Direct surface reconstruction from a moving sensor

A technique is described for surface reconstruction from a sequence of images when the sensor motion is known. The reconstruction takes the form of an orthographic depth map in a viewer-independent frame. Following a mathematical justification, the technique is illustrated with examples using noisy synthetic images.     

Combining Shape from Shading and Stereo: A Joint Variational Method for Estimating Depth,Illumination and Albedo

Shape from shading (SfS) and stereo are two fundamentally different strategies for image-based 3-D reconstruction. While approaches for SfS infer the depth solely from pixel intensities, methods for stereo are based on a matching process that establishes correspondences across images. This difference in approaching the reconstruction problem yields complementary advantages that are worthwhile being combined. So far, however, most “joint” approaches are based on an initial stereo mesh that is subsequently refined using shading information. In this paper we follow a completely different approach. We propose a joint variational method that combines both cues within a single minimisation framework. To this end, we fuse a Lambertian SfS approach with a robust stereo model and supplement the resulting energy functional with a detail-preserving anisotropic second-order smoothness term. Moreover, we extend the resulting model in such a way that it jointly estimates depth, albedo and illumination. This in turn makes the approach applicable to objects with non-uniform albedo as well as to scenes with unknown illumination. Experiments for synthetic and real-world images demonstrate the benefits of our combined approach: They not only show that our method is capable of generating very detailed reconstructions, but also that joint approaches are feasible in practice.     

A Generic and Provably Convergent Shape-from-Shading Method for Orthographic and Pinhole Cameras

We describe a mathematical and algorithmic study of the Lambertian “Shape-From-Shading” problem for orthographic and pinhole cameras. Our approach is based upon the notion of viscosity solutions of Hamilton-Jacobi equations. This approach provides a mathematical framework in which we can show that the problem is well-posed (we prove the existence of a solution and we characterize all the solutions). Our contribution is threefold. First, we model the camera both as orthographic and as perspective (pinhole), whereas most authors assume an orthographic projection (see Horn and Brooks (1989) for a survey of the SFS problem up to 1989 and Zhang et al. (1999), Kozera (1998), Durou et al. (2004) for more recent ones); thus we extend the applicability of shape from shading methods to more realistic acquisition models. In particular it extends the work of Prados et al. (2002a) and Rouy and Tourin (1992). We provide some novel mathematical formulations of this problem yielding new partial differential equations. Results about the existence and uniqueness of their solutions are also obtained. Second, by introducing a “generic” Hamiltonian, we define a general framework allowing to deal with both models (orthographic and perspective), thereby simplifying the formalization of the problem. Thanks to this unification, each algorithm we propose can compute numerical solutions corresponding to all the modeling. Third, our work allows us to come up with two new generic algorithms for computing numerical approximations of the “continuous solution of the “Shape-From-Shading” problem as well as a proof of their convergence toward that solution. Moreover, our two generic algorithms are able to deal with discontinuous images as well as images containing black shadows. First online version published in October, 2005     

Analysis and Approximation of Some Shape-from-Shading Models for Non-Lambertian Surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal-type equation when the light source is on the vertical axis. In this paper, we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary non-linear partial differential equations of Hamilton–Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren–Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren–Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.     

What can two images tell us about a third one?

This paper discusses the problem of predicting image features in an image from image features in two other images and the epipolar geometry between the three images. We adopt the most general camera model of perspective projection and show that a point can be predicted in the third image as a bilinear function of its images in the first two cameras, that the tangents to three corresponding curves are related by a trilinear function, and that the curvature of a curve in the third image is a linear function of the curvatures at the corresponding points in the other two images. Our analysis relies heavily on the use of the fundamental matrix which has been recently introduced (Faugeras et al, 1992) and on the properties of a special plane which we call the trifocal plane. Though the trinocular geometry of points and lines has been very recently addressed, our use of the differential properties of curves for prediction is unique.We thus completely solve the following problem: given two views of an object, predict what a third view would look like. The problem and its solution bear upon several areas of computer vision, stereo, motion analysis, and model-based object recognition. Our answer is quite general since it assumes the general perspective projection model for image formation and requires only the knowledge of the epipolar geometry for the triple of views. We show that in the special case of orthographic projection our results for points reduce to those of Ullman and Basri (Ullman and Basri, 1991). We demonstrate on synthetic as well as on real data the applicability of our theory.     

