Quincunx Sampling Research Articles

Design of two-dimensional PR quincunx filter banks with Euler–Frobenius polynomial and lifting scheme

Two-dimensional (2-D) filter banks (FBs) have played a significant role in retrieving the directional information of images. In this paper, we propose a technique to design 2-D two-channel perfect reconstruction (PR) FBs with quincunx sampling. The proposed design method comprises two stages. In the first stage, we propose the design of a new halfband polynomial using Euler–Frobenius polynomial (EFP). This is constructed by imposing vanishing moment and PR constraints on EFP. The resulting new polynomial is a maximally flat Euler–Frobenius halfband polynomial (EFHBP). Later, in the second stage, EFHBP is used in a modified 2-D lifting scheme to design 2-D filters. The design examples for 2-D filters are presented and compared to existing filters. The performance shows that proposed filters have better regularity, symmetry and less energy of the error compared with existing FBs. Finally, performance of designed filters is evaluated in image denoising application.

梅花形抽样不可分小波的多聚焦图像融合

本文提出的方法是针对离散小波变换(DWT)缺少移不变性和无下采样离散小波变换(UDWT)的高冗余及较大的计算量提出的一种方法，本文提出的方法是一种低冗余近似移不变的小波变换，通过介绍梅花形抽样与其它几种融合方法的比较，显现出梅花形抽样的多聚焦图像融合的优越性，即有较好的融合质量和相对较小的计算量。 This paper aims to provide a new approach to deal with the lack of shift invariance in Discrete Wavelet Transform (DWT), the high redundancy and the large amount of calculation involved in Undecimated-DWT (UDWT). In the paper, the author proposes a new method to achieve DWT with low redundancy and shows the advantages of multi-focus image fusion of quincunx sampling by comparing this sampling with other fusion methods. This novel method could arrive at the results with better fusion and reduce the amount of calculation in the process at the same time.

2차원 고밀도 이산 웨이브렛 변환의 성능 향상을 위한 Quincunx 표본화 기법

The quincunx lattice is a non-separable sampling method in image processing. It treats the different directions more homogeneously and good frequency property than the separable two dimensional schemes. The high density discrete wavelet transformation is one that expands an N point signal to M transform coefficients with M > N. In two dimensions, this transform outperforms the standard discrete wavelet transformation in terms of shift-invariant. Although the transformation utilizes more wavelets, sampling rates are high costs. This paper proposed the high density discrete wavelet transform using quincunx sampling, which is a discrete wavelet transformation that combines the high density discrete transformation and non-separable processing method, each of which has its own characteristics and advantages. Proposed wavelet transformation can service good performance in image processing fields.

Comparison and Simulation of Subpixel Imaging Modes for Linear CCD

Subpixel technique of linear CCD is effective to enhance the spatial resolution without increasing the focal length of optics and reducing the pixel size. To compare image quality of two main subpixel imaging modes, quincunx sampling and four-point sampling, a method to quantitatively evaluate image quality of subpixel based on MTF was proposed. The MTF of quincunx and four-point sampling modes were derived. Analytical results shows that theoretical limiting resolution of quincunx sampling and four-point sampling is improved to 1.4 and 1.86 times respectively, and MTF values at Nyquist frequency of two modes are increased by 0.1106 and 0.1679, respectively. MTFA in (0, 0.5) of two subpixel imaging modes were calculated and results illustrates that four-point sampling offers much more improvement with image quality than quincunx sampling, at the cost of double amount of data. A model for simulating subpixel imaging using Matlab was established, and simulation results of spoke target verify the theoretical analysis.

방향의 선택성 향상을 통한 이중 밀도 이산 웨이브렛 변환의 성능 개선

The double-density discrete wavelet transform(DWT) is an improvement upon the critically sampled DWT with important additional properties. It employs one scaling function and two distinct wavelets, which are designed to be offset from one another by one half. And it is overcomplete by a factor of two. Also, this transformation is nearly shift-invariant. But there is room for improvement because not all of the wavelets are directional. That is, although the double-density DWT utilizes more wavelets, some lack a dominant spatial orientation, which prevents them from being able to isolate those directions. Proposed method is a DWT that combines the double-density DWT and quincunx sampling, each of which has its own characteristics and advantages. Especially, the quincunx sampling treats the different directions more homogeneously. As a result, since proposed method can generate sub-images of multiple degrees rotated versions, this method provides an improved performance in image processing fields.

