Image interpolation by two-dimensional parametric cubic convolution.

Cubic convolution is a popular method for image interpolation. Traditionally, the piecewise-cubic kernel has been derived in one dimension with one parameter and applied to two-dimensional (2-D) images in a separable fashion. However, images typically are statistically nonseparable, which motivates this investigation of nonseparable cubic convolution. This paper derives two new nonseparable, 2-D cubic-convolution kernels. The first kernel, with three parameters (designated 2D-3PCC), is the most general 2-D, piecewise-cubic interpolator defined on [-2, 2] x [-2, 2] with constraints for biaxial symmetry, diagonal (or 90 degrees rotational) symmetry, continuity, and smoothness. The second kernel, with five parameters (designated 2D-5PCC), relaxes the constraint of diagonal symmetry, based on the observation that many images have rotationally asymmetric statistical properties. This paper also develops a closed-form solution for determining the optimal parameter values for parametric cubic-convolution kernels with respect to ensembles of scenes characterized by autocorrelation (or power spectrum). This solution establishes a practical foundation for adaptive interpolation based on local autocorrelation estimates. Quantitative fidelity analyses and visual experiments indicate that these new methods can outperform several popular interpolation methods. An analysis of the error budgets for reconstruction error associated with blurring and aliasing illustrates that the methods improve interpolation fidelity for images with aliased components. For images with little or no aliasing, the methods yield results similar to other popular methods. Both 2D-3PCC and 2D-5PCC are low-order polynomials with small spatial support and so are easy to implement and efficient to apply.     

High-accuracy interpolator design with reduced first-order derivative constraints

In this article, we propose a new frequency-domain weighted-least-squares method for designing high-accuracy interpolator (interpolation kernel) that reduces the number of the first-order derivative continuity constraints. By reducing the number of the first-order derivative constraints, we can increase the degree of freedom in the design, and thus have more flexibility to get more accurate design results. The interpolator consists of 6 piecewise polynomials of the third degree (cubic), and it is performed in the frequency-domain through minimising the weighted integrated-squared-error of the spectrum (frequency response). The weighting function is adjusted so as to ignore some insignificant frequency bands and put more emphasis on the important frequency bands. By imposing the continuity constraints on the interpolator itself as well as the reduced first-order derivative constraints at the contacting points, we get three free parameters of the interpolator. These three parameters are then optimised in such a way that the weighted integrated-squared-error of the frequency response is minimised. We will utilise a narrow-band example to demonstrate the performance improvement over other existing interpolators.     

Image reconstruction by convolution with symmetrical piecewisenth-order polynomial kernels

The reconstruction of images is an important operation in many applications. From sampling theory, it is well known that the sine-function is the ideal interpolation kernel which, however, cannot be used in practice. In order to be able to obtain an acceptable reconstruction, both in terms of computational speed and mathematical precision, it is required to design a kernel that is of finite extent and resembles the sinc-function as much as possible. In this paper, the applicability of the sine-approximating symmetrical piecewise nth-order polynomial kernels is investigated in satisfying these requirements. After the presentation of the general concept, kernels of first, third, fifth and seventh order are derived. An objective, quantitative evaluation of the reconstruction capabilities of these kernels is obtained by analyzing the spatial and spectral behavior using different measures, and by using them to translate, rotate, and magnify a number of real-life test images. From the experiments, it is concluded that while the improvement of cubic convolution over linear interpolation is significant, the use of higher order polynomials only yields marginal improvement.     

Frequency-domain weighted-least-squares design of quadratic interpolators

Existing finite-support interpolators are derived from continuities in the time-domain. In this study, the authors optimally design a quadratic interpolator using two second-degree piecewise polynomials in the frequency-domain. The optimal coefficients of the piecewise polynomials are found by minimising the weighted least-squares error between the ideal and actual frequency responses of the quadratic interpolator subject to a few constraints. Adjusting the weighting functions in different frequency bands can yield accurate frequency response in a specified passband and even can ignore `don`t care` bands so that various quadratic interpolators can be designed for interpolating various discrete signals containing different frequency components. One-dimensional and two-dimensional examples have shown that the quadratic interpolator can achieve much higher interpolation accuracy than the existing interpolators for wide-band signals, and various images have been tested to verify that the quadratic interpolator can achieve comparable interpolation accuracy as the Catmull-Rom cubic for narrow-band signals (images), but the computational complexity can be reduced to about 70%. Therefore both narrow-band and wide-band signals can be interpolated with high accuracy.     

Two-dimensional cubic convolution

The paper develops two-dimensional (2D), nonseparable, piecewise cubic convolution (PCC) for image interpolation. Traditionally, PCC has been implemented based on a one-dimensional (1D) derivation with a separable generalization to two dimensions. However, typical scenes and imaging systems are not separable, so the traditional approach is suboptimal. We develop a closed-form derivation for a two-parameter, 2D PCC kernel with support [-2,2]/spl times/[-2,2] that is constrained for continuity, smoothness, symmetry, and flat-field response. Our analyses, using several image models, including Markov random fields, demonstrate that the 2D PCC yields small improvements in interpolation fidelity over the traditional, separable approach. The constraints on the derivation can be relaxed to provide greater flexibility and performance.     

基于非均匀采样重构的图像插值算法

在大规模集成电路实现图像缩放时,通常使用的算法是插值。基于信号的非均匀采样重构算法,提出了一种新的图像插值算法[1]。该插值算法所使用的基函数是基于余弦和函数。本算法同其他算法相比,在插值基函数长度相同的情况下,本算法的性能最好。相较其他算法,本算法保持了图像中更多的高频分量,同时减少了插值过程中的图像混淆效应。同时并没有增加插值所需的运算资源。     

A constrained weighted least squares approach for time-frequencydistribution kernel design

In most applications of time-frequency (t-f) distributions, the t-f kernel is of finite extent and applied to discrete time signals. This paper introduces a matrix-based approach for t-f distribution kernel design. In this new approach, the optimum kernel is obtained as the solution of a linearly constrained weighted least squares minimization problem in which the kernel is vectorial and the constraints form a linear subspace. Similar to FIR temporal and spatial constrained least squares (LS) design methods, the passband, stopband, and transition band of an ideal kernel are first specified. The optimum kernel that best approximates the ideal kernel in the LS error sense, and simultaneously satisfies the multiple linear constraints, is then obtained using closed-form expressions. This proposed design method embodies a well-structured procedure for obtaining fixed and data-dependent kernels that are difficult to obtain using other design approaches     

Restoration and reconstruction of AVHRR images

Describes the design of small convolution kernels for the restoration and reconstruction of Advanced Very High Resolution Radiometer (AVHRR) images. The kernels are small enough to be implemented efficiently by convolution, yet effectively correct degradations and increase apparent resolution. The kernel derivation is based on a comprehensive, end-to-end system model that accounts for scene statistics, image acquisition blur, sampling effects, sensor noise, and postfilter reconstruction. The design maximizes image fidelity subject to explicit constraints on the spatial support and resolution of the kernel. The kernels can be designed with finer resolution than the image to perform partial reconstruction for geometric correction and other remapping operations. Experiments demonstrate that small kernels yield fidelity comparable to optimal unconstrained filters with less computation     

Short kernel fifth-order interpolation

An explicit analytical formula for a short kernel fifth-order polynomial interpolator is obtained. It is also possible to obtain the explicit forms of even higher order interpolation kernels with the method of calculation used, but it is seen that these local kernels become “remainder-dominated” as the order increases. The frequency domain properties and the accuracies of the obtained kernel and the known convolution kernels are compared. Frequency domain comparison with the cubic B-spline interpolators is also given. Some cases of proper use of the calculated kernels have been pointed out     

一族新的布尔核函数及其应用

核机器（Kernel Machine）已成为机器学习领域的热点研究问题。针对只具有离散属性的分类问题，在对合取范式进行深入分析的基础上提出了一族新的布尔核函数。利用这些布尔核函数，可以在布尔逻辑学习、决策树，决策规则学习以及基于项集的学习中，引入核机器技术。实验结果指出，使用结构简单而符合训练数据集特征的布尔核函数，有助于显著提高分类器的性能。     

Sampling signals with finite rate of innovation

The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems     

Polynomial-Based Interpolation Filters—Part I: Filter Synthesis

This paper introduces a generalized design method for polynomial-based interpolation filters. These filters can be implemented by using a modified Farrow structure, where the fixed finite impulse response (FIR) sub-filters possess either symmetrical or anti-symmetrical impulse responses. In the proposed approach, the piecewise polynomial impulse response of the interpolation filter is optimized directly in the frequency domain using either the minimax or least mean square criterion subject to the given time domain constraints. The length of the impulse response and the degree of the approximating polynomial in polynomial intervals can be arbitrarily selected. The optimization in the frequency domain makes the proposed design scheme more suitable for various digital signal processing applications and enables one to synthesize interpolation filters for arbitrary desired and weighting functions. Most importantly, the interpolation filters can be optimized in a manner similar to that of conventional linear-phase FIR filters.     

