Direct Diffusion Bridge using Data Consistency for Inverse Problems

Diffusion model-based inverse problem solvers have shown impressive performance, but are limited in speed, mostly as they require reverse diffusion sampling starting from noise. Several recent works have tried to alleviate this problem by building a diffusion process, directly bridging the clean and the corrupted for specific inverse problems. In this paper, we first unify these existing works under the name Direct Diffusion Bridges (DDB), showing that while motivated by different theories, the resulting algorithms only differ in the choice of parameters. Then, we highlight a critical limitation of the current DDB framework, namely that it does not ensure data consistency. To address this problem, we propose a modified inference procedure that imposes data consistency without the need for fine-tuning. We term the resulting method data Consistent DDB (CDDB), which outperforms its inconsistent counterpart in terms of both perception and distortion metrics, thereby effectively pushing the Pareto-frontier toward the optimum. Our proposed method achieves state-of-the-art results on both evaluation criteria, showcasing its superiority over existing methods. Code is available at https://github.com/HJ-harry/CDDBComment: NeurIPS 2023 camera-ready. 16 pages, 6 figure

Source Separation in the Presence of Side-information

The source separation problem involves the separation of unknown signals from their mixture. This problem is relevant in a wide range of applications from audio signal processing, communication, biomedical signal processing and art investigation to name a few. There is a vast literature on this problem which is based on either making strong assumption on the source signals or availability of additional data. This thesis proposes new algorithms for source separation with side information where one observes the linear superposition of two source signals plus two additional signals that are correlated with the mixed ones. The first algorithm is based on two ingredients: first, we learn a Gaussian mixture model (GMM) for the joint distribution of a source signal and the corresponding correlated side information signal; second, we separate the signals using standard computationally efficient conditional mean estimators. This also puts forth new recovery guarantees for this source separation algorithm. In particular, under the assumption that the signals can be perfectly described by a GMM model, we characterize necessary and sufficient conditions for reliable source separation in the asymptotic regime of low-noise as a function of the geometry of the underlying signals and their interaction. It is shown that if the subspaces spanned by the innovation components of the source signals with respect to the side information signals have zero intersection, provided that we observe a certain number of linear measurements from the mixture, then we can reliably separate the sources; otherwise we cannot. The second algorithms is based on deep learning where we introduce a novel self-supervised algorithm for the source separation problem. Source separation is intrinsically unsupervised and the lack of training data makes it a difficult task for artificial intelligence to solve. The proposed framework takes advantage of the available data and delivers near perfect separation results in real data scenarios. Our proposed frameworks – which provide new ways to incorporate side information to aid the solution of the source separation problem – are also employed in a real-world art investigation application involving the separation of mixtures of X-Ray images. The simulation results showcase the superiority of our algorithm against other state-of-the-art algorithms

Adapting image processing and clustering methods to productive efficiency analysis and benchmarking: A cross disciplinary approach

This dissertation explores the interdisciplinary applications of computational methods in quantitative economics. Particularly, this thesis focuses on problems in productive efficiency analysis and benchmarking that are hardly approachable or solvable using conventional methods. In productive efficiency analysis, null or zero values are often produced due to the wrong skewness or low kurtosis of the inefficiency distribution as against the distributional assumption on the inefficiency term. This thesis uses the deconvolution technique, which is traditionally used in image processing for noise removal, to develop a fully non-parametric method for efficiency estimation. Publications 1 and 2 are devoted to this topic, with focus being laid on the cross-sectional case and panel case, respectively. Through Monte-Carlo simulations and empirical applications to Finnish electricity distribution network data and Finnish banking data, the results show that the Richardson-Lucy blind deconvolution method is insensitive to the distributio-nal assumptions, robust to the data noise levels and heteroscedasticity on efficiency estimation. In benchmarking, which could be the next step of productive efficiency analysis, the 'best practice' target may not perform under the same operational environment with the DMU under study. This would render the benchmarks impractical to follow and adversely affects the managers to make the correct decisions on performance improvement of a DMU. This dissertation proposes a clustering-based benchmarking framework in Publication 3. The empirical study on Finnish electricity distribution network reveals that the proposed framework novels not only in its consideration on the differences of the operational environment among DMUs, but also its extreme flexibility. We conducted a comparison analysis on the different combinations of the clustering and efficiency estimation techniques using computational simulations and empirical applications to Finnish electricity distribution network data, based on which Publication 4 specifies an efficient combination for benchmarking in energy regulation.  This dissertation endeavors to solve problems in quantitative economics using interdisciplinary approaches. The methods developed benefit this field and the way how we approach the problems open a new perspective

