Horseshoe regularization for wavelet-based lensing inversion

Gravitational lensing, a phenomenon in astronomy, occurs when the gravitational field of a massive object, such as a galaxy or a black hole, bends the path of light from a distant object behind it. This bending results in a distortion or magnification of the distant object's image, often seen as arcs or rings surrounding the foreground object. The Starlet wavelet transform offers a robust approach to representing galaxy images sparsely. This technique breaks down an image into wavelet coefficients at various scales and orientations, effectively capturing both large-scale structures and fine details. The Starlet wavelet transform offers a robust approach to representing galaxy images sparsely. This technique breaks down an image into wavelet coefficients at various scales and orientations, effectively capturing both large-scale structures and fine details. The horseshoe prior has emerged as a highly effective Bayesian technique for promoting sparsity and regularization in statistical modeling. It aggressively shrinks negligible values while preserving important features, making it particularly useful in situations where the reconstruction of an original image from limited noisy observations is inherently challenging. The main objective of this thesis is to apply sparse regularization techniques, particularly the horseshoe prior, to reconstruct the background source galaxy from gravitationally lensed images. By demonstrating the effectiveness of the horseshoe prior in this context, this thesis tackles the challenging inverse problem of reconstructing lensed galaxy images. Our proposed methodology involves applying the horseshoe prior to the wavelet coefficients of lensed galaxy images. By exploiting the sparsity of the wavelet representation and the noise-suppressing behavior of the horseshoe prior, we achieve well-regularized reconstructions that reduce noise and artifacts while preserving structural details. Experiments conducted on simulated lensed galaxy images demonstrate lower mean squared error and higher structural similarity with the horseshoe prior compared to alternative methods, validating its efficacy as an efficient sparse modeling technique.Les lentilles gravitationnelles se produisent lorsque le champ gravitationnel d'un objet massif dévie la trajectoire de la lumière provenant d'un objet lointain, entraînant une distorsion ou une amplification de l'image de l'objet lointain. La transformation Starlet fournit une méthode robuste pour obtenir une représentation éparse des images de galaxies, capturant efficacement leurs caractéristiques essentielles avec un minimum de données. Cette représentation réduit les besoins de stockage et de calcul, et facilite des tâches telles que le débruitage, la compression et l'extraction de caractéristiques. La distribution a priori de fer à cheval est une technique bayésienne efficace pour promouvoir la sparsité et la régularisation dans la modélisation statistique. Elle réduit de manière agressive les valeurs négligeables tout en préservant les caractéristiques importantes, ce qui la rend particulièrement utile dans les situations où la reconstruction d'une image originale à partir d'observations bruitées est difficile. Étant donné la nature mal posée de la reconstruction des images de galaxies à partir de données bruitées, l'utilisation de la distribution a priori devient cruciale pour résoudre les ambiguïtés. Les techniques utilisant une distribution a priori favorisant la sparsité ont été efficaces pour relever des défis similaires dans divers domaines. L'objectif principal de cette thèse est d'appliquer des techniques de régularisation favorisant la sparsité, en particulier la distribution a priori de fer à cheval, pour reconstruire les galaxies d'arrière-plan à partir d'images de lentilles gravitationnelles. Notre méthodologie proposée consiste à appliquer la distribution a priori de fer à cheval aux coefficients d'ondelettes des images de galaxies lentillées. En exploitant la sparsité de la représentation en ondelettes et le comportement de suppression du bruit de la distribution a priori de fer à cheval, nous obtenons des reconstructions bien régularisées qui réduisent le bruit et les artefacts tout en préservant les détails structurels. Des expériences menées sur des images simulées de galaxies lentillées montrent une erreur quadratique moyenne inférieure et une similarité structurelle plus élevée avec la distribution a priori de fer à cheval par rapport à d'autres méthodes, validant son efficacité

Machine Learning and its Application in Automatic Change Detection in Medical Images

Change detection is a fundamental problem in various fields, such as image surveillance, remote sensing, medical imaging, etc. The challenge of change detection in medical images is to detect disease-related changes while rejecting changes caused by noise, patient position change, and imaging acquisition artifacts such as field inhomogeneity. In this thesis, first, we overview the existing change detection methods, their underlying mathematical frameworks and limitations. Second, we present our contributions in solving the problem. We design optimal subspaces to approximate the background image in more efficient fashion. This is based on our structure principal component analysis, aiming to capture the structural similarity between scans in the context of change detection. We theoretically and numerically discuss the proper choices of norms used in the subspace approximation. The mathematical frameworks developed in this thesis consist of: (i) a new mathematical model to change detection by defining it as an optimization problem involving a cost function, input and output image sets, projection onto a subspace, and a similarity measure; (ii) development and implementation of numerical pipelines to compute the clinical changes by designing four mathematical algorithms; (iii) refining our algorithms by introducing the co-registration step utilizing the local dictionaries; and (iv) two new structure subspace learning models that are robust to outliers and noise, reduce the dimensionality of the dataset, and computationally efficient. We defined the co-registration step as a minimization problem involving a cost function, input and output image sets, a set of transform functions, projection onto a subspace, and a similarity measure. Based on the mathematical frameworks discussed above, numerical schemes are developed to automatically filter out clinically unrelated changes and identify true structure changes that may be of clinical importance. Our approaches are data-driven and utilize the knowledge of machine learning. We quantitatively analyze the performance of these algorithms using both synthetic and real human data. Our work has the potential to be used in computer aided diagnosis

