a geometrically based criterion to avoid infimum gaps in optimal control abstract in optimal control theory infimum gap means a non zero difference between the infimum values of a given minimum problem and an extended problem obtained by embedding the original family v of controls in a larger family w for some embeddings like standard convex relaxations or impulsive extensions the normality of an extended minimizer has been shown to be sufficient for the avoidance of infimum gaps a natural issue is then the search of a general hypothesis under which the criterium normality implies no gap holds true we prove that this criterium is actually valid as soon as v is abundant in w without any convexity assumption on the extended dynamics abundance which was introduced by j warga in a convex context and was later generalized by b kaskosz strengthens density the latter being not sufficient for the mentioned criterium to hold true

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

shape from recognition a novel approach for 3 d face shape recovery in this paper we develop a novel framework for robust recovery of three dimensional 3 d surfaces of faces from single images the underlying principle is shape from recognition i e the idea that pre recognizing face parts can constrain the space of possible solutions to the image irradiance equation thus allowing robust recovery of the 3 d structure of a specific part parts of faces like nose lips and eyes are recognized and localized using robust expansion matching filter templates under varying pose and illumination specialized backpropagation based neural networks are then employed to recover the 3 d shape of particular face parts representation using principal components allows to efficiently encode classes of objects such as nose lips etc the specialized networks are designed and trained to map the principal component coefficients of the part images to another set of principal component coefficients that represent the corresponding 3 d surface shapes to achieve robustness to viewing conditions the network is trained with a wide range of illumination and viewing directions a method for merging recovered 3 d surface regions by minimizing the sum squared error in overlapping areas is also derived quantitative analysis of the reconstruction of the surface parts in varying illumination and pose show relatively small errors indicating that the method is robust and accurate several examples showing recovery of the complete face also illustrate the efficacy of the approach

nonlinear multiresolution a shape from shading example a method for image resolution reduction that simulates the reduction of shape resolution and is to be used with shape from shading algorithms is presented resolution reduction is used for accelerating shape from shading algorithms using multiresolution pyramids the goal of this correspondence is to show that resolution reduction based only on image intensities can be inferior to resolution reduction using knowledge of the surface reflectance this is true for all cases of parameter estimation from intensity images when the intensity is not a linear function of the parameters since optical reduction of images as when moving a camera away from the object is a simple gray level averaging a question is cast on the general scheme of shape from shading it is concluded that for accurate shape from shading one needs to know the distance from the object and the shape should also be known in advance

dynamic programming for nonlinear systems driven by ordinaryand impulsive controls a dynamic programming approach is considered for a class of minimum problems with impulses the minimization domain consists of trajectories satisfying an ordinary differential equation whose right hand side depends not only on a measurable control v but also on a second control u and on its time derivative dot u for this reason the control u and the differential equation are called impulsive the value function of the considered minimum problem turns out to depend on the time the state the u variable and the variation allowed to the impulsive control it is shown that the value function satisfies in a generalized sense a dynamic programming equation dpe which is obtained from a dynamic programming principle involving space time trajectories moreover the value function is the unique map solving equation dpe satisfying either an inequality condition or a supersolution condition at each point of the boundary incidentally this extends a result by barron jensen and menaldi nonlinear anal 21 1993 pp 241 268 where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable next a maximum principle is proved and the well known relationship between adjoint variables and value function is suitably extended to impulsive control systems a fully elaborated example concludes the paper

on the intersection of a clarke cone with a boltyanskii cone we provide an example of two closed sets s 1 s 2 subset r 4 such that s 1 cap s 2 0 yet at the origin a boltyanskii tangent cone c 1 to s 1 and the clarke tangent cone c 2 to s 2 are strongly transversal this settles a question originally proposed by h sussmann

pontryagin s maximum principle for constrained impulsive control problems abstract necessary conditions in the form of pontryagin s maximum principle are derived for impulsive control problems with mixed constraints a new mathematical concept of impulsive control is introduced as a requirement for the consistency of the impulsive framework additionally this control concept enables the incorporation of the engineering needs to consider conventional control action while the impulse develops the regularity assumptions under which the maximum principle is proved are weaker than those in the known literature ekeland s variational principle and lebesgue s discontinuous time variable change are used in the proof the article also contains an example showing how such impulsive controls could be relevant in actual applications

multiple illuminant direction detection with application to image synthesis pentland observed 1982 1984 that the human eye is sensitive to the change of intensities on an image of a smooth surface the change of intensities is maximal whenever the illuminant direction is perpendicular to the normal of the surface this motivates us to introduce the concept of critical points where the surface normal is perpendicular to some light source direction apparently the illuminant direction has a simple geometric relationship with the corresponding critical points in this paper for simplicity reasons we restrict our discussions to the shading of a lambertian sphere of known size in a multiple distant light source environment a novel global representation of the intensity function is derived based on this intensity characterization the least squares and iteration techniques are used to determine critical points and thus the light source directions and their intensities if certain conditions are satisfied the performance of this new approach is evaluated using both synthetic images and real images as an application we use it as a tool to determine light sources in real image synthesis the experimental results show that this technique can be used to superimpose synthetic objects with a real scene

point light source estimation from two images and its limits in this paper the performance of parameter estimation of a single static distant point light source from two video images is analyzed in terms of estimation theory the illumination parameters are the intensity and direction of the light source in the first part of this paper estimators from the literature are reviewed most recent estimators evaluate as input data two video images as well as the 3d shape and the 3d motion of the visible moving objects in the second part of the paper the performance of these recent methods is analyzed the input data to estimation as well as the inherent input data errors are described by a stochastic observation model based on this model the performance is analyzed regarding the cramer rao theoretical lower bound of estimation error variances the bound is derived for a variety of cases of scene illumination object motion and errors in input data for simplification purpose the bound is valid only for object motions with the rotation axis lying in the image plane the analysis shows in which cases which estimation accuracy can be expected with current methods finally a comparison of the bound with one of the recent estimators shows that recent estimators are suboptimal in case of errors in the 3d shape of the objects in future work the stochastic observation model presented in this paper can be used to improve illumination estimation

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

where and why local shading analysis works an apparent contradiction in the shape from shading literature is examined although practical experience suggests that shape can be inferred from the local analysis of shading mathematical analyses support the opposite a criterion for exact surface recovery is derived and it is shown that as a result of surface geometry elliptic hyperbolic and doubly curved surfaces could be recovered to reasonable accuracy provided that surface curvature and or foreshortening were limited in other words by relaxing the requirement that surfaces be recovered exactly it is found that local analysis of shading can provide useful descriptions of shape as evidenced by the results presented

on schwarz s domain decomposition methods for elliptic boundary value problems summary we study the additive and multiplicative schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness r 1 the finite element method reduces an elliptic boundary value problem to a linear system of equations it is well known that as the number of triangles in the underlying triangulation is increased which is indispensable for increasing the accuracy of the approximate solution the size and condition number of the linear system increases the schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size we shall show in this paper that the approximate solutions from the multiplicative schwarz domain decomposition method converge to the exact solution of the linear system geometrically we also show that the additive schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method we tested these methods for the biharmonic equation with dirichlet boundary condition over an arbitrary polygonal domain using c 1 cubic spline functions over a quadrangulation of the given domain the computer experiments agree with our theoretical results

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

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

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

necessary conditions for optimal impulsive control problems necessary conditions of optimality in the form of a maximum principle are derived for a class of optimal control problems certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation state trajectories are interpreted as robust solutions of the dynamic equations a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps the maximum principle which is derived improves on earlier optimality conditions for problems of this nature by allowing nonsmooth data measurable time dependence and a possibly time varying constraint set for the conventional controls

