International Journal of Wavelets, Multiresolution and Information Processing

In this paper, under a mild condition, the construction of compactly supported -wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

Wavelet transform has shown a strong capability to filter out noise that existed in images. The main problem to improve denoising performance lies in the fact that it is difficult to select suitable wavelet transform and related thresholds. In this paper, we completely explore characteristics of dyadic wavelet transform and its possible use to image denoising. An improved threshold method is elaborately designed to establish the self-adjusted layer threshold function and wavelet reconstruction. Theoretical analysis shows that denoising precision can be improved by way of adopting different thresholds according to the different scales of the wavelet coefficients of image and noise. Experiment results demonstrated that our method is obviously superior to that of fixed threshold regarding the effect of image denoising.

Estimations and optimal experimental designs for two-dimensional Haar-wavelet regression models are discussed. It is shown that the eigenvalues of the covariance matrix of the best linear unbiased estimator of the unknown parameters in a two-dimensional linear Haar-wavelet model can be represented in closed form. Some common discrete optimal designs for the model are constructed analytically from the eigenvalues. Some equivalences among these optimal designs are also given, and an example is demonstrated.

Since Donoho et al. proposed the wavelet thresholding method for signal denoising, many different denoising approaches have been suggested. In this paper, we present three different wavelet shrinkage methods, namely NeighShrink, NeighSure and NeighLevel. NeighShrink thresholds the wavelet coefficients based on Donoho's universal threshold and the sum of the squares of all the wavelet coefficients within a neighborhood window. NeighSure adopts Stein's unbiased risk estimator (SURE) instead of the universal threshold of NeighShrink so as to obtain the optimal threshold with minimum risk for each subband. NeighLevel uses parent coefficients in a coarser level as well as neighbors in the same subband. We also apply a multiplying factor for the optimal universal threshold in order to get better denoising results. We found that the value of the constant is about the same for different kinds and sizes of images. Experimental results show that our methods give comparatively higher peak signal to noise ratio (PSNR), are much more efficient and have less visual artifacts compared to other methods.

For arbitrary matrix dilation M whose determinant is odd or equal to ±2, we describe all symmetric interpolatory masks generating dual compactly supported wavelet systems with vanishing moments up to arbitrary order n. For each such mask, we give explicit formulas for a dual refinable mask and for wavelet masks such that the corresponding wavelet functions are real and symmetric/antisymmetric. We proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0. For matrix dilations M with |det M| = 2, we also give an explicit method for construction of masks (non-interpolatory) m₀ symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n. It is proved that for some matrix dilations (in particular, for the quincunx matrix) such a mask does not have a dual mask. Some of the constructed masks were successfully applied for signal processes.

The main purpose of this study is to present a novel personal authentication approach with the electrocardiogram (ECG) signal. The electrocardiogram is a recording of the electrical activity of the heart and the recorded signals can be used for individual verification because ECG signals of one person are never the same as those of others. The discrete wavelet transform was applied for extracting features that are the wavelet coefficients derived from digitized signals sampled from one-lead ECG signal. By the proposed approach applied on 35 normal subjects and 10 arrhythmia patients, the verification rate was 100% for normal subjects and 81% for arrhythmia patients. Furthermore, the performance of the ECG verification system was evaluated by the false acceptance rate (FAR) and false rejection rate (FRR). The FAR was 0.83% and FRR was 0.86% for a database containing only 35 normal subjects. When 10 arrhythmia patients were added into the database, FAR was 12.50% and FRR was 5.11%. The experimental results demonstrated that the proposed approach worked well for normal subjects. For this reason, it can be concluded that ECG used as a biometric measure for personal identity verification is feasible.

In this article, we exploit the adaptive properties of wavelets to develop some procedures for testing the equality of nonlinear and nonparametric mean response curves which are assumed by an experimenter to be the underlying functions generating several groups of data with possibly hetereoscedastic errors. The essential feature of the techniques is the transformation of the problem from the domain of the input variable to the wavelet domain through an orthogonal discrete wavelet transformation or a multiresolution expansion. We shall see that this greatly simplifies the testing problem into either a wavelet thresholding problem or a linear wavelet regression problem. The size and power performances of the tests are reported and compared to some existing methods. The tests are also applied to data on dose response curves for vascular relaxation in the absence or presence of a nitric oxide inhibitor.

In many classification problems, objects should be rejected when the confidence in their classification is too low. In this paper, we consider a new classification algorithm with a reject option. Based on the majority vote strategy and plug-in rules, we provide error analysis for this algorithm in ideal and realistic settings, respectively. In addition, some discussions of semi-supervised classification are given to demonstrate our theoretical analysis.