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Shape From Shading for Non-Lambertian Surfaces
Abstract
The human visual system has the capacity to infer the shape of a three-dimensional object using only two-dimensional information. One of the cues the human visual system uses to perform this inference is shading information. An object's shading is its variation in brightness over a given domain. Computer algorithms have been developed that recover the three-dimensional shape of objects depicted in images. These algorithms are called shape-from-shading algorithms. Most shape-from-shading algorithms rely on idealistic conditions. They usually assume orthographic projection, a distant point light source, and Lambertian reflectance. It is known that most real surfaces are neither perfectly diffuse (Lambertian) nor ideally specular (mirror-like); however, most shape-from-shading algorithms assume Lambertian reflectance. It is necessary to develop new techniques to recover the shape of objects whose surfaces are not necessarily Lambertian. In this thesis two algorithms are proposed that c...
... This equation can be generalised to consider perspective projections (Lee, Kuo, 1994), a light source no longer at infinity but at the centre of projection (Prados, Faugeras, 2005) or non Lambertian scene (Bakshi, Yang, 1994). It has been proved in (Belhumeur et al., 1999) that when the lighting direction and the albedo of the surface are unknown, the same image can be obtained by a continuous family of surfaces. ...
The storage and management of stockpiles of materials is a fundamental process in large scale activities such as mining, civil engineering, and in the management of waste landfill sites. Following the evolution of stocks has always been important, and advancements in remote sensing technology are not only facilitating this, but also making it possible in near real-time. Nowadays, this monitoring appears to be performed almost exclusively using UAV based techniques. This paper proposes to apply a simple Shape-from-Shading method on low resolution satellite images provided by PlanetScope to monitor the evolution of the volume of stockpiles. The proposed Shape-from-Shading formulation makes it possible to handle occluding objects in the scene. The loss in accuracy due to the low resolution of PlanetScope images is well compensated by the daily revisit frequency and by the fact that spaceborne acquisitions require no human supervision. With satellites it is also easy to follow simultaneously several stocking sites all over the world. We test our method on two coal storage sites and demonstrate that the stockpiles are well detected.
... Points í µí¼í µí¼ ∈ Ω such that I(x) is maximal correspond to the particular situation where x and n(x) point in the same direction: these points are usually called ''singular points''. Let us mention that Eq. (3) is not the most general equation of SFS[23]: since real materials are not purely Lambertian, some publications are concerned with non-Lambertian SFS problems [24][25][26]; moreover, the situation is more complex in the presence of other lighting models[27,28] or when the interreflections are taken into account[29,30] . We will also consider the equation which appears in most of the papers and American Journal of Computational and Applied Mathematics: 2011; 1(1): 33-40 35 corresponds to frontal light source at infinity, i.e. í µí½í µí½ = (0,0,1). ...
In this paper, a method for reconstructing a 3-D shape of an object from a 2-D shading image using a Genetic Algorithm (GA), which is an optimizing technique based on mechanisms of natural selection. The 3D-shape is recovered through the analysis of the gray levels in a single image of the scene. This problem is ill-posed except if some additional assumptions are made. In the proposed method, shape from shading is addressed as an energy minimization problem. The traditional deterministic approach provides efficient algorithms to solve this problem in terms of time but reaches its limits since the energy associated with shape from shading can contain multiple deep local minima. Genetic Algorithm is used as an alternative approach which is efficient at exploring the entire search space. The Algorithm is tested in both synthetic and real image and is found to perform accurate and efficient results.
There is considerable evidence demonstrating that erythrocyte deformability is playing an important role in the filerability of blood and consequently in the pathology of a variety of blood-related diseases. The shape of red blood cell is also a determining factor for its deformability and filerability. The original input red blood cell image we are aiming to deal with is captured by Scanned Electronic Microscope (SEM). We map the intensity level image into a 3-D depth through Shape from Shading (SFS) first to get the height field of each pixel, then each cell on top level is extracted from the whole image by means of region growing algorithm based on boundary contour tracing, which are used for feature extraction and statistics computation. The experimental result shows that this approach is easily to implement and promising.
