LogDet Rank Minimization with Application to Subspace Clustering

Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. In this paper, we propose to use a log-determinant (LogDet) function as a smooth and closer, though non-convex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based non-convex objective function on potentially large-scale data. By making use of the angular information of principal directions of the resultant low-rank representation, an affinity graph matrix is constructed for spectral clustering. Experimental results on motion segmentation and face clustering data demonstrate that the proposed method often outperforms state-of-the-art subspace clustering algorithms.Comment: 10 pages, 4 figure

Jumping Manifolds: Geometry Aware Dense Non-Rigid Structure from Motion

Given dense image feature correspondences of a non-rigidly moving object across multiple frames, this paper proposes an algorithm to estimate its 3D shape for each frame. To solve this problem accurately, the recent state-of-the-art algorithm reduces this task to set of local linear subspace reconstruction and clustering problem using Grassmann manifold representation \cite{kumar2018scalable}. Unfortunately, their method missed on some of the critical issues associated with the modeling of surface deformations, for e.g., the dependence of a local surface deformation on its neighbors. Furthermore, their representation to group high dimensional data points inevitably introduce the drawbacks of categorizing samples on the high-dimensional Grassmann manifold \cite{huang2015projection, harandi2014manifold}. Hence, to deal with such limitations with \cite{kumar2018scalable}, we propose an algorithm that jointly exploits the benefit of high-dimensional Grassmann manifold to perform reconstruction, and its equivalent lower-dimensional representation to infer suitable clusters. To accomplish this, we project each Grassmannians onto a lower-dimensional Grassmann manifold which preserves and respects the deformation of the structure w.r.t its neighbors. These Grassmann points in the lower-dimension then act as a representative for the selection of high-dimensional Grassmann samples to perform each local reconstruction. In practice, our algorithm provides a geometrically efficient way to solve dense NRSfM by switching between manifolds based on its benefit and usage. Experimental results show that the proposed algorithm is very effective in handling noise with reconstruction accuracy as good as or better than the competing methods.Comment: New version with corrected typo. 10 Pages, 7 Figures, 1 Table. Accepted for publication in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2019. Acknowledgement added. Supplementary material is available at https://suryanshkumar.github.io

Non-Rigid Structure from Motion

This thesis revisits a challenging classical problem in geometric computer vision known as "Non-Rigid Structure-from-Motion" (NRSfM). It is a well-known problem where the task is to recover the 3D shape and motion of a non-rigidly moving object from image data. A reliable solution to this problem is valuable in several industrial applications such as virtual reality, medical surgery, animation movies etc. Nevertheless, to date, there does not exist any algorithm that can solve NRSfM for all kinds of conceivable motion. As a result, additional constraints and assumptions are often employed to solve NRSfM. The task is challenging due to the inherent unconstrained nature of the problem itself as many 3D varying configurations can have similar image projections. The problem becomes even more challenging if the camera is moving along with the object. The thesis takes on a modern view to this challenging problem and proposes a few algorithms that have set a new performance benchmark to solve NRSfM. The thesis not only discusses the classical work in NRSfM but also proposes some powerful elementary modification to it. The foundation of this thesis surpass the traditional single object NRSFM and for the first time provides an effective formulation to realise multi-body NRSfM. Most techniques for NRSfM under factorisation can only handle sparse feature correspondences. These sparse features are then used to construct a scene using the organisation of points, lines, planes or other elementary geometric primitive. Nevertheless, sparse representation of the scene provides an incomplete information about the scene. This thesis goes from sparse NRSfM to dense NRSfM for a single object, and then slowly lifts the intuition to realise dense 3D reconstruction of the entire dynamic scene as a global as rigid as possible deformation problem. The core of this work goes beyond the traditional approach to deal with deformation. It shows that relative scales for multiple deforming objects can be recovered under some mild assumption about the scene. The work proposes a new approach for dense detailed 3D reconstruction of a complex dynamic scene from two perspective frames. Since the method does not need any depth information nor it assumes a template prior, or per-object segmentation, or knowledge about the rigidity of the dynamic scene, it is applicable to a wide range of scenarios including YouTube Videos. Lastly, this thesis provides a new way to perceive the depth of a dynamic scene which essentially trivialises the notion of motion estimation as a compulsory step to solve this problem. Conventional geometric methods to address depth estimation requires a reliable estimate of motion parameters for each moving object, which is difficult to obtain and validate. In contrast, this thesis introduces a new motion-free approach to estimate the dense depth map of a complex dynamic scene for successive/multiple frames. The work show that given per-pixel optical flow correspondences between two consecutive frames and the sparse depth prior for the reference frame, we can recover the dense depth map for the successive frames without solving for motion parameters. By assigning the locally rigid structure to the piece-wise planar approximation of a dynamic scene which transforms as rigid as possible over frames, we can bypass the motion estimation step. Experiments results and MATLAB codes on relevant examples are provided to validate the motion-free idea