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
2163352848 2035997599 1866173756 2900731076
Presents a theoretically very simple, yet efficient, multiresolution approach to gray-scale and rotation invariant texture classification based on local binary patterns and nonparametric discrimination of sample and prototype distributions. The method is based on recognizing that certain local binary patterns, termed "uniform," are fundamental properties of local image texture and their occurrence histogram is proven to be a very powerful texture feature. We derive a generalized gray-scale and rotation invariant operator presentation that allows for detecting the "uniform" patterns for any quantization of the angular space and for any spatial resolution and presents a method for combining multiple operators for multiresolution analysis. The proposed approach is very robust in terms of gray-scale variations since the operator is, by definition, invariant against any monotonic transformation of the gray scale. Another advantage is computational simplicity as the operator can be realized with a few operations in a small neighborhood and a lookup table. Experimental results demonstrate that good discrimination can be achieved with the occurrence statistics of simple rotation invariant local binary patterns. In order to solve multi-view point clouds registration in large non-feature marked scenes, a new registration and stitching method was proposed based on 2D SIFT(Scale-invariant feature transform) features. Firstly, we used texture mapping method to generate 2D effective texture image, and then extracted and match SIFT features, obtained accurate key points and registration relationship between the effective texture images. Secondly, we reflected the SIFT key points and registration relationship to the 3D point clouds data, obtained key points and registration relationship of multi-view point clouds, we can achieve multi-view point clouds stitching. Our algorithm used texture mapping method to generate 2D effective texture image, it can eliminate interference of the holes and ineffective points, and can eliminate unnecessary mistake matching. Our algorithm used correct extracted matching point pairs to stitch, avoiding stepwise iterated of ICP algorithm, so our algorithm is simple to calculate, and it's matching precision and matching efficiency are improved to some extent. We carried experiments on two-view point clouds in two large indoor; the experiment results verified the validity of our algorithm. This paper presents generalizations to the gray scale and rotation invariant texture classification method based on local binary patterns that we have recently introduced. We derive a generalized presentation that allows for realizing a gray scale and rotation invariant LBP operator for any quantization of the angular space and for any spatial resolution, and present a method for combining multiple operators for multiresolution analysis. The proposed approach is very robust in terms of gray scale variations, since the operator is by definition invariant against any monotonic transformation of the gray scale. Another advantage is computational simplicity, as the operator can be realized with a few operations in a small neighborhood and a lookup table. Excellent experimental results obtained in a true problem of rotation invariance, where the classifier is trained at one particular rotation angle and tested with samples from other rotation angles, demonstrate that good discrimination can be achieved with the occurrence statistics of simple rotation invariant local binary patterns. These operators characterize the spatial configuration of local image texture and the performance can be further improved by combining them with rotation invariant variance measures that characterize the contrast of local image texture. The joint distributions of these orthogonal measures are shown to be very powerful tools for rotation invariant texture analysis. Unlike images which are represented in regular dense grids, 3D point clouds are irregular and unordered, hence applying convolution on them can be difficult. In this paper, we extend the dynamic filter to a new convolution operation, named PointConv. PointConv can be applied on point clouds to build deep convolutional networks. We treat convolution kernels as nonlinear functions of the local coordinates of 3D points comprised of weight and density functions. With respect to a given point, the weight functions are learned with multi-layer perceptron networks and density functions through kernel density estimation. The most important contribution of this work is a novel reformulation proposed for efficiently computing the weight functions, which allowed us to dramatically scale up the network and significantly improve its performance. The learned convolution kernel can be used to compute translation-invariant and permutation-invariant convolution on any point set in the 3D space. Besides, PointConv can also be used as deconvolution operators to propagate features from a subsampled point cloud back to its original resolution. Experiments on ModelNet40, ShapeNet, and ScanNet show that deep convolutional neural networks built on PointConv are able to achieve state-of-the-art on challenging semantic segmentation benchmarks on 3D point clouds. Besides, our experiments converting CIFAR-10 into a point cloud showed that networks built on PointConv can match the performance of convolutional networks in 2D images of a similar structure.