9 research outputs found
A Compact Linear Programming Relaxation for Binary Sub-modular MRF
We propose a novel compact linear programming (LP) relaxation for binary
sub-modular MRF in the context of object segmentation. Our model is obtained by
linearizing an -norm derived from the quadratic programming (QP) form of
the MRF energy. The resultant LP model contains significantly fewer variables
and constraints compared to the conventional LP relaxation of the MRF energy.
In addition, unlike QP which can produce ambiguous labels, our model can be
viewed as a quasi-total-variation minimization problem, and it can therefore
preserve the discontinuities in the labels. We further establish a relaxation
bound between our LP model and the conventional LP model. In the experiments,
we demonstrate our method for the task of interactive object segmentation. Our
LP model outperforms QP when converting the continuous labels to binary labels
using different threshold values on the entire Oxford interactive segmentation
dataset. The computational complexity of our LP is of the same order as that of
the QP, and it is significantly lower than the conventional LP relaxation
Efficient SDP Inference for Fully-connected CRFs Based on Low-rank Decomposition
Conditional Random Fields (CRF) have been widely used in a variety of
computer vision tasks. Conventional CRFs typically define edges on neighboring
image pixels, resulting in a sparse graph such that efficient inference can be
performed. However, these CRFs fail to model long-range contextual
relationships. Fully-connected CRFs have thus been proposed. While there are
efficient approximate inference methods for such CRFs, usually they are
sensitive to initialization and make strong assumptions. In this work, we
develop an efficient, yet general algorithm for inference on fully-connected
CRFs. The algorithm is based on a scalable SDP algorithm and the low- rank
approximation of the similarity/kernel matrix. The core of the proposed
algorithm is a tailored quasi-Newton method that takes advantage of the
low-rank matrix approximation when solving the specialized SDP dual problem.
Experiments demonstrate that our method can be applied on fully-connected CRFs
that cannot be solved previously, such as pixel-level image co-segmentation.Comment: 15 pages. A conference version of this work appears in Proc. IEEE
Conference on Computer Vision and Pattern Recognition, 201
Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems
In this paper, we propose an efficient semidefinite programming (SDP)
approach to worst-case linear discriminant analysis (WLDA). Compared with the
traditional LDA, WLDA considers the dimensionality reduction problem from the
worst-case viewpoint, which is in general more robust for classification.
However, the original problem of WLDA is non-convex and difficult to optimize.
In this paper, we reformulate the optimization problem of WLDA into a sequence
of semidefinite feasibility problems. To efficiently solve the semidefinite
feasibility problems, we design a new scalable optimization method with
quasi-Newton methods and eigen-decomposition being the core components. The
proposed method is orders of magnitude faster than standard interior-point
based SDP solvers.
Experiments on a variety of classification problems demonstrate that our
approach achieves better performance than standard LDA. Our method is also much
faster and more scalable than standard interior-point SDP solvers based WLDA.
The computational complexity for an SDP with constraints and matrices of
size by is roughly reduced from to
( in our case).Comment: 14 page
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
A fast semidefinite approach to solving binary quadratic problems
Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spectral methods and semi definite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semi definite relaxation has a tighter bound, but its computational complexity is high for large scale problems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmentation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs.Peng Wang, Chunhua Shen, Anton van den Hengelhttp://www.pamitc.org/cvpr13
Calculating Sparse and Dense Correspondences for Near-Isometric Shapes
Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching