The Research Of 3D Reconstruction For Non-rigid Object In Trajectory Space

Non-rigid Structure from Motion(NRSFM) is a research on how to recover the motion parameters of camera and 3D structure of non-rigid object from 2D images sequences. The technique can be widely applied on fields like military reconnaissance, medical imaging and athletic and so on. NRSFM has always been a popular research in the field of computer vision and pattern recognition. And because of the complexity and uncertainty of the 3D motion, the study is also a difficult problem in the field. Firstly, NRSFM is solved in shape space. However, shape basis has its own specificity, it can’t generally apply to recover all non-rigid motion. So this method has some limitations and specificity. According to the duality of shape space and trajectory space, recently years, the research has been extended to trajectory space, and non-rigid structure can be represented as a linear combination of some predefined trajectory basis. The method not only overcome the difficulty of the instability and basis chosen appeared in shape basis methods, but also reduce the computational scale of the algorithm. Many researches show that the accuracy of 3D reconstruction can be improved by choosing optimization algorithms legitimately and efficiently. Can use less time or not to search the optimal solution in trajectory space is also a difficult problem of non-rigid 3D reconstruction. In order to solve this problem, this paper focuses on following research work which based on trajectory basis method:(1) A trace minimization problem of gram matrix(the gram matrix of correction matrix) in trajectory space, which is taken as a Semi-Definite Programming(SDP) problem to solve. According to the duality of shape space and trajectory space, the trace minimization problem of correction matrix is also a standard SDP problem. The gram matrix could be solved by SDP algorithm, then the correction matrix can be calculated by Cholesky Factorization. In order to further improve the accuracy of NRSFM, a new constraint is proposed, trace minimize constraint, and it’s combined with orthogonal constraint. Levenberg-Marquardt(LM) algorithm is used to optimize the correction matrix, which is obtained by Cholesky Factorization. And the algorithm satisfy the trace minimize constraint and orthogonal constraint. Once the correction matrix is known, the rotation matrix can be calculated, thus the 3D structure matrix can be calculated by Pseudo Inverse Method. Compared to PTA method, the results show that the precision of NRSFM improved effectively by using the proposed SDP method.(2) The nuclear-norm minimization problem about the structure matrix can be solved by Accelerated Proximal Gradient(APG) in the ideal time. The structure matrix is a low rank matrix, and it meet the rank minimization problem. Generally, however, the rank minimization problem is a NP-hard problem, it is difficult to accurately solve. So relax the above rank minimization to a nuclear-norm minimization form. In order to further improve the precision and convergence speed when solving this problem, APG algorithm is used to solve the nuclear norm minimization problem. And the structure matrix obtained by SDP method is taken as the initial value of APG algorithm. Compared to PTA method and SDP method, the results show that the APG algorithm runs quickly, and improves the precision of 3D reconstruction effectively.