Deformable 3-D Modelling from Uncalibrated Video Sequences

Submitted for the degree of Doctor of Philosophy, Queen Mary, University of Londo

Robust and Accurate Structure from Motion of Rigid and Nonrigid Objects

As a central theme in computer vision, the problem of 3D structure and motion recovery from image sequences has been widely studied during the past three decades, and considerable progress has been made in theory, as well as in prac- tice. However, there are still several challenges remaining, including algorithm robustness and accuracy, especially for nonrigid modeling. The thesis focuses on solving these challenges and several new robust and accurate algorithms have been proposed. The first part of the thesis reviews the state-of-the-art techniques of structure and motion factorization. First, an introduction of structure from motion and some mathematical background of the technique is presented. Then, the general idea and different formulations of structure from motion for rigid and nonrigid objects are discussed. The second part covers the proposed quasi-perspective projection model and its application to structure and motion factorization. Previous algorithms are based on either a simplified affine assumption or a complicated full perspective projection model. The affine model is widely adopted due to its simplicity, whereas the extension to full perspective suffers from recovering projective depths. A quasi-perspective model is proposed to fill the gap between the two models. It is more accurate than the affine model from both theoretical analysis and experimental studies. More geometric properties of the model are investigated in the context of one- and two-view geometry. Finally, the model was applied to structure from motion and a framework of rigid and nonrigid factorization under quasi-perspective assumption is established. The last part of the thesis is focused on the robustness and three new al- gorithms are proposed. First, a spatial-and-temporal-weighted factorization algorithm is proposed to handle significant image noise, where the uncertainty of image measurement is estimated from a new perspective by virtue of repro- jection residuals. Second, a rank-4 affine factorization algorithm is proposed to avoid the difficulty of image alignment with erroneous data, followed by a robust factorization scheme that can work with missing and outlying data. Third, the robust algorithm is extended to nonrigid scenarios and a new augmented nonrigid factorization algorithm is proposed to handle imperfect tracking data. The main contributions of the thesis are as follows: The proposed quasi- perspective projection model fills the gap between the simplicity of the affine model and the accuracy of the perspective model. Its application to structure and motion factorization greatly increases the efficiency and accuracy of the algorithm. The proposed robust algorithms do not require prior information of image measurement and greatly improve the overall accuracy and robustness of previous approaches. Moreover, the algorithms can also be applied directly to structure from motion of nonrigid objects

3D Non-Rigid Reconstruction with Prior Shape Constraints

3D non-rigid shape recovery from a single uncalibrated camera is a challenging, under-constrained problem in computer vision. Although tremendous progress has been achieved towards solving the problem, two main limitations still exist in most previous solutions. First, current methods focus on non-incremental solutions, that is, the algorithms require collection of all the measurement data before the reconstruction takes place. This methodology is inherently unsuitable for applications requiring real-time solutions. At the same time, most of the existing approaches assume that 3D shapes can be accurately modelled in a linear subspace. These methods are simple and have been proven effective for reconstructions of objects with relatively small deformations, but have considerable limitations when the deformations are large or complex. The non-linear deformations are often observed in highly flexible objects for which the use of the linear model is impractical. Note that specific types of shape variation might be governed by only a small number of parameters and therefore can be well-represented in a low dimensional manifold. The methods proposed in this thesis aim to estimate the non-rigid shapes and the corresponding camera trajectories, based on both the observations and the prior learned manifold. Firstly, an incremental approach is proposed for estimating the deformable objects. An important advantage of this method is the ability to reconstruct the 3D shape from a newly observed image and update the parameters in 3D shape space. However, this recursive method assumes the deformable shapes only have small variations from a mean shape, thus is still not feasible for objects subject to large scale deformations. To address this problem, a series of approaches are proposed, all based on non-linear manifold learning techniques. Such manifold is used as a shape prior, with the reconstructed shapes constrained to lie within the manifold. Those non-linear manifold based approaches significantly improve the quality of reconstructed results and are well-adapted to different types of shapes undergoing significant and complex deformations. Throughout the thesis, methods are validated quantitatively on 2D points sequences projected from the 3D motion capture data for a ground truth comparison, and are qualitatively demonstrated on real example of 2D video sequences. Comparisons are made for the proposed methods against several state-of-the-art techniques, with results shown for a variety of challenging deformable objects. Extensive experiments also demonstrate the robustness of the proposed algorithms with respect to measurement noise and missing data

Deformable and articulated 3D reconstruction from monocular video sequences

PhDThis thesis addresses the problem of deformable and articulated structure from motion from monocular uncalibrated video sequences. Structure from motion is defined as the problem of recovering information about the 3D structure of scenes imaged by a camera in a video sequence. Our study aims at the challenging problem of non-rigid shapes (e.g. a beating heart or a smiling face). Non-rigid structures appear constantly in our everyday life, think of a bicep curling, a torso twisting or a smiling face. Our research seeks a general method to perform 3D shape recovery purely from data, without having to rely on a pre-computed model or training data. Open problems in the field are the difficulty of the non-linear estimation, the lack of a real-time system, large amounts of missing data in real-world video sequences, measurement noise and strong deformations. Solving these problems would take us far beyond the current state of the art in non-rigid structure from motion. This dissertation presents our contributions in the field of non-rigid structure from motion, detailing a novel algorithm that enforces the exact metric structure of the problem at each step of the minimisation by projecting the motion matrices onto the correct deformable or articulated metric motion manifolds respectively. An important advantage of this new algorithm is its ability to handle missing data which becomes crucial when dealing with real video sequences. We present a generic bilinear estimation framework, which improves convergence and makes use of the manifold constraints. Finally, we demonstrate a sequential, frame-by-frame estimation algorithm, which provides a 3D model and camera parameters for each video frame, while simultaneously building a model of object deformation

Accurate Optimization of Weighted Nuclear Norm for Non-Rigid Structure from Motion

Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when applying more general singular value penalties, such as weighted nuclear norm priors, direct optimization over the elements of the matrix is typically used. Due to non-differentiability of the resulting objective function, first order sub-gradient or splitting methods are predominantly used. While these offer rapid iterations it is well known that they become inefficent near the minimum due to zig-zagging and in practice one is therefore often forced to settle for an approximate solution. In this paper we show that more accurate results can in many cases be achieved with 2nd order methods. Our main result shows how to construct bilinear formulations, for a general class of regularizers including weighted nuclear norm penalties, that are provably equivalent to the original problems. With these formulations the regularizing function becomes twice differentiable and 2nd order methods can be applied. We show experimentally, on a number of structure from motion problems, that our approach outperforms state-of-the-art methods