experiments on estimating egomotion and structure parameters using long monocular image sequences this paper presents a simple but robust model based approach to estimating the kinematics of a moving camera and the structure of the objects in a stationary environment using long noisy monocular image sequences both batch and recursive algorithms are presented and the problem due to occlusion is addressed the approach is based on representing the constant translational velocity and constant angular velocity of the camera motion using nine rectilinear motion parameters which are 3 d vectors of the position of the rotation center linear and angular velocities the structure parameters are 3 d coordinates of the salient feature points in the inertial coordinate system due to redundancies in parameterization the total number of independent parameters to be estimated is 3m 7 wherem is the number of feature points the image plane coordinates of these feature points in each frame are first detected and matched over the frames these noisy image coordinates serve as the input to our algorithms due to the nonlinear nature of perspective projection a nonlinear least squares method is formulated for the batch algorithm and a conjugate gradient method is then applied to find the solution a recursive method using an iterated extended kalman filter iekf for incremental estimation of motion and structure is also presented since the plant model is simple in our formulation closed form solutions for the state and covariance transition equations are easily derived experimental results for simulated imagery as well as several real image sequences are included

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

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

mixture of spherical distributions for single view relighting we present a method for simultaneously estimating the illumination of a scene and the reflectance property of an object from single view images a single image or a small number of images taken from the same viewpoint we assume that the illumination consists of multiple point light sources and the shape of the object is known first we represent the illumination on the surface of a unit sphere as a finite mixture of von mises fisher distributions based on a novel spherical specular reflection model that well approximates the torrance sparrow reflection model next we estimate the parameters of this mixture model including the number of its component distributions and the standard deviation of them which correspond to the number of light sources and the surface roughness respectively finally using these results as the initial estimates we iteratively refine the estimates based on the original torrance sparrow reflection model the final estimates can be used to relight single view images such as altering the intensities and directions of the individual light sources the proposed method provides a unified framework based on directional statistics for simultaneously estimating the intensities and directions of an unknown number of light sources as well as the specular reflection parameter of the object in the scene

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

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

appearance characterization of linear lambertian objects generalized photometric stereo and illumination invariant face recognition traditional photometric stereo algorithms employ a lambertian reflectance model with a varying albedo field and involve the appearance of only one object in this paper we generalize photometric stereo algorithms to handle all appearances of all objects in a class in particular the human face class by making use of the linear lambertian property a linear lambertian object is one which is linearly spanned by a set of basis objects and has a lambertian surface the linear property leads to a rank constraint and consequently a factorization of an observation matrix that consists of exemplar images of different objects e g faces of different subjects under different unknown illuminations integrability and symmetry constraints are used to fully recover the subspace bases using a novel linearized algorithm that takes the varying albedo field into account the effectiveness of the linear lambertian property is further investigated by using it for the problem of illumination invariant face recognition using just one image attached shadows are incorporated in the model by a careful treatment of the inherent nonlinearity in lambert s law this enables us to extend our algorithm to perform face recognition in the presence of multiple illumination sources experimental results using standard data sets are presented

rational finite elements of class ck over a quadrangulation abstract in this paper we give a constructive method for rational c k finite element method in r 2 for quadrilateral for k 2 the usual polynomials splines require some supersmoothness at the vertices of the quadrilateral even when dealing with macro elements this is no more a requirement for our rational approach the process of the construction is illustrated with the help of two examples namely the c 1 case and the c 2 case

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

on spiking models for synaptic activity and impulsive differential equations we illustrate the problems that can arise in writing differential equations that include dirac delta functions to model equations with state dependent impulsive forcing specifically difficulties arise in the interpretation of the products of distributions with discontinuous functions we suggest several methods to resolve these ambiguities such as using limiting sequences and asymptotic analysis with applications of the results given for discrete maps these suggestions are applied to a popular model describing synaptic connections in the brain

l1 limit solutions for control systems abstract for a control cauchy problem x f t x u v 1 m g x u x a x on an interval a b we propose a notion of limit solution x verifying the following properties i x is defined for l 1 impulsive inputs u and for standard bounded measurable controls v ii in the commutative case i e when g g 0 for all 1 m x coincides with the solution one can obtain via a change of coordinates that makes the g simultaneously constant iii x subsumes former concepts of solution valid for the generic noncommutative case in particular when u has bounded variation we investigate the relation between limit solutions and single valued graph completion solutions furthermore we prove consistency with the classical caratheodory solution when u and x are absolutely continuous even though some specific problems are better addressed by means of special representations of the solutions we believe that various theoretical issues call for a unified notion of trajectory for instance this is the case of optimal control problems possibly with state and endpoint constraints for which no extra assumptions like e g coercivity bounded variation commutativity are made in advance

a solution of the dichromatic model for multispectral photometric invariance in this paper we address the problem of photometric invariance in multispectral imaging making use of an optimisation approach based upon the dichromatic model in this manner we cast the problem of recovering the spectra of the illuminant the surface reflectance and the shading and specular factors in a structural optimisation setting making use of the additional information provided by multispectral imaging and the structure of image patches we recover the dichromatic parameters of the scene to do this we formulate a target cost function combining the dichromatic error and the smoothness priors for the surfaces under study the dichromatic parameters are recovered through minimising this cost function in a coordinate descent manner the algorithm is quite general in nature admitting the enforcement of smoothness constraints and extending in a straightforward manner to trichromatic settings moreover the objective function is convex with respect to the subset of variables to be optimised in each alternating step of the minimisation strategy this gives rise to an optimal closed form solution for each of the iterations in our algorithm we illustrate the effectiveness of our method for purposes of illuminant spectrum recovery skin recognition material clustering and specularity removal we also compare our results to a number of alternatives

shape from shading under perspective projection shape from shading sfs is a fundamental problem in computer vision a very common assumption in this field is that image projection is orthographic this paper re examines the basis of sfs the image irradiance equation under a perspective projection assumption the resultant equation does not depend on the depth function directly but rather on its natural logarithm as such it is invariant to scale changes of the depth function a reconstruction method based on the perspective formula is then suggested it is a modification of the fast marching method of kimmel and sethian following that a comparison of the orthographic fast marching perspective fast marching and the perspective algorithm of prados and faugeras on synthetic images is presented the two perspective methods show better reconstruction results than the orthographic the algorithm of prados and faugeras equates with the perspective fast marching following that a comparison of the orthographic and perspective versions of the fast marching method on endoscopic images is introduced the perspective algorithm outperformed the orthographic one these findings suggest that the more realistic set of assumptions of perspective sfs improves reconstruction significantly with respect to orthographic sfs the findings also provide evidence that perspective sfs can be used for real life applications in fields such as endoscopy

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

recovery of shgcs from a single intensity view generalized cylinders are a flexible loosely defined class of parametric shapes capable of modeling many real world objects straight homogeneous generalized cylinders are an important subclass of generalized cylinders whose cross sections are scaled versions of a reference curve although there has been considerable research into recovering the shape of shgcs from their contour this work has almost exclusively involved methods that couple contour and heuristic constraints a rigorous approach to the problem of recovering solid parametric shape from a single intensity view should involve at least two stages 1 deriving the contour constraints and 2 determining if additional image constraints e g intensity can be used to uniquely determine the 3d object shape in this paper the authors follow the approach just described this methodology is also important for the recovery of object classes like tubes where contour and heuristic constraints are shown to be insufficient for shape recovery first the authors prove that shgc contours generated under orthography have exactly two degrees of freedom next the authors show that the remaining free parameters can be resolved using reflectance based constraints without knowledge of the number of light sources their positions intensities the amount of ambient light or the surface albedo finally the reflectance based recovery algorithm is demonstrated on both synthetic and real shgc images

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

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

a nondegenerate maximum principle for the impulse control problem with state constraints in this article a free time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered a weakened maximum principle is obtained for problems with data measurable in the time variable being the time transversality conditions deduced with the help of some extra convexity assumption on the state constraints in the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method