Most of the shape from shading (SFS) algorithms have been developed under the simplifying assumptions of a Lambertian surface, an orthographic projection, and a distant light source. Due to the difficulty of the SFS problem, only a small number of algorithms have been proposed for surfaces with non-Lambertian reflectance, and among those, only very few algorithms are applicable for surfaces with specular and diffuse reflectance. In this paper we propose a unified framework that is capable of solving the SFS problem under various settings of imaging conditions i.e., Lambertian or non-Lambertian, orthographic or perspective projection, and distant or nearby light source. The proposed algorithm represents the image irradiance equation of each setting as an explicit Partial Differential Equation (PDF). In our implementation we use the Lax-Friedrichs sweeping method to solve this PDF. To demonstrate the efficiency of the proposed algorithm, several comparisons with the state of the art of the SFS literature are given.
Due to their improved capability to handle realistic illumination scenarios, non- Lambertian reflectance models are becoming increasingly more popular in the Shape from Shading (SfS) community. One of these advanced models is the Oren-Nayar model which is particularly suited to handle rough surfaces. However, not only the proper selection of the model is important, also the validation of stable and efficient algorithms plays a fundamental role when it comes to the practical applicability. While there are many works dealing with such algorithms in the case of Lambertian SfS, no such analysis has been performed so far for the Oren-Nayar model. In our paper we address this problem and present an in-depth study for such an advanced SfS model. To this end, we investigate under which conditions, i.e. model parameters, the Fast Marching (FM) method can be applied - A method that is known to be one of the most efficient algorithms for solving the underlying partial differential equations of Hamilton-Jacobi type. In this context, we do not only perform a general investigation of the model using Osher's criterion for verifying the suitability of the FM method. We also conduct a parameter dependent analysis that shows, that FM can safely be used for the model for a wide range of settings relevant for practical applications. Thus, for the first time, it becomes possible to theoretically justify the use of the FM method as solver for the Oren-Nayar model which has been applied so far on a purely empirical basis only. Numerical experiments demonstrate the validity of our theoretical analysis. They show a stable behaviour of the FM method for the predicted range of model parameters.
In this work, we extend the applicability of perspective Shape from Shading to images incorporating non-Lambertian surfaces.
To this end, we derive a new model inspired by the perspective model for Lambertian surfaces recently studied by Prados et
al. and the Phong reflection model incorporating ambient, diffuse and specular components. Besides the detailed description
of the modeling process, we propose an efficient and stable semi-implicit numerical realisation of the resulting Hamilton-Jacobi
equation. Numerical experiments on both synthetic and simple real-world images show the benefits of our new model: While computational
times stay modest, a large qualitative gain can be achieved.
The shape deformability of Red Blood Cell plays an important role in blood filterability. This paper proposed a combined approach for complex surface segmentation of Red Blood Cell. Firstly the individual cell was extracted using contour guiding method. Each pixel's shading variation information was then used for 3D shape reconstruction. The recovered depth map of each image contributed to segment the shape through multi-scale surface fitting. The experimental result shown good results and obtained high classification accuracy.
Many algorithms have been suggested for the shape-from-shading problem, and some years have passed since the publication of the survey paper by Zhang et al. [R. Zhang, P.-S. Tsai, J.E. Cryer, M. Shah, Shape from shading: a survey, IEEE Transactions on Pattern Analysis and Machine Intelligence 21 (8) (1999) 690–706]. In this new survey paper, we try to update their presentation including some recent methods which seem to be particularly representative of three classes of methods: methods based on partial differential equations, methods using optimization and methods approximating the image irradiance equation. One of the goals of this paper is to set the comparison of these methods on a firm basis. To this end, we provide a brief description of each method, highlighting its basic assumptions and mathematical properties. Moreover, we propose some numerical benchmarks in order to compare the methods in terms of their efficiency and accuracy in the reconstruction of surfaces corresponding to synthetic, as well as to real images.
