3,341 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
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure
We present an exploration of the rich theoretical connections between several
classes of regularized models, network flows, and recent results in submodular
function theory. This work unifies key aspects of these problems under a common
theory, leading to novel methods for working with several important models of
interest in statistics, machine learning and computer vision.
In Part 1, we review the concepts of network flows and submodular function
optimization theory foundational to our results. We then examine the
connections between network flows and the minimum-norm algorithm from
submodular optimization, extending and improving several current results. This
leads to a concise representation of the structure of a large class of pairwise
regularized models important in machine learning, statistics and computer
vision.
In Part 2, we describe the full regularization path of a class of penalized
regression problems with dependent variables that includes the graph-guided
LASSO and total variation constrained models. This description also motivates a
practical algorithm. This allows us to efficiently find the regularization path
of the discretized version of TV penalized models. Ultimately, our new
algorithms scale up to high-dimensional problems with millions of variables
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