a direct relative depth perception here the maximum uphill principle for inclined and perpendicular light source is derived instead of using the existing minimum downhill criteria in order to perceive the normal shape of the object also the convergence conditions for the algorithms are derived and proven under zero initial condition we show that the learning rate is linearly proportional to the minimum value of the input image in the discrete case furthermore in order to increase the rate of convergence the momentum factor is introduced

a generic and provably convergent shape from shading method for orthographic and pinhole cameras we describe a mathematical and algorithmic study of the lambertian shape from shading problem for orthographic and pinhole cameras our approach is based upon the notion of viscosity solutions of hamilton jacobi equations this approach provides a mathematical framework in which we can show that the problem is well posed we prove the existence of a solution and we characterize all the solutions our contribution is threefold first we model the camera both as orthographic and as perspective pinhole whereas most authors assume an orthographic projection see horn and brooks 1989 for a survey of the sfs problem up to 1989 and zhang et al 1999 kozera 1998 durou et al 2004 for more recent ones thus we extend the applicability of shape from shading methods to more realistic acquisition models in particular it extends the work of prados et al 2002a and rouy and tourin 1992 we provide some novel mathematical formulations of this problem yielding new partial differential equations results about the existence and uniqueness of their solutions are also obtained second by introducing a generic hamiltonian we define a general framework allowing to deal with both models orthographic and perspective thereby simplifying the formalization of the problem thanks to this unification each algorithm we propose can compute numerical solutions corresponding to all the modeling third our work allows us to come up with two new generic algorithms for computing numerical approximations of the continuous solution of the shape from shading problem as well as a proof of their convergence toward that solution moreover our two generic algorithms are able to deal with discontinuous images as well as images containing black shadows

a multigrid method for shape from shading the shape from shading problem has received much attention in the computer vision literature in recent years the basic problem is to recover the em shape z x y of a surface from a given map of its em shading i e its variation of brightness over a given domain mathematically one has to solve approximately the em image irradiance equation begin center e x y end center relating a given image irradiance em e x y to the radiance of the surface at each point em x y with em r p q a given em reflectance z x and q z y na possible presence of noise and lack of adequate boundary conditions adds to the difficulty of this problem a number of different approaches towards its solution have been proposed in the vision literature including various regularization models however a reliable efficient solution method for practical instances has remained elusive so far nin this paper we analyze the various solution models proposed with the aim of applying an efficient multigrid solver a combination of an fmg continuation technique with an appropriate discretization of one such solution model proposed by b horn yields an efficient solver our results are demonstrated by examples

shape estimation of transparent objects by using inverse polarization ray tracing few methods have been proposed to measure three dimensional shapes of transparent objects such as those made of glass and acrylic in this paper we propose a novel method for estimating the surface shapes of transparent objects by analyzing the polarization state of the light existing methods do not fully consider the reflection refraction and transmission of the light occurring inside a transparent object we employ a polarization raytracing method to compute both the path of the light and its polarization state polarization raytracing is a combination of conventional raytracing which calculates the trajectory of light rays and mueller calculus which calculates the polarization state of the light first we set an initial value of the shape of the transparent object then by changing the shape the method minimizes the difference between the input polarization data and the rendered polarization data calculated by polarization raytracing finally after the iterative computation is converged the shape of the object is obtained we also evaluate the method by measuring some real transparent objects

power bounded prolongations and picard lindelof iteration the possibility of balancing the iteration and discretization errors in iterative solution of large systems of initial value problems is discussed the main result answers the question affirmatively by stating that in the convergence process in a model situation the decay rate of iteration errors is essentially independent of the step size refinement process

a multiresolution approach for shape from shading coupling deterministic and stochastic optimization shape from shading is an ill posed inverse problem for which there is no completely satisfactory solution in the existing literature in this paper we address shape from shading as an energy minimization problem we first show that the deterministic approach provides efficient algorithms in terms of cpu time but reaches its limits since the energy associated with shape from shading can contain multiple deep local minima we derive an alternative stochastic approach using simulated annealing the obtained results strongly outperform the results of the deterministic approach the shortcoming is an extreme slowness of the optimization therefore we propose a hybrid approach which combines the deterministic and stochastic approaches in a multiresolution framework

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

recovery of surfaces with discontinuities by fusing shading and range data within a variational framework this work provides a variational framework for fusing range and intensity data for recovering regularized surfaces it is shown that this framework provides natural boundary conditions for the shape from shading problem results in a new shape from shading formulation in the absence of range data and provides a new fusion paradigm when range data is incorporated the approach is demonstrated on simulated range and intensity images error analysis with respect to the ground truth surface is presented it is shown that the formulation performs well even in very noisy images

fast facial shape recovery from a single image with general unknown lighting by using tensor representation in this paper we propose a fast 3 d facial shape recovery algorithm from a single image with general unknown lighting in order to derive the algorithm we formulate a nonlinear least square problem with two parameter vectors which are related to personal identity and light conditions we then combine the spherical harmonics for the surface normals of a human face with tensor algebra and show that in a certain condition the dimensionality of the least square problem can be further reduced to one tenth of the regular subspace based model by using tensor decomposition n mode svd which greatly speeds up the computations in order to enhance the shape recovery performance we have incorporated prior information in updating the parameters in the experiment the proposed algorithm takes less than 0 4s to reconstruct a face and shows a significant performance improvement over other reported schemes

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

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

impossible and ambiguous shading patterns a smooth object depicted in a monochrome image will often exhibit brightness variation or shading a problem much studied in computer vision has been that of how object shape may be recovered from image shading when the imaging conditions are such that an overhead point source illuminates a smooth lambertian surface the problem may be formulated as that of finding a solution to an eikonal equation this article will focus on the existence and uniqueness of such solutions reporting recent results obtained with regard to existence shading patterns are exhibited for which there is no corresponding object shape specifically a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions additionally a sufficient condition is given for an eikonal equation to have no solution whatsoever in connection with uniqueness we consider eikonal equations defined over a disc such that the euclidean norm of the gradient of any solution is circularly symmetric vanishes exactly at the disc center and diverges to infinity as the circumference of the disc is approached contrary to earlier influential work a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions

recovery of the 3 d location and motion of a rigid object through camera image an extended kalman filter approach this paper deals with the problem of locating a rigid object and estimating its motion in three dimensions this involves determining the position and orientation of the object at each instant when an image is captured by a camera and recovering the motion of the object between consecutive frames in the implementation scheme used here a sequence of camera images digitized at the sample instants is used as the initial input data measurements are made of the locations of certain features e g maximum curvature points of an image contour corners edges etc on the 2 d images to measure the feature locations a matching algorithm is used which produces correspondences between the features in the image and the object using the measured feature locations on the image an algorithm is developed to solve the location and motion problem the algorithm is an extended kalman filter modeled for this application

direct analytical methods for solving poisson equations in computer vision problems direct analytical methods are discussed for solving poisson equations of the general form delta u f on a rectangular domain some embedding techniques that may be useful when boundary conditions obtained from stereo and occluding boundary are defined on arbitrary contours are described the suggested algorithms are computationally efficient owing to the use of fast orthogonal transforms applications to shape from shading lightness and optical flow problems are also discussed a proof for the existence and convergence of the flow estimates is given experiments using synthetic images indicate that results comparable to those using multigrid can be obtained in a very small number of iterations

subpixel photometric stereo conventional photometric stereo recovers one normal direction per pixel of the input image this fundamentally limits the scale of recovered geometry to the resolution of the input image and cannot model surfaces with subpixel geometric structures in this paper we propose a method to recover subpixel surface geometry by studying the relationship between the subpixel geometry and the reflectance properties of a surface we first describe a generalized physically based reflectance model that relates the distribution of surface normals inside each pixel area to its reflectance function the distribution of surface normals can be computed from the reflectance functions recorded in photometric stereo images a convexity measure of subpixel geometry structure is also recovered at each pixel through an analysis of the shadowing attenuation then we use the recovered distribution of surface normals and the surface convexity to infer subpixel geometric structures on a surface of homogeneous material by spatially arranging the normals among pixels at a higher resolution than that of the input image finally we optimize the arrangement of normals using a combination of belief propagation and mcmc based on a minimum description length criterion on 3d textons over the surface the experiments demonstrate the validity of our approach and show superior geometric resolution for the recovered surfaces

