4,285 research outputs found
Labeled Interleaving Distance for Reeb Graphs
Merge trees, contour trees, and Reeb graphs are graph-based topological
descriptors that capture topological changes of (sub)level sets of scalar
fields. Comparing scalar fields using their topological descriptors has many
applications in topological data analysis and visualization of scientific data.
Recently, Munch and Stefanou introduced a labeled interleaving distance for
comparing two labeled merge trees, which enjoys a number of theoretical and
algorithmic properties. In particular, the labeled interleaving distance
between merge trees can be computed in polynomial time. In this work, we define
the labeled interleaving distance for labeled Reeb graphs. We then prove that
the (ordinary) interleaving distance between Reeb graphs equals the minimum of
the labeled interleaving distance over all labelings. We also provide an
efficient algorithm for computing the labeled interleaving distance between two
labeled contour trees (which are special types of Reeb graphs that arise from
simply-connected domains). In the case of merge trees, the notion of the
labeled interleaving distance was used by Gasparovic et al. to prove that the
(ordinary) interleaving distance on the set of (unlabeled) merge trees is
intrinsic. As our final contribution, we present counterexamples showing that,
on the contrary, the (ordinary) interleaving distance on (unlabeled) Reeb
graphs (and contour trees) is not intrinsic. It turns out that, under mild
conditions on the labelings, the labeled interleaving distance is a metric on
isomorphism classes of Reeb graphs, analogous to the ordinary interleaving
distance. This provides new metrics on large classes of Reeb graphs
Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)
This paper presents a computational framework for the Principal Geodesic
Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated
Principal Component Analysis (PCA) framework [87] to the Wasserstein metric
space of merge trees [92]. We formulate MT-PGA computation as a constrained
optimization problem, aiming at adjusting a basis of orthogonal geodesic axes,
while minimizing a fitting energy. We introduce an efficient, iterative
algorithm which exploits shared-memory parallelism, as well as an analytic
expression of the fitting energy gradient, to ensure fast iterations. Our
approach also trivially extends to extremum persistence diagrams. Extensive
experiments on public ensembles demonstrate the efficiency of our approach -
with MT-PGA computations in the orders of minutes for the largest examples. We
show the utility of our contributions by extending to merge trees two typical
PCA applications. First, we apply MT-PGA to data reduction and reliably
compress merge trees by concisely representing them by their first coordinates
in the MT-PGA basis. Second, we present a dimensionality reduction framework
exploiting the first two directions of the MT-PGA basis to generate
two-dimensional layouts of the ensemble. We augment these layouts with
persistence correlation views, enabling global and local visual inspections of
the feature variability in the ensemble. In both applications, quantitative
experiments assess the relevance of our framework. Finally, we provide a
lightweight C++ implementation that can be used to reproduce our results
Flow-based Influence Graph Visual Summarization
Visually mining a large influence graph is appealing yet challenging. People
are amazed by pictures of newscasting graph on Twitter, engaged by hidden
citation networks in academics, nevertheless often troubled by the unpleasant
readability of the underlying visualization. Existing summarization methods
enhance the graph visualization with blocked views, but have adverse effect on
the latent influence structure. How can we visually summarize a large graph to
maximize influence flows? In particular, how can we illustrate the impact of an
individual node through the summarization? Can we maintain the appealing graph
metaphor while preserving both the overall influence pattern and fine
readability?
To answer these questions, we first formally define the influence graph
summarization problem. Second, we propose an end-to-end framework to solve the
new problem. Our method can not only highlight the flow-based influence
patterns in the visual summarization, but also inherently support rich graph
attributes. Last, we present a theoretic analysis and report our experiment
results. Both evidences demonstrate that our framework can effectively
approximate the proposed influence graph summarization objective while
outperforming previous methods in a typical scenario of visually mining
academic citation networks.Comment: to appear in IEEE International Conference on Data Mining (ICDM),
Shen Zhen, China, December 201
Taming Horizontal Instability in Merge Trees: On the Computation of a Comprehensive Deformation-based Edit Distance
Comparative analysis of scalar fields in scientific visualization often
involves distance functions on topological abstractions. This paper focuses on
the merge tree abstraction (representing the nesting of sub- or superlevel
sets) and proposes the application of the unconstrained deformation-based edit
distance. Previous approaches on merge trees often suffer from instability:
small perturbations in the data can lead to large distances of the
abstractions. While some existing methods can handle so-called vertical
instability, the unconstrained deformation-based edit distance addresses both
vertical and horizontal instabilities, also called saddle swaps. We establish
the computational complexity as NP-complete, and provide an integer linear
program formulation for computation. Experimental results on the TOSCA shape
matching ensemble provide evidence for the stability of the proposed distance.
