Optimal Filters with Multiresolution Apertures

This paper presents an extension to the recently introduced class of nonlinear filters known as Aperture Filters. By taking a multiresolution approach, it can be shown that more accurate filtering results (in terms of mean absolute error) may be achieved compared to the standard aperture filter given the same size of training set. Most optimisation techniques for nonlinear filters require a knowledge of the conditional probabilities of the output. These probabilities are estimated from observations of a representative training set. As the size of the training set is related to the number of input combinations of the filter, it increases very rapidly as the number of input variables increases. It can be impossibly large for all but the simplest binary filters. In order to design nonlinear filters of practical use, it is necessary to limit the size of the search space i.e. the number of possible filters (and hence the training set size) by the application of filter constraints. Filter constraints take several different forms, the most general of which is the window constraint where the output filter value is estimated from only a limited range of input variables.Aperture filters comprise a special case of nonlinear filters in which the input window is limited not only in its domain (or duration) but also in its amplitude. The reduced range of input signal leads directly to a reduction in the size of training set required to produce accurate output estimates. However in order to solve complex filtering problems, it is necessary for the aperture to be sufficiently large so as to observe enough of the signal to estimate its output accurately.In this paper it is shown how the input range of the aperture may be expanded without increasing the size of the search space by adopting a multiresolution approach. The constraint applied in this case is the resolution constraint. This paper presents both theoretical and practical results to demonstrate and quantify the improvement.     

Multiresolution Design of Aperture Operators

The design of an aperture operator is based on adequately constraining the spatial domain and the graylevel range in order to diminish the space of operators and, consequently, the estimation error. The design of a resolution constrained operator is based on adequately combining information from two or more different resolutions and has the same motivation, that is, diminish the space of operators to facilitate design. This paper joins these approaches and studies multiresolution design of aperture operators for grayscale images. Spatial resolution constraint, range resolution constraint and the combination of both constraints are characterized, and the error increase by using the constrained filter in place of the optimal unconstrained one is analyzed. Pyramidal multiresolution design involves applying the resolution constraint approach hierarchically, from the higher to the lower resolution space. These approaches are also characterized and their error increase analyzed. The system that has been implemented to design pyramidal multiresolution operators is described and has its complexity (memory and runtime) analyzed. Several simulations and two applications for deblurring are shown and compared to optimal linear filters. The results confirm the usefulness of the approach.     

Image Interpolation Using Enhanced Multiresolution Critical-Point Filters

There are increasing demands for image interpolation in various fields such as virtual reality, computer animation, and video transmission. The critical-point filters (CPF) we have proposed previously enable completely automatic matching of two images. In our previous method, however, it has taken a long time to compute the optimal parameter values by iterative searches. This paper proposes an improved algorithm called the enhanced critical-point filters (ECPF) where the parameters are stably computed using together the inverse mapping from the destination to the source. It takes only a second to match images whose size is 64 × 64. The algorithm is also improved in its precision by directly handling color images while the previous algorithm has taken only the intensity value into account. We apply ECPF to keyframe interpolation of video sequences. We also apply ECPF to interpolating two different views of an object to generate any intermediate views. In this case, there are many constraints that can be used to determine the mapping between the images. We propose a method to use such constraints to improve the accuracy of the mappings. As the results, image-based pseudo-3D models are easily created from a set of views without any special devices.     

Optimal Binary Morphological Bandpass Filters Induced by Granulometric Spectral Representation

Euclidean granulometries are used to decompose a binary image into a disjoint union based on interaction between image shape and the structuring elements generating the granulometry. Each subset of the resulting granulometric spectral bands composing the union defines a filter by passing precisely the bands in the subset. Given an observed image and an ideal image to be estimated, an optimal filter must minimize the expected symmetric-difference error between the ideal image and filtered observed image. For the signal-union-noise model, and for both discrete and Euclidean images, given a granulometry, a procedure is developed for finding a filter that optimally passes bands of the observed noisy image. The key is characterization of an optimal filter in the Euclidean case. Optimization is achieved by decomposing the mean functions of the signal and noise size distributions into singular and differentiable parts, deriving an error representation based on the decomposition, and describingoptimality in terms of generalized derivatives for the singular parts and ordinary derivatives for the differentiable parts. Owing to the way in which spectral bands are optimally passed, there are strong analogies with the Wiener filter.     