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

Robust Algorithms for Low-Rank and Sparse Matrix Models

Data in statistical signal processing problems is often inherently matrix-valued, and a natural first step in working with such data is to impose a model with structure that captures the distinctive features of the underlying data. Under the right model, one can design algorithms that can reliably tease weak signals out of highly corrupted data. In this thesis, we study two important classes of matrix structure: low-rankness and sparsity. In particular, we focus on robust principal component analysis (PCA) models that decompose data into the sum of low-rank and sparse (in an appropriate sense) components. Robust PCA models are popular because they are useful models for data in practice and because efficient algorithms exist for solving them. This thesis focuses on developing new robust PCA algorithms that advance the state-of-the-art in several key respects. First, we develop a theoretical understanding of the effect of outliers on PCA and the extent to which one can reliably reject outliers from corrupted data using thresholding schemes. We apply these insights and other recent results from low-rank matrix estimation to design robust PCA algorithms with improved low-rank models that are well-suited for processing highly corrupted data. On the sparse modeling front, we use sparse signal models like spatial continuity and dictionary learning to develop new methods with important adaptive representational capabilities. We also propose efficient algorithms for implementing our methods, including an extension of our dictionary learning algorithms to the online or sequential data setting. The underlying theme of our work is to combine ideas from low-rank and sparse modeling in novel ways to design robust algorithms that produce accurate reconstructions from highly undersampled or corrupted data. We consider a variety of application domains for our methods, including foreground-background separation, photometric stereo, and inverse problems such as video inpainting and dynamic magnetic resonance imaging.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143925/1/brimoor_1.pd

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Sparse Coding Based Feature Representation Method for Remote Sensing Images

In this dissertation, we study sparse coding based feature representation method for the classification of multispectral and hyperspectral images (HSI). The existing feature representation systems based on the sparse signal model are computationally expensive, requiring to solve a convex optimization problem to learn a dictionary. A sparse coding feature representation framework for the classification of HSI is presented that alleviates the complexity of sparse coding through sub-band construction, dictionary learning, and encoding steps. In the framework, we construct the dictionary based upon the extracted sub-bands from the spectral representation of a pixel. In the encoding step, we utilize a soft threshold function to obtain sparse feature representations for HSI. Experimental results showed that a randomly selected dictionary could be as effective as a dictionary learned from optimization. The new representation usually has a very high dimensionality requiring a lot of computational resources. In addition, the spatial information of the HSI data has not been included in the representation. Thus, we modify the framework by incorporating the spatial information of the HSI pixels and reducing the dimension of the new sparse representations. The enhanced model, called sparse coding based dense feature representation (SC-DFR), is integrated with a linear support vector machine (SVM) and a composite kernels SVM (CKSVM) classifiers to discriminate different types of land cover. We evaluated the proposed algorithm on three well known HSI datasets and compared our method to four recently developed classification methods: SVM, CKSVM, simultaneous orthogonal matching pursuit (SOMP) and image fusion and recursive filtering (IFRF). The results from the experiments showed that the proposed method can achieve better overall and average classification accuracies with a much more compact representation leading to more efficient sparse models for HSI classification. To further verify the power of the new feature representation method, we applied it to a pan-sharpened image to detect seafloor scars in shallow waters. Propeller scars are formed when boat propellers strike and break apart seagrass beds, resulting in habitat loss. We developed a robust identification system by incorporating morphological filters to detect and map the scars. Our results showed that the proposed method can be implemented on a regular basis to monitor changes in habitat characteristics of coastal waters

Content-based Information Retrieval via Nearest Neighbor Search

Content-based information retrieval (CBIR) has attracted significant interest in the past few years. When given a search query, the search engine will compare the query with all the stored information in the database through nearest neighbor search. Finally, the system will return the most similar items. We contribute to the CBIR research the following: firstly, Distance Metric Learning (DML) is studied to improve retrieval accuracy of nearest neighbor search. Additionally, Hash Function Learning (HFL) is considered to accelerate the retrieval process. On one hand, a new local metric learning framework is proposed - Reduced-Rank Local Metric Learning (R2LML). By considering a conical combination of Mahalanobis metrics, the proposed method is able to better capture information like data\u27s similarity and location. A regularization to suppress the noise and avoid over-fitting is also incorporated into the formulation. Based on the different methods to infer the weights for the local metric, we considered two frameworks: Transductive Reduced-Rank Local Metric Learning (T-R2LML), which utilizes transductive learning, while Efficient Reduced-Rank Local Metric Learning (E-R2LML)employs a simpler and faster approximated method. Besides, we study the convergence property of the proposed block coordinate descent algorithms for both our frameworks. The extensive experiments show the superiority of our approaches. On the other hand, *Supervised Hash Learning (*SHL), which could be used in supervised, semi-supervised and unsupervised learning scenarios, was proposed in the dissertation. By considering several codewords which could be learned from the data, the proposed method naturally derives to several Support Vector Machine (SVM) problems. After providing an efficient training algorithm, we also study the theoretical generalization bound of the new hashing framework. In the final experiments, *SHL outperforms many other popular hash function learning methods. Additionally, in order to cope with large data sets, we also conducted experiments running on big data using a parallel computing software package, namely LIBSKYLARK