This thesis is concerned with inferring scene shape by combining two specific techniques: shape-from-shading and stereopsis. Shape-from-shading calculates shape using the lighting equation, which takes surface orientation and lighting information to irradiance. As irradiance and lighting information are provided this is the problem of inverting a many to one function to get surface orientation.
Surface orientation may be integrated to get depth. Stereopsis matches pixels between two images taken from different locations of the same scene - this is the correspondence problem. Depth can then be calculated using camera calibration information, via triangulation. These methods both fail for certain inputs; the advantage of combining them is that where one fails the other may continue to work. Notably, shape-from-shading requires a smoothly shaded surface, without texture, whilst stereopsis requires texture - each works where the other does not.
The first work of this thesis tackles the problem directly.
A novel modular solution is proposed to combine both methods; combining is itself done using Gaussian belief propagation. This modular approach highlights missing and weak modules; the rest of the thesis is then concerned with providing a new module and an improved module. The improved module is given in the second research chapter and consists of a new shape-from-shading algorithm. It again uses belief propagation, but this time with directional statistics to represent surface orientation. Message passing is performed using a novel method; it is analytical, which makes this algorithm particularly fast. In the final research chapter a new module is provided, to estimate the light source direction. Without such a module the user of the system has to provide it; this is tedious and error prone, and impedes automation. It is a probabilistic method that uniquely estimates the light source direction using a stereo pair as input.
Face verification is a challenging pattern recognition problem. The face is a biometric that, we as humans, know can be recognised. However, the face is highly deformable and its appearance alters significantly when the pose, illumination or expression changes. These changes in appearance are most notable for texture images, or two-dimensional (2D) data. But the underlying structure of the face, or three dimensional (3D) data, is not changed by pose or illumination variations. Over the past five years methods have been investigated to combine 2D and 3D face data to improve the accuracy and robustness of face verification. Much of this research has examined the fusion of a 2D verification system and a 3D verification system, known as multi-modal classifier score fusion. These verification systems usually compare two feature vectors (two image representations), a and b, using distance or angular-based similarity measures. However, this does not provide the most complete description of the features being compared as the distances describe at best the covariance of the data, or the second order statistics (for instance Mahalanobis based measures). A more complete description would be obtained by describing the distribution of the feature vectors. However, feature distribution modelling is rarely applied to face verification because a large number of observations is required to train the models. This amount of data is usually unavailable and so this research examines two methods for overcoming this data limitation: 1. the use of holistic difference vectors of the face, and 2. by dividing the 3D face into Free-Parts. The permutations of the holistic difference vectors is formed so that more observations are obtained from a set of holistic features. On the other hand, by dividing the face into parts and considering each part separately many observations are obtained from each face image; this approach is referred to as the Free-Parts approach. The extra observations from both these techniques are used to perform holistic feature distribution modelling and Free-Parts feature distribution modelling respectively. It is shown that the feature distribution modelling of these features leads to an improved 3D face verification system and an effective 2D face verification system. Using these two feature distribution techniques classifier score fusion is then examined. This thesis also examines methods for performing classifier fusion score fusion. Classifier score fusion attempts to combine complementary information from multiple classifiers. This complementary information can be obtained in two ways: by using different algorithms (multi-algorithm fusion) to represent the same face data for instance the 2D face data or by capturing the face data with different sensors (multimodal fusion) for instance capturing 2D and 3D face data. Multi-algorithm fusion is approached as combining verification systems that use holistic features and local features (Free-Parts) and multi-modal fusion examines the combination of 2D and 3D face data using all of the investigated techniques. The results of the fusion experiments show that multi-modal fusion leads to a consistent improvement in performance. This is attributed to the fact that the data being fused is collected by two different sensors, a camera and a laser scanner. In deriving the multi-algorithm and multi-modal algorithms a consistent framework for fusion was developed. The consistent fusion framework, developed from the multi-algorithm and multimodal experiments, is used to combine multiple algorithms across multiple modalities. This fusion method, referred to as hybrid fusion, is shown to provide improved performance over either fusion system on its own. The experiments show that the final hybrid face verification system reduces the False Rejection Rate from 8:59% for the best 2D verification system and 4:48% for the best 3D verification system to 0:59% for the hybrid verification system; at a False Acceptance Rate of 0:1%.
The directional distribution of radiant flux reflected from roughened surfaces is analyzed on the basis of geometrical optics. The analytical model assumes that the surface consists of small, randomly disposed, mirror-like facets. Specular reflection from these facets plus a diffuse component due to multiple reflections and/or internal scattering are postulated as the basic mechanisms of the reflection process. The effects of shadowing and masking of facets by adjacent facets are included in the analysis. The angular distributions of reflected flux predicted by the analysis are in very good agreement with experiment for both metallic and nonmetallic surfaces. Moreover, the analysis successfully predicts the off-specular maxima in the reflection distribution which are observed experimentally and which emerge as the incidence angle increases. The model thus affords a rational explanation for the off-specular peak phenomenon in terms of mutual masking and shadowing of mirror-like, specularly reflecting surface facets.
The orientation of patches on the surface of an object can be determined from multiple images taken with different illumination, but from the same viewing position. The method, referred to as photometric stereo, can be implemented using table lookup based on numerical inversion of reflectance maps. Here we concentrate on objects with specularly reflecting surfaces, since these are of importance in industrial applications. Previous methods, intended for diffusely reflecting surfaces, employed point source illumination, which is quite unsuitable in this case. Instead, we use a distributed light source obtained by uneven illumination of a diffusely reflecting planar surface. Experimental results are shown to verify analytic expressions obtained for a method employing three light source distributions.
We develop a systematic approach to the discovery of parallel iterative schemes for solving the shape-from-shading problem on a grid. A standard procedure for finding such schemes is outlined, and subsequently used to derive several new ones. The shape-from-shading problem is known to be mathematically equivalent to a nonlinear first-order partial differential equation in surface elevation. To avoid the problems inherent in methods used to solve such equations, we follow previous work in reformulating the problem as one of finding a surface orientation field that minimizes the integral of the brightness error. The calculus of variations is then employed to derive the appropriate Euler equations on which iterative schemes can be based. The problem of minimizing the integral of the brightness error term is ill posed, since it has an infinite number of solutions in terms of surface orientation fields. A previous method used a regularization technique to overcome this difficulty. An extra term was added to the integral to obtain an approximation to a solution that was as smooth as possible. We point out here that surface orientation has to obey an integrability constraint if it is to correspond to an underlying smooth surface. Regularization methods do not guarantee that the surface orientation recovered satisfies this constraint. Consequently, we attempt to develop a method that enforces integrability, but fail to find a convergent iterative scheme based on the resulting Euler equations. We show, however, that such a scheme can be derived if, instead of strictly enforcing the constraint, a penalty term derived from the constraint is adopted. This new scheme, while it can be expressed simply and elegantly using the surface gradient, unfortunately cannot deal with constraints imposed by occluding boundaries. These constraints are crucial if ambiguities in the solution of the shape-from-shading problem are to be avoided. Differrent schemes result if one uses different parameters to describe surface orientation. We derive two new schemes, using unit surface normals, that facilitate the incorporation of the occluding boundary information. These schemes, while more complex, have several advantages over previous ones.
Traditionally, image intensities have been processed to segment an image into regions or to find edge-fragments. Image intensities carry a great deal more information about three-dimensional shape, however. To exploit this information, it is necessary to understand how images are formed and what determines the observed intensity in the image. The gradient space, popularized by Huffman and Mackworth in a slightly different context, is a helpful tool in the development of new methods.