基于图像分割的三维重构

三维重构方法是医学图像可视化系统、治疗计划系统的重要技术。基于图像分割的三维重构方法结合了图像分割、等值面抽取、网格简化三种技术,是不同于传统Marching Cubes算法的一种三维重构方法。它首先将医学图像分割为二值图,然后利用Marching Cubes方法进行等值面抽取,最后对得到的网格模型进行简化。实验结果表明,基于图像分割的三维重构方法加快了Marching Cubes的运算速度,改善了重构的效果,有利于实现对基于三维重构的大型几何模型的实时绘制和交互。     

一种面向三维点集的快速表面重构算法

在对目前比较流行的空间三角化算法进行对比研究的基础上 ,对 Hugues Hoppe提出的算法进行了改进 ,即借鉴 Marching Cubes算法的基本思想 ,首先通过自动选取适当的参数 ,用包围盒方法将三维散乱点划分为数据区域 ;然后求取点的切平面及法向 ,同时采用广度优先算法遍历数据点来调整法向和快速地求取 Marching Cubes的等势函数 ;最后用基于查表法的 Marching Cubes来输出三角面片 ,即得到表面模型 .实验结果表明 ,改进后的算法效率有较大的提高 .新算法不仅适用于表面三维散乱点数据 ,也可以对体数据进行重构 ,具有一定的通用性 .     

Self-calibration of an affine camera from multiple views

A key limitation of all existing algorithms for shape and motion from image sequences under orthographic, weak perspective and para-perspective projection is that they require the calibration parameters of the camera. We present in this paper a new approach that allows the shape and motion to be computed from image sequences without having to know the calibration parameters. This approach is derived with the affine camera model, introduced by Mundy and Zisserman (1992), which is a more general class of projections including orthographic, weak perspective and para-perspective projection models. The concept of self-calibration, introduced by Maybank and Faugeras (1992) for the perspective camera and by Hartley (1994) for the rotating camera, is then applied for the affine camera.This paper introduces the 3 intrinsic parameters that the affine camera can have at most. The intrinsic parameters of the affine camera are closely related to the usual intrinsic parameters of the pin-hole perspective camera, but are different in the general case. Based on the invariance of the intrinsic parameters, methods of self-calibration of the affine camera are proposed. It is shown that with at least four views, an affine camera may be self-calibrated up to a scaling factor, leading to Euclidean (similarity) shape réconstruction up to a global scaling factor. Another consequence of the introduction of intrinsic and extrinsic parameters of the affine camera is that all existing algorithms using calibrated affine cameras can be assembled into the same framework and some of them can be easily extented to a batch solution.Experimental results are presented and compared with other methods using calibrated affine cameras.     

基于透视和正投影图像的3D运动与结构恢复

论文首次研究了由一幅正投影图像和一幅透视投影图像的特征点对应来进行刚体3D运动重建与结构恢复的问题,给出了有效的线性算法。以往的由运动恢复结构的工作主要集中于一组透视图像或一组正投影(通常为仿射)图像,文中采用了透视模型和正投影模型的组合。数据模拟实验结果显示该方法是比较有效和稳定的。     

A paraperspective factorization method for shape and motionrecovery

The factorization method, first developed by Tomasi and Kanade (1992), recovers both the shape of an object and its motion from a sequence of images, using many images and tracking many feature points to obtain highly redundant feature position information. The method robustly processes the feature trajectory information using singular value decomposition (SVD), taking advantage of the linear algebraic properties of orthographic projection. However, an orthographic formulation limits the range of motions the method can accommodate. Paraperspective projection, first introduced by Ohta et al. (1981), is a projection model that closely approximates perspective projection by modeling several effects not modeled under orthographic projection, while retaining linear algebraic properties. Our paraperspective factorization method can be applied to a much wider range of motion scenarios, including image sequences containing motion toward the camera and aerial image sequences of terrain taken from a low-altitude airplane