영상 잡음 제거와 수중 영상 복원을 위한 정규화 방법

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 강명주.In this thesis, we discuss regularization methods for denoising images corrupted by Gaussian or Cauchy noise and image dehazing in underwater. In image denoising, we introduce the second-order extension of structure tensor total variation and propose a hybrid method for additive Gaussian noise. Furthermore, we apply the weighted nuclear norm under nonlocal framework to remove additive Cauchy noise in images. We adopt the nonconvex alternating direction method of multiplier to solve the problem iteratively. Subsequently, based on the color ellipsoid prior which is effective for restoring hazy image in the atmosphere, we suggest novel dehazing method adapted for underwater condition. Because attenuation rate of light varies depending on wavelength of light in water, we apply the color ellipsoid prior only for green and blue channels and combine it with intensity map of red channel to refine the obtained depth map further. Numerical experiments show that our proposed methods show superior results compared with other methods both in quantitative and qualitative aspects.본 논문에서 우리는 가우시안 또는 코시 분포를 따르는 잡음으로 오염된 영상과 물 속에서 얻은 영상을 복원하기 위한 정규화 방법에 대해 논의한다. 영상 잡음 문제에서 우리는 덧셈 가우시안 잡음의 해결을 위해 구조 텐서 총변이의 이차 확장을 도입하고 이것을 이용한 혼합 방법을 제안한다. 나아가 덧셈 코시 잡음 문제를 해결하기 위해 우리는 가중 핵 노름을 비국소적인 틀에서 적용하고 비볼록 교차 승수법을 통해서 반복적으로 문제를 푼다. 이어서 대기 중의 안개 낀 영상을 복원하는데 효과적인 색 타원면 가정에 기초하여, 우리는 물 속의 상황에 알맞은 영상 복원 방법을 제시한다. 물 속에서 빛의 감쇠 정도는 빛의 파장에 따라 달라지기 때문에, 우리는 색 타원면 가정을 영상의 녹색과 청색 채널에 적용하고 그로부터 얻은 깊이 지도를 적색 채널의 강도 지도와 혼합하여 개선된 깊이 지도를 얻는다. 수치적 실험을 통해서 우리가 제시한 방법들을 다른 방법과 비교하고 질적인 측면과 평가 지표에 따른 양적인 측면 모두에서 우수함을 확인한다.1 Introduction 1 1.1 Image denoising for Gaussian and Cauchy noise 2 1.2 Underwater image dehazing 5 2 Preliminaries 9 2.1 Variational models for image denoising 9 2.1.1 Data-fidelity 9 2.1.2 Regularization 11 2.1.3 Optimization algorithm 14 2.2 Methods for image dehazing in the air 15 2.2.1 Dark channel prior 16 2.2.2 Color ellipsoid prior 19 3 Image denoising for Gaussian and Cauchy noise 23 3.1 Second-order structure tensor and hybrid STV 23 3.1.1 Structure tensor total variation 24 3.1.2 Proposed model 28 3.1.3 Discretization of the model 31 3.1.4 Numerical algorithm 35 3.1.5 Experimental results 37 3.2 Weighted nuclear norm minimization for Cauchy noise 46 3.2.1 Variational models for Cauchy noise 46 3.2.2 Low rank minimization by weighted nuclear norm 52 3.2.3 Proposed method 55 3.2.4 ADMM algorithm 56 3.2.5 Numerical method and experimental results 58 4 Image restoration in underwater 71 4.1 Scientific background 72 4.2 Proposed method 73 4.2.1 Color ellipsoid prior on underwater 74 4.2.2 Background light estimation 78 4.3 Experimental results 80 5 Conclusion 87 Appendices 89Docto

Geodesic Active Fields:A Geometric Framework for Image Registration

Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

Higher-order Losses and Optimization for Low-level and Deep Segmentation

Regularized objectives are common in low-level and deep segmentation. Regularization incorporates prior knowledge into objectives or losses. It represents constraints necessary to address ill-posedness, data noise, outliers, lack of supervision, etc. However, such constraints come at significant costs. First, regularization priors may lead to unintended biases, known or unknown. Since these can adversely affect specific applications, it is important to understand the causes & effects of these biases and to develop their solutions. Second, common regularized objectives are highly non-convex and present challenges for optimization. As known in low-level vision, first-order approaches like gradient descent are significantly weaker than more advanced algorithms. Yet, variants of the gradient descent dominate optimization of the loss functions for deep neural networks due to their size and complexity. Hence, standard segmentation networks still require an overwhelming amount of precise pixel-level supervision for training. This thesis addresses three related problems concerning higher-order objectives and higher-order optimizers. First, we focus on a challenging application—unsupervised vascular tree extraction in large 3D volumes containing complex ``entanglements" of near-capillary vessels. In the context of vasculature with unrestricted topology, we propose a new general curvature-regularizing model for arbitrarily complex one-dimensional curvilinear structures. In contrast, the standard surface regularization methods are impractical for thin vessels due to strong shrinking bias or the complexity of Gaussian/min curvature modeling for two-dimensional manifolds. In general, the shrinking bias is one well-known example of bias in the standard regularization methods. The second contribution of this thesis is a characterization of other new forms of biases in classical segmentation models that were not understood in the past. We develop new theories establishing data density biases in common pair-wise or graph-based clustering objectives, such as kernel K-means and normalized cut. This theoretical understanding inspires our new segmentation algorithms avoiding such biases. The third contribution of the thesis is a new optimization algorithm addressing the limitations of gradient descent in the context of regularized losses for deep learning. Our general trust-region algorithm can be seen as a high-order chain rule for network training. It can use many standard low-level regularizers and their powerful solvers. We improve the state-of-the-art in weakly-supervised semantic segmentation using a well-motivated low-level regularization model and its graph-cut solver

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

What's new and what's next in diffusion MRI preprocessing

Diffusion MRI (dMRI) provides invaluable information for the study of tissue microstructure and brain connectivity, but suffers from a range of imaging artifacts that greatly challenge the analysis of results and their interpretability if not appropriately accounted for. This review will cover dMRI artifacts and preprocessing steps, some of which have not typically been considered in existing pipelines or reviews, or have only gained attention in recent years: brain/skull extraction, B-matrix incompatibilities w.r.t the imaging data, signal drift, Gibbs ringing, noise distribution bias, denoising, between- and within-volumes motion, eddy currents, outliers, susceptibility distortions, EPI Nyquist ghosts, gradient deviations, bias fields, and spatial normalization. The focus will be on “what’s new” since the notable advances prior to and brought by the Human Connectome Project (HCP), as presented in the predecessing issue on “Mapping the Connectome” in 2013. In addition to the development of novel strategies for dMRI preprocessing, exciting progress has been made in the availability of open source tools and reproducible pipelines, databases and simulation tools for the evaluation of preprocessing steps, and automated quality control frameworks, amongst others. Finally, this review will consider practical considerations and our view on “what’s next” in dMRI preprocessing

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences