Daubechies Orthogonal Wavelets Research Articles

Algorithm Design for Edge Detection of High-Speed Moving Target Image under Noisy Environment.

For some measurement and detection applications based on video (sequence images), if the exposure time of camera is not suitable with the motion speed of the photographed target, fuzzy edges will be produced in the image, and some poor lighting condition will aggravate this edge blur phenomena. Especially, the existence of noise in industrial field environment makes the extraction of fuzzy edges become a more difficult problem when analyzing the posture of a high-speed moving target. Because noise and edge are always both the kind of high-frequency information, it is difficult to make trade-offs only by frequency bands. In this paper, a noise-tolerant edge detection method based on the correlation relationship between layers of wavelet transform coefficients is proposed. The goal of the paper is not to recover a clean image from a noisy observation, but to make a trade-off judgment for noise and edge signal directly according to the characteristics of wavelet transform coefficients, to realize the extraction of edge information from a noisy image directly. According to the wavelet coefficients tree and the Lipschitz exponent property of noise, the idea of neural network activation function is adopted to design the activation judgment method of wavelet coefficients. Then the significant wavelet coefficients can be retained. At the same time, the non-significant coefficients were removed according to the method of judgment of isolated coefficients. On the other hand, based on the design of Daubechies orthogonal compactly-supported wavelet filter, rational coefficients wavelet filters can be designed by increasing free variables. By reducing the vanishing moments of wavelet filters, more high-frequency information can be retained in the wavelet transform fields, which is benefit to the application of edge detection. For a noisy image of high-speed moving targets with fuzzy edges, by using the length 8-4 rational coefficients biorthogonal wavelet filters and the algorithm proposed in this paper, edge information could be detected clearly. Results of multiple groups of comparative experiments have shown that the edge detection effect of the proposed algorithm in this paper has the obvious superiority.

Fast compression and reconstruction of astronomical images based on compressed sensing

With the fast increase in the resolution of astronomical images, the question of how to process and transfer such large images has become a key issue in astronomy. We propose a new real-time compression and fast reconstruction algorithm for astronomical images based on compressive sensing techniques. We first reconstruct the original signal with fewer measurements, according to its compressibility. Then, based on the characteristics of astronomical images, we apply Daubechies orthogonal wavelets to obtain a sparse representation. A matrix representing a random Fourier ensemble is used to obtain a sparse representation in a lower dimensional space. For reconstructing the image, we propose a novel minimum total variation with block adaptive sensing to balance the accuracy and computation time. Our experimental results show that the proposed algorithm can efficiently reconstruct colorful astronomical images with high resolution and improve the applicability of compressed sensing.

On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets

In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.

B-Spline Wavelet on the Interval Element for the Solution of Hamilton Canonical Equation

Based on the modified Hellinger-Reissner (H-R) variational principle for the elastic material, the formulation of B-spline wavelet on the interval (BSWI) element of the Hamilton canonical equation was derived through the use of the interpolation scaling function of BSWI. To demonstrate the excellent predictive capability of the formulation, numerical studies were conducted on a thick plate and a three-layer plate with various common boundary conditions. The basic steps from which the formulation of the BSWI element of the Hamilton canonical equation was derived can also be extended to deduce the expressions of other wavelet elements in the Hamiltonian system, such as Daubechies orthogonal wavelet, biorthogonal wavelet based on lifting scheme, and so on.

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi-banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS-form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.

STENCIL COEFFICIENT COMPUTATIONS FOR THE MULTIRESOLUTION TIME DOMAIN METHOD - A FILTERBANK APPROACH

Multiresolution Time Domain (MRTD) techniques based on wavelet expansions can be used for adaptive refinement of computations to economize the resources in regions of space and time where the fields or circuit parameters or their derivatives are large. Hitherto, standard wavelets filter coefficients have been used with the MRTD method but the design of such filter itself may enable to incorporate desired properties for different applications. Towards this, in this paper, a new set of stencil coefficients in terms of scaling coefficients starting from a half band filter, designed by window method and deriving a physically realizable filter by spectral factorization using cepstral technique, for the MRTD method is presented. These stencil coefficients for the MRTD are found to give good agreement with similar MRTD schemes such as those obtained using Daubechies orthogonal wavelets.

Wavelet analysis applied to magnetograms: Singularity detections related to geomagnetic storms

When the interplanetary magnetic field carried by the solar wind opposes to the Earth intrinsic magnetic field, a substantial transfer of energy into the terrestrial magnetosphere takes place. If this condition persists for several hours, the magnetosphere becomes very disturbed. As a result, at mid-to-low latitudes a ring current starts to develop and at high latitudes ionospheric currents (electrojet and field-aligned currents) dominate. The ring current provides the geomagnetic conditions for magnetic storms to settle down. Wavelet analysis is becoming an usual tool since they allow the decomposition of data, functions or operators into different frequency or scale components. Accordingly, wavelet transforms seem to be suited to analyze short-lived high-frequency phenomena such as discontinuities ( shocks) in signals and transient structures. In this work, the remarkable ability of wavelets to highlight the singularities associated with discontinuities present in the horizontal component of the Earth's magnetic field is explored. Magnetograms obtained at five magnetic stations for two geomagnetic storms have been analyzed by a Daubechies orthogonal wavelet transform. The wavelet coefficient magnitudes at three levels have been studied. In both cases, the physical discontinuities in the horizontal component of the geomagnetic field are clearly detected by means of these coefficients identifying the disturbed interval related to geomagnetic storms. Wavelet analysis has proved to be a useful tool in the identification of the geomagnetic storms using non-processed data.

Armlets and balanced multiwavelets: flipping filter construction

In the scalar-valued setting, it is well-known that the two-scale sequences {q/sub k/} of Daubechies orthogonal wavelets can be given explicitly by the two-scale sequences {p/sub k/} of their corresponding orthogonal scaling functions, such as q/sub k/=(-1)/sup k/p/sub 1-k/. However, due to the noncommutativity of matrix multiplication, there is little such development in the multiwavelet literature to express the two-scale matrix sequence {Q/sub k/} of an orthogonal multiwavelet in terms of the two-scale matrix sequence {P/sub k/} of its corresponding scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multiwavelets of dimension r=2. In particular, the two lowpass filters are flipping filters, whereas the two highpass filters are linear phase. These results will be applied to constructing both a family of the most recently introduced notion of armlet of order n and a family of n-balanced orthogonal multiwavelets.

Multiwavelets and Integer Transforms

In many applications of image processing, the given data are integer-valued. It is therefore desirable to construct transformations that map data of this type to an integer (or rational) ring. Calderbank, Daubechies, Sweldens, and Yeo [1] devised two methods for modifying orthogonal and biorthogonal wavelets so that they map integers to integers. The first method involves appropriately scaling the transform so that data that has been transformed and truncated can be recovered via the inverse wavelet transform. In developing this method, the authors of [1] created a useful factorization of the 4-tap Daubechies orthogonal wavelet transform [2]. We have observed that this factorization can be extended to 4-tap multiwavelets of arbitrary size. In this paper we will discuss this generalization and illustrate the factorization on two multiwavelets. In particular, the well-known Donovan, Geronimo, Hardin, and Massopust (DGHM) [3] multiwavelet transform can be scaled so that it maps integers to integers. Since this transform is (anti)symmetric in addition to orthogonal, regular, and compactly supported, the ability to modify it so that it maps integers to integers should be useful in image processing applications.

