83 research outputs found
Riemannian Flows for Supervised and Unsupervised Geometric Image Labeling
In this thesis we focus on the image labeling problem, which is used as a subroutine in many image processing applications. Our work is based on the assignment flow which was recently introduced as a novel geometric approach to the image labeling problem. This flow evolves over time on the manifold of row-stochastic matrices, whose elements represent label assignments as assignment probabilities. The strict separation of assignment manifold and feature space enables the data to lie in any metric space, while a smoothing operation on the assignment manifold results in an unbiased and spatially regularized labeling.
The first part of this work focuses on theoretical statements about the asymptotic behavior of the assignment flow. We show under weak assumptions on the parameters that the assignment flow for data in general position converges towards integral probabilities and thus ensures unique assignment decisions. Furthermore, we investigate the stability of possible limit points depending on the input data and parameters. For stable limits, we derive conditions that allow early evidence of convergence towards these limits and thus provide convergence guarantees.
In the second part, we extend the assignment flow approach in order to impose global convex constraints on the labeling results based on linear filter statistics of the assignments. The corresponding filters are learned from examples using an eigendecomposition. The effectiveness of the approach is numerically demonstrated in several academic labeling scenarios.
In the last part of this thesis we consider the situation in which no labels are given and therefore these prototypical elements have to be determined from the data as well. To this end we introduce an additional flow on the feature manifold, which is coupled to the assignment flow. The resulting flow adapts the prototypes in time to the assignment probabilities. The simultaneous adaptation and assignment of prototypes not only provides suitable prototypes, but also improves the resulting image segmentation, which is demonstrated by experiments.
For this approach it is assumed that the data lie on a Riemannian manifold. We elaborate the approach for a range of manifolds that occur in applications and evaluate the resulting approaches in numerical experiments
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
Motzkin path decompositions of functionals in noncommutative probability
The decomposition of free random variables into series of orthogonal replicas
gives random variables associated with important notions of noncommutative
independence: boolean, monotone, orthogonal, free and free with subordination.
In this paper, we study this decomposition from a new perspective. First, we
show that the mixed moments of orthogonal replicas define functionals indexed
by the elements of the lattices of Motzkin paths. By a duality relation, we
then obtain a family of path-dependent linear functionals on the free product
of algebras. They play the role of a generating set of the space of product
functionals in which the boolean product corresponds to constant Motzkin paths
and the free product to the sums of all Motzkin paths. Similar Motzkin path
decompositions can be derived for other product functionals. In simplified
terms, this paper initiates the `Motzkin path approach to noncommutative
probability', in which functionals convolving variables according to different
models of noncommutative independence are obtained by taking suitable subsets
of the set of Motzkin paths and perhaps conditioning them on the algebra labels
in nonsymmetric models. Consequently, we obtain a unified framework that allows
us to look at functionals inherent to different notions of independence
together rather than separately.Comment: 39 pages, 4 figure
Nonsmooth Convex Variational Approaches to Image Analysis
Variational models constitute a foundation for the formulation and understanding of models in many areas of image processing and analysis. In this work, we consider a generic variational framework for convex relaxations of multiclass labeling problems, formulated on continuous domains. We propose several relaxations for length-based regularizers, with varying expressiveness and computational cost. In contrast to graph-based, combinatorial approaches, we rely on a geometric measure theory-based formulation, which avoids artifacts caused by an early discretization in theory as well as in practice. We investigate and compare numerical first-order approaches for solving the associated nonsmooth discretized problem, based on controlled smoothing and operator splitting. In order to obtain integral solutions, we propose a randomized rounding technique formulated in the spatially continuous setting, and prove that it allows to obtain solutions with an a priori optimality bound. Furthermore, we present a method for introducing more advanced prior shape knowledge into labeling problems, based on the sparse representation framework
An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number
Quantum topology provides various frameworks for defining and computing
invariants of manifolds. One such framework of substantial interest in both
mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum
construction, which uses the data of a spherical fusion category to define
topological invariants of triangulated 3-manifolds via tensor network
contractions. In this work we consider a restricted class of state sum
invariants of 3-manifolds derived from Tambara-Yamagami categories. These
categories are particularly simple, being entirely specified by three pieces of
data: a finite abelian group, a bicharacter of that group, and a sign .
Despite being one of the simplest sources of state sum invariants, the
computational complexities of Tambara-Yamagami invariants are yet to be fully
understood.
We make substantial progress on this problem. Our main result is the
existence of a general fixed parameter tractable algorithm for all such
topological invariants, where the parameter is the first Betti number of the
3-manifold with coefficients. We also explain that
these invariants are sometimes #P-hard to compute (and we expect that this is
almost always the case).
Contrary to other domains of computational topology, such as graphs on
surfaces, very few hard problems in 3-manifold topology are known to admit FPT
algorithms with a topological parameter. However, such algorithms are of
particular interest as their complexity depends only polynomially on the
combinatorial representation of the input, regardless of size or combinatorial
width. Additionally, in the case of Betti numbers, the parameter itself is
easily computable in polynomial time.Comment: 24 pages, including 3 appendice
On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype
Given a Baumslag-Solitar group, we study its space of subgroups from a
topological and dynamical perspective. We first determine its perfect kernel
(the largest closed subset without isolated points). We then bring to light a
natural partition of the space of subgroups into one closed subset and
countably many open subsets that are invariant under the action by conjugation.
One of our main results is that the restriction of the action to each piece is
topologically transitive. This partition is described by an arithmetically
defined function, that we call the phenotype, with values in the positive
integers or infinity. We eventually study the closure of each open piece and
also the closure of their union. We moreover identify in each phenotype a (the)
maximal compact invariant subspace.Comment: 60 pages, companion webpage available at
https://doi.org/10.5281/zenodo.7225585 . Comments welcome
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