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.
This paper presents a new reflectance model for rendering computer synthesized images. The model accounts for the relative brightness of different materials and light sources in the same scene. It describes the directional distribution of the reflected light and a color shift that occurs as the reflectance changes with incidence angle. The paper presents a method for obtaining the spectral energy distribution of the light reflected from an object made of a specific real material and discusses a procedure for accurately reproducing the color associated with the spectral energy distribution. The model is applied to the simulation of a metal and a plastic.
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.
It appears that the development of machine vision may benefit from a detailed understanding of the imaging process. The reflectance map, showing scene radiance as a function of surface gradient, has proved to be helpful in this endeavor. The reflectance map depends both on the nature of the surface layers of the objects being imaged and the distribution of light sources. Recently, a unified approach to the specification of surface reflectance in terms of both incident and reflected beam geometry has been proposed. The reflecting properties of a surface are specified in terms of the bidirectional reflectance-distribution function (BRDF). Here we derive the reflectance map in terms of the BRDF and the distribution of source radiance. A number of special cases of practical importance are developed in detail. The significance of this approach to the understanding of image formation is briefly indicated.
Well-known methods for solving the shape-from-shading problem require knowledge of the reflectance map. Here we show how the shape-from-shading problem can be solved when the reflectance map is not available, but is known to have a given form with some unknown parameters. This happens, for example, when the surface is known to be Lambertian, but the direction to the light source is not known. We give an iterative algorithm that alternately estimates the surface shape and the light source direction. Use of the unit normal in parameterizing the reflectance map, rather than the gradient or stereographic coordinates, simpliflies the analysis. Our approach also leads to an iterative scheme for computing shape from shading that adjusts the current estimates of the focal normals toward or away from the direction of the light source. The amount of adjustment is proportional to the current difference between the predicted and the observed brightness. We also develop generalizations to less constrained forms of reflectance maps.
A robust approach to recovery of shape from shading information is
presented. Assuming uniform albedo and Lambertian surface for the
imaging model, methods are presented for the estimation of illuminant
direction and surface albedo. The illuminant azimuth is estimated by
averaging local estimates. The illuminant elevation and surface albedo
are estimated from image statistics. Using the estimated reflectance map
parameters, the surface shape is computed using a procedure that
implements the smoothness constraint by enforcing the gradients of
reconstructed intensity to be close to the gradients of the input image.
Typical results on real images are given to illustrate the usefulness of
this approach
A method is presented for determining the shapes of hybrid
surfaces without prior knowledge of the relative strengths of the
Lambertian and specular components of reflection. The object surface is
illuminated using extended light sources and is viewed from a single
direction. Surface illumination using extended sources makes it possible
to ensure the detection of both Lambertian and specular reflections.
Uniformly distributed source directions are used to obtain an image
sequence of the object. This method of obtaining photometric
measurements is called photometric sampling. An extraction algorithm
uses the set of image intensity values measured at each surface point to
compute orientation as well as relative strengths of the Lambertian and
specular reflection components. The simultaneous recovery of shape and
reflectance parameters enables the method to adapt to variations in
reflectance properties from one scene point to another. Experiments were
conducted on Lambertian surfaces, specular surfaces, and hybrid surfaces
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
Reflectance models based on physical optics and geometrical optics are studied. Specifically, the authors consider the Beckmann-Spizzichino (physical optics) model and the Torrance-Sparrow (geometrical optics) model. These two models were chosen because they have been reported to fit experimental data well. Each model is described in detail, and the conditions that determine the validity of the model are clearly stated. By studying reflectance curves predicted by the two models, the authors propose a reflectance framework comprising three components: the diffuse lobe, the specular lobe, and the specular spike. The effects of surface roughness on the three primary components are analyzed in detail
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
The application of the reformulated current method to scattering from smooth moving surfaces was investigated. The applicability of several methods for the study of scattering from rough, stationary surfaces was also examined. In the reformulated current method, the moving surface was replaced by equivalent current sources; the motion was retained in these sources. Both deterministic and non-deterministic motion were assumed. The surfaces considered were a sphere and infinitely long cylinder. For non-deterministic motion, the mean of the electromagnetic field was obtained. An infinite integral transform was used to investigate wave scattering from a rough cylinder in which the surface was characterized in terms of its surface impedance. Two types of polarization are treated. A similar procedure is applied to a rough sphere.
