449 research outputs found
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
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
Nearness to Local Subspace Algorithm for Subspace and Motion Segmentation
There is a growing interest in computer science, engineering, and mathematics
for modeling signals in terms of union of subspaces and manifolds. Subspace
segmentation and clustering of high dimensional data drawn from a union of
subspaces are especially important with many practical applications in computer
vision, image and signal processing, communications, and information theory.
This paper presents a clustering algorithm for high dimensional data that comes
from a union of lower dimensional subspaces of equal and known dimensions. Such
cases occur in many data clustering problems, such as motion segmentation and
face recognition. The algorithm is reliable in the presence of noise, and
applied to the Hopkins 155 Dataset, it generates the best results to date for
motion segmentation. The two motion, three motion, and overall segmentation
rates for the video sequences are 99.43%, 98.69%, and 99.24%, respectively
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
Spectral clustering of linear subspaces for motion segmentation
International audienceThis paper studies automatic segmentation of multiple motions from tracked feature points through spectral embedding and clustering of linear subspaces. We show that the dimension of the ambient space is crucial for separability, and that low dimensions chosen in prior work are not optimal. We suggest lower and upper bounds together with a data-driven procedure for choosing the optimal ambient dimension. Application of our approach to the Hopkins155 video benchmark database uniformly outperforms a range of state-of-the-art methods both in terms of segmentation accuracy and computational speed
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