Efficient and Robust Camera-independent Multiview 3D Geometric Reconstruction from Noisy Monocular Depth Estimation and Multiple Point Matching

We present an efficient and robust method for 3D geometric reconstruction that is based solely on the camera-independent linear relationships among a given set of points, which are stable over time and robustly estimated using multiple point matches. We essentially learn, from correspondences between points across several frames, a linear geometric auto-regression matrix W, which establishes how a point in 3D can be expressed as a linear combination of all the others. This matrix is constant and does not depend on the world coordinate system or the camera pose---it is an intrinsic property of the point set. We also show that the principal eigenvectors of W, which all have eigenvalue 1, provide a homogeneous representation of the 3D point configuration. The first version of our method takes advantage of noisy monocular depth maps in order to obtain, from multiple frames, a robust geometric auto-regression matrix W of linear relationships between the 3D points. Thus, we build on recent advances in deep learning, which now provide monocular depth estimation models that are fast but very often noisy. Our approach handles noise through robust linear estimation over several frames. The second version of our method does not need monocular depth estimation maps. It applies in cases of weak-perspective projection, when the linear combinations between the 3D points can be robustly estimated from their 2D projections in the image. Note that the camera projection matrix is never used in our derivations. Consequently, our method does not recover camera pose, but only 3D structure. This is a key difference between our method and the related literature on 3D geometric reconstruction.