Conditions are established under which the scattering of individual modes of an external field by a rough surface can be considered quasi-local. Also considered is the feasibility of using the local coefficients to calculate the scattered field. The scattering of radio waves by the rough sea surface is considered as an example.
Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. All rights reserved.
Shading is important for estimation of three-dimensional shape from the two-dimensional image, for instance, for distinguishing between the smooth occluding contour generated by the edge of a sphere and the sharp occluding contour generated by the edge of a disk. In order to use shading information to solve such problems, one must know the illuminant direction L. This is because variations in image intensity (shading) are caused by changes in surface orientation relative to the illuminant. Each illuminant direction L has a unique effect on the distribution of changes in image intensity dI, potentially potentially permitting the estimation of L. However, because dI is a function of bothL and the surface curvature, L can be estimated from the image only by making an assumption about the imaged surface curvature. One assumption that is sufficient to disentangle L and surface curvature is that changes in surface orientation are isotropically distributed. This condition is true of images of convex objects bounded by a smooth occluding contour and is true on average over all scenes. Estimates made by using this assumption agree with estimates of illuminant direction given by human subjects for images of natural objects, even when both are objectively wrong. Further, there is a significant correlation between the variance of these estimates and the variance of the human subjects’ estimates.
The quality of computer generated images of three-dimensional scenes depends on the shading technique used to paint the objects on the cathode-ray tube screen. The shading algorithm itself depends in part on the method for modeling the object, which also determines the hidden surface algorithm. The various methods of object modeling, shading, and hidden surface removal are thus strongly interconnected. Several shading techniques corresponding to different methods of object modeling and the related hidden surface algorithms are presented here. Human visual perception and the fundamental laws of optics are considered in the development of a shading rule that provides better quality and increased realism in generated images.
The objective of the paper is to describe a technique that produces shaded renderings of solids; and to describe an assembly procedure which may be used to assemble representations of 3-D objects in such a way that they can be photographically rendered. The algorithm that is used to produce photographic renderings may be subdivided into two parts -- (1) the visible surface procedure that determines which surfaces are visible and which are invisible and retains the description of only the visible surfaces, and (2) the shader which varies the tonal gradation on all visible surfaces. The assembly procedure is a tool that allows a user to form, manipulate and photographically render 3-D objects. A set of primitive structures described by planar triangles are used as the 3-D object building-set. Both the rendering and assembly procedures are presently operational software programs written in a combination of Fortran V and UNIVAC 1108 Assembly Language. (Author)
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.
Thephotometric stereo method is extended to four-source photometry for obtaining 3-dimensional shapes of visually textured and specular surfaces from intensity values. The table look-up approach (based on reflectance maps) for determining surface normals is shown to be of limited value in this context and a more direct method of computing these normals is used. Our method also eliminates the requirement of a preliminary system calibration procedure. The method is applied to real images as a test of its applicability.
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.
We show that highlights in images of objects with specularly reflecting surfaces provide significant information about the surfaces which generate them. A brief survey is given of specular reflectance models which have been used in computer vision and graphics. For our work, we adopt the Torrance-Sparrow specular model which, unlike most previous models, considers the underlying physics of specular reflection from rough surfacers. From this model we derive powerful relationships between the properties of a specular feature in an image and local properties of the corresponding surface. We show how this analysis can be used for both prediction and interpretation in a vision system. A shape from specularity system has been implemented to test our approach. The performance of the system is demonstrated by careful experiments with specularly reflecting objects.
