Bi-impulse response design of isotropic quadratic filters

The bi-impulse response of a 1-D or 2-D quadratic Volterra filter is introduced as a mathematical tool able to completely describe the nonlinear operator. The conditions that must be satisfied in order to obtain isotropic input/output relations are studied. The result is a formal framework that allows simple but effective operators (particularly for image enhancement and preprocessing) to be designed and a deeper insight into the properties of Volterra filters to be acquired. Examples of design and of the performance obtained by processing natural and synthetic images are presented     

Quadratic weighted median filters for edge enhancement of noisy images.

Quadratic Volterra filters are effective in image sharpening applications. The linear combination of polynomial terms, however, yields poor performance in noisy environments. Weighted median (WM) filters, in contrast, are well known for their outlier suppression and detail preservation properties. The WM sample selection methodology is naturally extended to the quadratic sample case, yielding a filter structure referred to as quadratic weighted median (QWM) that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers arising in the higher order statistics of environment noise. Through statistical analysis of higher order samples, it is shown that, although the parent Gaussian distribution is light tailed, the higher order terms exhibit heavy-tailed distributions. The optimal combination of terms contributing to a quadratic system, i.e., cross and square, is approached from a maximum likelihood perspective which yields the WM processing of these terms. The proposed QWM filter structure is analyzed through determination of the output variance and breakdown probability. The studies show that the QWM exhibits lower variance and breakdown probability indicating the robustness of the proposed structure. The performance of the QWM filter is tested on constant regions, edges and real images, and compared to its weighted-sum dual, the quadratic Volterra filter. The simulation results show that the proposed method simultaneously suppresses the noise and enhances image details. Compared with the quadratic Volterra sharpener, the QWM filter exhibits superior qualitative and quantitative performance in noisy image sharpening.     

图像边缘提取的自适应Volterra滤波器设计

Ｖｏｌｔｅｒｒａ滤波是研究信号高阶统计冗余性的一种有效途径。我们提出了一种用于提取图像边缘的自适应二次Ｖｏｌｔｅｒｒａ滤波器设计方法。这种滤波器是推广型Ｔｅａｇｅｒ基滤子的线性组合，其系数用基于极小化最小均方能量函数的共轭梯度法研究；它兼有局域平均和高通特性，因而可均衡去除噪声和增强图像边缘。文章还给出了计算实例。     

Subpixel rendering for diamond-shaped PenTile displays using patch-based adaptive filters

We propose a novel subpixel rendering algorithm for diamond-shaped PenTile displays, which reduces color distortions while improving apparent resolutions. We develop two types of subpixel rendering filters: main filter and color distortion reduction (CDR) filters. To derive the filters, we formulate a quadratic program to minimize the difference between an original input image and a virtual image that the human visual system perceives. By imposing two constraints for filter size and coefficients, we obtain the main filter, which has a suitable size and is normalized. Then, we design the CDR filters based on the analysis of various patch patterns for image areas. We define the patch patterns to classify local areas with possible color distortions. By imposing additional constraints according to the patch patterns, we derive the CDR filters. Lastly, by matching local areas in the input image into the pre-defined patch patterns, we render the image using the main filter and the CDR filters, which are applied adaptively to the local areas. Experimental results demonstrate that the proposed subpixel rendering algorithm improves apparent resolutions and suppresses color distortions effectively, thereby outperforming conventional algorithms.     

Quadratic-linear filters for signal detection

The second-order Volterra series (quadratic-linear) filter was synthesized under three different optimization criteria for signal detection. The performance of the filters is shown to exceed that of quadratic-only or linear-only filters when the noise process is characterized by both separation in the mean and separation in variance. The applications are in object detection and image segmentation, especially in model-drive approaches to these problems. For 2-D applications, the mapping order of pixels into the signal vector is shown to have no effect on the optimality of the filters, regardless of nonlinearity     

Localized 2-D filter-based linear coherent noise attenuation

A novel localized two-dimensional (2-D) filter is proposed. The proposed filter derived from the frequency-wavenumber filter and Radon transform filter, with the filtering operation applied at the stage of Fourier projection, has good local property and less filtering distortion. An example of the proposed method to attenuate linear coherent noise in a seismic image is given. Comparisons of the results between our method and the conventional 2-D filters (including frequency-wavenumber filter and Radon transform filter) show that the new method outperforms both frequency-wavenumber method and Radon transform method.     

Decision-directed nonlinear filter for image processing

A complete quadratic Volterra filter working under the control of a decision algorithm is proposed for image restoration and enhancement. A particular application in the enhancement of images having reduced luminance dynamics is considered.     

Blurred fingerprint image enhancement: algorithm analysis and performance evaluation

A conventional automatic fingerprint matching process uses similarity score to quantify similarity between fingerprint images to be matched, and the similarity score can be determined with a minutiae extraction algorithm (MEA) which extracts minutiae from fingerprint images. The performance of MEA relies on the quality of fingerprint images. In case of blurred fingerprint images, it becomes difficult to obtain a reliable similarity score. As the result, an image enhancement algorithm should be incorporated with MEA when the fingerprint image is blurred. In this study, Volterra filter is proposed to enhance blurred fingerprints and compared against different enhancement algorithms. Experimental results show that Volterra filter outperforms other techniques such as Laplacian, Wiener, and Gabor filters for enhancing blurred images and its calculation complexity is moderate among techniques considered in this study.     

A Linear Phase Condition for Volterra Filters

In linear finite-impulse-response filter design, it is desirable that the filter frequency response has linear phase (LP). In this brief, we investigate the LP concept for nonlinear Volterra filters. The LP condition of Volterra filters is defined in terms of its output spectrum in which the phase term introduced by the Volterra kernels is linear. It is shown that under certain symmetry conditions, the LP condition is satisfied by Volterra filters. Moreover, the LP condition for Volterra filters can be considered as an extension to the linear filter case.     

Nonlinear image operators for the evaluation of local intrinsicdimensionality

Local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters. We derive a class of polynomial operators for the detection of intrinsically 2-D image features like curved edges and lines, junctions, line ends, etc. Although it is a deterministic concept, intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal. Furthermore, there is an intimate connection to multidimensional surface geometry and to the concept of ;Gaussian curvature'. Nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are, by the superposition principle, restricted to OR-combinations of their intrinsically 1-D eigenfunctions. The essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components. Therefore, such operators can provide the AND-combination of complex exponentials, which is required for the exploitation of intrinsic dimensionality. Using frequency design methods, we obtain a generalized class of quadratic Volterra operators that are selective to intrinsically 2-D signals. These operators can be adapted to the requirements of the signal processing task. For example, one can control the "curvature tuning" by adjusting the width of the stopband for intrinsically 1-D signals, or the operators can be provided in isotropic and in orientation-selective versions. We first derive the quadratic Volterra kernel involved in the computation of Gaussian curvature and then present examples of operators with other arrangements of stop and passbands. Some of the resulting operators show a close relationship to the end-stopped and dot-responsive neurons of the mammalian visual cortex.     

Geometrical properties of optimal Volterra filters for signaldetection

Linear-quadratic filters are a special example of Volterra filters that are limited to the second order. It is shown that all the results recently published which are valid in the linear-quadratic case can be extended with the appropriate notations to Volterra filters of arbitrary order. Particularly, the optimum Volterra filter giving the maximum of the deflection for detecting a signal in noise is wholly calculated. In addition, several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular, the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced, allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation. Extensions to the infinite case and relations with the likelihood ratio are also investigated     

Equivariant holomorphic filters for contour denoising and rapid object detection.

It is well known that linear filters are not powerful enough for many low-level image processing tasks. However, it is also very difficult to design robust nonlinear filters that respond exclusively to features of interest and that are, at the same time, equivariant with respect to translation and rotation. This paper proposes a new class of rotation-equivariant nonlinear filters that is based on the principle of group integration. These filters become efficiently computable by an iterative scheme based on repeated differentiation of products and summations of intermediate results. The relations of the proposed approach to Volterra filters and steerable filters are shown. In the context of detection problems, the filter may be interpreted as some kind of generalized Hough transform. The experiments show that the new filter can be used for enhancing noisy contours and rapid object detection in microscopical images. In the detection context, our experiments show that the proposed filter is definitely superior to alternative approaches, when high localization accuracy is required.     

On filtering by means of generalized integral images: a review and applications

In this paper the method of multidimensional (n-D) filtering based on prior signal integration is analyzed. This method has the advantage that the computational complexity for filtering is independent of the filter kernel size. An overview of recent 2-D image processing systems is presented where these types of filters are applied. Based on this overview a framework that covers this class of filters is derived using repeated integration. These filters include for example rect and triangle-filters which can be used to approximate Gaussian derivative filters. Furthermore the normalization of the filters, computational complexity, and storage cost are discussed. Finally, two image processing systems which benefit from the application of the filters are presented. They belong to the topic of advanced driver assistance systems.     

