Wavelet Subspaces Research Articles

Sampling Theorems of Wavelet Subspaces with M Band in Higher Dimensions

Sampling theorem has a key role in signal processing and image processing. In this paper, the scaling functions with cardinal property are discussed in the dimensions and their symmetry property is classified. Therefore, sampling theorems of wavelet subspaces are obtained. Then, the cardinal orthogonal scaling function with cardinal property is characterized in the dimensions and an equation between the highpass filter coefficients and wavelet samples are got. The existing results are generalized to the case of M band.

Four Band Cardinal Orthogonal Scaling Functions and Wavelets in Higher Dimensions

Sampling theorem plays an important role in the engineering such as signal processing, image processing, digital communications, and so on. In this paper, the symmetry property of cardinal orthogonal scaling function is discussed. Then, a 4-band cardinal orthogonal scaling function from the relation between the highpass filter coefficients and wavelet is provided. Thus, sampling theorem in the wavelet subspace is obtained.

Recovery of the optimal approximation from samples in wavelet subspace

The success of the typical sampling theories for a wavelet subspace mostly benefits from the fact that the sampling operation is an isomorphism of a wavelet subspace onto l 2 ( R ) . However, this operation is not an isometry of a general wavelet subspace onto l 2 ( R ) . As a result, many sampling theories only concentrate on the recovery of a signal in a single wavelet subspace. In this paper, some theorems are proposed to discuss the isometric isomorphism of a wavelet subspace and a convolved l 2 ( R ) space. We show that the sampling operation is an isometric isomorphism of a wavelet subspace onto a convolved l 2 ( R ) space only if the sampling operation is an isomorphism of a wavelet subspace onto l 2 ( R ) . Based on the isometric isomorphism, we further verify the existence of the mapping from the samples to the projection of a signal on an approximation space. At last, we propose the corresponding algorithm to construct this mapping so that the optimal approximations of a signal at the different resolution can be recovered from the samples. The simulation shows that our algorithm is more suitable to recover the projection of a signal than Shannon sampling theorem in a general multiresolution analysis.

Wavelets from Laguerre Polynomials and Toeplitz-Type Operators

We study Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true polyanalytic Bergman spaces. Isomorphisms between the Calder\'on-Toeplitz operator algebras and functional algebras are described and their consequences are discussed.

Scalable Video Compression Framework With Adaptive Orientational Multiresolution Transform and Nonuniform Directional Filterbank Design

Although wavelet-based scalable video coding becomes the state-of-the-art video compression engine for its adaptability to heterogeneous networks and clients, a large number of attempts have been made to integrate local directionality onto discrete wavelet transform to explore the intrinsic geometrical structures. Taking into consideration that the contours and textures scattered in different scales change their directional resolutions as their curvatures change, we investigate adaptive directional resolutions along scales to achieve the dual (scale and orientation) multiresolution transform. This paper proposes nonuniform directional frequency decompositions for video representation and approximation, and exploits the nonuniformity of orientation multiresolution distribution and designs nonuniform directional filter banks to make the geometrical transform more sparse and efficient. The nonuniform directional frequency decomposition under arbitrary scales is fulfilled by a non-symmetric binary tree (NSBT) topology structure with nonuniform directional filterbank design. In turn, the proposed scalable video coding framework, called DMSVC, is enriched with the dual multiresolution transform. Each temporal subband through motion compensated temporal filtering is further decomposed into multiscale subbands, and the highpass wavelet subspaces are divided into an arbitrary number of directional subspaces in alignment with the orientation distribution via phase congruency to establish NSBT. The paraunitary perfect reconstruction condition is provided through a polyphase identical form of filter bank. Comparing with the isolated wavelet basis, our transform provides a greater correlated set of localized and anisotropic basis functions. The spatio-temporal subband coefficients are coded by a 3-D ESCOT entropy coding algorithm which is adopted to match the structure of NSBT. Experimental results show that the reconstructed video frames DMSVC in the proposed DMSVC scheme have better visual quality than existing scalable video coding schemes. It could produce higher compression ratio on video sequences full of directional edges and textures.

Classifying Cardinal Orthogonal Scaling Function with Dilation Factor 3

Sampling theorem plays an important role in many fields such as signal processing and image processing. In this paper, the cardinal orthogonal scaling function with dilation factor 3 is classified by the highpass filter coefficient, thus, the sampling theorem in the wavelet subspace is obtained. Then, the symmetry property of cardinal orthogonal scaling function is discussed.

On Boundedness of Calderón–Toeplitz Operators

We study the boundedness of Toeplitz-type operators defined in the context of the Calderón reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calderón–Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.

A Donoho–Stark criterion for stable signal recovery in discrete wavelet subspaces

We derive a sufficient condition by means of which one can recover a scale-limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by Donoho and Stark (1989) [4]. The proof follows from simple computations involving the Zak transform, well-known in solid-state physics. Geometric harmonics (in the terminology of Coifman and Lafon (2006) [22]) for scale-limited subspaces of L 2 ( R ) are also displayed for several test-cases. Finally, some algorithms are studied for the treatment of zero-angle problems.

Oversampling Expansion in Wavelet Subspaces

We find necessary and sufficient conditions for the (shifted) oversampling expansions to hold in wavelet subspaces. In particular, we characterize scaling functions with the (shifted) oversampling property. We also obtain L2 and L∞ norm estimates for the truncation and aliasing errors of the oversampling expansion.

Detection and 2-Dimensional display of short tandem repeats based on signal decomposition

This paper presents a wavelet-based Empirical Mode Decomposition (EMD) to detect short tandem repeats in DNA sequences. A wavelet subspace algorithm combined with EMD is introduced as a pre-processor and a Cross-Correlation Analysis (CCA) is applied as a post-processor to create subspaced Intrinsic Mode Functions (IMFs). The new proposed method is called the Empirical Mode and Wavelet Decomposition (EMWD). The algorithms can display the power spectral density in the two-dimensional frequency-time (f-t) plane efficiently for both very long signals and short signals. Simulations are applied on the real human DNA sequences from public data source Genbank (http://www.ncbi.nlm.nih.gov/Genbank/). Application of the EMWD algorithms to the short tandem repeat detection has achieved an averaged accuracy of 98.5%.

An Adaptive Run Length Encoding method for the compression of electrocardiograms

A compression method, based on the choice of a wavelet that minimizes the distortion of compression for each electrocardiogram considered, is proposed in this paper. The scaling filter used on the determination of the wavelet function is obtained from the resolution of an optimization problem, which is unconstrained since the scaling filter is parametrized in a way that the constraints applied to the scaling filter are embedded on the parameters. The coefficients of projection of the signal over the wavelet subspaces are calculated and only the most significant ones are retained, being the significant coefficients determined in order to satisfy a pre-specified distortion measure. The bitmap that informs the positions of the retained coefficients is encoded along with the values of the coefficients by using an improved version of the Run Length Encoding technique. Experiments that compare the proposed approach with other techniques illustrate the efficiency of the method.

Vector sampling theorem for wavelet subspaces

The vector sampling theorem has been investigated and widely used by multi-channel deconvolution, multi-source separation and multi-input multi-output (MIMO) systems. Commonly, for most of the results on MIMO systems, the input signals are supposed to be band-limited. In this paper, we study the vector sampling theorem for the wavelet subspaces with reproducing kernel. The case of uniform sampling is discussed, and the necessary and sufficient conditions for reconstruction are given. Examples are also presented.

A procedure for denoising dual-axis swallowing accelerometry signals

Dual-axis swallowing accelerometry is an emerging tool for the assessment of dysphagia (swallowing difficulties). These signals however can be very noisy as a result of physiological and motion artifacts. In this note, we propose a novel scheme for denoising those signals, i.e. a computationally efficient search for the optimal denoising threshold within a reduced wavelet subspace. To determine a viable subspace, the algorithm relies on the minimum value of the estimated upper bound for the reconstruction error. A numerical analysis of the proposed scheme using synthetic test signals demonstrated that the proposed scheme is computationally more efficient than minimum noiseless description length (MNDL)-based denoising. It also yields smaller reconstruction errors than MNDL, SURE and Donoho denoising methods. When applied to dual-axis swallowing accelerometry signals, the proposed scheme exhibits improved performance for dry, wet and wet chin tuck swallows. These results are important for the further development of medical devices based on dual-axis swallowing accelerometry signals.

Analysis and short-time extrapolation of stock market indexes through projection onto discrete wavelet subspaces

We consider the problem of short-time extrapolation of blue chips’ stocks indexes in the context of wavelet subspaces following the theory proposed by X.-G. Xia and co-workers in a series of articles [10–13]. The idea is first to approximate the oscillations of the corresponding stock index at some scale by means of the scaling function which is part of a given multi-resolution analysis of L 2 ( R ) . Then, since oscillations at a finer scale are discarded, it becomes possible to extend such a signal up to a certain time in the future; the finer the approximation, the shorter this extrapolation interval. At the numerical level, a so-called Generalized Gerchberg–Papoulis (GGP) algorithm is set up which is shown to converge toward the minimum L 2 norm solution of the extrapolation problem. When it comes to implementation, an acceleration by means of a Conjugate Gradient (CG) routine is necessary in order to obtain quickly a satisfying accuracy. Several examples are investigated with different international stock market indexes.

Wavelet matrix transform for time-series similarity measurement

A time-series similarity measurement method based on wavelet and matrix transform was proposed, and its anti-noise ability, sensitivity and accuracy were discussed. The time-series sequences were compressed into wavelet subspace, and sample feature vector and orthogonal basics of sample time-series sequences were obtained by K-L transform. Then the inner product transform was carried out to project analyzed time-series sequence into orthogonal basics to gain analyzed feature vectors. The similarity was calculated between sample feature vector and analyzed feature vector by the Euclid distance. Taking fault wave of power electronic devices for example, the experimental results show that the proposed method has low dimension of feature vector, the anti-noise ability of proposed method is 30 times as large as that of plain wavelet method, the sensitivity of proposed method is 1/3 as large as that of plain wavelet method, and the accuracy of proposed method is higher than that of the wavelet singular value decomposition method. The proposed method can be applied in similarity matching and indexing for lager time series databases.

A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration

We present a multilevel extension of the popular "thresholded Landweber" algorithm for wavelet-regularized image restoration that yields an order of magnitude speed improvement over the standard fixed-scale implementation. The method is generic and targeted towards large-scale linear inverse problems, such as 3-D deconvolution microscopy. The algorithm is derived within the framework of bound optimization. The key idea is to successively update the coefficients in the various wavelet channels using fixed, subband-adapted iteration parameters (step sizes and threshold levels). The optimization problem is solved efficiently via a proper chaining of basic iteration modules. The higher level description of the algorithm is similar to that of a multigrid solver for PDEs, but there is one fundamental difference: the latter iterates though a sequence of multiresolution versions of the original problem, while, in our case, we cycle through the wavelet subspaces corresponding to the difference between successive approximations. This strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation. We establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer. We also provide experimental evidence that the improvement in convergence rate is essentially determined by the (unconstrained) linear part of the algorithm, irrespective of the type of wavelet. Finally, we illustrate the technique with some image deconvolution examples, including some real 3-D fluorescence microscopy data.

A note on wavelet subspaces

The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from Vasilevski (Integral Equ. Operator Theory 33:471-488, 1999) with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.

Determining an unknown source in the heat equation by a wavelet dual least squares method

We consider the problem of determining an unknown source, which depends only on the spatial variable, in a heat equation. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. For a reconstruction of the unknown source from measured data the dual least squares method generated by a family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found, under which the convergence result with the error estimate is obtained.

Studies of spectral properties of short genes using the wavelet subspace Hilbert–Huang transform (WSHHT)

This paper presents a new algorithm for the analysis of spectral properties of short genes using the wavelet transform and the Hilbert–Huang transform (HHT). A wavelet subspace algorithm combined with the empirical mode decomposition (EMD) is introduced to create subdivided intrinsic mode functions (IMFs) and a cross-correlation analysis is applied to remove pseudo-spectral components. Experiments are carried out on DNA sequences with the double-base (DB) curve representation and the results show that the signal-to-noise ratio of buried signals can be enhanced using the proposed method, yielding significant patterns that are rarely observed with conventional methods. The wavelet subspace Hilbert–Huang transform (WSHHT) algorithm is able to correctly identify spectral patterns of very short genes (below 70 bp) in DNA sequences.

Aliasing Error of Sampling Series in Wavelet Subspaces

Alising error arises whenever a sampling formula, valid for a prescribed space, is applied to a function in a bigger space. In this work, we estimate the aliasing error of classic and average sampling expansions in wavelet subspaces of a multiresolution analysis.