Median K-flats for hybrid linear modeling with many outliers

We describe the Median K-Flats (MKF) algorithm, a simple online method for hybrid linear modeling, i.e., for approximating data by a mixture of flats. This algorithm simultaneously partitions the data into clusters while finding their corresponding best approximating l1 d-flats, so that the cumulative l1 error is minimized. The current implementation restricts d-flats to be d-dimensional linear subspaces. It requires a negligible amount of storage, and its complexity, when modeling data consisting of N points in D-dimensional Euclidean space with K d-dimensional linear subspaces, is of order O(n K d D+n d^2 D), where n is the number of iterations required for convergence (empirically on the order of 10^4). Since it is an online algorithm, data can be supplied to it incrementally and it can incrementally produce the corresponding output. The performance of the algorithm is carefully evaluated using synthetic and real data

Randomized hybrid linear modeling by local best-fit flats

The hybrid linear modeling problem is to identify a set of d-dimensional affine sets in a D-dimensional Euclidean space. It arises, for example, in object tracking and structure from motion. The hybrid linear model can be considered as the second simplest (behind linear) manifold model of data. In this paper we will present a very simple geometric method for hybrid linear modeling based on selecting a set of local best fit flats that minimize a global l1 error measure. The size of the local neighborhoods is determined automatically by the Jones' l2 beta numbers; it is proven under certain geometric conditions that good local neighborhoods exist and are found by our method. We also demonstrate how to use this algorithm for fast determination of the number of affine subspaces. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the algorithm on synthetic and real hybrid linear data.Comment: To appear in the proceedings of CVPR 201

Kernel Spectral Curvature Clustering (KSCC)

Multi-manifold modeling is increasingly used in segmentation and data representation tasks in computer vision and related fields. While the general problem, modeling data by mixtures of manifolds, is very challenging, several approaches exist for modeling data by mixtures of affine subspaces (which is often referred to as hybrid linear modeling). We translate some important instances of multi-manifold modeling to hybrid linear modeling in embedded spaces, without explicitly performing the embedding but applying the kernel trick. The resulting algorithm, Kernel Spectral Curvature Clustering, uses kernels at two levels - both as an implicit embedding method to linearize nonflat manifolds and as a principled method to convert a multiway affinity problem into a spectral clustering one. We demonstrate the effectiveness of the method by comparing it with other state-of-the-art methods on both synthetic data and a real-world problem of segmenting multiple motions from two perspective camera views.Comment: accepted to 2009 ICCV Workshop on Dynamical Visio

Kernel Truncated Regression Representation for Robust Subspace Clustering

Subspace clustering aims to group data points into multiple clusters of which each corresponds to one subspace. Most existing subspace clustering approaches assume that input data lie on linear subspaces. In practice, however, this assumption usually does not hold. To achieve nonlinear subspace clustering, we propose a novel method, called kernel truncated regression representation. Our method consists of the following four steps: 1) projecting the input data into a hidden space, where each data point can be linearly represented by other data points; 2) calculating the linear representation coefficients of the data representations in the hidden space; 3) truncating the trivial coefficients to achieve robustness and block-diagonality; and 4) executing the graph cutting operation on the coefficient matrix by solving a graph Laplacian problem. Our method has the advantages of a closed-form solution and the capacity of clustering data points that lie on nonlinear subspaces. The first advantage makes our method efficient in handling large-scale datasets, and the second one enables the proposed method to conquer the nonlinear subspace clustering challenge. Extensive experiments on six benchmarks demonstrate the effectiveness and the efficiency of the proposed method in comparison with current state-of-the-art approaches.Comment: 14 page

A Sparsity-Inducing Optimization-Based Algorithm for Planar Patches Extraction from Noisy Point-Cloud Data

Currently, much of the manual labor needed to generate as-built Building Information Models (BIMs) of existing facilities is spent converting raw Point Cloud Datasets (PCDs) to BIMs descriptions. Automating the PCD conversion process can drastically reduce the cost of generating as-built BIMs. Due to the widespread existence of planar structures in civil infrastructures, detecting and extracting planar patches from raw PCDs is a fundamental step in the conversion pipeline from PCDs to BIMs. However, existing methods cannot effectively address both automatically detecting and extracting planar patches from infrastructure PCDs. The existing methods cannot resolve the problem due to the large scale and model complexity of civil infrastructure, or due to the requirements of extra constraints or known information. To address the problem, this paper presents a novel framework for automatically detecting and extracting planar patches from large-scale and noisy raw PCDs. The proposed method automatically detects planar structures, estimates the parametric plane models, and determines the boundaries of the planar patches. The first step recovers existing linear dependence relationships amongst points in the PCD by solving a group-sparsity inducing optimization problem. Next, a spectral clustering procedure based on the recovered linear dependence relationships segments the PCD. Then, for each segmented group, model parameters of the extracted planes are estimated via Singular Value Decomposition (SVD) and Maximum Likelihood Estimation Sample Consensus (MLESAC). Finally, the α-shape algorithm detects the boundaries of planar structures based on a projection of the data to the planar model. The proposed approach is evaluated comprehensively by experiments on two types of PCDs from real-world infrastructures, one captured directly by laser scanners and the other reconstructed from video using structure-from-motion techniques. In order to evaluate the performance comprehensively, five evaluation metrics are proposed which measure different aspects of performance. Experimental results reveal that the proposed method outperforms the existing methods, in the sense that the method automatically and accurately extracts planar patches from large-scaled raw PCDs without any extra constraints nor user assistance.This is the accepted manuscript. The final version is available from Wiley at http://onlinelibrary.wiley.com/doi/10.1111/mice.12063/abstract

Community detection method based on mixed-norm sparse subspace clustering

Community or group is an important structure in disciplines such as social networks, biology gene expression, and physics systems. Community detections for different types of networks have attracted considerable interest. However, it is still challenging to find meaningful community structures in various networks. In particular, accurate community description and implementation of effective detection algorithms with huge datasets are still not solved. In this paper, we present a novel community detection algorithm based on the theory of sparse subspace clustering (SSC) with mixed-norm constraints. Inspired by the sparse representation of subspace, each community in a given network can span a subspace in some similarity measure space. If the basis of subspaces can be solved, all of the nodes can be represented as a linear combination of the nodes that span the same subspace. By introducing a novel mixed-norm constraint in SCC, the connections of nodes among different communities are modeled as noise to improve the clustering accuracy. The formulation of the basis of subspaces is derived from the self-representation property of data by using SSC. Then, the alternating directions method of multipliers (ADMM) framework is used to solve the formulation. Finally, communities are detected by subspace clustering method. The proposed method is compared with state-of-the-art algorithms on synthetic networks and real-world networks. The experimental results show the effectiveness of the proposed algorithm in accurately describing the community. The results also show that the mixed-norm SSC is a practical approach for detecting communities in huge datasets