Fast inversion of matrices arising in image processing

In recent years, new nonlinear partial differential equation (PDE) based approaches have become popular for solving image processing problems. Although the outcome of these methods is often very promising, their actual realization is in general computationally intensive. Therefore, accurate and efficient schemes are needed. The aim of this paper is twofold. First, we will show that the three dimensional alignment problem of a histological data set of the human brain may be phrased in terms of a nonlinear PDE. Second, we will devise a fast direct solution technique for the associated structured large systems of linear equations. In addition, the potential of the derived method is demonstrated on real-life data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Application of learning algorithms for colour recognition on underwater images

Objects look very different in the underwater environment compared to their appearance in sunlight. High quality images with correct colouring simplify the detection of underwater objects. Hence, image processing is required to obtain images of high quality and correct colouring. Current algorithms focus on the colour reconstruction of scenery at diving depth where a significant part of sunlight is still present and different colours can still be distinguished. At greater depth the filtering is much stronger such that this is not possible. In this study it is investigated whether machine learning can be used to transform image data obtained in a controlled laboratory setup. The images are fed through learning machines with or without pre-filters. It is shown that k-nearest neighbour and support vector machines are most suitable for the given task and yield excellent results. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast incremental algorithm for speeding up the computation of binarization

Binarization is an important basic operation in image processing community. Based on the thresholded value, the gray image can be segmented into a binary image, usually consisting of background and foreground. Given the histogram of input gray image, based on minimizing the within-variance (or maximizing the between-variance), the Otsu method can obtain a satisfactory binary image. In this paper, we first transfer the within-variance criterion into a new mathematical formulation, which is very suitable to be implemented in a fast incremental way, and it leads to the same thresholded value. Following our proposed incremental computation scheme, an efficient heap- and quantization-based (HQ-based) data structure is presented to realize its implementation. Under eight real gray images, experimental results show that our proposed HQ-based incremental algorithm for binarization has 36% execution-time improvement ratio in average when compared to the Otsu method. Besides this significant speedup, our proposed HQ-based incremental algorithm can also be applied to speed up the Kittler and Illingworth method for binarization.     

图像处理中全变差正则化数据拟合问题算法回顾

全变差正则化数据拟合问题产生于许多图像处理任务,如图像去噪、去模糊、图像修复、磁共振成像、压缩图像感知等.近年来,求解此类问题的快速高效算法发展很快.以最小二乘、最小一乘等为例简要回顾求解此类问题的主要算法,并讨论一个全变差正则化非凸数据拟合模型在脉冲噪声图像去模糊问题中的应用.     

Multiwavelets and Integer Transforms

In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthogonal wavelets so that they map integers to integers. The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size. In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.     

基于L~1度量分析的图象分解与重构算法

Mallat‘s decompositon and reconstruction algorithms are very important in the the field of wavelet theory and its applications to signal processing.Wavelet Anal-ysis,which is based on L^2(R) space,can eliminate redundancy of signals with the help of orthogonality and characterize the processing precision with the meansquare error.In the recent years,it is understood that the mean square measuredoes not match human visual sensitivity well.From the point of view,R.DeVore studied L^1 measure instead.Similarly,considering the principles of image com-pression,Yang introduced and dealt with orthogonality in L^1 space based on thebest approximation theory,and consequently established the corresponding decom-position and reconstruction algorithms for signals.In this paper,error analyses for the algorithms above are taken and the selection of the best parameters in the algorithms are discussed in detail.Finally,the algorithms are compared with the classical Haar and Daubechies‘‘s orthogonal wavelets based on the singal-to-noiseratio data computed.     

A unified framework for 3D reconstruction

3D reconstruction is a branch of computer vision with a broad range of applications like computer aided design, animation, medicine and many others. In this talk we use continuous linear functionals for characterizing different kinds of 3D data. These problems can be tackled in the framework of Reproducing Kernel Hilbert Spaces and regularization of inverse problems. It can be said that 3D reconstruction is the general problem of estimating or finding functional dependencies from a three-dimensional data set. The origin of these data covers a wide range that includes tomography, surface reconstruction from point clouds or image and signal processing. Usually the problem of reconstruction has a very close relationship with scientific visualization and computer graphics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multilevel approach for nonnegative matrix factorization

Nonnegative matrix factorization (NMF), the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices, has been shown to be useful in many applications, such as text mining, image processing, and computational biology. In this paper, we explain how algorithms for NMF can be embedded into the framework of multilevel methods in order to accelerate their initial convergence. This technique can be applied in situations where data admit a good approximate representation in a lower dimensional space through linear transformations preserving nonnegativity. Several simple multilevel strategies are described and are experimentally shown to speed up significantly three popular NMF algorithms (alternating nonnegative least squares, multiplicative updates and hierarchical alternating least squares) on several standard image datasets.     

On a class of ill-posed minimization problems in image processing

In this paper, we show that minimization problems involving sublinear regularizing terms are ill-posed, in general, although numerical experiments in image processing give very good results. The energies studied here are inspired by image restoration and image decomposition. Rewriting the nonconvex sublinear regularizing terms as weighted total variations, we give a new approach to perform minimization via the well-known Chambolle's algorithm. The approach developed here provides an alternative to the well-known half-quadratic minimization one.     

Implementing an image based control algorithm for an optomechanical image derotator

In this paper an image-based control for an optomechanical image derotator is implemented. A derotator is an optical system to support measurements on rotating components by tracking their rotational movement. As a consequence, the position and rotational velocity of the measurement object has to be known continuously. In general this would be accomplished by measuring these variables using a rotary encoder. However, not all measuring objects are equipped for this task. As a solution universally applicable to a wide range of measuring objects, an image-based approach is developed in the scope of this work. The object is captured with a high-speed camera to determine its position and velocity by image processing algorithms. To proof the applicability of this concept, a controller using the data acquired with the camera and a controller using data of the rotary encoder are compared. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A cyclic projected gradient method

In recent years, convex optimization methods were successfully applied for various image processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently in Grewenig et al. (2010) that the simple idea of so-called “superstep cycles” leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. The “superstep cycles” approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years. In this paper, we investigate the incorporation of superstep cycles into the projected gradient method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in ?² with projections onto the Euclidean ball.     

Fast data extrapolating

Data-extrapolating (extension) technique has important applications in image processing on implicit surfaces and in level set methods. The existing data-extrapolating techniques are inefficient because they are designed without concerning the specialities of the extrapolating equations. Besides, there exists little work on locating the narrow band after data extrapolating—a very important problem in narrow band level set methods. In this paper, we put forward the general Huygens’ principle, and based on the principle we present two efficient data-extrapolating algorithms. The algorithms can easily locate the narrow band in data extrapolating. Furthermore, we propose a prediction–correction version for the data-extrapolating algorithms and the corresponding band locating method for a special case where the direct band locating method is hard to apply. Experiments demonstrate the efficiency of our algorithms and the convenience of the band locating method.     

基于PDE和几何曲率流驱动扩散的图像分析与处理

本文介绍由变分优化模型导出的偏微分方程(PDEs)模型与几何曲率流驱动扩散在图像恢复方面的应用，以及多种非线性异质扩散模型，讨论了PDEs模型在图像分析与处理方面的优点，理论与实验结果表明，要恢复得到商质量的图像，PDEs模型的利用是极为必要的．文中还介绍了求解PDEs模型的数值方案．其中，曲率计算是一个关键问题，其结果直接参与自适应扩散的控制．详细总结了基于有限差分和水平集方法，求解藕合非线性异质扩散模型方程的数值方案，追求高质量图像、高精度计算方法、降低计算复杂性是本文处理方法不断进步的发展动力。     

Using Conventional Edge Detectors and Postsmoothing for Segmentation of Spotted Microarray Images

Segmentation of spotted microarray images is important in generating gene expression data. It aims to distinguish foreground pixels from background pixels for a given spot of a microarray image. Edge detection in the image processing literature is a closely related research area, because spot boundary curves separating foregrounds from backgrounds in a microarray image can be treated as edges. However, for generating gene expression data, segmentation methods for handling spotted microarray images are required to classify each pixel as either a foreground or a background pixel; most conventional edge detectors in the image processing literature do not have this classification property, because their detected edge pixels are often scattered in the whole design space and consequently the foreground or background pixels are not defined. In this article, we propose a general postsmoothing procedure for estimating spot boundary curves from the detected edge pixels of conventional edge detectors, such that these conventional edge detectors together with the proposed postsmoothing procedure can be used for segmentation of spotted microarray images. Numerical studies show that this proposal works well in applications.

Datasets and computer code are available in the online supplements.     

The most-energetic traveltime of seismic waves

In tomographic image processing of seismic data, the first-arrival traveltime (FATT) is often different from those of more energetic wavefronts in realistic media. Since the traveltime of most-energetic wavefront (METT) dominates the data, computing the METT is recognized as an essential element in modern seismic imaging techniques. Solving the full wave equation is extremely expensive to be impractical even for large-size computers to carry out; the solution of the eikonal equation for which the corresponding amplitude is continuous is conjectured to be the METT.     

On continuous wavelet transforms of distributions

Wavelet analysis is a universal and promising tool with very rich mathematical content and great potential for applications in various scientific fields, in particular, in signal (image) processing and the theory of differential equations. On the other hand distributions are widely used in these fields. And to apply wavelet analysis in these areas it is important to define and investigate wavelet transforms of distributions. In this paper we introduce continuous wavelet transforms of distributions and study convergence properties of these transforms.     

一种基于遗传算法的图像增强方法

遗传算法是借鉴生物的自然选择和遗传进化机制而开发出的一种全局优化自适应概率搜索算法，目前已在包括图象处理在内的很多领域得到了很好的应用。在图像处理领域，目前的研究主要集中在将遗传算法用于图像分割，图像分类，模式识别等方面。本文将遗传算法用于图像增强，使用图像的参数模型。将图像增强转化为参数的优化，实验结果表明，该方法是一种有效的图像增强方法。     

n维数据统计信息增益的度量及应用

采样数据的增加究竟有多少相应的有效Fisher信息增益,这是测量数据处理、图像数据融合等应用领域中关心的问题.以(共轭)正态分布为基础,利用统计推断理论,导出一定相关性下样本数据的增加与统计信息(Fisher信息)增益之间的关系,并经一维航天测量数据和二维图像超分辨仿真算例验证.     

ROI-based procedures for progressive transmission of digital images: A comparison

Nowadays, problems arise when handling large-sized images (i.e. medical image such as Computed Tomographies or satellite images) of 10, 50, 100 or more Megabytes, due to the amount of time required for transmitting and displaying, this time being even worse when a narrow bandwidth transmission medium is involved (i.e. dial-up or mobile network), because the receiver must wait until the entire image has arrived. To solve this issue, progressive transmission schemes are used. These schemes allow the image sender to encode the image data in such a way that it is possible for the receiver to perform a reconstruction of the original image from the very beginning of transmission. Despite this reconstruction being, of course, partial, it is possible to improve the reconstruction on the fly, as more and more data of the original image are received. There are many progressive transmission methods available, such as it planes, TSVQ, DPCM, and, more recently, matrix polynomial interpolation, Discrete Cosine Transform (DCT, used in JPEG) and wavelets (used in JPEG 2000). However, none of them is well suited, or perform poorly, when, in addition to progressive transmission, we want to include also ROIs (Region Of Interest) handling. In the progressive transmission of ROIs, we want not only to reconstruct the image as we receive image data, but also to be able to select which part or parts of the emerging image we think are relevant and want to receive first, and which part or parts are of no interest. In this context we present an algorithm for lossy adaptive encoding based on singular value decomposition (SVD). This algorithm turns out to be well suited for progressive transmission and ROI selection of 2D and 3D images, as it is able to avoid redundancy in data transmission and does not require any sort of data recodification, even if we select arbitrary ROIs on the fly. We compare the performing of SVD with DCT and wavelets and show the results.     

Fixed point algorithm based on adapted metric method for convex minimization problem with application to image deblurring

Recently, optimization algorithms for solving a minimization problem whose objective function is a sum of two convex functions have been widely investigated in the field of image processing. In particular, the scenario when a non-differentiable convex function such as the total variation (TV) norm is included in the objective function has received considerable interests since many variational models encountered in image processing have this nature. In this paper, we propose a fast fixed point algorithm based on the adapted metric method, and apply it in the field of TV-based image deblurring. The novel method is derived from the idea of establishing a general fixed point algorithm framework based on an adequate quadratic approximation of one convex function in the objective function, in a way reminiscent of Quasi-Newton methods. Utilizing the non-expansion property of the proximity operator we further investigate the global convergence of the proposed algorithm. Numerical experiments on image deblurring problem demonstrate that the proposed algorithm is very competitive with the current state-of-the-art algorithms in terms of computational efficiency.