the feasibility of motion and structure from noisy time varying image velocity information this research addresses the problem of noise sensitivity inherent in motion and structure algorithms the motion and structure paradigm is a two step process first we measure image velocities and perhaps their spatial and temporal derivatives are obtained from time varying image intensity data and second we use these data to compute the motion of a moving monocular observer in a stationary environment under perspective projection relative to a single 3 d planar surface the first contribution of this article is an algorithm that uses time varying image velocity information to compute the observer s translation and rotation and the normalized surface gradient of the 3 d planar surface the use of time varying image velocity information is an important tool in obtaining a more robust motion and structure calculation the second contribution of this article is an extensive error analysis of the motion and structure problem any motion and structure algorithm that uses image velocity information as its input should exhibit error sensitivity behavior compatible with the results reported here we perform an average and worst case error analysis for four types of image velocity information full and normal image velocities and full and normal sets of image velocity and its derivatives these derivatives are simply the coefficients of a truncated taylor series expansion about some point in space and time the main issues we address here are just how sensitive is a motion and structure computation in the presence of noisy input or alternately how accurate must our image velocity information be how much and what type of input data is needed and under what circumstances is motion and structure feasible that is when can we be sure that a motion and structure computation will produce usable results we base our answers on a numerical error analysis we conduct for a large number of motions

a robust real time algorithm for facial shape recovery from a single image containing cast shadow under general unknown lighting it is difficult to find a non iterative solution for statistical shape from shading sfs for a single image containing cast shadow under unknown arbitrary light conditions this is because it is not trivial to find a mapping from a convex cone in the image space to a point in the depth space to find a non iterative solution for the statistical sfs which has not been done yet to our knowledge we show that it is possible to approximate a cone of images illumination cone to a polytope by a nonlinear function which is a transform from cartesian coordinates to hyperspherical coordinates the images of a subject form a convex cone in the image space and it can be better to use the direction of an image in hyperspherical coordinates as an input feature for the mapping rather than the image itself the maximum error occurs on one of the vertices of the polytope if we choose the objective function to be the squared error of a linear function in the hyperspherical space hence we can solve the least square problem using the cost only for the vertices in deriving the solution we use the generalized rayleigh quotient canonical correlation analysis cca and prior information on input images to reduce the solution space and improve 3 d reconstruction performance in experiments the proposed scheme performs robustly under light variations and is fast enough for real time applications

recursive estimation of time varying motion and structure parameters we present a computational framework for recovering both first order motion parameters observer direction of translation and observer rotation second order motion parameters observer rotational acceleration and relative depth maps from time varying optical flow we recover translation speed and acceleration in units which are scaled relative to the distance to the object our assumption is that the observer rotational motion is no more than second order in other words observer motion is either constant or has at most constant acceleration we examine the effect of noise on the solution of the motion and structure parameters this ensemble of unknowns comprises a solution to the classical structure and motion from optic flow problem our complete framework utilizes a method for interpreting the bilinear image velocity equation by solving simple systems of linear equations since our noise analysis yields uncertainty measures for each parameter a kalman filter is employed to incrementally integrate new measurements as they become available as each additional frame in the sequence is processed we conclude by analysing this reduction of uncertainty over time as the system converges to a stable solution for both synthetic and real image sequences

optical flow using spatiotemporal filters a model is presented consonant with current views regarding the neurophysiology and psychophysics of motion perception that combines the outputs of a set of spatiotemporal motion energy filters to estimate image velocity a parallel implementation computes a distributed representation of image velocity a measure of image flow uncertainty is formulated preliminary results indicate that this uncertainty measure may be used to recognize ambiguity due to the aperture problem the model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for some patterns that have large differences in contrast at different spatial orientations

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

bayesian helmholtz stereopsis with integrability prior helmholtz stereopsis is a 3d reconstruction method uniquely independent of surface reflectance yet its sub optimal maximum likelihood formulation with drift prone normal integration limits performance via three contributions this paper presents a complete novel pipeline for helmholtz stereopsis first we propose a bayesian formulation replacing the maximum likelihood problem by a maximum a posteriori one second a tailored prior enforcing consistency between depth and normal estimates via a novel metric related to optimal surface integrability is proposed third explicit surface integration is eliminated by taking advantage of the accuracy of prior and high resolution of the coarse to fine approach the pipeline is validated quantitatively and qualitatively against alternative formulations reaching sub millimetre accuracy and coping with complex geometry and reflectance

some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive image frames this paper describes the results of some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive tv images and discusses several factors which affect the accuracy of the results these factors include the sizes of the motion parameters themselves the accuracy of the raw data and the number of point correspondences in addition we address two related topics determining corner positions to subpixel accuracy and matching point patterns with different scales

some remarks on the shape from shading problem in computer vision the shape from shading problem in computer vision see horn 11 deals with the reconstruction of the shape of an object from a two dimensional image we consider an object illuminated by a point source the reflected light being recorded on a photographic plate at a fixed position the intensity of the reflected light is a function of the position of the source the position of the plate and the normal to the surface of the object if the surface is given as u u x jj this leads see horn i to a first order partial differential equation of the form

shape from shading level set propagation and viscosity solutions we present a new implementation of an algorithm aimed at recovering a 3d shape from its 2d gray level picture in order to reconstruct the shape of the object an almost arbitrarily initialized 3d function is propagated on a rectangular grid so that a level set of this function tracks the height contours of the shape the method imports techniques from differential geometry fluid dynamics and numerical analysis and provides an accurate shape from shading algorithm the method solves some topological problems and gracefully handles cases of non smooth surfaces that give rise to shocks in the propagating contours real and synthetic images of 3d profiles were submitted to the algorithm and the reconstructed surfaces are presented demonstrating the effectiveness of the proposed method

maximum likelihood estimation of depth maps using photometric stereo photometric stereo and depth map estimation provide a way to construct a depth map from images of an object under one viewpoint but with varying illumination directions while estimating surface normals using the lambertian model of reflectance is well established depth map estimation is an ongoing field of research and dealing with image noise is an active topic using the zero mean gaussian model of image noise this paper introduces a method for maximum likelihood depth map estimation that accounts for the propagation of noise through all steps of the estimation process solving for maximum likelihood depth map estimates involves an independent sequence of nonlinear regression estimates one for each pixel followed by a single large and sparse linear regression estimate the linear system employs anisotropic weights which arise naturally and differ in value to related work the new depth map estimation method remains efficient and fast making it practical for realistic image sizes experiments using synthetic images demonstrate the method s ability to robustly estimate depth maps under the noise model practical benefits of the method on challenging imaging scenarios are illustrated by experiments using the extended yale face database b and an extensive data set of 500 reflected light microscopy image sequences

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 cal c 2 surface proofs for the existence of a unique solution in the cases of unknown constant and non constant albedos are provided

a method for enforcing integrability in shape from shading algorithms an approach for enforcing integrability a particular implementation of the approach an example of its application to extending an existing shape from shading algorithm and experimental results showing the improvement that results from enforcing integrability are presented a possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes the integrability projection constraint was applied to extending an iterative shape from shading algorithm of m j brooks and b k p horn 1985 experimental results show that the extended algorithm converges faster and with less error than the original version good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces

parabolic approximations of diffusive dispersive equations abstract we consider a lower order approximation for a third order diffusive dispersive conservation law with nonlinear flux it consists of a system of two second order parabolic equations a coupling parameter is also added if the flux has an inflection point it is well known on the one hand that the diffusive dispersive law admits traveling wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law on the other hand if the diffusive dispersive regularization vanishes the solutions of the corresponding initial value problem converge to a weak solution of the hyperbolic conservation law we show that both of these properties also hold for the lower order approximation furthermore when the coupling parameter tends to infinity we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive dispersive law the proofs rely on new a priori energy estimates for higher order derivatives and the technique of compensated compactness

new constraints on data closeness and needle map consistency for shape from shading this paper makes two contributions to the problem of needle map recovery using shape from shading first we provide a geometric update procedure which allows the image irradiance equation to be satisfied as a hard constraint this not only improves the data closeness of the recovered needle map but also removes the necessity for extensive parameter tuning second we exploit the improved ease of control of the new shape from shading process to investigate various types of needle map consistency constraint the first set of constraints are based on needle map smoothness the second avenue of investigation is to use curvature information to impose topographic constraints third we explore ways in which the needle map is recovered so as to be consistent with the image gradient field in each case we explore a variety of robust error measures and consistency weighting schemes that can be used to impose the desired constraints on the recovered needle map we provide an experimental assessment of the new shape from shading framework on both real world images and synthetic images with known ground truth surface normals the main conclusion drawn from our analysis is that the data closeness constraint improves the efficiency of shape from shading and that both the topographic and gradient consistency constraints improve the fidelity of the recovered needle map

