A 3D Imaging Framework Based on High-Resolution Photometric-Stereo and Low-Resolution Depth

This paper introduces a 3D imaging framework that combines high-resolution photometric stereo and low-resolution depth. Our approach targets imaging scenarios based on either macro-lens photography combined with focal stacking or a large-format camera that are able to image objects with more than 600 samples per mm $^2$ . These imaging techniques allow photometric stereo algorithms to obtain surface normals at resolutions that far surpass corresponding depth values obtained with traditional approaches such as structured-light, passive stereo, or depth-from-focus. Our work offers two contributions for 3D imaging based on these scenarios. The first is a multi-resolution, patched-based surface reconstruction scheme that can robustly handle the significant resolution difference between our surface normals and depth samples. The second is a method to improve the initial normal estimation by using all the available focal information for images obtained using a focal stacking technique.     

Reconstructing a 3-D depth map from one or more images

Several algorithms are suggested for recovering depth and orientation maps of a surface from its image intensities. They combine the advantages of stereo vision and shape-from-shading (SFS) methods. These algorithms generate accurate, unambiguous and dense surface depth and orientation maps. Most of the existing SFS algorithms cannot be directly extended to combine stereo images because the recovery of surface depth and that of orientation are separated in these formulations. We first present an SFS algorithm that couples the generation of depth and orientation maps. This formulation also ensures that the reconstructed surface depth and its orientation are consistent. The SFS algorithm for a single image is then extended to utilize stereo images. The correspondence over stereo images is established simultaneously with the generation of surface depth and orientation. An alternative approach is also suggested for combining stereo and SFS techniques. This approach can be used to combine needle maps which are directly available from other sources such as photometric stereo. Finally we present an algorithm to combine sparse depth measurements with an orientation map to reconstruct a surface. The same algorithm can be combined with the above algorithms for solving the SFS problem with sparse depth measurements. Thus various information sources can be used to accurately reconstruct a surface.     

Two-dimensional BRDF estimation from polarisation

This paper presents a novel technique for reflectance function (BRDF) estimation, which uses polarisation information and photometric stereo. The first stage of the technique is standard and involves the acquisition of polarisation information (angle and degree of polarisation) using a linear polariser and a digital camera. This yields a field of ambiguous surface normal estimates for an arbitrarily shaped object. A photometric stereo algorithm is then used with three different light source directions to disambiguate the surface normals. Next, the proposed algorithm constructs a 3D histogram of the surface normals and pixel brightnesses. A surface, representing the BRDF, is then fitted to the histogram data using simulated annealing optimisation. The result is a set of Cartesian triples that relate the surface normals to the observed pixel brightnesses. Unlike most previous techniques for BRDF estimation, the technique is image-based and does not require sophisticated equipment or intrusive light sources. Although the technique is restricted to smooth and slightly rough dielectric objects, no prior knowledge about the surface geometry is assumed.     

Evaluating the effect of diffuse light on photometric stereo reconstruction

Photometric stereo surface reconstruction requires each input image to be associated with a particular 3D illumination vector. This signifies that the subject should be illuminated in turn by various directional illumination sources. In real life, this directionality may be reduced by ambient illumination, which is typically present as a diffuse component of the incident light. This work assesses the photometric stereo reconstruction quality for various ratios of ambient to directional illuminance and provides a reference for the robustness of photometric stereo with respect to that illuminance ratio. In our analysis, we focus on the face reconstruction application of photometric stereo, as faces are convex objects with rich surface variation, thus providing a suitable platform for photometric stereo reconstruction quality evaluation. Results demonstrate that photometric stereo renders realistic reconstructions of the given surface for ambient illuminance as high as nine times the illuminance of the directional light component.     

Stereo Image Analysis of Non-Lambertian Surfaces

Stereo image analysis is based on establishing correspondences between a pair of images by determining similarity measures for potentially corresponding image parts. Such similarity criteria are only strictly valid for surfaces with Lambertian (diffuse) reflectance characteristics. Specular reflections are viewpoint dependent and may thus cause large intensity differences at corresponding image points. In the presence of specular reflections, traditional stereo approaches are often unable to establish correspondences at all, or the inferred disparity values tend to be inaccurate, or the established correspondences do not belong to the same physical surface point. The stereo image analysis framework for non-Lambertian surfaces presented in this contribution combines geometric cues with photometric and polarimetric information into an iterative scheme that allows to establish stereo correspondences in accordance with the specular reflectance behaviour and at the same time to determine the surface gradient field based on the known photometric and polarimetric reflectance properties. The described approach yields a dense 3D reconstruction of the surface which is consistent with all observed geometric and photopolarimetric data. Initially, a sparse 3D point cloud of the surface is computed by traditional blockmatching stereo. Subsequently, a dense 3D profile of the surface is determined in the coordinate system of camera 1 based on the shape from photopolarimetric reflectance and depth technique. A synthetic image of the surface is rendered in the coordinate system of camera 2 using the illumination direction and reflectance properties of the surface material. Point correspondences between the rendered image and the observed image of camera 2 are established with the blockmatching technique. This procedure yields an increased number of 3D points of higher accuracy, compared to the initial 3D point cloud. The improved 3D point cloud is used to compute a refined dense 3D surface profile. These steps are iterated until convergence of the 3D reconstruction. An experimental evaluation of our method is provided for areas of several square centimetres of forged and cast iron objects with rough surfaces displaying both diffuse and significant specular reflectance components, where traditional stereo image analysis largely fails. A comparison to independently measured ground truth data reveals that the root-mean-square error of the 3D reconstruction results is typically of the order 30–100 μm at a lateral pixel resolution of 86 μm. For two example surfaces, the number of stereo correspondences established by the specular stereo algorithm is several orders of magnitude higher than the initial number of 3D points. For one example surface, the number of stereo correspondences decreases by a factor of about two, but the 3D point cloud obtained with the specular stereo method is less noisy, contains a negligible number of outliers, and shows significantly more surface detail than the initial 3D point cloud. For poorly known reflectance parameters we observe a graceful degradation of the accuracy of 3D reconstruction.     

Determining surface orientations of specular surfaces by using the photometric stereo method

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.     

A photometric sampling method for facial shape recovery

The authors propose a photometric method to recover facial shape that is consistent with expected facial proportions. The method borrows ideas from photometric sampling, a technique that estimates shape from continuous variations of a light source around a single circular path. This approach aims at enriching photometric information by including variations of the light source along its zenith angle. To this end, a luminance matrix describing lighting response along both azimuth and zenith angles of the light source is built for each pixel. A method based on fitting sine functions onto the singular vectors of the collected luminance matrices is proposed for estimating a surface normal map. The estimated surface normals are later refined to maximize a facial proportion criterion and finally be integrated. Experiments demonstrate that our approach successfully approximates 3D face shape while preserving facial proportions within the limits of expected depth.     

关于融合多针图确定物体三维表面绝对深度的研究

利用PS(Photometric Stereo)系统很易确定物体表面方向及相对深度,但不能确定绝对深度.为确定绝对深度,本文提出的算法首先利用BPS(Binocular Photometric Stereo)系统获得一对表面方向图,然后,基于geodesic dome分割这对方向图并计算图中对应区域间的视差.最后,通过施加多种约束,经适当融合及精确的视差匹配,确定景物物体3D表面绝对深度.这一方法对进一步研究怎样确定任意3D表面深度并复原景物结构有着十分重要的意义.     

Shape Reconstruction of 3D Bilaterally Symmetric Surfaces

The paper presents a new approach for shape recovery based on integrating geometric and photometric information. We consider 3D bilaterally symmetric objects, that is, objects which are symmetric with respect to a plane (e.g., faces), and their reconstruction from a single image. Both the viewpoint and the illumination are not necessarily frontal. Furthermore, no correspondence between symmetric points is required.The basic idea is that an image taken from a general, non frontal viewpoint, under non-frontal illumination can be regarded as a pair of images. Each image of the pair is one half of the object, taken from different viewing positions and with different lighting directions. Thus, one-image-variants of geometric stereo and of photometric stereo can be used. Unlike the separate invocation of these approaches, which require point correspondence between the two images, we show that integrating the photometric and geometric information suffice to yield a dense correspondence between pairs of symmetric points, and as a result, a dense shape recovery of the object. Furthermore, the unknown lighting and viewing parameters, are also recovered in this process.Unknown distant point light source, Lambertian surfaces, unknown constant albedo, and weak perspective projection are assumed. The method has been implemented and tested experimentally on simulated and real data.     

Helmholtz Stereopsis: Exploiting Reciprocity for Surface Reconstruction

We present a method—termed Helmholtz stereopsis—for reconstructing the geometry of objects from a collection of images. Unlike existing methods for surface reconstruction (e.g., stereo vision, structure from motion, photometric stereopsis), Helmholtz stereopsis makes no assumptions about the nature of the bidirectional reflectance distribution functions (BRDFs) of objects. This new method of multinocular stereopsis exploits Helmholtz reciprocity by choosing pairs of light source and camera positions that guarantee that the ratio of the emitted radiance to the incident irradiance is the same for corresponding points in the two images. The method provides direct estimates of both depth and surface normals, and consequently weds the advantages of both conventional stereopsis and photometric stereopsis. Results from our implementation lend empirical support to our technique.     

Photometric stereo using LCD displays

This paper considers the problem of shape-from-shading using nearby extended light sources. The paper reviews a number of methods that employ nearby illuminants, and describes a new technique that assumes a rectangular planar nearby distributed uniform isotropic illuminant. It is shown that such a light source illuminating a small Lambertian surface patch is equivalent to a single isotropic point light source at infinity, in the absence of shadowing. A closed-form solution is given for the equivalent point light source direction in terms of the illuminant corner locations. Equivalent point light sources can be obtained for distinct illuminant patterns allowing standard photometric stereo algorithms to be used. An extension is given to the case of a rectangular planar illuminant with arbitrary radiance distribution. Experimental results are shown demonstrating the application of the theory to photometric stereo using illumination from a LCD computer monitor. Details on the photometric calibration of the illumination source and image acquisition device are provided.     

Higher-order nonlinear priors for surface reconstruction

For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.     

Robust 3D face capture using example-based photometric stereo

We show that using example-based photometric stereo, it is possible to achieve realistic reconstructions of the human face. The method can handle non-Lambertian reflectance and attached shadows after a simple calibration step. We use spherical harmonics to model and de-noise the illumination functions from images of a reference object with known shape, and a fast grid technique to invert those functions and recover the surface normal for each point of the target object. The depth coordinate is obtained by weighted multi-scale integration of these normals, using an integration weight mask obtained automatically from the images themselves. We have applied these techniques to improve the PhotoFace system of Hansen et al. (2010).     

Uncalibrated flatfielding and illumination vector estimationfor photometric stereo face reconstruction

Within the context of photometric stereo reconstruction, flatfielding may be used to compensate for the effect of the inverse-square law of light propagation on the pixel brightness. This would require capturing a set of reference images at an off-line imaging session, which employs a calibrating device that should be captured under the exact conditions as the main session. Similarly, the illumination vectors, on which photometric stereo relies, are typically precomputed based on another dedicated calibration session. In practice, implementing such off-line sessions is inconvenient and often infeasible. This work aims at enabling accurate photometric stereo reconstruction for the case of non-interactive on-line capturing of human faces. We propose unsupervised methodologies, which extract all information that is required for accurate face reconstruction from the images of interest themselves. Specifically, we propose an uncalibrated flatfielding and an uncalibrated illumination vector estimation methodology, and we assess their effect on photometric stereo face reconstruction. Results demonstrate that incorporating our methodologies into the photometric stereo framework halves the reconstruction error, while eliminating the need of off-line calibration.     

A New Gradient Fidelity Term for Avoiding Staircasing Effect

Image denoising with some second order nonlinear PDEs often leads to a staircasing effect, which may produce undesirable blocky image. In this paper, we present a new gradient fidelity term and couple it with these PDEs to solve the problem. At first, we smooth the normal vector fields (i.e., the gradient fields) of the noisy image by total variation (TV) minimization and make the gradient of desirable image close to the smoothed normals, which is the idea of our gradient fidelity term. Then, we introduce the Euler-Lagrange equation of the gradient fidelity term into nonlinear diffusion PDEs for noise and staircasing removal. To speed up the computation of the vectorial TV minimization, the dual approach proposed by Bresson and Chan is employed. Some numerical experiments demonstrate that our gradient fidelity term can help to avoid the staircasing effect effectively, while preserving sharp discontinuities in images.     

