19,383 research outputs found
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
Hierarchical image simplification and segmentation based on Mumford-Shah-salient level line selection
Hierarchies, such as the tree of shapes, are popular representations for
image simplification and segmentation thanks to their multiscale structures.
Selecting meaningful level lines (boundaries of shapes) yields to simplify
image while preserving intact salient structures. Many image simplification and
segmentation methods are driven by the optimization of an energy functional,
for instance the celebrated Mumford-Shah functional. In this paper, we propose
an efficient approach to hierarchical image simplification and segmentation
based on the minimization of the piecewise-constant Mumford-Shah functional.
This method conforms to the current trend that consists in producing
hierarchical results rather than a unique partition. Contrary to classical
approaches which compute optimal hierarchical segmentations from an input
hierarchy of segmentations, we rely on the tree of shapes, a unique and
well-defined representation equivalent to the image. Simply put, we compute for
each level line of the image an attribute function that characterizes its
persistence under the energy minimization. Then we stack the level lines from
meaningless ones to salient ones through a saliency map based on extinction
values defined on the tree-based shape space. Qualitative illustrations and
quantitative evaluation on Weizmann segmentation evaluation database
demonstrate the state-of-the-art performance of our method.Comment: Pattern Recognition Letters, Elsevier, 201
Submodular relaxation for inference in Markov random fields
In this paper we address the problem of finding the most probable state of a
discrete Markov random field (MRF), also known as the MRF energy minimization
problem. The task is known to be NP-hard in general and its practical
importance motivates numerous approximate algorithms. We propose a submodular
relaxation approach (SMR) based on a Lagrangian relaxation of the initial
problem. Unlike the dual decomposition approach of Komodakis et al., 2011 SMR
does not decompose the graph structure of the initial problem but constructs a
submodular energy that is minimized within the Lagrangian relaxation. Our
approach is applicable to both pairwise and high-order MRFs and allows to take
into account global potentials of certain types. We study theoretical
properties of the proposed approach and evaluate it experimentally.Comment: This paper is accepted for publication in IEEE Transactions on
Pattern Analysis and Machine Intelligenc
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
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