accuracy and stability of multistage multistep formulas we continue our research concerning relations between accuracy and stability of time discretizations here we concentrate rather than on the size and shape of the stability region on the obtainable accuracy when the number of parameters in the formulas is large

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

vector transport for shape from shading in this paper we describe a new shape from shading method we show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error we commence by showing how to make local estimates of the hessian matrix from surface normal information with the local hessian matrix to hand we develop an em like algorithm for updating the surface normal directions at each image location parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions from this sample a local weighted estimate of the image brightness is made the transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation the revised surface normals obtained in this way may in turn be used to re estimate the hessian matrix and the process iterated until stability is reached we experiment with the method on a variety of real world and synthetic data here we explore the properties of the fields of surface normals and the height data delivered by the method

height and gradient from shading the method described here for recovering the shape of a surface from a shaded image can deal with complex wrinkled surfaces integrability can be enforced easily because both surface height and gradient are represented the robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell the new scheme can find an exact solution of a given shape from shading problem even though a regularizing term is included this is a reflection of the fact that shape from shading problems are it not ill posed when boundary conditions are available or when the image contains singular points

a two level generative model for cloth representation and shape from shading in this paper we present a two level generative model for representing the images and surface depth maps of drapery and clothes the upper level consists of a number of folds which will generate the high contrast ridge areas with a dictionary of shading primitives for 2d images and fold primitives for 3d depth maps these primitives are represented in parametric forms and are learned in a supervised learning phase using 3d surfaces of clothes acquired through photometric stereo the lower level consists of the remaining flat areas which fill between the folds with a smoothness prior markov random field we show that the classical ill posed problem shape from shading sfs can be much improved by this two level model for its reduced dimensionality and incorporation of middle level visual knowledge i e the dictionary of primitives given an input image we first infer the folds and compute a sketch graph using a sketch pursuit algorithm as in the primal sketch guo et al 2003 the 3d folds are estimated by parameter fitting using the fold dictionary and they form the skeleton of the drapery cloth surfaces then the lower level is computed by conventional sfs method using the fold areas as boundary conditions the two levels interact at the final stage by optimizing a joint bayesian posterior probability on the depth map we show a number of experiments which demonstrate more robust results in comparison with state of the art work in a broader scope our representation can be viewed as a two level inhomogeneous mrf model which is applicable to general shape from x problems our study is an attempt to revisit marr s idea marr and freeman 1982 of computing the 2frac12d sketch from primal sketch in a companion paper barbu and zhu 2005 we study shape from stereo based on a similar two level generative sketch representation

a graph spectral approach to shape from shading in this paper we explore how graph spectral methods can be used to develop a new shape from shading algorithm we characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction modeling the blocks of the weight matrix as distinct surface patches we use a graph seriation method to find a surface integration path that maximizes the sum of curvature dependent weights and that can be used for the purposes of height reconstruction to smooth the reconstructed surface we fit quadrics to the height data for each patch the smoothed surface normal directions are updated ensuring compliance with lambert s law the processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained we provide results on synthetic and real world imagery

practical matrix completion and corruption recovery using proximal alternating robust subspace minimization low rank matrix completion is a problem of immense practical importance recent works on the subject often use nuclear norm as a convex surrogate of the rank function despite its solid theoretical foundation the convex version of the problem often fails to work satisfactorily in real life applications real data often suffer from very few observations with support not meeting the randomness requirements ubiquitous presence of noise and potentially gross corruptions sometimes with these simultaneously occurring this paper proposes a proximal alternating robust subspace minimization method to tackle the three problems the proximal alternating scheme explicitly exploits the rank constraint on the completed matrix and uses the ell 0 l 0 pseudo norm directly in the corruption recovery step we show that the proposed method for the non convex and non smooth model converges to a stationary point although it is not guaranteed to find the global optimal solution in practice we find that our algorithm can typically arrive at a good local minimizer when it is supplied with a reasonably good starting point based on convex optimization extensive experiments with challenging synthetic and real data demonstrate that our algorithm succeeds in a much larger range of practical problems where convex optimization fails and it also outperforms various state of the art algorithms

recovering three dimensional shape from a single image of curved objects an algorithm to recover three dimensional shape i e surface orientation and relative depth from a single segmented image is presented it is assumed that the scene is composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or textures it is also assumed that the reflectance map r n is known for the canonical case of lambertian surfaces illuminated by a point light source this implies knowing the light source direction a variational formulation of line drawing and shading constraints in a common framework is developed the global constraints are partitioned into constraint sets corresponding to the faces edges and vertices in the scene for a face the constraints are given by horn s image irradiance equation a variational formulation of the constraints at an edge both from the known direction of the image curve corresponding to the edge and shading is developed at a vertex the constraints are modeled by a system of nonlinear equations an algorithm is presented to solve this system of constraints

a non local low rank approach to enforce integrability we propose a new approach to enforce integrability using recent advances in non local methods our formulation consists in a sparse gradient data fitting term to handle outliers together with a gradient domain non local low rank prior this regularization has two main advantages 1 the low rank prior ensures similarity between non local gradient patches which helps recovering high quality clean patches from severe outliers corruption and 2 the low rank prior efficiently reduces dense noise as it has been shown in recent image restoration works we propose an efficient solver for the resulting optimization formulation using alternate minimization experiments show that the new method leads to an important improvement compared with previous optimization methods and is able to efficiently handle both outliers and dense noise mixed together

a computational framework and an algorithm for the measurement of visual the robust measurement of visual motion from digitized image sequences has been an important but difficult problem in computer vision this paper describes a hierarchical computational framework for the determination of dense displacement fields from a pair of images and an algorithm consist ent with that framework our framework is based on the separation of the image intensity information as well as the process of measuring motion according to scale the large scale intensity information is first used to obtain rough estimates of image motion which are then refined by using intensity information at smaller scales the estimates are in the form of displacement or velocity vectors for pixels and are accompanied by a direction dependent confidence measure a smoothness constraint is employed to propagate the measurements with high confidence to their neigboring areas where the confidences are low at all levels the computations are pixel parallel uniform across the image and based on information from a small neighborhood of a pixel for our algorithm the local displacement vectors are determind by mini mizing the sum of squared differences ssd of intensities the confidence measures are derived from the shape of the ssd surface and the smoothness constraint is cast in the form of energy minimization results of applying our algorithm to pairs of real images are included in addition to our own

accuracy bounds for semidiscretizations of hyperbolic problems bounds are given for the error constant of stable finite difference methods for first order hyperbolic equations in one space dimension which use r downwind and s upwind points in the discretization of the space derivatives and which are of optimal order p min r s 2 r 2 2 s it is known that this order can be obtained by interpolatory methods examples show however that their error constants can be improved

a graph spectral method for surface height recovery this paper describes a graph spectral method for 3d surface integration the algorithm takes as its input a 2d field of surface normal estimates delivered for instance by a shape from shading or shape from texture procedure we commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature the weight matrix is used to compute a row normalized transition probability matrix and we pose the recovery of the integration path as that of finding the steady state random walk for the markov chain defined by this matrix the steady state random walk is given by the leading eigenvector of the original affinity weight matrix by threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration the height increments along the path are simply related to the traversed path length and the slope of the local tangent plane the method is evaluated on needle maps delivered by a shape from shading algorithm applied to real world data and also on synthetic data the method is compared with the local geometric height reconstruction method of bors hancock and wilson and the global methods of horn and brooks and frankot and chellappa

photometric motion the author compares the photometric effects of motion which is defined as the variation of a point s imaged intensity as a consequence of motion and the geometric effects of motion which is defined as the variation in projected surface geometry as a consequence of motion it is shown that photometric motion provides a cue to surface shape that is potentially as useful as that provided by geometric motion a simple technique for using this photometric motion information to extract both surface shape and reflectance is developed and a biological implementation is proposed how this photometric motion mechanism can be integrated with and used to enhance existing structure motion algorithms is discussed intensity information is sometimes even more important than geometric distortion when estimating the shape of a single continuous surface that is rotating relative to the observer s frame of reference

