Wavelets and partial differential equations for image denoising

In this paper a wavelet based model for image de-noising is presented. Wavelet coefficients are modelled as waves that grow while dilating along scales. The model establishes a precise link between corresponding modulus maxima in the wavelet domain and then allows to predict wavelet coefficients at each scale from the first one. This property combined with the theoretical results about the characterization of singularities in the wavelet domain enables to discard noise. Significant structures of the image are well recovered while some annoying artifacts along image edges are reduced. Some experimental results show that the proposed approach outperforms the most recent and effective wavelet based denoising schemes

A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection

We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian distribution. The highly flexible variational structure of the corresponding regulariser encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM). A suitable preliminary variable splitting together with a novel result in multivariate non-convex proximal calculus yield a very efficient minimisation algorithm. Several numerical results showing significant quality-improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular in terms of texture and detail preservation

Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data

Regularization promotes well-posedness in solving an inverse problem with incomplete measurement data. The regularization term is typically designed based on a priori characterization of the unknown signal, such as sparsity or smoothness. The standard inhomogeneous regularization incorporates a spatially changing exponent $p$ of the standard $\ell_p$ norm-based regularization to recover a signal whose characteristic varies spatially. This study proposes a weighted inhomogeneous regularization that extends the standard inhomogeneous regularization through new exponent design and weighting using spatially varying weights. The new exponent design avoids misclassification when different characteristics stay close to each other. The weights handle another issue when the region of one characteristic is too small to be recovered effectively by the $\ell_p$ norm-based regularization even after identified correctly. A suite of numerical tests shows the efficacy of the proposed weighted inhomogeneous regularization, including synthetic image experiments and real sea ice recovery from its incomplete wave measurements

Left-invariant evolutions of wavelet transforms on the Similitude Group

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

시공간 자료의 다중척도 분석

학위논문(박사)--서울대학교 대학원 :자연과학대학 통계학과,2019. 8. 오희석.This thesis presents a multiscale analysis of spatio-temporal data. The content of this thesis consists of three chapters. First, we suggest an enhancement of the lifting scheme, one of the popular multiscale method, by using clustering-based network design. The proposed method is originally developed for enhancement of graph signal data, and the simulation and real data analysis results show that the proposed method has the advantage to reconstruct the noisy data compared to conventional lifting scheme method. Moreover, the advantage of the proposed method is not limited to the graph signal denoising. It is also shown that the proposed method the proposed neighborhood selection is able to combine with lifting one coefficient at a time (LOCAAT) algorithm, which is a lifting scheme algorithm frequently used in signal denoising. Second, we suggest a new lifting scheme concept which could be applied for streamflow data. It is impossible to apply the original lifting scheme to streamflow data directly because of its complex structure. In this thesis, to adapt the concept of lifting scheme to streamflow data, we suggest a new lifting scheme algorithm for streamflow data with flow-adaptive neighborhood selection, flow proportional weight generation, and flow-length adaptive removal point selection. By using the proposed method, we can successfully construct a multiscale analysis of streamflow data. Simulation study supports the performance of the lifting scheme for streamflow data is competitive for signal denoising. Besides, the proposed methods can visualize the multiscale structure of the network by adding or subtracting observations. Third, multiscale analysis for particulate matter data in Seoul is provided as a case study. We suggest a new method, which is a novel combi- nation of multiscale analysis and extreme value theory. The study starts from the idea that every climate event has its spatial or temporal event lengths. By changing the event area and duration time, we can estimate multiple extreme value parameters using generalized extreme value (GEV) distribution. Besides, we suggest a new property, called piecewise scaling property to combine multiple GEV estimators into a single equation. By using the proposed method, we can construct a return level map with ar- bitrary duration time and event area.이 논문은 다중 척도 분석을 시공간 자료에 응용한 방법들을 제시한다. 첫째, 그래프 신호 자료에서의 다중 척도 분석 방법 중 하나인 리프팅 스킴을 군집 에 기반한 이웃 재설정을 통해 기대 예측 오차를 줄일 수 있음을 보이고 이를 통해 리프팅 스킴의 성능을 향상하였다. 둘째, 공간적으로 복잡하고 방향성이 있는 구조에서 생성된 유량 네트워크 자료에 알맞는 리프팅 스킴을 구성하기 위해 네트워크의 특성을 반영한 이웃 선택, 예측 필터 구성 및 영역 설정으로 유량 네트워크 자료에 대한 리프팅 스킴을 구성하고 시공간 자료에의 확장 가능성을 살펴보았다. 마지막으로 서울특별시 고농도 미세먼지 자료를 다양 한 시간, 공간 및 시공간 집적을 통해 변환한 후 얻어진 일반화 극단값 모형의 모수들의 관계를 수문학에서 사용하는 강도-지속시간-발생빈도 곡선에 매듭 을 추가한 변형된 형태의 강도-지속시간-발생빈도 곡선을 따라 모델링하였고 사례 연구를 통해 원 자료의 복귀 수준 지도를 좀 더 정확히 묘사할 수 있음을 보였다.Abstract i 1 Introduction 1 2 Review: Multiscale analysis 4 2.1 Wavelets 4 2.1.1 Haar transforms 5 2.2 Multiresolution analysis 6 2.3 Lifting scheme 7 2.3.1 Lifting one coefficient at a time (LOCAAT) 11 2.3.2 Other lifting scheme methods 12 3 Enhancement of lifting scheme on graph signal data via clustering-based network design 14 3.1 Graph notations 15 3.2 Previous works 16 3.3 The use of clustering under the piecewise generalized moving average model 17 3.3.1 Piecewise generalized moving average model 18 3.3.2 Optimal UPA assignment under the piecewise homogeneous model 21 3.3.3 Extension to the spatio-temporal data 23 3.4 Simulation study 24 3.4.1 Stochastic block model 24 3.4.2 Image data analysis 26 3.4.3 Blocks signal denoising 29 3.5 Real data analysis 31 3.6 Summary and discussion 32 4 Streamflow lifting scheme 34 4.1 Dataset 36 4.2 Streamflow lifting scheme 37 4.2.1 Neighborhood selection 38 4.2.2 Prediction filter construction 39 4.2.3 Removal point selection 41 4.3 Simulation study 42 4.4 Real data analysis 47 4.5 Summary and further works 50 5 Multiscale analysis for PM10 extremes in Seoul 51 5.1 Data description 51 5.2 Temporal analysis of Seoul extreme PM10 data 54 5.2.1 Temporal aggregation and conventional scale property 54 5.2.2 Temporal multiscale modeling and modified scaling property 56 5.2.3 Result 1: GEV parameter estimation via piecewise linear approximation 57 5.2.4 Result 2: Return level map by the proposed modified scaling approach 62 5.3 Spatio-temporal multiscale analysis of Seoul extreme PM10 data 65 5.3.1 Spatio-temporal aggregation of Seoul extreme PM10 data 65 5.3.2 Result: Areal aggregation of Seoul extreme PM10 data 70 5.4 Summary and discussion 73 6 Concluding remarks 76 A Generalized extreme value distribution 77 B Scaling property theory 79 Abstract (in Korean) 85Docto

On a differential equation involving a new kind of variable exponents

In this paper, we are concerned with some new first order differential equation defined on the whole real axis R. The principal part of the equation involves an operator with variable exponent p depending on the variable x ∈ R through the unknown solution while the nonlinear part involves the classical variable exponent p(x). Such kind of situation is very related to the presence of the variable exponent and has not been treated before. Our existence result of nontrivial solution cannot be reached using standard variational or topological methods of nonlinear analysis and some sophisticated arguments have to be employed

Piecewise parameterised Markov random fields for semi-local Hurst estimation

Semi-local Hurst estimation is considered by incorporating a Markov random field model to constrain a wavelet-based pointwise Hurst estimator. This results in an estimator which is able to exploit the spatial regularities of a piecewise parametric varying Hurst parameter. The pointwise estimates are jointly inferred along with the parametric form of the underlying Hurst function which characterises how the Hurst parameter varies deterministically over the spatial support of the data. Unlike recent Hurst regularistion methods, the proposed approach is flexible in that arbitrary parametric forms can be considered and is extensible in as much as the associated gradient descent algorithm can accommodate a broad class of distributional assumptions without any significant modifications. The potential benefits of the approach are illustrated with simulations of various first-order polynomial forms

Advanced Restoration Techniques for Images and Disparity Maps

With increasing popularity of digital cameras, the field of Computa- tional Photography emerges as one of the most demanding areas of research. In this thesis we study and develop novel priors and op- timization techniques to solve inverse problems, including disparity estimation and image restoration. The disparity map estimation method proposed in this thesis incor- porates multiple frames of a stereo video sequence to ensure temporal coherency. To enforce smoothness, we use spatio-temporal connec- tions between the pixels of the disparity map to constrain our solution. Apart from smoothness, we enforce a consistency constraint for the disparity assignments by using connections between the left and right views. These constraints are then formulated in a graphical model, which we solve using mean-field approximation. We use a filter-based mean-field optimization that perform efficiently by updating the dis- parity variables in parallel. The parallel updates scheme, however, is not guaranteed to converge to a stationary point. To compare and demonstrate the effectiveness of our approach, we developed a new optimization technique that uses sequential updates, which runs ef- ficiently and guarantees convergence. Our empirical results indicate that with proper initialization, we can employ the parallel update scheme and efficiently optimize our disparity maps without loss of quality. Our method ranks amongst the state of the art in common benchmarks, and significantly reduces the temporal flickering artifacts in the disparity maps. In the second part of this thesis, we address several image restora- tion problems such as image deblurring, demosaicing and super- resolution. We propose to use denoising autoencoders to learn an approximation of the true natural image distribution. We parametrize our denoisers using deep neural networks and show that they learn the gradient of the smoothed density of natural images. Based on this analysis, we propose a restoration technique that moves the so- lution towards the local extrema of this distribution by minimizing the difference between the input and output of our denoiser. Weii demonstrate the effectiveness of our approach using a single trained neural network in several restoration tasks such as deblurring and super-resolution. In a more general framework, we define a new Bayes formulation for the restoration problem, which leads to a more efficient and robust estimator. The proposed framework achieves state of the art performance in various restoration tasks such as deblurring and demosaicing, and also for more challenging tasks such as noise- and kernel-blind image deblurring. Keywords. disparity map estimation, stereo matching, mean-field optimization, graphical models, image processing, linear inverse prob- lems, image restoration, image deblurring, image denoising, single image super-resolution, image demosaicing, deep neural networks, denoising autoencoder