Adaptive 2-D Wavelet Transform Based on the Lifting Scheme With Preserved Vanishing Moments

In this paper, we propose novel adaptive wavelet filter bank structures based on the lifting scheme. The filter banks are nonseparable, based on quincunx sampling, with their properties being pixel-wise adapted according to the local image features. Despite being adaptive, the filter banks retain a desirable number of primal and dual vanishing moments. The adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel, based on the intersection of confidence intervals (ICI) rule. The image denoising results are presented for both synthetic and real-world images. It is shown that the obtained wavelet decompositions perform well, especially for synthetic images that contain periodic patterns, for which the proposed method outperforms the state of the art in image denoising.

Directional filter bank-based segmentation for improved evaluation of nondestructive evaluation images

This paper introduces a directional filter bank (DFB) for segmentation of NDE images containing directional information. The DFB is used to split an image into a desired number of sub-band images with each sub-band image containing features belonging only to a given angular range. The DFB is a two-channel decomposition employing the Quincunx sampling matrix and the diamond half band filter pair. The DFB is also designed to incorporate the property of perfect reconstruction or alias free reconstruction. Applications of the DFB towards segmenting C-scan images of fiber-reinforced composites, magnetic flux leakage (MFL) images of seamless tubes, IR images of solar cell panels and optical images for the computation of area coverage in a shot-peening process are discussed.

On Filter Symmetries in a Class of Tree-Structured 2-D Nonseparable Filter-Banks

In one-dimensional (1-D) filter-banks (FBs), symmetries (or anti-symmetries) in the filter impulse-responses (which implies linear-phase filters) are required for symmetric signal extension schemes for finite-extent signals. In two-dimensional (2-D) separable FBs, essentially, 1-D processing is done independently along each dimension. When 2-D nonseparable FBs are considered, the 2-D filters (2-D signals in general) can have a much larger variety of symmetries (anti-symmetries) than the 1-D case. Some examples of 2-D symmetries possible are quadrantal, diagonal, centro, 4-fold rotational, etc. In this letter, we analyze the filter symmetries in a subclass of tree-structured 2-D nonseparable FBs, whose sampling matrices can be factored as a product of Quincunx sampling matrix and a diagonal matrix. Within this subclass, we distinguish between two types and show that we can have diagonally symmetric filters in Type-I FBs and quadrantally symmetric filters in Type-II FBs. We then discuss how these FBs with quadrantally and diagonally symmetric filters can be used with a symmetric signal extension scheme on finite-extent signals

Oriented Wavelet Transform for Image Compression and Denoising

In this paper, we introduce a new transform for image processing, based on wavelets and the lifting paradigm. The lifting steps of a unidimensional wavelet are applied along a local orientation defined on a quincunx sampling grid. To maximize energy compaction, the orientation minimizing the prediction error is chosen adaptively. A fine-grained multiscale analysis is provided by iterating the decomposition on the low-frequency band. In the context of image compression, the multiresolution orientation map is coded using a quad tree. The rate allocation between the orientation map and wavelet coefficients is jointly optimized in a rate-distortion sense. For image denoising, a Markov model is used to extract the orientations from the noisy image. As long as the map is sufficiently homogeneous, interesting properties of the original wavelet are preserved such as regularity and orthogonality. Perfect reconstruction is ensured by the reversibility of the lifting scheme. The mutual information between the wavelet coefficients is studied and compared to the one observed with a separable wavelet transform. The rate-distortion performance of this new transform is evaluated for image coding using state-of-the-art subband coders. Its performance in a denoising application is also assessed against the performance obtained with other transforms or denoising methods.