Advanced VLBI Imaging

Very Long Baseline Interferometry (VLBI) is an observational technique developed in astronomy for combining multiple radio telescopes into a single virtual instrument with an effective aperture reaching up to many thousand kilometers and enabling measurements at highest angular resolutions. The celebrated examples of applying VLBI to astrophysical studies include detailed, high-resolution images of the innermost parts of relativistic outflows (jets) in active galactic nuclei (AGN) and recent pioneering observations of the shadows of supermassive black holes (SMBH) in the center of our Galaxy and in the galaxy M87. Despite these and many other proven successes of VLBI, analysis and imaging of VLBI data still remain difficult, owing in part to the fact that VLBI imaging inherently constitutes an ill-posed inverse problem. Historically, this problem has been addressed in radio interferometry by the CLEAN algorithm, a matching-pursuit inverse modeling method developed in the early 1970-s and since then established as a de-facto standard approach for imaging VLBI data. In recent years, the constantly increasing demand for improving quality and fidelity of interferometric image reconstruction has resulted in several attempts to employ new approaches, such as forward modeling and Bayesian estimation, for application to VLBI imaging. While the current state-of-the-art forward modeling and Bayesian techniques may outperform CLEAN in terms of accuracy, resolution, robustness, and adaptability, they also tend to require more complex structure and longer computation times, and rely on extensive finetuning of a larger number of non-trivial hyperparameters. This leaves an ample room for further searches for potentially more effective imaging approaches and provides the main motivation for this dissertation and its particular focusing on the need to unify algorithmic frameworks and to study VLBI imaging from the perspective of inverse problems in general. In pursuit of this goal, and based on an extensive qualitative comparison of the existing methods, this dissertation comprises the development, testing, and first implementations of two novel concepts for improved interferometric image reconstruction. The concepts combine the known benefits of current forward modeling techniques, develop more automatic and less supervised algorithms for image reconstruction, and realize them within two different frameworks. The first framework unites multiscale imaging algorithms in the spirit of compressive sensing with a dictionary adapted to the uv-coverage and its defects (DoG-HiT, DoB-CLEAN). We extend this approach to dynamical imaging and polarimetric imaging. The core components of this framework are realized in a multidisciplinary and multipurpose software MrBeam, developed as part of this dissertation. The second framework employs a multiobjective genetic evolutionary algorithm (MOEA/D) for the purpose of achieving fully unsupervised image reconstruction and hyperparameter optimization. These new methods are shown to outperform the existing methods in various metrics such as angular resolution, structural sensitivity, and degree of supervision. We demonstrate the great potential of these new techniques with selected applications to frontline VLBI observations of AGN jets and SMBH. In addition to improving the quality and robustness of image reconstruction, DoG-HiT, DoB-CLEAN and MOEA/D also provide such novel capabilities as dynamic reconstruction of polarimetric images on minute time-scales, or near-real time and unsupervised data analysis (useful in particular for application to large imaging surveys). The techniques and software developed in this dissertation are of interest for a wider range of inverse problems as well. This includes such versatile fields such as Ly-alpha tomography (where we improve estimates of the thermal state of the intergalactic medium), the cosmographic search for dark matter (where we improve forecasted bounds on ultralight dilatons), medical imaging, and solar spectroscopy

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented