53 research outputs found

    Non-convex optimization for 3D point source localization using a rotating point spread function

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    We consider the high-resolution imaging problem of 3D point source image recovery from 2D data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by S.~Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new nonconvex regularization method with a data-fitting term based on Kullback-Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can be readily applied to other kinds of depth-encoding PSFs as well.Comment: 28 page

    Discrete Riemannian Calculus and A Posteriori Error Control on Shape Spaces

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    In this thesis, a novel discrete approximation of the curvature tensor on Riemannian manifolds is derived, efficient methods to interpolate and extrapolate images in the context of the time discrete metamorphosis model are analyzed, and an a posteriori error estimator for the binary Mumford–Shah model is examined. Departing from the variational time discretization on (possibly infinite-dimensional) Riemannian manifolds originally proposed by Rumpf and Wirth, in which a consistent time discrete approximation of geodesic curves, the logarithm, the exponential map and parallel transport is analyzed, we construct the discrete curvature tensor and prove its convergence under certain smoothness assumptions. To this end, several time discrete parallel transports are applied to suitably rescaled tangent vectors, where each parallel transport is computed using Schild’s ladder. The associated convergence proof essentially relies on multiple Taylor expansions incorporating symmetry and scaling relations. In several numerical examples we validate this approach for surfaces. The by now classical flow of diffeomorphism approach allows the transport of image intensities along paths in time, which are characterized by diffeomorphisms, and the brightness of each image particle is assumed to be constant along each trajectory. As an extension, the metamorphosis model proposed by TrouvĂ©, Younes and coworkers allows for intensity variations of the image particles along the paths, which is reflected by an additional penalization term appearing in the energy functional that quantifies the squared weak material derivative. Taking into account the aforementioned time discretization, we propose a time discrete metamorphosis model in which the associated time discrete path energy consists of the sum of squared L2-mismatch functionals of successive square-integrable image intensity functions and a regularization functional for pairwise deformations. Our main contributions are the existence proof of time discrete geodesic curves in the context of this model, which are defined as minimizers of the time discrete path energy, and the proof of the Mosco-convergence of a suitable interpolation of the time discrete to the time continuous path energy with respect to the L2-topology. Using an alternating update scheme as well as a multilinear finite element respectively cubic spline discretization for the images and deformations allows to efficiently compute time discrete geodesic curves. In several numerical examples we demonstrate that time discrete geodesics can be robustly computed for gray-scale and color images. Taking into account the time discretization of the metamorphosis model we define the discrete exponential map in the space of images, which allows image extrapolation of arbitrary length for given weakly differentiable initial images and variations. To this end, starting from a suitable reformulation of the Euler–Lagrange equations characterizing the one-step extrapolation a fixed point iteration is employed to establish the existence of critical points of the Euler–Lagrange equations provided that the initial variation is small in L2. In combination with an implicit function type argument requiring H1-closeness of the initial variation one can prove the local existence as well as the local uniqueness of the discrete exponential map. The numerical algorithm for the one-step extrapolation is based on a slightly modified fixed point iteration using a spatial Galerkin scheme to obtain the optimal deformation associated with the unknown image, from which the unknown image itself can be recovered. To prove the applicability of the proposed method we compute the extrapolated image path for real image data. A common tool to segment images and shapes into multiple regions was developed by Mumford and Shah. The starting point to derive a posteriori error estimates for the binary Mumford–Shah model, which is obtained by restricting the original model to two regions, is a uniformly convex and non-constrained relaxation of the binary model following the work by Chambolle and Berkels. In particular, minimizers of the binary model can be exactly recovered from minimizers of the relaxed model via thresholding. Then, applying duality techniques proposed by Repin and Bartels allows deriving a consistent functional a posteriori error estimate for the relaxed model. Afterwards, an a posteriori error estimate for the original binary model can be computed incorporating a suitable cut-out argument in combination with the functional error estimate. To calculate minimizers of the relaxed model on an adaptive mesh described by a quadtree structure, we employ a primal-dual as well as a purely dual algorithm. The quality of the error estimator is analyzed for different gray-scale input images

    Space-time residual minimization for parabolic partial differential equations

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    Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE)

    Canonical Correlation Analysis And Network Data Modeling: Statistical And Computational Properties