Comments on “Multi-dimensional filter banks and wavelets—A system theoretic perspective”

In [S. Basu, Multi-dimensional filter banks and wavelets—a system theoretic perspective, J. Franklin Inst. 335B(8) (1998) 1367–1409], the problems of constructing and parameterizing multi-dimensional (MD) filter banks (FB) and wavelets were studied extensively, especially in the situation of two-dimensional (2D) and quincunx-downsampled systems. Finding the other filter complementary to a given 2D filter for quincunx sampling and linear phase perfect reconstruction (LPPR) system is based on the fact that the construction is possible if and only if the polyphase components of the given filter do not share any common zero. The construction is elaborated in another book chapter [S. Basu, Multidimensional signals, circuits and systems, in: K. Galkowski, J. Wood (Eds.), On the Structure of Linear Phase Perfect Reconstruction Quincunx Filter Banks, Taylor & Francis, London, 2001, pp. 193–208 (Chapter 10)] by the same author. This approach further relies on the fact that the resultant of the polyphase components is monomial. In this comment, we show that both above mentioned facts are not necessarily true for the quincunx FB as well as for general MD FB.

An orthogonal family of quincunx wavelets with continuously adjustable order

We present a new family of two-dimensional and three-dimensional orthogonal wavelets which uses quincunx sampling. The orthogonal refinement filters have a simple analytical expression in the Fourier domain as a function of the order lamda, which may be noninteger. We can also prove that they yield wavelet bases of L2(R2) for any lambda > 0. The wavelets are fractional in the sense that the approximation error at a given scale a decays like O(a(lamda)); they also essentially behave like fractional derivative operators. To make our construction practical, we propose a fast Fourier transform-based implementation that turns out to be surprisingly fast. In fact, our method is almost as efficient as the standard Mallat algorithm for separable wavelets.

Design of Signal-Adapted Multidimensional Lifting Scheme for Lossy Coding

This paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform. Our approach is stated in the general case, and is illustrated for the 2-D separable and for the quincunx images. Results are shown for the JPEG2000 database and for satellite images acquired on a quincunx sampling grid. The design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases. Our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case. It is based on a two-step lifting scheme and joins the lifting theory with Wiener's optimization. The prediction step is designed in order to minimize the variance of the signal, and the update step is designed in order to minimize a reconstruction error. Application for lossy compression shows the performances of the method.

A new two-dimensional signal decomposition scheme using the extreme-lifting scheme

This paper discusses the signal decomposition method using the extreme-lifting scheme and two two-dimensional decomposition schemes: separable one-dimensional scheme and two-dimensional scheme with quincunx sampling. The structure of the relation “∼” between E x and E y of these two schemes is symmetrical and both these two schemes have shortcomings. An unsymmetrical scheme of the extreme-lifting scheme is proposed in this paper, which can be directly used to decompose two-dimensional image and can get better decomposition result than the two schemes with little computation cost.

Hierarchical representation and coding of surfaces using 3-D polygon meshes

This paper presents a novel procedure for the representation and coding of three-dimensional (3-D) surfaces using hierarchical adaptive triangulation. The proposed procedure is based on pyramidal analysis using the quincunx sampling minimum variance interpolation (QMVINT) filters. These are reduced pyramids with quincunx sampling applied to the parametric representation of the surface, chosen so as to minimize the variance of the interpolation error, and thus, when combined with the appropriate encoding of the coefficients, optimize the compression of the mesh information transmitted. At the same time, it produces a hierarchy of meshes based on quincunx sampling where coarse meshes are as similar to their finer versions as possible. This is very much desirable in progressive transmission. Depending on its interpolation error and the available bitrate, each filtered sample is a candidate for becoming a vertex of the mesh. The result is a progressive sequence of meshes consisting of more triangles wherever large variations exist and fewer in uniform regions. Complete correspondence between triangles at each level is identified, resulting in an efficient hierarchical representation of the mesh. The algorithm can be also used for the triangulation of a specific region of interest. Experimental results demonstrate that the proposed scheme provides an improvement in quality (MSE) by a factor of two when compared with other well known adaptive triangulation schemes.

Lossless image compression based on optimal prediction, adaptive lifting, and conditional arithmetic coding

The optimal predictors of a lifting scheme in the general n-dimensional case are obtained and applied for the lossless compression of still images using first quincunx sampling and then simple row-column sampling. In each case, the efficiency of the linear predictors is enhanced nonlinearly. Directional postprocessing is used in the quincunx case, and adaptive-length postprocessing in the row-column case. Both methods are seen to perform well. The resulting nonlinear interpolation schemes achieve extremely efficient image decorrelation. We further investigate context modeling and adaptive arithmetic coding of wavelet coefficients in a lossless compression framework. Special attention is given to the modeling contexts and the adaptation of the arithmetic coder to the actual data. Experimental evaluation shows that the best of the resulting coders produces better results than other known algorithms for multiresolution-based lossless image coding.

Optimal construction of reduced pyramids for lossless and progressive image coding

Reduced pyramids, including in particular pyramids without analysis filters, are known to produce excellent results when used for lossless signal and image compression. The present paper presents a methodology for the optimal construction of such pyramids by selecting the interpolation synthesis post-filters so as to minimize the error variance at each level of the pyramid. This establishes optimally efficient interpolative pyramidal lossless compression. It also has the added advantage of producing lossy replicas of the original which, at lower resolutions, retain as much similarity to the original as possible. This is highly useful for the progressive coding of signals or images needed for many applications such as fast browsing through image databases or hybrid lossless/lossy medical image coding. The general optimization methodology is developed first, for a general family of reduced pyramids. Subsequently, this is applied to the optimization of pyramids in this family formed using separable, two-dimensional (2-D) quincunx and three-dimensional (3-D) face-centered orthorhombic lattice sampling matrices. It is shown that this family includes in particular the well known 2-D and 3-D "hierarchical interpolation" (HINT) techniques which have been particularly popular for the lossless compression of medical records. Optimal versions of these techniques are determined for 2-D and 3-D images characterized by separable or isotropic correlation functions. The advantages of the developed methods are demonstrated by experimental evaluation. It is shown that the method outperforms the HINT method for the lossless compression of 3-D images. It is also shown to outperform all other known interpolative coders and to produce results comparable to the best predictive lossless coder of 2-D images.

Two-Dimensional Wavelets with Complementary Filter Banks

In this paper, the Two-Dimensional (2D) Complementary Filter (CF) Banks, a signal processing technique which can be used to get scaling and wavelets functions, are presented. The 2D multirate signal processing theory and complementary filters properties are the base of CF Banks, permitting us to consider different types of sampling and filters. Two-dimensional nonseparable quincunx, rectangular and circular complementary filters are designed for an alias free decimation and interpolation. When CF bank is implemented with quincunx sampling and filters, perfect reconstruction is achieved and although it is not reached in others cases, the analysis and synthesis are performed without aliasing. The CF banks are related with wavelet theory and procedures to get 2D scaling and bandpass wavelets functions from the L-level filter bank tree structure iteratively are shown. We illustrate scaling and wavelets functions convergence as the level of decomposition increases.

Practical design rules for minimax linear phase diamond-shaped 2-D FIR low-pass filters

Diamond-shaped filters are used as antialiasing filters in the conversion between images sampled on the rectangular and quincunx sampling grids. Here we investigate some characteristics of the linear phase diamond-shaped low-pass filters and present a set of design rules applicable to the design of such filters optimal in the minimax error sense. The design rules provide good estimates for the minimum filter support size required to satisfy a given set of specifications. In most cases, the difference between the estimated minimum filter support size and the actual minimum filter support size is less than 5%.

Design of 2-D perfect reconstruction filter banks using transformations of variables: IIR case

We have previously (1993) presented a design technique for 2-channel multidimensional filter banks. The technique is based on the transformation of variables and provides the flexibility of controlling the frequency characteristics of the resulting subband filters with ease. In this paper we extend the design technique to yield IIR filters. We emphasize the 2-D diamond subband case (with quincunx sampling) but the technique can be used for the parallelogram and diagonally quadrant subband cases as well. The IIR nature of the filters occurs from the use of IIR transformation functions instead of FIR. Two cases are considered. The first is with zero-phase IIR transformations which yield linear phase subband filters but require noncausal filtering. The second is with nonlinear phase IIR transformations which yield nonlinear phase subband filters that can be implemented in a causal manner.

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. >