shape from shading using the heat equation this paper offers two new directions to shape from shading namely the use of the heat equation to smooth the field of surface normals and the recovery of surface height using a low dimensional embedding turning our attention to the first of these contributions we pose the problem of surface normal recovery as that of solving the steady state heat equation subject to the hard constraint that lambert s law is satisfied we perform our analysis on a plane perpendicular to the light source direction where the z component of the surface normal is equal to the normalized image brightness the x y or azimuthal component of the surface normal is found by computing the gradient of a scalar field that evolves with time subject to the heat equation we solve the heat equation for the scalar potential and hence recover the azimuthal component of the surface normal from the average image brightness making use of a simple finite difference method the second contribution is to pose the problem of recovering the surface height function as that of embedding the field of surface normals on a manifold so as to preserve the pattern of surface height differences and the lattice footprint of the surface normals we experiment with the resulting method on a variety of real world image data where it produces qualitatively good reconstructed surfaces

incorporating illumination constraints in deformable models for shape from shading and light direction estimation we present a method for the integration of nonlinear holonomic constraints in deformable models and its application to the problems of shape and illuminant direction estimation from shading experimental results demonstrate that our method performs better than previous shape from shading algorithms applied to images of lambertian objects under known illumination it is also more general as it can be applied to non lambertian surfaces and it does not require knowledge of the illuminant direction in this paper 1 we first develop a theory for the numerically robust integration of nonlinear holonomic constraints within a deformable model framework in this formulation we use lagrange multipliers and a baumgarte stabilization approach 1972 2 we also describe a fast new method for the computation of constraint based forces in the case of high numbers of local parameters 3 we demonstrate how any type of illumination constraint from the simple lambertian model to more complex highly nonlinear models can be incorporated in a deformable model framework 4 we extend our method to work when the direction of the light source is not known we couple our shape estimation method with a method for light estimation in an iterative process where improved shape estimation results in improved light estimation and vice versa 5 we perform a series of experiments

stability and accuracy of difference schemes for hyperbolic problems abstract bounds for the error order and error constant of full and semidiscretizations of hyperbolic problems are given in order to be able to compare fulldiscretizations a scaling is introduced this scaling reflects the amount of work needed to perform one time step using the semidiscretization associated with the fulldiscretization lower bounds for these scaled error constants are given these results are derived using the order star technique

shape from shading using graph cuts in this paper we investigate the applicability of graph cuts to the sfs shape from shading problem we propose a new semi global method for sfs using graph cuts the new algorithm combines the local method proposed by lee and rosenfeld c h lee a rosenfeld improved methods of estimating shape from shading using the light source coordinate system artif intell 26 1985 125 143 and a global method using an energy minimization technique by employing a new global energy minimization formulation the convex concave ambiguity problem of lee and rosenfeld s method can be resolved efficiently a new combinatorial optimization technique the graph cuts method is used for the minimization of the proposed energy functional experimental results on a variety of synthetic and real world images show that the proposed algorithm reconstructs the 3 d shape of objects very efficiently

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

local shading analysis local analysis of image shading in the absence of prior knowledge about the viewed scene may be used to provide information about the scene the following has been proved every image point has the same image intensity and first and second derivatives as the image of some point on a lambertian surface with principal curvatures of equal magnitude further if the principal curvatures are assumed to be equal there is a unique combination of image formation parameters up to a mirror reversal that will produce a particular set of image intensity and first and second derivatives a solution for the unique combination of surface orientation etc is presented this solution has been extended to natural imagery by using general position and regional constraints to obtain estimates of the following surface orientation at each image point the qualitative type of the surface i e whether the surface is planar cylindrical convex concave or saddle the illuminant direction within a region algorithms to recover illuminant direction and estimate surface orientation have been evaluated on both natural and synthesized images and have been found to produce useful information about the scene