Complete parameterization of piecewise-polynomial interpolation kernels

Every now and then, a new design of an interpolation kernel appears in the literature. While interesting results have emerged, the traditional design methodology proves laborious and is riddled with very large systems of linear equations that must be solved analytically. We propose to ease this burden by providing an explicit formula that can generate every possible piecewise-polynomial kernel given its degree, its support, its regularity, and its order of approximation. This formula contains a set of coefficients that can be chosen freely and do not interfere with the four main design parameters; it is thus easy to tune the design to achieve any additional constraints that the designer may care for.     

Closed-form solutions for lowpass filtering of 3D digital geometry

Closed-form solutions for lowpass filtering of 3D digital geometry are presented. Conventional signal processing is difficult to apply since 3D digital geometry does not usually satisfy the prerequisites of regular sampling and global parameterisation. Representing digital geometry as a level set of a 3D scalar field fitted with Gaussians and polynomials, derived are closed-form solutions of convolution with Gaussian smoothing kernels and lowpass filtering without the pitfalls of other approaches are analytically achieved     

Polyphase antialiasing in resampling of images.

Changing resolution of images is a common operation. It is also common to use simple, i.e., small, interpolation kernels satisfying some "smoothness" qualities that are determined in the spatial domain. Typical applications use linear interpolation or piecewise cubic interpolation. These are popular since the interpolation kernels are small and the results are acceptable. However, since the interpolation kernel, i.e., impulse response, has a finite and small length, the frequency domain characteristics are not good. Therefore, when we enlarge the image by a rational factor of (L/M), two effects usually appear and cause a noticeable degradation in the quality of the image. The first is jagged edges and the second is low-frequency modulation of high-frequency components, such as sampling noise. Both effects result from aliasing. Enlarging an image by a factor of (L/M) is represented by first interpolating the image on a grid L times finer than the original sampling grid, and then resampling it every M grid points. While the usual treatment of the aliasing created by the resampling operation is aimed toward improving the interpolation filter in the frequency domain, this paper suggests reducing the aliasing effects using a polyphase representation of the interpolation process and treating the polyphase filters separately. The suggested procedure is simple. A considerable reduction in the aliasing effects is obtained for a small interpolation kernel size. We discuss separable interpolation and so the analysis is conducted for the one-dimensional case.     

尺度核函数支撑矢量机

本文提出了一种可容许的支撑矢量机核—尺度核.该尺度核函数可以被看作是一个具有平移因子的多维尺度函数,它能作为平方可积空间的子空间上一组完备的基函数.在此意义上,采用尺度核函数的支撑矢量机,可以认为是在尺度空间中寻找最佳的尺度系数.因此在理论上尺度核函数支撑矢量机能够以零误差逼近某一空间上的任何目标函数,文中给出的仿真实验进一步验证了它的可行性和有效性.     

用于文本相似度计算的新核函数

为了提高文本相似检测的综合表现,在文本文档相似特征的基础上构造了新的核函数S_Wang核函数。结合文本相似计算过程中的实际情况,将待比对的文本表示成向量,考虑通过2个向量间的乘积和欧氏距离来描述向量之间的相似程度,从而构造了适合文本相似度计算的新核函数。并根据Mercer定理证明了所构造函数可以作为核函数。实验验证了新构造的核函数在文本文档相似度计算中的表现,实验结果表明S_Wang核其相似度计算精度和综合指标均分别优于Cauchy核,潜在语义核（LSK）以及CLA复合核。S_Wang核适用于文本相似度计算。     

基于分段线性插值法的高精度测温研究

本文针对深海热流计中铂电阻测温误差问题,提出了一种软硬件相结合的方法补偿误差。在硬件上,通过给热敏电阻并联合适的电阻初步实现线性化;软件方面,应用分段线性插值法进行测量误差补偿,从而达到较高的测量精度。本文详细的介绍了并联电阻阻值的确定方法,完整的论述了利用分段线性插值法进行误差补偿的过程。经测试,测量结果可以达到较高的精度,有效的减少了误差。     

基于PLC的线性插值模糊控制器的设计

通过对喷淋式除尘系统采用的常规模糊控制器在实际应用中存在的问题进行分析,提出一种分段线性插值的设计方法,并将该方法在模糊控制器中应用。该设计采用PLC及其常规DI/DO模块组成的模糊控制器,用STL指令实现其控制功能。最后在喷淋式除尘系统中进行应用,并和常规模糊控制器的控制效果进行比较。实验结果证明:分段线性插值模糊控制器能缩短系统调节时间,提高模糊控制器的控制精度,具有一定的实用价值。