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.
The shape-from-shading problem is known to be mathematically equivalent to a nonlinear first-order partial differential equation in surface elevation. Many different algorithms have been developed for determining shape-from-shading information. Horn proposed an algorithm for recovering height from a single image, on a data-directed curve in the (x,y) plane called a characteristic strip. He concentrated on the importance of singular points, surface models and occluding boundaries in providing initialization for the algorithm. Bruckstein proposed a new method for obtaining the height information considering the behavior of equal-height contours. When one such contour is available his proposed algorithm reconstructs all the equal-height curves of the surface of interest in a well-defined region. This paper compares these two algorithms experimentally. It was found that the Horn's characteristic strip algorithm appears to behave better.
Almost nothing can be deduced about a general 3-D surface given only its occluding contours in an image, yet contour information is easily and effectively used by us to infer the shape of a surface. Therefore, implicit in the perceptual analysis of occluding contour must lie various assumptions about the viewed surfaces. The assumptions that seem most natural are (a) that the distinction between convex and concave segments reflects real properties of the viewed surface; and (b) that contiguous portions of contour arise from contiguous parts of the viewed surface ??e. there are no invisible obscuring edges. It is proved that, for smooth surfaces, these assumptions are essentially equivalent to assuming that the viewed surface is a generalized cone. Methods are defined for finding the axis of such a cone, and for segmenting a surface constructed of several cones into its components, whose axes can then be found separately. These methods, together with the algorithms for implementing them devised by Vatan & Marr (1977), provide one link between an uninterpreted figure extracted from an image, and the 3-D representation theory of Marr and Nishihara (1977).
The quality of computer generated images of three-dimensional scenes depends on the shading technique used to paint the objects on the cathode-ray tube screen. The shading algorithm itself depends in part on the method for modeling the object, which also determines the hidden surface algorithm. The various methods of object modeling, shading, and hidden surface removal are thus strongly interconnected. Several shading techniques corresponding to different methods of object modeling and the related hidden surface algorithms are presented here. Human visual perception and the fundamental laws of optics are considered in the development of a shading rule that provides better quality and increased realism in generated images
This is a discussion of the physics of illumination and the associated techniques for modeling global and local illumination in computer generated imagery. It was state-of-the-art in 1988, but is now rather outdated. It does include discussions of physics and color theory basics that have not changed, and a discussion of illumination models through ray tracing models using various specular reflectance functions and including Fresnel effects. This text is currently out of print. However, we still receive numerous requests for an electronic version of the source code in the book.
Bui-Tuong Phong published his illumination model in 1973, in the paper titled "Illumination for Computer-Generated Images". Phong's model is a local illumination model, which means only direct reflections are taken into account. Light that bounces off more than one surface before reaching the eye is not accounted for. While this may not be very realistic, it allows the lighting to be computed efficiently. To properly handle indirect lighting, a global illumination method such as radiosity is required, which is much more expensive. In addition to Phong's basic lighting equation, we will look at a variation invented by Jim Blinn. Blinn changed the way specular is calculated, making the computations slightly cheaper. Blinn published his approach in his paper "Models of Light Reflection for Computer Synthesised Pictures" in 1977.
Over the past 20 years, a great deal of interest has developed in the use of computer graphics and numerical methods for three-dimensional design. Significant progress in geometric modeling is being made, predominantly for objects best represented by lists of edges, faces, and vertices. One long-term goal of this work is a unified mathematical formalism, to form the basis of an interactive and intuitive design environment in which designers can simulate three-dimensional scenes with shading and texture, produce usable design images, verify numerical machining-control commands, and set up finite-element meshwork for structural and dynamic analysis.
