recovering surface shape and orientation from texture texture provides an important source of information about the three dimensional structure of visible surfaces particularly for stationary monocular views to recover 3d structure the distorting effects of projection must be distinguished from properties of the texture on which the distortion acts this requires that assumptions must be made about the texture yet the unpredictability of natural textures precludes the use of highly restrictive assumptions the recovery method reported in this paper exploits the minimal assumption that textures do not mimic projective effects this assumption determines the strategy of attributing as much as possible of the variation observed in the image to projection equivalently the interpretation is chosen for which the texture prior to projection is made as uniform as possible this strategy was implemented using statistical methods first for the restricted case of planar surfaces and then by extension for curved surfaces the technique was applied successfully to natural images

improved methods of estimating shape from shading using the light source coordinate system abstract this paper develops improved methods of estimating local surface orientation from shading information with the aid of a coordinate system having one axis in the assumed direction of the light source in this system the surface tilt is related to the direction of the gray level gradient at the given point an alternative formulation of the surface slant probability density is developed that takes the discrete nature of digital images into account and that yields a better estimate of the light source direction

shape from texture estimation isotropy and moments a theory is proposed for the interpretation of 3d textures with oriented elements it builds on two previous theories a statistical one due to witkin and kanatani s buffon transform we show that the statistical estimation of orientation of a textured plane is one of a general family of optimal backprojection problems which also includes maximal compactness general uniqueness and convergence results are obtained for an efficient algorithm employing second moment feedback this framework greatly enhances the usefulness of surface orientation estimation for texture in two ways first an error distribution is derived for the estimator and this is crucial for integration of shape information second hypotheses about intrinsic texture statistics can be verified by a 2 test which if failed warns that the orientation estimator is not to be believed finally it is argued that these results suggest a general moment tensor approach to analysis of 3d texture

shape from regular patterns niis paper describes an aigoritlitr1 to obtaitl local surface orierltatiorr frofti the apparetlt surface pattertl distortion in an image we propose a spherical projection to model perspctiue imaging a trtappitlg is debtled based of1 the measurement of the local distortions of a repeated known rexture pattern due io the image projection 77tis mapping maps an apparent shape on the image sphere lo a locus of possible surface orienrations on the gaussian sphere an ireratiw constraint propagation algorithm with the orientations at occluding boundaries reduces possible surface orientations to a unique orientation 7lis algorithm can recover local surface orientation as well as interpolate surjace orientations where no information is auailable xis algorithm is applied to a real image ro demonstrate its perfontlance

qualitative features and the generalized hough transform abstract in this paper we show how the use of qualitative features can enhance the performance of recognition and localization techniques in particular the generalized hough transform qualitative features i e scene features with qualitative attributes assigned to them are shown to be effective in pruning the search space of possible scene interpretations and also reducing the number of spurious interpretations explored by the recognition and localization technique the redundancy of the computed transform and the probability of spurious peaks of significant magnitude due to random accumulation of evidence are two criteria by which the performance of the generalized hough transform is judged the straightforward generalized hough transform shows a high probability of spurious peaks of significant magnitude even for small values of redundancy and small magnitude of the search space of scene interpretations the use of qualitative features enables us to come up with a weighted generalized hough transform where each match of a scene feature with a model feature is assigned a weight based on the qualitative attributes assigned to the scene feature these weights could be looked upon as membership function values for the fuzzy sets defined by these qualitative attributes analytic expressions for the probability of accumulation of random events within a bucket are derived for the weighted generalized hough transform and compared with the corresponding expression for the straightforward generalized hough transform the weighted generalized hough transform is shown to perform better than the straightforward generalized hough transform an experiment for the recognition of polyhedral objects from range images is described using dihedral junctions as features for matching and pose computation the experimental results bring out the advantages of the weighted generalized hough transform over the straightforward generalized hough transform

planar surface orientation from texture spatial frequencies abstract this paper presents a computational model and a practical algorithm for determining the three dimensional orientation of a planar surface from visual texture information the model consists of three parts 1 a local spatial frequency based texture representation 2 a model describing the projection of surface texture to image texture and 3 a solution of the texture projection model for the surface orientation under an assumption of surface texture homogeneity the algorithm first measures the dominant frequency at each image point using three wavelet like transforms and then finds the surface orientation that minimizes the variance of the image frequencies backprojections the algorithm is tested on photographs of real world surfaces exhibiting an average accuracy of better than 3 in slant and 4 in tilt the current model and algorithm are more accurate yet substantially simpler than earlier versions of this approach

segmentation through variable order surface fitting the solution of the segmentation problem requires a mechanism for partitioning the image array into low level entities based on a model of the underlying image structure a piecewise smooth surface model for image data that possesses surface coherence properties is used to develop an algorithm that simultaneously segments a large class of images into regions of arbitrary shape and approximates image data with bivariate functions so that it is possible to compute a complete noiseless image reconstruction based on the extracted functions and regions surface curvature sign labeling provides an initial coarse image segmentation which is refined by an iterative region growing method based on variable order surface fitting experimental results show the algorithm s performance on six range images and three intensity images