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    Classical decision theory evaluates an estimator mostly by its statistical properties, either the closeness to the underlying truth or the predictive ability for new observations. The goal is to find estimators to achieve statistical optimality. Modern Big Data applications, however, necessitate efficient processing of large-scale ( big-n-big-p\u27 ) datasets, which poses great challenge to classical decision-theoretic framework which seldom takes into account the scalability of estimation procedures. On the one hand, statistically optimal estimators could be computationally intensive and on the other hand, fast estimation procedures might suffer from a loss of statistical efficiency. So the challenge is to kill two birds with one stone. This thesis brings together statistical and computational perspectives to study canonical correlation analysis (CCA) and network data modeling, where we investigate both the optimality and the scalability of the estimators. Interestingly, in both cases, we find iterative estimation procedures based on non-convex optimization can significantly reduce the computational cost and meanwhile achieve desirable statistical properties. In the first part of the thesis, motivated by the recent success of using CCA to learn low-dimensional feature representations of high-dimensional objects, we propose novel metrics which quantify the estimation loss of CCA by the excess prediction loss defined through a prediction-after-dimension-reduction framework. These new metrics have rich statistical and geometric interpretations, which suggest viewing CCA estimation as estimating the subspaces spanned by the canonical variates. We characterize, with minimal assumptions, the non-asymptotic minimax rates under the proposed error metrics, especially how the minimax rates depend on the key quantities including the dimensions, the condition number of the covariance matrices and the canonical correlations. Finally, by formulating sample CCA as a non-convex optimization problem, we propose an efficient (stochastic) first order algorithm which scales to large datasets. In the second part of the thesis, we propose two universal fitting algorithms for networks (possibly with edge covariates) under latent space models: one based on finding the exact maximizer of a convex surrogate of the non-convex likelihood function and the other based on finding an approximate optimizer of the original non-convex objective. Both algorithms are motivated by a special class of inner-product models but are shown to work for a much wider range of latent space models which allow the latent vectors to determine the connection probability of the edges in flexible ways. We derive the statistical rates of convergence of both algorithms and characterize the basin-of-attraction of the non-convex approach. The effectiveness and efficiency of the non-convex procedure is demonstrated by extensive simulations and real-data experiments

    Structured Learning with Parsimony in Measurements and Computations: Theory, Algorithms, and Applications

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    University of Minnesota Ph.D. dissertation. July 2018. Major: Electrical Engineering. Advisor: Jarvis Haupt. 1 computer file (PDF); xvi, 289 pages.In modern ``Big Data'' applications, structured learning is the most widely employed methodology. Within this paradigm, the fundamental challenge lies in developing practical, effective algorithmic inference methods. Often (e.g., deep learning) successful heuristic-based approaches exist but theoretical studies are far behind, limiting understanding and potential improvements. In other settings (e.g., recommender systems) provably effective algorithmic methods exist, but the sheer sizes of datasets can limit their applicability. This twofold challenge motivates this work on developing new analytical and algorithmic methods for structured learning, with a particular focus on parsimony in measurements and computation, i.e., those requiring low storage and computational costs. Toward this end, we make efforts to investigate the theoretical properties of models and algorithms that present significant improvement in measurement and computation requirement. In particular, we first develop randomized approaches for dimensionality reduction on matrix and tensor data, which allow accurate estimation and inference procedures using significantly smaller data sizes that only depend on the intrinsic dimension (e.g., the rank of matrix/tensor) rather than the ambient ones. Our next effort is to study iterative algorithms for solving high dimensional learning problems, including both convex and nonconvex optimization. Using contemporary analysis techniques, we demonstrate guarantees of iteration complexities that are analogous to the low dimensional cases. In addition, we explore the landscape of nonconvex optimizations that exhibit computational advantages over their convex counterparts and characterize their properties from a general point of view in theory

    Convex regression and its extensions to learning a Bregman divergence and difference of convex functions

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    Nonparametric convex regression has been extensively studied over the last two decades. It has been shown any Lipschitz convex function can be approximated with arbitrarily accuracy with a max of linear functions. Using this framework, in this thesis we generalize convex regression to learning an arbitrary Bregman divergence and learning a difference of convex functions. We provide approximation guarantees and sample complexity bounds for both these extensions. Furthermore, we provide algorithms to solve the resulting optimization problems based on 2-block alternative direction method of multipliers (ADMM). For this algorithm, we provide convergence guarantees with iteration complexity of O(n√d/) for a dataset X ℝ^n,d and arbitrary positive . Finally we provide experiments for both the Bregman divergence learning and difference of convex functions learning based on UCI datasets that demonstrate the state of the art on regression and classification datasets

    4D imaging in tomography and optical nanoscopy

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    Diese Dissertation gehört zu den Gebieten mathematische Bildverarbeitung und inverse Probleme. Ein inverses Problem ist die Aufgabe, Modellparameter anhand von gemessenen Daten zu berechnen. Solche Probleme treten in zahlreichen Anwendungen in Wissenschaft und Technik auf, z.B. in medizinischer Bildgebung, Biophysik oder Astronomie. Wir betrachten Rekonstruktionsprobleme mit Poisson Rauschen in der Tomographie und optischen Nanoskopie. Bei letzterer gilt es Bilder ausgehend von verzerrten und verrauschten Messungen zu rekonstruieren, wohingegen in der Positronen-Emissions-Tomographie die Aufgabe in der Visualisierung physiologischer Prozesse eines Patienten besteht. Standardmethoden zur 3D Bildrekonstruktion berĂŒcksichtigen keine zeitabhĂ€ngigen Informationen oder Dynamik, z.B. Herzschlag oder Atmung in der Tomographie oder Zellmigration in der Mikroskopie. Diese Dissertation behandelt Modelle, Analyse und effiziente Algorithmen fĂŒr 3D und 4D zeitabhĂ€ngige inverse Probleme. This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics or astronomy. We mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In the latter case, the task is to reconstruct images from blurred and noisy measurements, whereas in positron emission tomography the task is to visualize physiological processes of a patient. In 3D static image reconstruction standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in tomography or cell motion in microscopy. This thesis is a treatise on models, analysis and efficient algorithms to solve 3D and 4D time-dependent inverse problems
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