In this paper we propose and study an optimization problem over a matrix group orbit that we call (GOO). We prove that GOO can be used to induce matrix decomposition techniques such as singular value decomposition (SVD), LU decomposition, QR decomposition, Schur decomposition and Cholesky decomposition, etc. This gives rise to a unified framework for matrix decomposition and allows us to bridge these matrix decomposition methods. Moreover, we generalize GOO for tensor decomposition. As a concrete application of GOO, we devise a new data decomposition method over a special linear group to normalize point cloud data. Experiment results show that our normalization method is able to obtain recovery well from distortions like shearing, rotation and squeezing.
In @cite_11 , a non-linear GOO is used to find texture invariant to rotation for 2D point cloud @math : [ | ( ( )) |_* . ] As @math is a unit group, the optimization is well defined and the induced matrix decomposition is found to be useful as a rotation-invariant representation for texture. The same paper also considers finding homography-invariant representation for texture for 2D point cloud @math : [ , ; ( ( ))= | [ ( )] |_* . ] Note that here a coefficient @math is intentionally added to ensure @math measure of the point cloud be preserved the action of @math .
@cite_11
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In this paper, we propose a new tool to efficiently extract a class of "low-rank textures" in a 3D scene from user-specified windows in 2D images despite significant corruptions and warping. The low-rank textures capture geometrically meaningful structures in an image, which encompass conventional local features such as edges and corners as well as many kinds of regular, symmetric patterns ubiquitous in urban environments and man-made objects. Our approach to finding these low-rank textures leverages the recent breakthroughs in convex optimization that enable robust recovery of a high-dimensional low-rank matrix despite gross sparse errors. In the case of planar regions with significant affine or projective deformation, our method can accurately recover both the intrinsic low-rank texture and the unknown transformation, and hence both the geometry and appearance of the associated planar region in 3D. Extensive experimental results demonstrate that this new technique works effectively for many regular and near-regular patterns or objects that are approximately low-rank, such as symmetrical patterns, building facades, printed text, and human faces.