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

numerical shape from shading and occluding boundaries an iterative method for computing shape from shading using occluding boundary information is proposed some applications of this method are shown we employ the stereographic plane to express the orientations of surface patches rather than the more commonly used gradient space use of the stereographic plane makes it possible to incorporate occluding boundary information but forces us to employ a smoothness constraint different from the one previously proposed the new constraint follows directly from a particular definition of surface smoothness we solve the set of equations arising from the smoothness constraints and the image irradiance equation iteratively using occluding boundary information to supply boundary conditions good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution numerical experiments show that the method is effective and robust finally we analyze scanning electron microscope sem pictures using this method other applications are also proposed

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

minimal time sequential batch reactors with bounded and impulse controls for one or more species we consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource with the objective being to reach a given level of the resource we allow controls to be bounded measurable functions of time plus possible impulses for the one species case we show that the immediate one impulse strategy filling the whole reactor with one single impulse at the initial time is optimal when the growth function is monotonic for nonmonotonic growth functions with one maximum we show that a particular singular arc strategy precisely defined in section 3 is optimal these results extend and improve former ones obtained for the class of measurable controls only for the two species case with monotonic growth functions we give conditions under which the immediate one impulse strategy is optimal we also give optimality conditions for the singular arc strategy at a level that depends on the initial condition to be optimal the possibility for the immediate one impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations

3d shape recovery of smooth surfaces dropping the fixed viewpoint assumption we present a new method for recovering the 3d shape of a featureless smooth surface from three or more calibrated images illuminated by different light sources three of them are independent this method is unique in its ability to handle images taken from unconstrained perspective viewpoints and unconstrained illumination directions the correspondence between such images is hard to compute and no other known method can handle this problem locally from a small number of images our method combines geometric and photometric information in order to recover dense correspondence between the images and accurately computes the 3d shape only a single pass starting at one point and local computation are used this is in contrast to methods that use the occluding contours recovered from many images to initialize and constrain an optimization process the output of our method can be used to initialize such processes in the special case of fixed viewpoint the proposed method becomes a new perspective photometric stereo algorithm nevertheless the introduction of the multiview setup self occlusions and regions close to the occluding boundaries are better handled and the method is more robust to noise than photometric stereo experimental results are presented for simulated and real images

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

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

shape from patterns regularization we present a theory for the recovery of the shape of a surface covered with small elements texels the theory is based on the apparent surface pattern distortion in the image and fits the regularization paradigm recently introduced in computer vision by poggio et al 1985 a mapping is defined on the basis of measurement of the local distortions of a repeated unknown texture pattern due to the image projection from this a constraint on the gradient orientation is determined from the apparent area of a texture element the analysis is done under an approximation of the perspective projection called paraperspective the resulting algorithm is applied to several synthetic and real images to demonstrate its performance

linear shape from shading in many situations the reflectance function of a surface is approximately linear and there is an effielent closed form solution to the shape from shading problem when boundary conditions e g edges singular points are not available good estimates of shape may still be extracted by using the assumption of general viewing position an improved method for estimating the illuminant direction is also presented

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

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

minimizers that are not also relaxed minimizers relaxation is a widely used regularization procedure in optimal control involving the replacement of velocity sets by their convex hulls to ensure the existence of a minimizer it can be an important step in the construction of suboptimal controls for the original unrelaxed optimal control problem which may not have a minimizer based on obtaining a minimizer for the relaxed problem and approximating it in some cases the infimum cost of the unrelaxed problem is strictly greater than the infimum cost over relaxed state trajectories we need to identify such situations because then the above procedure fails the noncoincidence of these two infima leads also to a breakdown of the dynamic programming method because typically solving the hamilton jacobi equation yields the minimum cost of the relaxed not the original optimal control problem following on from earlier work by warga we explore the relation between on the one hand noncoincidence of the minimum cost of the optimal control and its re

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

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

measure driven differential inclusions measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data we introduce the concept of a robust solution to a measure driven inclusion which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by dal maso and rampazzo and others closure properties of sets of robust solutions are established and notions of relaxation investigated implications for optimality conditions for impulsive control problems are pursued in a companion paper

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