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   -> Volume 3, Issue 12

Preprint: (p)reprints from X. Xia and co-workers available
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Author Message (Xianggen Xia)

PostPosted: Mon Dec 02, 2002 1:17 pm    
Subject: Preprint: (p)reprints from X. Xia and co-workers available
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Preprint: (p)reprints from X. Xia and co-workers available

0. Preprint: Vector-Valued Wavelets and Vector Filter Banks

Authors: Xiang-Gen Xia and Bruce W. Suter
Dept. of Electrical and Computer Engineering
Air Force Institute of Technology
Wright-Patterson AFB, OH 45433-7765

Phone: (513)255-6565 ext. 4413
FAX: (513)476-4055


In this paper, we introduce vector-valued multiresolution
analysis and vector-valued wavelets for vector-valued signal
spaces. We construct vector-valued wavelets by using
paraunitary vector filter bank theory. In particular, we
construct vector-valued Meyer wavelets that are band-limited.
We classify and construct vector-valued wavelets with sampling
property. As an application of vector-valued wavelets, multiwavelets
can be constructed from vector-valued wavelets. We show that certain
linear combinations of known scalar-valued wavelets may yield
multiwavelets. The difference between vector-valued wavelets and
multiwavelets is that vector-valued wavelets can not only decorrelate
vector-valued signals in the time domain but also in the spatial-domain.
We also present discrete vector-valued wavelet
transforms for discrete vector-valued (or blocked) signals, which
can be thought of as a family of unitary vector transforms.

1. Reprint: On Sampling Theorem, Wavelets, Wavelet Transforms,
IEEE Trans. on Signal Processing, vol. 41,
pp. 3524-3535, Dec. 1993.

Authors: Xiang-Gen Xia and Zhen Zhang


In this paper, we classify all orthogonal interpolating
wavelets (OIW) and prove that OIW with compact support
have and only have one possibility that is the Haar wavelet.
We also present a family of OIW that have exponential decay.
With these OIW, an application is the computation of wavelet
series coefficients of a signal by the Mallat algorithm.
For signals that are not in multiresolution spaces, we estimate
the aliasing error in the sampling theorem by using uniform

Remark: In the paper, the OIW was called cardinal orthogonal
scaling functions (COSF).

2. Reprint: The Backus-Gilbert Method for Signals in Reproducing
Kernel Hilbert Spaces and Wavelet Subspaces,
Inverse Problems, vol. 10, pp. 785-804, 1994.

Authors: Xiang-Gen Xia and M. Z. Nashed


The Backus-Gilbert (BG) method is an inversion procedure for
a moment problem when moments of a function and related kernel
functions are known. In this paper, we consider the BG method
when, in addition, the signal to be recovered is known a priori
to be in certain reproducing kernel Hilbert spaces (RKHS), such as
wavelet subspaces. We show that better performance may be achieved
over the original BG method. In particular, under the D-criterion
the BG method with RKHS information for a sampled signal in wavelet
subspaces can be completely recover the original signal, while
the one without any additional information can only provide a
constant-valued signal.

3. Preprint: Signal Extrapolation in Wavelet Subspaces,
to appear in SIAM J. on Scientific Computation,
Jan. 1995.

Authors: Xiang-Gen Xia, C.-C. Jay Kuo and Zhen Zhang


The Papoulis-Gerchberg (PG) algorithm is well-known for band-limited
signal extrapolation. We consider the generalization of the PG
algorithm to signals in the wavelet subspaces in this research.
The uniqueness of the extrapolation for continuous-time signals
is examined, and sufficient conditions on signals and wavelet
bases for the generalized PG (GPG) algorithm to converge are given.
We prove that all Meyer wavelets satisfy the conditions, where
a family of interpolating Meyer wavelets are also given. We also
propose a discrete GPG algorithm for discrete-time signal extrapolation,
and investigate its convergence. Numerical examples are given to
illustrate the performance of the discrete GPG algorithm.

4. Preprint: On Wavelet Coefficient Computation with Optimal
to appear in IEEE Trans. on Signal Processing,
Aug. 1994.

Authors: Xiang-Gen Xia, C.-C. Jay Kuo and Zhen Zhang


Several issues on signal sampling and wavelet coefficient computation
for a continuous time signal with orthogonal or biorthogonal wavelet
bases are studied in this research. Discrete wavelet transform (DWT)
is often used to approximate wavelet series transform (WST) and
continuous wavelet transform (CWT), since they can be computed numerically.
We first study the accuracy of the computed DWT coefficients obtained
from the Mallat and Shensa algorithms as an approximation of the WST
coefficients. Based on the accuracy analysis, we show a procedure to
design optimal FIR prefilters used in the Shensa algorithm to reduce
the approximation error. Then, by extending the sampling theorm in wavelet
spaces, we identify a class of signals whose exact WST coefficients can
be obtained by using the Shensa algorithm. We also derive formulas
characterizing the aliasing error resulted from general signals not
in the class. Finally, numerical examples are presented to show the
performance of the optimal FIR prefilters.

5. Preprint: On Orthogonal Wavelets with Oversampling Property,
to appear in J. of Fourier Analysis and Appl.

Authors: Xiang-Gen Xia


In this paper, we consider orthogonal wavelets with oversampling
property. We prove that if an orthogonal scaling function with exponential
decay has the oversampling property then it has the sampling property, i.e.,
it takes values 1 at 0 and o at other integers, and therefore, an orthogonal
scaling function with compact support has the oversampling property if and only
if it is the Haar function.

6. Preprint: On the Limit of Sampled Signal Extrapolation with a Wavelet
Signal Model

Authors: Xiang-Gen Xia, Li-Chien Lin and C.-C. Jay Kuo


The extrapolation of sampled signals from a given interval using
a wavelet model with various sampling rates is examined in this
paper. We present sufficient conditions on signals and wavelet bases
so that the discrete-time extrapolated signal converges to its
continuous-time counterpart when the sampling rate goes to infinity.
Thus, this work provides a practical procedure to implement
continuous-time signal extrapolation, which is important in wideband
radar and sonar signal processing, with a discrete one via carefully
choosing the sampling rate and wavelet basis. A numerical example
is given to illustarte the performance.
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