多分辨率图象锥的产生方法及分析

利用多分辨率技术处理和分析图象是近年来的一个研究热点。多分辨率图象锥为多分辨率处理和分析图象提供了一种帝用的、有铲的数据结构和工具,本文阐述了多分辨率图 象锥的概念,用统一的框架分析了现有的多分辨率图象锥产生方法,建立了产生多分辨率图象锥的统一模型,分析了利用多分辨率图象锥解决图象处理和分析任务中面临的问
问题。     

Optimal Conjunctive Granulometric Bandpass Filters

A conjunctive homothetic granulometry is an intersection of openings by independently scaled structuring elements. Like classical Euclidean granulometries, conjunctive granulometries possess size distributions and pattern spectra; however, they are based on intersections of openings by scaled structuring elements instead of unions of such openings. For a disjunctive granulometry, which is a union of openings, a grain (or part thereof) is passed if there exists a translate of at least one structuring element that is a subset of the grain; for a conjunctive granulometry, there must exist a translate of each structuring element that is a subset of the grain. Like disjunctive granulometries, they possess size distributions; however, unlike disjunctive granulometries, their pattern spectra are not probability densities. An optimal granulometric bandpass filter passes image components so as to minimize the expected area of the symmetric difference between the filtered and ideal images. This paper provides an analytic formulation of optimal conjunctive granulometric bandpass filters. The theory provides one of the few areas of nonlinear image processing in which three of the basic components of linear optimization apply: (1) there is an analytic expression determining the optimal filter; (2) there is an explicit error formula; (3) and there is a closed-form representation of the optimal filter based on a decomposition of the observed random image. These correspond to the Wiener-Hopf equation, mean-square-error formula, and filter representation via an integral canonical representation of the observed image in linear filtering.     

Design and Analysis of Robust Binary Filters in the Context of a Prior Distribution for the States of Nature

An optimal binary-image filter estimates an ideal random set by means of an observed random set. A fundamental and practically important question regards the robustness of a designed filter: to what extent does performance degrade when the filter is applied to a different model than the one for which it has been designed? By parameterizing the ideal and observation random sets, one can analyze the robustness of filter design relative to parameter states. Based on a prior distribution for the states, a robustness mesure is defined for each state in terms of how well its optimal filter performs on models for different states. Not only is filter performance on other states taken into account, but so too is the contribution of other states in terms of their mass relative to the prior state distribution. This paper characterizes maximally robust states, derives performance bounds, treats mean robustness (as opposed to robustness by state), introduces a global filter that is applied across all states, particularizes the entire analysis to a sparse noise model for which there are analytic robustness expressions, and proposes a simplified model for determination of robust states from data. Sufficient conditions are given under which the global filter is uniformly more robust than all state-specific optimal filters.     

滤波长度为4的双正交多尺度分析的构造

该文研究了滤波长度为4的双正交多尺度分析的一般构造，依据Lawton条件，通过解决一个线性代数问题，求出了其实滤波系数所在的范围，给出了一些构造的例子，并通过计算和分析这些小波用于图像压缩的熵和信噪比数据，研究了它们用于图像压缩的性能。     

Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.     

基于小波多分辨率分析的故障检测方法

提出了利用小波多分辨率分析技术对系统信号进行分析的故障检测方法。利用小波多分辨率分析技术对信号进行多尺度小波分解,得到信号的高频分量,对此高频分量进行去噪和特征提取,可以得到反映系统故障的特征,实现对系统的故障检测,并可根据阈值大小的不同区分故障程度,实现对不同故障的区分。为避免因阈值选择不当可能造成的故障漏检,对于去噪阈值的选择,提出了一个有效的解决方法。仿真结果证明该故障检测方法的有效性和可行性。     

Multiresolution Tiling

This paper describes an efficient method for constructing a tiling between a pair of planar contours. The problem is of interest in a number of domains, including medical imaging, biological research and geological reconstructions. Our method, based on ideas from multiresolution analysis and wavelets, requires O(n) space and appears to require O(n log n) time for average inputs, compared to the O(n²) space and O(n² log n) time required by the optimizing algorithm of Fuchs, Kedem and Uselton¹. The results computed by our algorithm are in many cases nearly the same as those of the optimizing algorithm, but at a small fraction of the computational cost. The performance improvement makes the algorithm usable for large contours in an interactive system. The use of multiresolution analysis provides an efficient mechanism for data compression by discarding wavelet coefficients smaller than a threshold value during reconstruction. The amount of detail lost can be controlled by appropriate choice of the threshold value. The use of lower resolution approximations to the original contours yields significant savings in the time required to display a reconstructed object, and in the space required to store it.     