一种全解耦的RLS自适应Volterra滤波器

本文研究了Volterra自适应滤波器解耦问题,提出了一种全解耦的RLS自适应Volterra滤波器.按照Volterra滤波器的伪线性组合结构,采用RLS滤波原理和约束优化理论,导出了满足最小累计平方误差指标的具有分块对角形输入相关矩阵的全解耦Volterra标准方程,据此设计了一种全解耦的RLS自适应Volterra滤波器,给出了滤波器权向量自适应修正的一套公式.仿真结果验证了本文方法的有效性.     

Adaptive McClellan transformations for quincunx filter banks

A technique is developed for the design of 2-D nonseparable two-channel filter banks for a quincunx sampling lattice, where the isopotentials of the frequency response can be optimized and adapted to the input signal's statistics. By employing known odd-length symmetric linear phase filter banks as the l-D prototype filters for 2-D filters parameterized by the McClellan transformation, conditions are derived such that the resulting 2-D two-channel filter bank retains the perfect-reconstruction or aliasing-free properties of the 1-D prototype two-channel filter bank. A particular two-parameter transformation function is developed that has sufficient flexibility to adapt its orientation in any direction and whose optimization involves a simple constrained least-squares problem in which the feasible set lies within a circle. The results have practical applications in many areas of image and video processing where multirate filter banks are used     

Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator

The complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. As the delta operator (??-domain) formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in ??-domain which utilize the various symmetries in the filter specifications. Furthermore, with the delta operator formulation, the discrete-time systems and results converge to their continuous-time counterparts as the sampling periods tend to zero. So a unifying theory can be established for both discrete- and continuous-time systems using the delta operator approach. With these motivations, we comprehensively establish the unifying symmetry theory for delta-operator formulated discrete-time complex-coefficient 2-D polynomials and functions, arising out of the many types of symmetries in their magnitude responses. The derived symmetry results merge with the s-domain results when the sampling periods tend to zero, and are more general than the real-coefficient results presented earlier. An example is provided to illustrate the use of the symmetry constraints in the design of a 2-D IIR filter with complex coefficients. For the narrow-band filter in the example, it can be seen that the ??-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.     

Quadratic Gabor filters for object detection

We present a new class of quadratic filters that are capable of creating spherical, elliptical, hyperbolic and linear decision surfaces which result in better detection and classification capabilities than the linear decision surfaces obtained from correlation filters. Each filter comprises of a number of separately designed linear basis filters. These filters are linearly combined into several macro filters; the output from these macro filters are passed through a magnitude square operation and are then linearly combined using real weights to achieve the quadratic decision surface. For detection, the creation of macro filters (linear combinations of multiple single filters) allows for a substantial computational saving by reducing the number of correlation operations required. In this work, we consider the use of Gabor basis filters; the Gabor filter parameters are separately optimized. The fusion parameters to combine the Gabor filter outputs are optimized using an extended piecewise quadratic neural network (E-PQNN). We demonstrate methods for selecting the number of macro Gabor filters, the filter parameters and the linear and nonlinear combination coefficients. We present preliminary results obtained for an infrared (IR) vehicle detection problem.     

Adaptive template filtering for signal-to-noise ratio enhancementin magnetic resonance imaging

In this paper, a local shape-adaptive template filtering is proposed for the enhancement of the signal-to-noise ratio (SNR) without the loss of resolution in magnetic resonance (MR) imaging. Unlike conventional filtering, where the template shape and coefficients are fixed, multiple templates are defined in the proposed algorithm. An optimal template is selected and optimal filtering, based on the template, is applied on a pixel-by-pixel basis. Using the proposed process, edge blurring is minimized and SNR enhancement is maximized by selecting the optimally matched template. Compared to existing two-dimensional (2-D) adaptive linear least square error (LLSE) filters or direction-adaptive recursive filters, the proposed adaptive template filter provides higher SNR and sharper edges for both MR and artificial resolution phantom images.     

Quadratic filters for signal processing

Polynomial (or Volterra) filters are introduced, and the quadratic filters are presented as the simplest example of such filters. The principle aspects and properties of quadratic filters are derived in the framework of the discrete Volterra expansion. Fixed as well as adaptive filters are considered in one-dimensional and multidimensional environments. Such issues as design and efficient realizations are thoroughly addressed, and standard and advanced adaptation algorithms are presented. Several examples of signal processing applications requiring quadratic filters are discussed