120 research outputs found
Nonlocal PdES on graphs for active contours models with applications to image segmentation and data clustering
International audienceWe propose a transcription on graphs of recent continuous global active contours proposed for image segmentation to address the problem of binary partitioning of data represented by graphs. To do so, using the framework of Partial difference Equations (PdEs), we propose a family of nonlocal regularization functionals that verify the co-area formula on graphs. The gradients of a sub-graph are introduced and their properties studied. Relations, for the case of a sub-graph, between the introduced nonlocal regularization functionals and nonlocal discrete perimeters are exhibited and the co-area formula on graphs is introduced. Finally, nonlocal global minimizers can be considered on graphs with the associated energies. Experiments show the benefits of the approach for nonlocal image segmentation and high dimensional data clustering
Combinatorial Continuous Maximal Flows
Maximum flow (and minimum cut) algorithms have had a strong impact on
computer vision. In particular, graph cuts algorithms provide a mechanism for
the discrete optimization of an energy functional which has been used in a
variety of applications such as image segmentation, stereo, image stitching and
texture synthesis. Algorithms based on the classical formulation of max-flow
defined on a graph are known to exhibit metrication artefacts in the solution.
Therefore, a recent trend has been to instead employ a spatially continuous
maximum flow (or the dual min-cut problem) in these same applications to
produce solutions with no metrication errors. However, known fast continuous
max-flow algorithms have no stopping criteria or have not been proved to
converge. In this work, we revisit the continuous max-flow problem and show
that the analogous discrete formulation is different from the classical
max-flow problem. We then apply an appropriate combinatorial optimization
technique to this combinatorial continuous max-flow CCMF problem to find a
null-divergence solution that exhibits no metrication artefacts and may be
solved exactly by a fast, efficient algorithm with provable convergence.
Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the
fact, already proved by Nozawa in the continuous setting, that the max-flow and
the total variation problems are not always equivalent.Comment: 26 page
A Two-stage Classification Method for High-dimensional Data and Point Clouds
High-dimensional data classification is a fundamental task in machine
learning and imaging science. In this paper, we propose a two-stage multiphase
semi-supervised classification method for classifying high-dimensional data and
unstructured point clouds. To begin with, a fuzzy classification method such as
the standard support vector machine is used to generate a warm initialization.
We then apply a two-stage approach named SaT (smoothing and thresholding) to
improve the classification. In the first stage, an unconstraint convex
variational model is implemented to purify and smooth the initialization,
followed by the second stage which is to project the smoothed partition
obtained at stage one to a binary partition. These two stages can be repeated,
with the latest result as a new initialization, to keep improving the
classification quality. We show that the convex model of the smoothing stage
has a unique solution and can be solved by a specifically designed primal-dual
algorithm whose convergence is guaranteed. We test our method and compare it
with the state-of-the-art methods on several benchmark data sets. The
experimental results demonstrate clearly that our method is superior in both
the classification accuracy and computation speed for high-dimensional data and
point clouds.Comment: 21 pages, 4 figure
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
PDEs level sets on weighted graphs
International audienceIn this paper we propose an adaptation of PDEs level sets over weighted graphs of arbitrary structure, based on PdEs and using a framework of discrete operators. A general PDEs level sets formulation is presented and an algorithm to solve such equation is described. Some transcriptions of well-known models under this formalism, as the mean-curvature-motion or active contours, are also provided. Then, we present several applications of our formalism, including image segmentation with active contours, using weighted graphs of arbitrary topologies
Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling
In this thesis, we focus on the image labeling problem which is the task of performing unique
pixel-wise label decisions to simplify the image while reducing its redundant information. We
build upon a recently introduced geometric approach for data labeling by assignment flows
[
APSS17
] that comprises a smooth dynamical system for data processing on weighted graphs.
Hereby we pursue two lines of research that give new application and theoretically-oriented
insights on the underlying segmentation task.
We demonstrate using the example of Optical Coherence Tomography (OCT), which is the
mostly used non-invasive acquisition method of large volumetric scans of human retinal tis-
sues, how incorporation of constraints on the geometry of statistical manifold results in a novel
purely data driven
geometric
approach for order-constrained segmentation of volumetric data
in any metric space. In particular, making diagnostic analysis for human eye diseases requires
decisive information in form of exact measurement of retinal layer thicknesses that has be done
for each patient separately resulting in an demanding and time consuming task. To ease the
clinical diagnosis we will introduce a fully automated segmentation algorithm that comes up
with a high segmentation accuracy and a high level of built-in-parallelism. As opposed to many
established retinal layer segmentation methods, we use only local information as input without
incorporation of additional global shape priors. Instead, we achieve physiological order of reti-
nal cell layers and membranes including a new formulation of ordered pair of distributions in an
smoothed energy term. This systematically avoids bias pertaining to global shape and is hence
suited for the detection of anatomical changes of retinal tissue structure. To access the perfor-
mance of our approach we compare two different choices of features on a data set of manually
annotated
3
D OCT volumes of healthy human retina and evaluate our method against state of
the art in automatic retinal layer segmentation as well as to manually annotated ground truth
data using different metrics.
We generalize the recent work [
SS21
] on a variational perspective on assignment flows and
introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs.
The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was
introduced in
J. Math. Imaging & Vision
58(2), 2017. Due to this parameterization, solving the
G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with re-
spect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions
(DC) decomposition of this potential and show that the basic geometric Euler scheme for inte-
grating the assignment flow is equivalent to solving the G-PDE by an established DC program-
ming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit
higher-order information of the vector field that drives the assignment flow, in order to devise a
novel accelerated DC programming scheme. A detailed convergence analysis of both numerical
schemes is provided and illustrated by numerical experiments
Sparse Representation on Graphs by Tight Wavelet Frames and Applications
In this paper, we introduce a new (constructive) characterization of tight
wavelet frames on non-flat domains in both continuum setting, i.e. on
manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet
frame transforms can be computed and how they can be effectively used to
process graph data. We start with defining the quasi-affine systems on a given
manifold \cM that is formed by generalized dilations and shifts of a finite
collection of wavelet functions .
We further require that is generated by some refinable function
with mask . We present the condition needed for the masks so that the associated quasi-affine system generated by is a
tight frame for L_2(\cM). Then, we discuss how the transition from the
continuum (manifolds) to the discrete setting (graphs) can be naturally done.
In order for the proposed discrete tight wavelet frame transforms to be useful
in applications, we show how the transforms can be computed efficiently and
accurately by proposing the fast tight wavelet frame transforms for graph data
(WFTG). Finally, we consider two specific applications of the proposed WFTG:
graph data denoising and semi-supervised clustering. Utilizing the sparse
representation provided by the WFTG, we propose -norm based
optimization models on graphs for denoising and semi-supervised clustering. On
one hand, our numerical results show significant advantage of the WFTG over the
spectral graph wavelet transform (SGWT) by [1] for both applications. On the
other hand, numerical experiments on two real data sets show that the proposed
semi-supervised clustering model using the WFTG is overall competitive with the
state-of-the-art methods developed in the literature of high-dimensional data
classification, and is superior to some of these methods
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
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