A new collection of smooth parametric objects and a new set of three-dimensional parametric modifiers show potential for helping to achieve this goal. The superquadric primitives and angle-preserving transformations extend the traditional geometric primitives-quadric surfaces and parametric patches-used in existing design packages, producing a new spectrum of flexible forms. Their chief advantage is that they allow complex solids and surfaces to be constructed and altered easily from a few interactive parameters.
A new approach for the reconstruction of a smooth three
dimensional object from its two dimensional gray level image is
presented. An algorithm based on topological properties of simple smooth
surfaces is provided to solve the problem of global reconstruction.
Classifying singular points in the shading image as maxima minima and
two kinds of saddle points, serves as the key to the solution of the
problem. This classification is performed globally with no assumptions
on the local behavior of characteristics near singular points. The
global reconstruction procedure, being deterministic and using
topological properties of the surface performs better than other
approaches proposed so far, based on classification of singular points
according to the local behavior of characteristics in their
neighborhood. The proposed algorithm is simple, easy to implement and
works remarkably fast on a parallel machine
In the use of comptuer graphics and numerical methods for three-dimensional design, significant progress in geometric modeling is being made, predominantly for objects best represented by lists of edges, faces, and vertices. A new collection of smooth parametric objects and a new set of three-dimensional parametric modifiers show potential for helping to achieve this goal. The superquadric primitives and angle-preserving transformations extend the traditional geometric primitives-quadric surfaces and parametric patches - used in existing design packages, producing a new spectrum of flexible forms. Their chief advantage is that they allow complex solids and surfaces to be constructed and altered easily from a few interactive parameters.
A heuristic-based algorithm known as the shading logic algorithm
is proposed for recovering shape from shading. The heuristics are
derived on the basis of a geometrical interpretation of the M.J. Brooks
and B.K.P. Horn (1985) algorithm. An experimental evaluation was
performed using synthesized objects, in particular, superquadrics. The
advantage of using the superquadrics is that the shape of the objects
can be varied incrementally and systematically. Despite the fact that
the shading logic algorithm is heuristic based, experimental results
show that the proposed algorithm has a better performance than the
Brooks and Horn algorithm. In addition, the proposed approach does not
seem to suffer the stability problem common to most variational-based
methods
A theory of photometric stereo is proposed for a large class of non-Lambertian reflectance maps. The authors review the different reflectance maps proposed in the literature for modeling reflection from real-world surfaces. From this, they obtain a mathematical class of reflectance maps to which the maps belong. They show that three lights can be sufficient for a unique inversion of the photometric stereo equation for the entire class of reflectance maps. They also obtain a constraint on the positions of light sources for obtaining this solution. They investigate the sufficiency of three light sources to estimate the surface normal and the illuminant strength. The issue of completeness of reconstruction is addressed. They show that if k lights are sufficient for a unique inversion, 2 k lights are necessary for a complete inversion
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
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Shape from color Center for Systems Science CSS/LCCR TR 92-07
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M. Drew. Shape from color. Center for Systems Science CSS/LCCR TR 92-07,
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A study of shape-from-shading algorithms
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S. Bakshi. A study of shape-from-shading algorithms. In Proceedings of the
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Saskatchewan.
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Default shape theory: With applications to the recovery of shape and light source from shading
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On shape from shading Computer Vision, Graphics, and Image Processing
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algorithm is lower than the A.I.E. for the V.A.N.L. The A.I.E. over all objects for the G.S.L. is only 3.611957, while the A.I.E. for the V.A.N.L. is 5.718210. It can also be visually con rmed that the resynthesized images resemble the original image. From the three-dimensional mesh diagrams
- A I E The
- G S L For The
The A.I.E. for the G.S.L. algorithm is lower than the A.I.E. for the V.A.N.L. The
A.I.E. over all objects for the G.S.L. is only 3.611957, while the A.I.E. for the V.A.N.L.
is 5.718210. It can also be visually con rmed that the resynthesized images resemble
the original image. From the three-dimensional mesh diagrams, it is observed that
both algorithms can recover the shape of object using real images. The height maps