Difference Wavelet - Theory and A Comparison Study

Wavelet methods with polynomial filters are usually favored in applications for their fast wavelet transforms and compact support. However, wavelet methods with rational filters have more freedom to achieve smaller condition numbers, more regularity and better efficiency. Such meth- ods can be attractive if they also possess fast algorithms and have fast decay (as if the corresponding wavelets had compact support). In the first part of this paper, we propose a new wavelet method with rational filters which do have these properties. We call it the difference wavelet method. It is a generalization of Butterworth wavelets. The analysis part is simply averaging and finite differencing. The wavelet coefficients measure the finite differences of the a of an input data sequence. Its synthesis part involves rational filters, which can be performed with linear computational complexity by the cyclic reduction method. Their Riesz basis property, biorthogonality, decay and regularity are investigated. In the second part of this paper, we perform comparison studies of the difference wavelet method (Diff) with three other popular wavelet methods: the Cohen-Daubechies-Feauveau biorthogonal wavelets (CDF), the Daubechies orthogonal wavelets (Daub) and the Chui-Wang semi-orthogonal wavelets (CW). Natural criteria in designing good wavelet methods for representing functions and operators are speed, stability and efficiency. Therefore, the items of our first comparison include (i) operation counts for performing transformations, (ii) condition numbers of the wavelet trans- formations, (iii) compression ratios, by some numerical experiments, for representing (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels). The results show that (i) Diff, Daub and CDF have about the same operation counts, and CW has more; (ii) Diff has about the same condition numbers as those of CDF and CW; (iii) Diff has better compression ratio for both (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels). The items of our second comparison include regularity, approximation power (the constant in the approximation estimate), approximation errors for non-smooth functions (where Gibbs phenomena appear) and the The results show that Diff has better regularity and better approximation ability with only slightly bigger essential supports. It is evident that the better efficiency of Diff for smooth functions is due to its regularity. It is surprising that, even for non- smooth functions, Diff is comparable to, sometimes even superior to, other methods, despite its infinite-support property. This paper is organized as follows. Sec. 1 is preliminary. Sec. 2 provides the theory of the difference wavelet method. Sec. 3 contains the comparison studies. Experts are suggested to read Sec. 3 directly.

On a choice of wavelet bases in the wavelet transform approach

The Daubechies orthogonal wavelet (DOW) is compared with the nonorthogonal cardinal spline wavelet (NCSW) in the wavelet transform approach and it is shown that the DOW is better than the NCSW in view of the computation cost. First, the computation cost required for the wavelet transform based on the DOW is less than that based on the NCSW because the DOW has smaller support provided the same number of vanishing moments of wavelets is used. Second, in contrast with the fact that the wavelet transform based on the DOW does not affect the condition number of the impedance matrix, that, based on the NCSW, has an effect to make it very large. As a result, even though the NCSW results in a sparser impedance matrix, it requires more computation cost for solving the resultant matrix equation in comparison with the DOW because the cost depends not only on the sparsity, but also on the condition number of the matrix.

Simultaneous quantitative analysis of overlapping spectrophotometric signals using wavelet multiresolution analysis and partial least squares

The mathematical bases and program algorithms of discrete wavelet transform (DWT), multiresolution and Mallat’s pyramid algorithm were described. The multiresolution analysis (MRA) based on Daubechies orthogonal wavelet basis was studied as a tool for removing noise and irrelevant information from spectrophotometric spectra. After wavelet MRA pre-treatment, eight error functions were calculated for deducing the number of factors. A partial least squares based on wavelet MRA (WPLS) method was developed to perform simultaneous spectrophotometric determination of Fe(II) and Fe(III) with overlapping peaks. Data reduction was performed using wavelet MRA and principal component analysis (PCA) algorithm. Two programs, SPWMRA and SPWPLS, were designed to perform wavelet MRA and simultaneous multicomponent determination. Experimental results showed the WPLS method to be successful even where there was severe overlap of spectra.

Perceptual-based hyperspectral image fusion using multiresolution analysis

A perceptual-based multiresolution image fusion technique is demonstrated using the Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) hyperspectral sensor data. The AVIRIS sensor, which simultaneously collects information in 224 spectral bands that range from 0.4 to 2.5 μm in approximately 10-nm increments, produces 224 images, each representing a single spectral band. The fusion algorithm consists of three stages. First, a Daubechies orthogonal wavelet basis set is used to perform a multiresolution decomposition of each spectral image. Next, the coefficients from each image are combined using a perceptual-based weighting. The weighting of each coefficient, from a given spectral band image, is determined by the spatial-frequency response (contrast sensitivity) of the human visual system. The spectral image with the higher saliency value, where saliency is based on a perceptual energy, will receive the larger weight. Finally, the fused coefficients are used for reconstruction to obtain the fused image. The image fusion algorithm is analyzed using test images with known image characteristics and image data from the AVIRIS hyperspectral sensor. By analyzing the signal-to-noise ratios and visual aesthetics of the fused images, contrast-sensitivity-based fusion is shown to provide excellent fusion results and to outperform previous fusion methods.