Adaptive application of feature detection operators based on image variance

It is well known that the strength of a feature in an image may depend on the scale at which the appropriate detection operator is applied. It is also the case that many features in images exist significantly over a limited range of scales, and, of particular interest here, that the most salient scale may vary spatially over the feature. Hence, when designing feature detection operators, it is necessary to consider the requirements for both the systematic development and adaptive application of such operators over scale- and image-domains. We present an overview to the design of scalable derivative edge detectors, based on the finite element method, that addresses the issues of method and scale-adaptability. The finite element approach allows us to formulate scalable image derivative operators that can be implemented using a combination of piecewise-polynomial and Gaussian basis functions. The general adaptive technique may be applied to a range of operators. Here we evaluate the approach using image gradient operators, and we present comparative qualitative and quantitative results for both first and second order derivative methods.     

Content-adaptive feature extraction using image variance

The problem of scale is of fundamental interest in image processing, as the features that we visually perceive and find meaningful vary significantly depending on their size and extent. It is well known that the strength of a feature in an image may depend on the scale at which the appropriate detection operator is applied. It is also the case that many features in images exist significantly over a limited range of scales, and, of particular interest here, that the most salient scale may vary spatially over the feature. Hence, when designing feature detection operators, it is necessary to consider the requirements for both the systematic development and adaptive application of such operators over scale- and image-domains.

We present a new approach to the design of scalable derivative edge detectors, based on the finite element method, that addresses the issues of method and scale adaptability. The finite element approach allows us to formulate scalable image derivative operators that can be implemented using a combination of piecewise-polynomial and Gaussian basis functions. The issue of scale is addressed by partitioning the image in order to identify local key scales at which significant edge points may exist. This is achieved by consideration of empirically designed functions of local image variance.     

Circularity — a new principle underlying the design of accurate edge orientation operators

Currently available fast edge detection operators are reviewed. Their rationale is studied with particular reference to the estimation of angular orientation of edges. The idea of ‘circular’ edge detection operators is developed, and used to explain the surprising accuracy of the Sobel operator. A family of circular operators is designed for use in neighbourhoods of various sizes, and general ideas on edge operator design are developed. The concept of ‘closed bands’ of pixels for making neighbourhoods close to circular should be widely applicable in digital image analysis, not just in the ambit of fast edge detection.     

Design of steerable filters for feature detection using canny-like criteria

We propose a general approach for the design of 2D feature detectors from a class of steerable functions based on the optimization of a Canny-like criterion. In contrast with previous computational designs, our approach is truly 2D and provides filters that have closed-form expressions. It also yields operators that have a better orientation selectivity than the classical gradient or Hessian-based detectors. We illustrate the method with the design of operators for edge and ridge detection. We present some experimental results that demonstrate the performance improvement of these new feature detectors. We propose computationally efficient local optimization algorithms for the estimation of feature orientation. We also introduce the notion of shape-adaptable feature detection and use it for the detection of image corners.     

A computational approach to edge detection

This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.     

Numerical Simulation of the Sedimentation of Rigid Bodies in an Incompressible Viscous Fluid by Lagrange Multiplier/Fictitious Domain Methods Combined with the Taylor–Hood Finite Element Approximation

In this work we discuss an application of a distributed Lagrange multiplier based fictitious domain method, to the numerical simulation of the motion of rigid bodies settling in an incompressible viscous fluid. The solution method combines a third order finite element approximation, and time integration by operator splitting. Convergence results are shown for a simple Stokes flow with a circular rigid body that rotates with constant angular velocity. Results of numerical experiments for two sedimenting cylinders in a two-dimensional channel are presented. We present also results for the sedimentation of 100 and 504 cylinders.     

Nonlinear Filter Design Using Envelopes

Machine design of a signal or image operator involves estimating the optimal filter from sample data. The optimal filter is the best filter, relative to the error measure used; however, owing to design error, the designed filter might not perform well. In general it is suboptimal. The envelope constraint involves using two humanly designed filters that form a lower and upper bound for the designed operator. The method has been employed for binary operators. This paper considers envelope design for gray-scale filters, in particular, aperture filters. Some basic theoretical properties are stated, including optimality of the design method relative to the constraint imposed by the envelope. Examples are given for noise reduction and de-blurring.     

一种计算图象形态梯度的多尺度算法

分水岭变换是一种非常适用于图象分割的形态算子,然而,基于分水岭变换的图象分割方法,其性能在很大程度上依赖于用来计算待分割图象梯度的算法。为了高效地进行分水岭变换,提出了一种计算图象形态梯度的多尺度算法,从而对阶跃边缘和“模糊”边缘进行了有效的处理,此外,还提出了一种去除因噪声或量化误差造成的局部“谷底”的算法,实验结果表明,图象采用本文算法处理后,再进行分水岭变换,即使不进行区域合并,也能产生有意义的分割,因而极大地减轻了计算负担。     

Extracting image orientation feature by using integration operator

This paper presents an orientation operator to extract image local orientation features. We show that a proper employment of image integration leads to an unbiased orientation estimate, based on which an orientation operator is proposed. The resulting discrete operator has flexibility in the scale selection as the scale change does not violate the bias minimization criteria. An analytical formula is developed to compare orientation biases of various discrete operators. The proposed operator shows lower bias than eight well-known gradient operators. Experiments further demonstrate higher orientation accuracy of the proposed operator than these gradient operators.     

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $Q_1$ element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.     

基于改进高斯-拉普拉斯算子的噪声图像边缘检测方法

针对现有梯度算子在图像边缘检测中存在的对噪声比较敏感的问题,提出了一种基于改进高斯-拉普拉斯算子的图像边缘检测方法。噪声图像中的边缘检测是一项关键任务,然而目前常用的几种梯度算子,包括已经提出的高斯-拉普拉斯算子都没能取得理想效果。提出的方法对传统的拉普拉斯边缘检测算子做了改进,并与高斯滤波器相结合。首先,应用高斯滤波器来平滑图像,抑制噪声。然后基于拉普拉斯梯度边缘检测器进行边缘检测。最后在标准图像上进行评估,评估结果显示,提出的边缘检测方法所获得的峰值信噪比(PSNR)和均方误差(MSE)均优于其他几种对比方法。     

The Superconvergent Cluster Recovery Method

A new gradient recovery technique SCR (Superconvergent Cluster Recovery) is proposed and analyzed for finite element methods. A linear polynomial approximation is obtained by a least-squares fitting to the finite element solution at certain sample points, which in turn gives the recovered gradient at recovering points. Compared with similar techniques such as SPR and PPR, our approach is cheaper and efficient, while having same or even better accuracy. In additional, it can be used as an a posteriori error estimator, which is relatively simple to implement, cheap in terms of storage and computational cost for adaptive algorithms. We present some numerical examples illustrating the effectiveness of our recovery procedure.     

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L²-norm.     

Operator Splittings,Bregman Methods and Frame Shrinkage in Image Processing

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view—as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.     

Shape sensitivity analysis using low- and high-order finite elements

A general procedure to perform the sensitivity analysis for the shape optimal design of elastic structures is proposed. The method is based on the implicit differentiation of the discretized equilibrium equations used in the finite element method (FEM). The so-called semianalytical approach is followed, that is, finite differences are used to differentiate the finite element matrices. The technique takes advantage of the geometric modeling concepts typical of the computer-aided design (CAD) technology used in the creation of a compact design model. This procedure is largely independent of the types of finite elements used in the analysis and has been implemented in ah-version andp-version finite element program. Very accurate and stable shape sensitivity derivatives were obtained from both programs over a wide range of finite difference step sizes. It is shown that the method is computationally efficient, general, and relatively easy to implement. Some classical shape optimal design problems have been solved using the CONLIN optimizer supplied with these gradients.     

一种基于数学形态学的边缘检测方法

基于微分算子的边缘检测方法存在抗噪性能弱等缺点,非线性的数学形态学边缘检测可以克服这些缺点,而使用单一结构元素对图像进行形态学处理会模糊很多细节。鉴于这些不足,运用多结构多尺度的思想,将形态学的滤波和边缘检测结合起来,提出一种边缘检测方法,首先采用多结构多尺度的方法对噪声图像进行串联开闭滤波,再利用改进的多结构元素的形态学梯度算子进行边缘提取。实验结果表明,该方法具有较好的抗噪效果且提取的边缘比较平滑。     

Multilevel Training of Binary Morphological Operators

The design of binary morphological operators that are translation-invariant and locally defined by a finite neighborhood window corresponds to the problem of designing Boolean functions. As in any supervised classification problem, morphological operators designed from training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multi-level design approach to deal with the issue of designing large neighborhood based operators. The main idea is inspired from stacked generalization (a multi-level classifier design approach) and consists in, at each training level, combining the outcomes of the previous level operators. The final operator is a multi-level operator that ultimately depends on a larger neighborhood than of the individual operators that have been combined. Experimental results show that two-level operators obtained by combining operators designed on subwindows of a large window consistently outperforms the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multi-level approach to obtain better results.     

Gradient Recovery for Singularly Perturbed Boundary Value Problems I: One-Dimensional Convection-Diffusion

We consider a Galerkin finite element method that uses piecewise linears on a class of Shishkin-type meshes for a model singularly perturbed convection-diffusion problem. We pursue two approaches in constructing superconvergent approximations of the gradient. The first approach uses superconvergence points for the derivative, while the second one combines the consistency of a recovery operator with the superconvergence property of an interpolant. Numerical experiments support our theoretical results. Received November 12, 1999; revised September 9, 2000     

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ? 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.     

Geometric camery calibration: the virtual camera approach

Accurate dimensional measurements made with TV cameras require a careful geometric calibration of the imaging device. This work presents a new calibration approach that makes use of the virtual camera paradigm, in which a calibrated surface (target) plays the role of the image plane of a distortionless, fictitious camera. The method yields a simple closed form solution for the calibration parameters. We use an error analysis to estimate the influence of the calibration set-up on the angular measures. We use simulation to verify the analysis. The results of experimental tests are also reported, indicating good accuracy of the proposed procedure compared to some of the best-known calibration methods.