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

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

parametric estimation of the orientation of textured planar surfaces this paper presents a parametric solution to the problem of estimating the orientation in space of a planar textured surface from a single noisy observed image of it the coordinate transformation from surface to image coordinates due to the perspective projection transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location the functional dependence of the sinusoid phase in location is uniquely determined by the tilt and slant angles of the surface using the phase differencing algorithm we fit a polynomial phase model to a sinusoidal component of the observed texture assuming the estimated polynomial coefficients are the coefficients of a taylor series expansion of the phase we establish a linear recursive relation between the model parameters and the unknown slant and tilt a linear least squares solution of the resulting system provides the slant and tilt estimates to improve accuracy an iterative refinement procedure is applied in a small neighborhood of these estimates the performance of the proposed algorithms is evaluated by applying them to images of different planar surfaces and by comparing their statistical performance with the cramer rao bound the combined two stage algorithm is shown to produce estimates that are close to the bound

shape from texture general principle abstract the 3d shape of a textured surface is recovered from its projected image on the assumption that the texture is homogeneously distributed our method does not require recognition of the structure regularity periodicity parallelism orthogonality etc of the texture distribution first the homogeneity of a discrete texture is precisely defined in terms of the theory of distributions next distortion of the observed texture due to perspective projection is described in terms of the first fundamental form expressed as a function of the image coordinates based on this result the 3d recovery equations for determining the surface shape are derived for both planar and curved surfaces some numerical schemes for solving these equations are proposed ambiguity in the interpretation of curved surfaces is also analyzed finally some numerical examples for synthetic data are presented and our method is compared with other existing methods

exploiting the generic viewpoint assumption the generic viewpoint assumption states that an observer is not in a special position relative to the scene it is commonly used to disqualify scene interpretations that assume special viewpoints following a binary decision that the viewpoint was either generic or accidental in this paper we apply bayesian statistics to quantify the probability of a view and so derive a useful tool to estimate scene parameters generic variables can include viewpoint object orientation and lighting position by considering the image as a differentiable function of these variables we derive the probability that a set of scene parameters created a given image this scene probability equation has three terms the fidelity of the scene interpretation to the image data the prior probability of the scene interpretation and a new genericity term which favors scenes likely to produce the observed image the genericity term favors image interpretations for which the image is stable with respect to changes in the generic variables it results from integration over the generic variables using a low noise approximation common in bayesian statistics this approach may increase the scope and accuracy of scene estimates it applies to a range of vision problems we show shape from shading examples where we rank shapes or reflectance functions in cases where these are otherwise unknown the rankings agree with the perceived values

shape from periodic texture using the eigenvectors of local affine distortion shows how the local slant and tilt angles of regularly textured curved surfaces can be estimated directly without the need for iterative numerical optimization we work in the frequency domain and measure texture distortion using the affine distortion of the pattern of spectral peaks the key theoretical contribution is to show that the directions of the eigenvectors of the affine distortion matrices can be used to estimate local slant and tilt angles of tangent planes to curved surfaces in particular the leading eigenvector points in the tilt direction although not as geometrically transparent the direction of the second eigenvector can be used to estimate the slant direction the required affine distortion matrices are computed using the correspondences between spectral peaks established on the basis of their energy ordering we apply the method to a variety of real world and synthetic imagery

obtaining a 3 d orientation of projective textures using a morphological method on the premise that the texture image is a planar surface we propose a new morphological approach to obtain a 3 d surface orientation using the variation of texture image caused by projective distortions centroids of texels can be obtained from the farthest texel to the nearest texel sequentially by recursive erosions so a texture image is segmented into several sub regions using major axes and sizes of structuring elements in segmented regions we can compute a vanishing point and obtain a 3 d surface orientation in the experiments we have ascertained the proposed algorithm was more effective than the aggregation transform

shape reconstruction on a varying mesh a central class of image understanding problems is concerned with reconstructing a shape from an incomplete data set such as fitting a surface to partially given contours a new theory for solving such problems is presented unlike the current heuristic methods the method used starts from fundamental principles that should be followed by any reconstruction method regardless of its mathematical or physical implementation a mathematical procedure which conforms to these principles is presented one major advantage of the method is the ability to handle shapes containing both smooth and sharp parts without using thresholds a sharp variation such as a corner requires a high resolution mesh for adequate representation while slowly varying sections can be represented with sparser mesh points unlike current methods this procedure fits the surface on a varying mesh the mesh is constructed automatically to be more dense at parts of the image that have more rapid variation analytical examples are given in simple cases followed by numerical experiments

on the geometric interpretation of image contours abstract in this paper we suggest a computational model for the 3d interpretation of a 2d view based on contour classification and contour interpretation we concentrate on those contours arising from discontinuities in surface orientation we combine a generic surface description well suited for visual tasks with a model of the image formation process in order to derive image contour configurations that are likely to be interpreted in terms of surface contours next we describe a computer algorithm which attempts to interpret image contours on the following grounds first an image analysis process produces a description in terms of contours and relationships between them second among these contours we select those which form a desired configuration third the selected contours are combined with constraints available with the image formation process in order to be interpreted in terms of discontinuities in surface orientation as a consequence there is a dramatic reduction in the number of possible orientations of the associated scene surfaces

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

computation of surface orientation and structure of objects using grid coding in this correspondence algorithms are introduced to infer surface orientation and structure of visible object surfaces using grid coding we adopt the active lighting technique to spatially encode the scene for analysis the observed objects which can have surfaces of arbitrary shape are assumed to rest on a plane base plane in a scene which is encoded with light cast through a grid plane two orthogonal grid patterns are used where each pattern is obtained with a set of equally spaced stripes marked on a glass pane the scene is observed through a camera and the object surface orientation is determined using the projected patterns on the object surface if the surfaces under consideration obey certain smoothness constraints a dense orientation map can be obtained through proper interpolation the surface structure can then be recovered given this dense orientation map both planar and curved surfaces can be handled in a uniform manner the algorithms we propose yield reasonably accurate results and are relatively tolerant to noise especially when compared to shape from shading techniques in contrast to other grid coding techniques reported which match the grid junctions for depth reconstruction under the stereopsis principle our techniques use the direction of the projected stripes to infer local surface orientation and do not require any correspondence relationship between either the grid lines or the grid junctions to be specified the algorithm has the ability to register images and can therefore be embedded in a system which integrates knowledge from multiple views

resolution of computational aeroacoustics problems on unstructured grids with a higher order finite volume scheme computational fluid dynamics cfd has become increasingly used in the industry for the simulation of flows nevertheless the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult from this point of view the development of higher order methods for unstructured grids is desirable the finite volume method can be used with unstructured grids but unfortunately it is difficult to achieve an order of accuracy higher than two and the common approach is a simple extension of the one dimensional case the increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives this problem is overcome with the application of the moving least squares mls technique on a finite volume framework in this work we present the application of this method fv mls to the solution of aeroacoustic problems

direct estimation of shape from texture witkin 1981 has proposed a maximum likelihood ml estimator of surface orientation based on the observed directional bias of projected texture elements however a drawback of this procedure is that the estimate is only defined indirectly in terms of a set of nonlinear equations an alternative method is proposed which allows an estimate of the surface orientation to be computed directly in a single step from certain simple statistics of the image data we also show that this direct estimate allows witkin s ml estimate to be computed to within 0 05 degrees in only two or three iterative steps the performance of the new estimator is demonstrated experimentally and compared to that of the ml estimator using both synthetic data and real gray level images

inexact newton methods with restricted additive schwarz based nonlinear elimination for problems with high local nonlinearity the classical inexact newton algorithm is an efficient and popular technique for solving large sparse nonlinear systems of equations when the nonlinearities in the system are well balanced a near quadratic convergence is often observed however if some of the equations are much more nonlinear than the others in the system the convergence is much slower the slow convergence or sometimes divergence is often determined by the small number of equations in the system with the highest nonlinearities the idea of nonlinear preconditioning has been proven to be very useful through subspace nonlinear solves the local high nonlinearities are removed and the fast convergence can then be restored when the inexact newton algorithm is called after the preconditioning recently a left preconditioned inexact newton method was proposed in which the nonlinear function is replaced by a preconditioned function with more balanced nonlinearities in this paper we combine an inexact newton method with a restricted additive schwarz based nonlinear elimination the new approach is easier to implement than the left preconditioned method since the nonlinear function does not have to be replaced and furthermore the nonlinear elimination step does not have to be called at every outer newton iteration we show numerically that it performs well for as an example solving the incompressible navier stokes equations with high reynolds numbers and on machines with large numbers of processors

