175 research outputs found
Fast Solvers and Simulation Data Compression Algorithms for Granular Media and Complex Fluid Flows
Granular and particulate flows are common forms of materials used in various physical and industrial applications. For instance, we model the soil as a collection of rigid particles with frictional contact in soil-vehicle simulations, and we simulate bacterial colonies as active rigid particles immersed in a viscous fluid. Due to the complex interactions in-between the particles and/or the particles and the fluid, numerical simulations are often the only way to study these systems apart from typically expensive physical experiments.
A standard method for simulating these systems is to apply simple physical laws to each of the particles using the discrete element method (DEM) and evolve the resulting multibody system in time. However, due to the sheer number of particles in even a moderate-scale real-world system, it quickly becomes expensive to timestep these systems unless we exploit fast algorithms and high-performance computing techniques. For instance, a big challenge in granular media simulations is resolving contact between the constituent particles. We use a cone-complementarity formulation of frictional contact to resolve collisions; this approach leads to a quadratic optimization problem whose solution gives us the contact forces between particles at each timestep. In this thesis, we introduce strategies for solving these optimization problems on distributed memory machines. In particular, by imposing a locality-preserving partitioning of the rigid bodies among the computing nodes, we minimize the communication cost and construct a scalable framework for collision detecting and resolution that can be easily scaled to handle hundreds of millions of particles.
For robust and efficient simulation of axisymmetric particles in viscous fluids, we introduce a fast method for solving Stokes boundary integral equations (BIEs) on surfaces of revolution. By first transforming the Stokes integral kernels into a rotationally invariant form and then decoupling the transformed kernels using the Fourier series, we reduce the dimensionality of the problem. This approach improves the time complexity of the BIE solvers by an order of magnitude; additionally we can use high-order one-dimensional singular quadrature schemes to construct highly accurate solvers. Finally, coupling our solver framework with the fast multipole method, we construct a fast solver for simulating Stokes flow past a system of axisymmetric bodies. Combining this with our complementarity collision resolution framework, we have the potential to simulate dense particulate suspensions.
Physics-based simulations similar to those described above generate large amounts of output data, often in the hundreds of gigabytes range. We introduce data compression techniques based on the tensor-train decomposition for DEM simulation outputs and demonstrate the high compressibility of these large datasets. This allows us to keep a reduced representation of simulated data for post-processing or use in learning tasks.
Finally, due to the high cost of physics-based models and limited computational budget, we can typically run only a limited number of simulations when exploring a high-dimensional parameter space. Formally, this can be posed as a matrix/tensor completion problem, and Bayesian inference coupled with a linear factorization model is often used in this setup. We use Markov chain Monte Carlo (MCMC) methods to sample from the unnormalized posteriors in these inference problems. In this thesis, we explore the properties of the posterior in a simple low-rank matrix factorization setup and develop strategies to break its symmetries. This leads to better quality MCMC samples and lowers the reconstruction errors with various synthetic and real-world datasets.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169614/1/saibalde_1.pd
Visuelle Detektion unabhängig bewegter Objekte durch einen bewegten monokularen Beobachter
The development of a driver assistant system supporting drivers in complex intersection situations would be a major achievement for traffic safety, since many traffic accidents happen in such situations. While this is a highly complex task, which is still not accomplished, this thesis focused on one important and obligatory aspect of such systems: The visual detection of independently moving objects. Information about moving objects can, for example, be used in an attention guidance system, which is a central component of any complete intersection assistant system. The decision to base such a system on visual input had two reasons: (i) Humans gather their information to a large extent visually and (ii) cameras are inexpensive and already widely used in luxury and professional vehicles for specific applications. Mimicking the articulated human head and eyes, agile camera systems are desirable. To avoid heavy and sensitive stereo rigs, a small and lightweight monocular camera system mounted on a pan-tilt unit has been chosen as input device. In this thesis information about moving objects has been used to develop a prototype of an attention guidance system. It is based on the analysis of sequences from a single freely moving camera and on measurements from inertial sensors rigidly coupled with the camera system.Die Entwicklung eines Fahrerassistenzsystems, welches den Fahrer in komplexen Kreuzungssituationen unterstützt, wäre ein wichtiger Beitrag zur Verkehrssicherheit, da sehr viele Unfälle in solchen Situationen passieren. Dies ist eine hochgradig komplexe Aufgabe und daher liegt der Fokus dieser Arbeit auf einen wichtigen und notwendigen Aspekt solcher Systeme: Die visuelle Detektion unabhängig bewegter Objekte. Informationen über bewegte Objekte können z.B. für ein System zur Aufmerksamkeitssteuerung verwendet werden. Solch ein System ist ein integraler Bestandteil eines jeden kompletten Kreuzungsassistenzssystems. Zwei Gründe haben zu der Entscheidung geführt, das System auf visuellen Daten zu stützen: (i) Der Mensch sammelt seine Informationen zum Großteil visuell und (ii) Kameras sind zum Einen günstig und zum Anderen bereits jetzt in vielen Fahrzeugen verfügbar. Agile Kamerasysteme sind nötig um den beweglichen menschlichen Kopf zu imitieren. Die Wahl einer kleinen und leichten monokularen Kamera, die auf einer Schwenk-Neige-Einheit montiert ist, vermeidet die Verwendung von schweren und empfindlichen Stereokamerasystemen. Mit den Informationen über bewegte Objekte ist in dieser Arbeit der Prototyp eines Fahrerassistenzsystems Aufmerksamkeitssteuerung entwickelt worden. Das System basiert auf der Analyse von Bildsequenzen einer frei bewegten Kamera und auf Messungen von der mit der Kamera starr gekoppelten Inertialsensorik