一种基于小波多分辨分析的多率采样系统辨识方法

基于小波多分辨分析理论并结合面向控制的辨识思想, 提出一种多率采样系统分频段加权辨识方法. 首先研究了采样信息的一致性重构问题, 然后给出一种新颖的分频段加权辨识方法. 此方法的最大特点是对噪声不敏感, 易实现对感兴趣频段的精确建模, 便于和控制系统设计相配合. 仿真结果验证了这种方法的可行性.     

基于投影的多尺度小波矩快速算法

多尺度小波变换矩采用几何矩定义，由信号在某一尺度下的小波逼近系数计算得到，它反映的是信号在该尺度下小波分解对应的视觉不变量。由于受到矩本身计算量的限制，小波矩的计算很难实时实现。本文提出了一种基于投影变换的二维多尺度小波变换矩快速算法。该快速算法通过投影变换将二维图像小波变换低频分量信息投影到一维空间中，从而降低了多尺度小波变换矩的计算复杂度，保证了计算精度。     

基于多尺度小波分析的激光探测系统消噪

根据激光测距探测系统中噪声的特点，分析了去噪阈值的选取，详细地讨论了小波的多尺度浮动阈值方法，它能较好地满足在强噪声下检测微弱信号的要求，并将这种方法应用于激光探测系统去噪，做到滤除噪声的同时有效地保留信号的细节，收到了较好的效果。     

非线性动态滤波的迭代算法

阐述了标称状态的线性化方法和扩展的卡尔曼滤波公式及迭代卡尔曼滤波,探讨了非线性动态滤波的近似处理方法,围绕标称状态将非线性模型进行线性化,将标准的卡尔曼滤波扩展到非线性模型,得到扩展的卡尔曼滤波公式,研究了迭代滤波计算方法。扩展的卡尔曼滤波方法已经有效地用于非线性模型。     

多分辨率形态学目标检测

用形态小波变换得到图像的塔式表示,并从最低分辨率到原始分辨率由粗到精地提取目标区域.在每个分辨率上,先应用分水岭变换分割该分辨率下的低频分量（或上一级分辨率得到的标记图）,得到一个标记图;再用一个区域搜索策略来更新该标记图.对多类目标的实验结果验证了该算法具有速度快、精度高和对噪声不敏感的优点.     

适于点绘制的层次多分辨率建模方法的研究

在基于点的图形绘制技术中,需要记录大量的场景采样点,如何有效地组织这些数据是实现快速高效绘制的关键问题之一。论文研究了适于点绘制的有效数据组织——层次多分辨率建模,分析了三种常用的模型结构及其构建算法,讨论了模型结构中的层次细节选择方案,提出了一种层次细节的顺序选择方法,实验证明这些方法是有效的。     

基于小波的B样条曲线多分辨表示及编辑

多分辨表示方法为曲线提供了更为灵活的表达方式,使得我们可以在不同分辨率下对曲线进行编辑,小波技术是实现曲线多分辨表示的一种新颖方法,已有许多论文从理论上论述了这项技术,文中从几何概念出发,由浅入深地论述了基于小波的准均匀三次B样条曲线多分辨表示的原理及其实现,并通过实例描述了B样条曲线的多分辨编辑。     

Horizon Picking on Subbottom Profiles Using Multiresolution Analysis

Maroni, C.-S., Quinquis, A., and Vinson, S., Horizon Picking on Subbottom Profiles Using Multiresolution Analysis, Digital Signal Processing11 (2001) 269–287A fully automatic algorithm is proposed for the mapping of sediment layers on subbottom profiles. This mapping should significantly speed up data analysis and sedimentary data base building. The proposed method combines two techniques: edge following algorithms and multiresolution analysis using the wavelet transform. Image analysis at different scales allows us to follow sediment layer contours without interruption, and with adequate accuracy. In order to reduce detection errors a thresholding with hysteresis is first performed. The thresholds are automatically set according to data distribution. Performances of the algorithm are discussed, from testing on a set of real subbottom profile data.     

A Wavelet-Based Multiresolution Method to Automatically Register Images

We present a method to automatically register images presenting bothglobal and local deformations. The image registration process isperformed by exploiting a multi-level/multi-image approach wherebyafter having wavelet-transformed the images, the subband images atdifferent levels are used in a non-feature-based way to determine themotion vectors between the reference and the target images. The crudemotion field determined by block matching at the coarsest level ofthe pyramid is successively refined by taking advantage of both theorientation sensitivity of the different subbands and thecontribution of the adjacent levels. The final registered image isobtained by applying the motion field to the lowest level of thepyramid and by inversely transforming it.