Abstract:
This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.

Abstract:
We show how periodized wavelet packet transforms and periodized wavelet transforms can be implemented on a quantum computer. Surprisingly, we find that the implementation of wavelet packet transforms is less costly than the implementation of wavelet transforms on a quantum computer.

Abstract:
This paper gives the definition of wavelet fractional filter and presents the me thod of constructing wavelets by the fractional filter. This method not only incl udes Daubechies method which is used to construct wavelet by polynomial filter,but also includes the method of the filter of B-spline wavelet which is a special case of the fractional filter.

Abstract:
in 1992 unser and colleagues proved that the sequence of normalized and scaled b-splines bm tends to the gaussian function as the order m increases, [1]. in this article the result of unser et al. is extended to the derivatives of the b-splines. as a consequence, a certain sequence of wavelets defined by b-splines, tends to the famous mexican hat wavelet. another consequence can be observed in the continuous wavelet transform (cwt) of a function analyzed with different b-spline wavelets.

Abstract:
In 1992 Unser and colleagues proved that the sequence of normalized and scaled B-splines Bm tends to the Gaussian function as the order m increases, [1]. In this article the result of Unser et al. is extended to the derivatives of the B-splines. As a consequence, a certain sequence of wavelets defined by B-splines, tends to the famous Mexican hat wavelet. Another consequence can be observed in the continuous wavelet transform (CWT) of a function analyzed with different B-spline wavelets.

Abstract:
We first analyse the effect of cutting off the Canny operator on the calculation speed for multiscale edge detection, and then propose the center B spline spline dyadic wavelet multiscale edge detection algorithm. Some properties of the Canny operator and center B function are studied in detail: 1) the center B spline function has compact support and the Canny operator does not; 2) the center B spline function converges to the Gaussian function and the derivative of center B spline {unction converges to Canny operator; 3) the derivative of the fourth order center B spline function is more like the optimal edge detection filter than the Canny operator is ; and 4) the fourth order center B spline function is the only solution to the second order smoothing problem and its time-frequency uncertainty value is very close to the optimal value. The result of the multiscale edge detection shows that the center B spline dyadic wavelet is superior to the Canny operator.

Abstract:
We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.

Abstract:
This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate. 1. Introduction For the last two decades, the wavelet transforms have become very useful tools in a variety of applications such as signal and image processing and numerical computation. The construction of classical wavelets is now well understood thanks to such pioneering works as [1–3]. Many properties, such as symmetry (or antisymmetry), vanishing moments, regularity, and short support, are required in a practical use for application areas. In particular, polynomial splines have been a common source for wavelet construction [1, 3–6]. A new class of compactly supported biorthogonal wavelet systems that are constructed from pseudosplines was introduced in [7]. Exponential B-splines and polynomials have been found to be quite useful in a number of applications such as computer-aided geometric design, shape-preserving curve fitting, and signal interpolation [8–10]. Exponential B-splines were used as a key ingredient for the construction of wavelets [11, 12] and particularly used in wavelet construction on and [13]. In particular, in the approximation and sparse representation of acoustic signals, polynomial-based (stationary) wavelet systems have an important limitation because they do not consider the spectral features (e.g., band limited) of a given signal. However, (non-stationary) wavelet systems based on the exponential B-spline can be tuned to the specific trait of the given signal, yielding better approximations and sparser representations than classical wavelets at strictly the same computational costs. Details on exponential splines can be found in the selected references [10, 11, 14–16]. Related studies on non-stationary wavelets can be found in [11, 12, 15, 17–22]. One natural and convenient way to introduce wavelets is to follow the notion of multiresolution analysis. However, because the refinement masks we are interested in are non-stationary (i.e.,

Abstract:
Given an i.i.d. sample from a distribution $F$ on $\mathbb{R}$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"older balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.

Abstract:
Investigating characteristics of spline wavelet, we found that if the two-order spline function, the derivative function of the three-order B spline function, is used as the wavelet base function, the spline wavelet transform has both the property of denoising and that of differential. As a result, the relation between the spline wavelet transform and the differential was studied in theory. Experimental results show that the spline wavelet transform can well be applied to the differential of the electroanalytical signal. Compared with other kinds of wavelet transform, the spline wavelet transform has a characteristic of differential. Compared with the digital differential and simulative differential with electronic circuit, the spline wavelet transform not only can carry out denoising and differential for a signal, but also has the advantages of simple operation and small quantity of calculation, because step length, RC constant and other kinds of parameters need not be selected. Compared with Alexander Kai-man Leung’s differential method, the differential method with spline wavelet transform has the characteristic that the differential order is not dependent on the number of data points in the original signal.