Graphical Model approaches for Biclustering
In many scientific areas, it is crucial to group (cluster) a set of objects, based on a set of observed features. Such operation is widely known as Clustering and it has been exploited in the most different scenarios ranging from Economics to Biology passing through Psychology. Making a step forward, there exist contexts where it is crucial to group objects and simultaneously identify the features that allow to recognize such objects from the others. In gene expression analysis, for instance, the identification of subsets of genes showing a coherent pattern of expression in subsets of objects/samples can provide crucial information about active biological processes. Such information, which cannot be retrieved by classical clustering approaches, can be extracted with the so called Biclustering, a class of approaches which aim at simultaneously clustering both rows and columns of a given data matrix (where each row corresponds to a different object/sample and each column to a different feature). The problem of biclustering, also known as co-clustering, has been recently exploited in a wide range of scenarios such as Bioinformatics, market segmentation, data mining, text analysis and recommender systems. Many approaches have been proposed to address the biclustering problem, each one characterized by different properties such as interpretability, effectiveness or computational complexity. A recent trend involves the exploitation of sophisticated computational models (Graphical Models) to face the intrinsic complexity of biclustering, and to retrieve very accurate solutions. Graphical Models represent the decomposition of a global objective function to analyse in a set of smaller/local functions defined over a subset of variables. The advantages in using Graphical Models relies in the fact that the graphical representation can highlight useful hidden properties of the considered objective function, plus, the analysis of smaller local problems can be dealt with less computational effort. Due to the difficulties in obtaining a representative and solvable model, and since biclustering is a complex and challenging problem, there exist few promising approaches in literature based on Graphical models facing biclustering. 3 This thesis is inserted in the above mentioned scenario and it investigates the exploitation of Graphical Models to face the biclustering problem. We explored different type of Graphical Models, in particular: Factor Graphs and Bayesian Networks. We present three novel algorithms (with extensions) and evaluate such techniques using available benchmark datasets. All the models have been compared with the state-of-the-art competitors and the results show that Factor Graph approaches lead to solid and efficient solutions for dataset of contained dimensions, whereas Bayesian Networks can manage huge datasets, with the overcome that setting the parameters can be not trivial. As another contribution of the thesis, we widen the range of biclustering applications by studying the suitability of these approaches in some Computer Vision problems where biclustering has been never adopted before. Summarizing, with this thesis we provide evidence that Graphical Model techniques can have a significant impact in the biclustering scenario. Moreover, we demonstrate that biclustering techniques are ductile and can produce effective solutions in the most different fields of applications
Robust motion segmentation with subspace constraints
Motion segmentation is an important task in computer vision with
many applications such as dynamic scene understanding and
multi-body structure from motion. When the point correspondences
across frames are given, motion segmentation can be addressed as
a subspace clustering problem under an affine camera model. In
the first two parts of this thesis, we target the general
subspace clustering problem and propose two novel methods, namely
Efficient Dense Subspace Clustering (EDSC) and the Robust Shape
Interaction Matrix (RSIM) method.
Instead of following the standard compressive sensing approach,
in EDSC we formulate subspace clustering as a Frobenius norm
minimization problem, which inherently yields denser connections
between data points. While in the noise-free case we rely on the
self-expressiveness of the observations, in the presence of noise
we recover a clean dictionary to represent the data. Our
formulation lets us solve the subspace clustering problem
efficiently. More specifically, for outlier-free observations,
the solution can be obtained in closed-form, and in the presence
of outliers, we solve the problem by performing a series of
linear operations. Furthermore, we show that our Frobenius norm
formulation shares the same solution as the popular nuclear norm
minimization approach when the data is free of any noise.
In RSIM, we revisit the Shape Interaction Matrix (SIM) method,
one of the earliest approaches for motion segmentation (or
subspace clustering), and reveal its connections to several
recent subspace clustering methods. We derive a simple, yet
effective algorithm to robustify the SIM method and make it
applicable to real-world scenarios where the data is corrupted by
noise. We validate the proposed method by intuitive examples and
justify it with the matrix perturbation theory. Moreover, we show
that RSIM can be extended to handle missing data with a
Grassmannian gradient descent method.
The above subspace clustering methods work well for motion
segmentation, yet they require that point trajectories across
frames are known {\it a priori}. However, finding point
correspondences is in itself a challenging task. Existing
approaches tackle the correspondence estimation and motion
segmentation problems separately. In the third part of this
thesis, given a set of feature points detected in each frame of
the sequence, we develop an approach which simultaneously
performs motion segmentation and finds point correspondences
across the frames. We formulate this problem in terms of Partial
Permutation Matrices (PPMs) and aim to match feature descriptors
while simultaneously encouraging point trajectories to satisfy
subspace constraints. This lets us handle outliers in both point
locations and feature appearance. The resulting optimization
problem is solved via the Alternating Direction Method of
Multipliers (ADMM), where each subproblem has an efficient
solution. In particular, we show that most of the subproblems can
be solved in closed-form, and one binary assignment subproblem
can be solved by the Hungarian algorithm.
Obtaining reliable feature tracks in a frame-by-frame manner is
desirable in applications such as online motion segmentation. In
the final part of the thesis, we introduce a novel multi-body
feature tracker that exploits a multi-body rigidity assumption to
improve tracking robustness under a general perspective camera
model. A conventional approach to addressing this problem would
consist of alternating between solving two subtasks: motion
segmentation and feature tracking under rigidity constraints for
each segment. This approach, however, requires knowing the number
of motions, as well as assigning points to motion groups, which
is typically sensitive to motion estimates. By contrast, we
introduce a segmentation-free solution to multi-body feature
tracking that bypasses the motion assignment step and reduces to
solving a series of subproblems with closed-form solutions.
In summary, in this thesis, we exploit the powerful subspace
constraints and develop robust motion segmentation methods in
different challenging scenarios where the trajectories are either
given as input, or unknown beforehand. We also present a general
robust multi-body feature tracker which can be used as the first
step of motion segmentation to get reliable trajectories
Robust Subspace Estimation via Low-Rank and Sparse Decomposition and Applications in Computer Vision
PhDRecent advances in robust subspace estimation have made dimensionality reduction and
noise and outlier suppression an area of interest for research, along with continuous
improvements in computer vision applications. Due to the nature of image and video
signals that need a high dimensional representation, often storage, processing, transmission,
and analysis of such signals is a difficult task. It is therefore desirable to obtain a
low-dimensional representation for such signals, and at the same time correct for corruptions,
errors, and outliers, so that the signals could be readily used for later processing.
Major recent advances in low-rank modelling in this context were initiated by the work of
Cand`es et al. [17] where the authors provided a solution for the long-standing problem of
decomposing a matrix into low-rank and sparse components in a Robust Principal Component
Analysis (RPCA) framework. However, for computer vision applications RPCA
is often too complex, and/or may not yield desirable results. The low-rank component
obtained by the RPCA has usually an unnecessarily high rank, while in certain tasks
lower dimensional representations are required. The RPCA has the ability to robustly
estimate noise and outliers and separate them from the low-rank component, by a sparse
part. But, it has no mechanism of providing an insight into the structure of the sparse
solution, nor a way to further decompose the sparse part into a random noise and a structured
sparse component that would be advantageous in many computer vision tasks. As
videos signals are usually captured by a camera that is moving, obtaining a low-rank
component by RPCA becomes impossible. In this thesis, novel Approximated RPCA
algorithms are presented, targeting different shortcomings of the RPCA. The Approximated
RPCA was analysed to identify the most time consuming RPCA solutions, and
replace them with simpler yet tractable alternative solutions. The proposed method is
able to obtain the exact desired rank for the low-rank component while estimating a
global transformation to describe camera-induced motion. Furthermore, it is able to
decompose the sparse part into a foreground sparse component, and a random noise
part that contains no useful information for computer vision processing. The foreground
sparse component is obtained by several novel structured sparsity-inducing norms, that
better encapsulate the needed pixel structure in visual signals. Moreover, algorithms for
reducing complexity of low-rank estimation have been proposed that achieve significant
complexity reduction without sacrificing the visual representation of video and image
information. The proposed algorithms are applied to several fundamental computer
vision tasks, namely, high efficiency video coding, batch image alignment, inpainting,
and recovery, video stabilisation, background modelling and foreground segmentation,
robust subspace clustering and motion estimation, face recognition, and ultra high definition
image and video super-resolution. The algorithms proposed in this thesis including
batch image alignment and recovery, background modelling and foreground segmentation,
robust subspace clustering and motion segmentation, and ultra high definition
image and video super-resolution achieve either state-of-the-art or comparable results to
existing methods
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