Dense photometric stereo: a Markov random field approach

We address the problem of robust normal reconstruction by dense photometric stereo, in the presence of complex geometry, shadows, highlight, transparencies, variable attenuation in light intensities, and inaccurate estimation in light directions. The input is a dense set of noisy photometric images, conveniently captured by using a very simple set-up consisting of a digital video camera, a reflective mirror sphere, and a handheld spotlight. We formulate the dense photometric stereo problem as a Markov network and investigate two important inference algorithms for Markov Random Fields (MRFs)--graph cuts and belief propagation--to optimize for the most likely setting for each node in the network. In the graph cut algorithm, the MRF formulation is translated into one of energy minimization. A discontinuity-preserving metric is introduced as the compatibility function, which allows alpha-expansion to efficiently perform the maximum a posteriori (MAP) estimation. Using the identical dense input and the same MRF formulation, our tensor belief propagation algorithm recovers faithful normal directions, preserves underlying discontinuities, improves the normal estimation from one of discrete to continuous, and drastically reduces the storage requirement and running time. Both algorithms produce comparable and very faithful normals for complex scenes. Although the discontinuity-preserving metric in graph cuts permits efficient inference of optimal discrete labels with a theoretical guarantee, our estimation algorithm using tensor belief propagation converges to comparable results, but runs faster because very compact messages are passed and combined. We present very encouraging results on normal reconstruction. A simple algorithm is proposed to reconstruct a surface from a normal map recovered by our method. With the reconstructed surface, an inverse process, known as relighting in computer graphics, is proposed to synthesize novel images of the given scene under user-specified light source and direction. The synthesis is made to run in real time by exploiting the state-of-the-art graphics processing unit (GPU). Our method offers many unique advantages over previous relighting methods and can handle a wide range of novel light sources and directions.     

An empirical study on the effects of translucency on photometric stereo

We present an empirical study on the effects of translucency on photometric stereo. Our study shows that the impact on the accuracy of the photometric normals is related to the relative size of the geometrical features and the mean free path. We show that under simplified conditions, the obtained photometric normals are a blurred version of the true surface normals, where the blur kernel is directly related to the subsurface scattering profile. We furthermore investigate the impact of scattering albedo, index of refraction, and single scattering on the accuracy. We perform our analysis using simulations, and demonstrate the validity on a real world example.     

Overcoming shadows in 3-source photometric stereo

Light occlusions are one of the most significant difficulties of photometric stereo methods. When three or more images are available without occlusion, the local surface orientation is overdetermined so that shape can be computed and the shadowed pixels can be discarded. In this paper, we look at the challenging case when only two images are available without occlusion, leading to a one degree of freedom ambiguity per pixel in the local orientation. We show that, in the presence of noise, integrability alone cannot resolve this ambiguity and reconstruct the geometry in the shadowed regions. As the problem is ill-posed in the presence of noise, we describe two regularization schemes that improve the numerical performance of the algorithm while preserving the data. Finally, the paper describes how this theory applies in the framework of color photometric stereo where one is restricted to only three images and light occlusions are common. Experiments on synthetic and real image sequences are presented.     

Photometric Stereo with General,Unknown Lighting

Work on photometric stereo has shown how to recover the shape and reflectance properties of an object using multiple images taken with a fixed viewpoint and variable lighting conditions. This work has primarily relied on known lighting conditions or the presence of a single point source of light in each image. In this paper we show how to perform photometric stereo assuming that all lights in a scene are distant from the object but otherwise unconstrained. Lighting in each image may be an unknown and may include arbitrary combination of diffuse, point and extended sources. Our work is based on recent results showing that for Lambertian objects, general lighting conditions can be represented using low order spherical harmonics. Using this representation we can recover shape by performing a simple optimization in a low-dimensional space. We also analyze the shape ambiguities that arise in such a representation. We demonstrate our method by reconstructing the shape of objects from images obtained under a variety of lightings. We further compare the reconstructed shapes against shapes obtained with a laser scanner.