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

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

direct passive navigation in this correspondence we show how to recover the motion of an observer relative to a planar surface from image brightness derivatives we do not compute the optical flow as an intermediate step only the spatial and temporal brightness gradients at a minimum of eight points we first present two iterative schemes for solving nine nonlinear equations in terms of the motion and surface parameters that are derived from a least squares fomulation an initial pass over the relevant image region is used to accumulate a number of moments of the image brightness derivatives all of the quantities used in the iteration are efficiently computed from these totals without the need to refer back to the image we then show that either of two possible solutions can be obtained in closed form we first solve a linear matrix equation for the elements of a 3 3 matrix the eigenvalue decomposition of the symmetric part of the matrix is then used to compute the motion parameters and the plane orientation a new compact notation allows us to show easily that there are at most two planar solutions

asymptotic stability barriers for natural runge kutta processes for delay equations this paper investigates the asymptotic stability properties of a class of numerical methods for delay differential equations ddes the so called natural runge kutta methods we first examine the behavior of these methods when applied to the neutral model equation y t a y t b y t 1 c y t 1 with a b c in mathbb r we also consider the case when a b c in mathbb c and provide a suitable geometric characterization of their asymptotic stability regions then by means of the obtained results in conjunction with those given in n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 we are able to give a final answer concerning the possible preservation of asymptotic stability of the considered class of methods when applied to systems of linear ddes of the form y t l y t m y t 1 with l m in mathbb r d times d d 1 we are interested here in methods that produce stable numerical solutions for all values of the parameters a b and c in the first equation and l and m in the second one for which the exact solution tends to zero to this aim we direct our attention to a novel stability setting recently introduced and investigated for the scalar nonneutral case see n guglielmi numer math 77 1997 pp 467 485 n guglielmi ima j numer anal 18 1998 pp 399 418 n guglielmi and e hairer numer math 83 1999 pp 371 383 n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 s maset numer math 87 2000 pp 355 371

estimation of illuminant direction albedo and shape from shading a robust approach to the recovery of shape from shading information is presented assuming uniform albedo and lambertian surface for the imaging model two methods for estimating the azimuth of the illuminant are presented one is based on local estimates on smooth patches and the other method uses shading information along image contours the elevation of the illuminant and surface albedo are estimated from image statistics taking into consideration the effect of self shadowing with the estimated reflectance map parameters the authors then compute the surface shape using a procedure that implements the smoothness constraint by requiring the gradients of reconstructed density to be close to the gradients of the input image the algorithm is data driven stable updates the surface slope and height maps simultaneously and significantly reduces the residual errors in irradiance and integrability terms a hierarchical implementation of the algorithm is presented typical results on synthetic and images are given to illustrate the usefulness of the approach

order stars and stiff integrators order stars introduced in g wanner e hairer s p norsett order stars and stability theorems bit 18 1978 475 489 have become a fundamental tool for the understanding of order and stability properties of numerical methods for stiff differential equations this survey retraces their discovery and their principal achievements we also sketch some later extensions and describe some recent developments

shape from shading with a linear triangular element surface model the authors propose to combine a triangular element surface model with a linearized reflectance map to formulate the shape from shading problem the main idea is to approximate a smooth surface by the union of triangular surface patches called triangular elements and express the approximating surface as a linear combination of a set of nodal basis functions since the surface normal of a triangular element is uniquely determined by the heights of its three vertices or nodes image brightness can be directly related to nodal heights using the linearized reflectance map the surface height can then be determined by minimizing a quadratic cost functional corresponding to the squares of brightness errors and solved effectively with the multigrid computational technique the proposed method does not require any integrability constraint or artificial assumptions on boundary conditions simulation results for synthetic and real images are presented to illustrate the performance and efficiency of the method

shape from shading using probability functions and belief propagation shape from shading sfs aims to reconstruct the three dimensional shape of an object from a single shaded image this article proposes an improved framework based on belief propagation for computing sfs the implementation of the well known brightness integrability and smoothness constraints inside this framework is shown we implement the constraints as probability density functions for example the brightness constraint is a two dimensional probability density function that relates all possible surface gradients at a pixel to their probability given the pixel intensity a straightforward extension of the framework to photometric stereo is presented where multiple images of the same scene taken under different lighting conditions are available the results are promising especially since the solution is obtained by iteratively applying simple operations on a regular grid of points the presented framework therefore can be implemented in parallel and is a reasonably likely biological scheme

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