We thereby showcase the potential of handling saddle swaps for comparison of
scalar fields through merge trees
Data complexity measured by principal graphs
How to measure the complexity of a finite set of vectors embedded in a
multidimensional space? This is a non-trivial question which can be approached
in many different ways. Here we suggest a set of data complexity measures using
universal approximators, principal cubic complexes. Principal cubic complexes
generalise the notion of principal manifolds for datasets with non-trivial
topologies. The type of the principal cubic complex is determined by its
dimension and a grammar of elementary graph transformations. The simplest
grammar produces principal trees.
We introduce three natural types of data complexity: 1) geometric (deviation
of the data's approximator from some "idealized" configuration, such as
deviation from harmonicity); 2) structural (how many elements of a principal
graph are needed to approximate the data), and 3) construction complexity (how
many applications of elementary graph transformations are needed to construct
the principal object starting from the simplest one).
We compute these measures for several simulated and real-life data
distributions and show them in the "accuracy-complexity" plots, helping to
optimize the accuracy/complexity ratio. We discuss various issues connected
with measuring data complexity. Software for computing data complexity measures
from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres
Customizable tubular model for n-furcating blood vessels and its application to 3D reconstruction of the cerebrovascular system
Understanding the 3D cerebral vascular network is one of the pressing issues impacting the diagnostics of various systemic disorders and is helpful in clinical therapeutic strategies. Unfortunately, the existing software in the radiological workstation does not meet the expectations of radiologists who require a computerized system for detailed, quantitative analysis of the human cerebrovascular system in 3D and a standardized geometric description of its components. In this study, we show a method that uses 3D image data from magnetic resonance imaging with contrast to create a geometrical reconstruction of the vessels and a parametric description of the reconstructed segments of the vessels. First, the method isolates the vascular system using controlled morphological growing and performs skeleton extraction and optimization. Then, around the optimized skeleton branches, it creates tubular objects optimized for quality and accuracy of matching with the originally isolated vascular data. Finally, it optimizes the joints on n-furcating vessel segments. As a result, the algorithm gives a complete description of shape, position in space, position relative to other segments, and other anatomical structures of each cerebrovascular system segment. Our method is highly customizable and in principle allows reconstructing vascular structures from any 2D or 3D data. The algorithm solves shortcomings of currently available methods including failures to reconstruct the vessel mesh in the proximity of junctions and is free of mesh collisions in high curvature vessels. It also introduces a number of optimizations in the vessel skeletonization leading to a more smooth and more accurate model of the vessel network. We have tested the method on 20 datasets from the public magnetic resonance angiography image database and show that the method allows for repeatable and robust segmentation of the vessel network and allows to compute vascular lateralization indices. Graphical abstract: [Figure not available: see fulltext.]</p
Depth from Monocular Images using a Semi-Parallel Deep Neural Network (SPDNN) Hybrid Architecture
Deep neural networks are applied to a wide range of problems in recent years.
In this work, Convolutional Neural Network (CNN) is applied to the problem of
determining the depth from a single camera image (monocular depth). Eight
different networks are designed to perform depth estimation, each of them
suitable for a feature level. Networks with different pooling sizes determine
different feature levels. After designing a set of networks, these models may
be combined into a single network topology using graph optimization techniques.
This "Semi Parallel Deep Neural Network (SPDNN)" eliminates duplicated common
network layers, and can be further optimized by retraining to achieve an
improved model compared to the individual topologies. In this study, four SPDNN
models are trained and have been evaluated at 2 stages on the KITTI dataset.
The ground truth images in the first part of the experiment are provided by the
benchmark, and for the second part, the ground truth images are the depth map
results from applying a state-of-the-art stereo matching method. The results of
this evaluation demonstrate that using post-processing techniques to refine the
target of the network increases the accuracy of depth estimation on individual
mono images. The second evaluation shows that using segmentation data alongside
the original data as the input can improve the depth estimation results to a
point where performance is comparable with stereo depth estimation. The
computational time is also discussed in this study.Comment: 44 pages, 25 figure
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