shape from texture using local spectral moments presents a non feature based solution to the problem of computing the shape of curved surfaces from texture information first the use of local spatial frequency spectra and their moments to describe texture is discussed and motivated a new more accurate method for measuring the local spatial frequency moments of an image texture using gabor elementary functions and their derivatives is presented also described is a technique for separating shading from texture information which makes the shape from texture algorithm robust to the shading effects found in real imagery second a detailed model for the projection of local spectra and spectral moments of any surface reflectance patterns not just textures is developed third the conditions under which the projection model can be solved for the orientation of the surface at each point are explored unlike earlier non feature based curved surface shape from texture approaches the assumption that the surface texture is isotropic is not required surface texture homogeneity can be assumed instead the algorithm s ability to operate on anisotropic and nondeterministic textures and on both smooth and rough textured surfaces is demonstrated

estimating the orientation of planar surfaces algorithms and bounds this paper presents a computationally and statistically efficient parametric solution to the problem of estimating the orientation in space of a planar textured surface from a single noisy observed image of it the coordinate transformation from surface to image coordinates due to the perspective projection transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location the functional dependence of the sinusoid phase in location is uniquely determined by the tilt and slant angles of the surface from the physical model of the perspective projection we derive the cramer rao lower bound on the error variance of estimating the tilt and slant of the observed surface in the presence of observation noise it is shown in this paper that the phase of each of the sinusoids can be expressed as a linear function of some variables that are related to the surface tilt and slant angles using the phase differencing algorithm we fit a polynomial phase model to a sinusoidal component of the observed texture substituting in the derived linear relation the unknown phase with the one estimated using the phase differencing algorithm we obtain a closed form analytic and computationally efficient solution to the problem of estimating the tilt and slant angles the algorithm performance is shown to be close to the cramer rao bound even for low signal to noise ratios at computational complexity which is considerably lower than that of any existing algorithm

volumetric descriptions from a single intensity image since the early days of computer vision research shape from contour has been one of the most challenging problems many researchers in the field have attempted to understand this problem and proposed different approaches to solve it shape from contour still remains one of the hardest problems in the field the problem has two major difficulties first 2d properties of contours of viewed objects are generally not sufficient by themselves to uniquely determine 3d shape as one dimension is lost in the projection second real images produce imperfect contours that make their interpretation particularly difficult the first problem has received some attention in the research community but in the context of perfect contours the second one however has received very little in this work we propose a promising methodology to address this last problem for a large class of objects generalized cylinders it is based on exploiting mathematical invariant properties of the contours of generalized cylinders in a perceptual grouping approach we show that using these properties greatly helps addressing the figure ground problem in a more rigorous way than previous intuitive perceptual grouping methods our approach exploits the interplay between local and global features by handling different levels of the feature hierarchy we have developed and implemented a method that handles shgcs in complex seenes with markings and occlusion we demonstrate the application of our method of shape description and scene segmentation on complex real images we also demonstrate the usage of the obtained descriptions for recovery of complete 3 d object centered descriptions of viewed objects from a single intensity image

the probabilistic peaking effect of viewed angles and distances with application to 3 d object recognition two novel probabilistic models for viewed angles and distances are derived using an observability sphere method the method which is based on the assumption that the prior probability density is isotropic for all viewing orientations can be used for the computation of observation probabilities for object s aspects features and probability densities of their quantitative attributes using the sphere it is discovered that the probability densities of viewed angles distances and even projected curvature have sharp peaks at their original values from this peaking effect it is concluded that in most cases the values of angles and distances are being altered only slightly by the imaging process and they can still serve as a strong cue for model based recognition the probabilistic models for 3 d object recognition from monocular images are used to form the angular elements that are needed the objects are represented by their linear features and specific points primitives using the joint density model of angles and distances the probabilities of initial matching hypotheses and mutual information coefficients are estimated these results are then used for object recognition by optimal matching search and stochastic labeling schemes various synthetic and real objects are recognized by this approach

an extremum principle for shape from contour an extremum principle is developed that determines three dimensional surface orientation from a two dimensional contour the principle maximizes the ratio of the area to the square of the perimeter a measure of the compactness or symmetry of the three dimensional surface the principle interprets regular figures correctly and it interprets skew symmetries as oriented real symmetries the maximum likelihood method approximates the principle on irregular figures but we show that it consistently overestimates the slant of an ellipse

the visual interpretation of surface contours this article examines the computational problems underlying the 3 d interpretation of surface contours a surface contour is the image of a curve across a physical surface such as the edge of a shadow cast across a surface a gloss contour wrinkle seam or pigmentation marking surface contours by and large are not as restricted as occluding contours and therefore pose a more difficult interpretation problem nonetheless we are adept at perceiving a definite 3 d surface from even simple line drawings e g graphical depictions of continuous functions of two variables the solution of a specific surface shape comes by assuming that the physical curves are particularly restricted in their geometric relationship to the underlying surface these geometric restrictions are examined

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

on symmetry and multiple view geometry structure pose and calibration from a single image in this paper we provide a principled explanation of how knowledge in global 3 d structural invariants typically captured by a group action on a symmetric structure can dramatically facilitate the task of reconstructing a 3 d scene from one or more images more importantly since every symmetric structure admits a canonical coordinate frame with respect to which the group action can be naturally represented the canonical pose between the viewer and this canonical frame can be recovered too which explains why symmetric objects e g buildings provide us overwhelming clues to their orientation and position we give the necessary and sufficient conditions in terms of the symmetry group admitted by a structure under which this pose can be uniquely determined we also characterize when such conditions are not satisfied to what extent this pose can be recovered we show how algorithms from conventional multiple view geometry after properly modified and extended can be directly applied to perform such recovery from all hidden images of one image of the symmetric structure we also apply our results to a wide range of applications in computer vision and image processing such as camera self calibration image segmentation and global orientation large baseline feature matching image rendering and photo editing as well as visual illusions caused by symmetry if incorrectly assumed

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

part ii 3 d object recognition and shape estimation from image contours using b splines shape invariant matching and neural network for pt i see ibid p 1 12 1994 this paper is the second part of a 3 d object recognition and shape estimation system that identifies particular objects by recognizing the special markings text symbols drawings etc on their surfaces the shape of the object is identified from the image curves using b spline curve modeling as described in part i as well as a binocular stereo imaging system this is achieved by first estimating the 3 d control points from the corresponding curves in each image in the stereo imaging system from the 3 d control points the 3 d object curves are generated and these are subsequently used for estimating the 3 d surface parameters a bayesian framework is used for classifying the image into one of c possible surfaces based on the extracted 3 d object curves this is complemented by a neural network nn that recognizes the surface as a particular object e g a pepsi can versus a peanut butter jar by reading the text markings on the surface to reduce the amount of training the nn has to undergo for recognition the object curves are unwarped into planar curves before the matching process this eliminates the need for templates that are surface shape dependent and results in a planar curve that might be a rotated translated and scaled version of the template hence for the matching process we need to use measures that are invariant to these transformations one such measure is the fourier descriptors fd derived from the control points associated with the unwarped parent curves the approach is tried on a variety of images of real objects and appears to hold great promise

detection of surface orientation and motion from texture by a stereoogical technique abstract a new approach is given to detect the surface orientation and motion from the texture on the surface by making use of a mathematical principle called stereology information about the surface orientation is contained in features computed by scanning the image by parallel lines and counting the number of intersections with the curves of the texture a synthetic example is given to illustrate the technique this scheme can also detect surface motions relative to the viewer by computing features of its texture at one time and a short time later the motion is specified by explicit formulae of the computed features

an algebraic approach to shape from image problems abstract this paper presents a new method for recovering three dimensional shapes of polyhedral objects from their single view images the problem of recovery is formulated in a constrained optimization problem in which the constraints reflect the assumption that the scene is composed of polyhedral objects and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture for practical purpose it is decomposed into the two more tractable problems a linear programming problem and an unconstrained optimization problem in the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra whereas similar constraints have usually been represented in terms of a gradient space moreover superstrictness of the constraints can be circumvented by a new concept position free incidence structure for this reason the present method has several advantages it can recover the polyhedral shape even if image data are incorrect due to vertex position errors it can deal with perspective projection as well as orthographic projection the number of variables in the optimization problem is very small three or a little greater than three and any kinds of surface information can be incorporated in a unifying manner