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## Über dieses Buch

This volume reflects the latest developments in the area of wavelet analysis and its applications. Since the cornerstone lecture of Yves Meyer presented at the ICM 1990 in Kyoto, to some extent, wavelet analysis has often been said to be mainly an applied area. However, a significant percentage of contributions now are connected to theoretical mathematical areas, and the concept of wavelets continuously stretches across various disciplines of mathematics.

Key topics:

Approximation and Fourier Analysis Construction of Wavelets and Frame Theory Fractal and Multifractal Theory Wavelets in Numerical Analysis Time-Frequency Analysis Adaptive Representation of Nonlinear and Non-stationary Signals Applications, particularly in image processing

Through the broad spectrum, ranging from pure and applied mathematics to real applications, the book will be most useful for researchers, engineers and developers alike.

## Inhaltsverzeichnis

### Local Smoothness Conditions on a Function Which Guarantee Convergence of Double Walsh-Fourier Series of This Function

The local smoothness conditions on a function are obtained, which guarantee convergence almost everywhere on some set of positive measure of the double Walsh-Fourier series of this function summed over rectangles.

S. K. Bloshanskaya, I. L. Bloshanskii

### Linear Transformations of ℝN and Problems of Convergence of Fourier Series of Functions Which Equal Zero on Some Set

Let

$$\mathfrak{M}$$

be a class of (all) linear transformations of ℝ

N

,

N

≥ 1. Let

$$\mathcal{A} = \mathcal{A}{\text{(}}\mathbb{T}^N {\text{),}}\mathbb{T}^N = [ - \pi ,\pi )^N$$

be some linear subspace of

$$L_{\text{1}} {\text{(}}\mathbb{T}^N {\text{)}}$$

, and let

$$\mathfrak{A}$$

be an arbitrary set of positive measure

$$\mathfrak{A} \subset \mathbb{T}^N$$

.

We consider the problem: how are the sets of convergence and divergence everywhere or almost everywhere (a.e.) of trigonometric Fourier series (in case

N

≥ 2 summed over rectangles) of function

$$(f \circ \mathfrak{m})(x) = f(\mathfrak{m}(x)),f \in \mathcal{A}$$

,

$$f(x) = 0{\text{ }}on \mathfrak{A}, \mathfrak{m} \in \mathfrak{M}$$

, changed depending on the smoothness of the function

f

(i.e. on the space

$$\mathcal{A}$$

), as well as on the transformation

$$\mathfrak{m}$$

.

In the paper a (wide) class of spaces

$$\mathcal{A}$$

is found such that for each

$$\mathcal{A}$$

the system of classes (of nonsingular linear transformations)

$$\Psi _k ,\Psi _k \subset \mathfrak{M}$$

(

k

= 0, 1 , . . .,

N

), which “change” the sets of convergence and divergence everywhere or a.e. of the indicated Fourier expansions is defined.

I. L. Bloshanskii

### Sidon Type Inequalities for Wavelets

In 1938, [

S. Sidon [9]

] proved an inequality for the complex trigonometric system on inter val [0, 1) known as Sidon type inequality. This inequality was generalized by [

Bojanic and Stanojevic [1]

]. The Walsh case was investigated by [

Moricz and Schipp [7]

]. Another generalization for trigonometric case was given by [

Buntinas and Tanovic-Miller [2]

]. Here in this paper we proved it for wavelet case and also obtained the convergence of wavelet series in

L

1

norm.

N. A. Sheikh

### Almansi Decomposition for Dunkl-Helmholtz Operators

We consider the iterated Dunkl-Helmholtz equation (Δ

h

γ

)

n

f

= 0 for nonzero

γ

in a domain of ℝ

N

. Here Δ

h

= Σ

j

=1

N

D

j

2

is the Dunkl Laplacian, and

D

j

is the Dunkl operator attached to the Coxeter group

G

associated with the reduced root system

R

,

$$\mathcal{D}_j f(x) = \frac{{\partial f}} {{\partial x_j }}(x) + \sum\limits_{v \in R_ + } {k_v \frac{{f(x) - f(\sigma _v x)}} {{\langle x,v\rangle }}} v_j ,$$

where

κ

v

is a multiplicity function on

R

and

σ

v

is the reflection with respect to the root

v

.

We prove that any solution

f

of the iterated Dunkl-Helmholtz equation has a decomposition of the form

$$f(x) = f_0 (x) + R_\mu f_1 (x) + \cdots + R_\mu ^{n - 1} f_{n - 1} (x),\forall x \in \Omega ,$$

where

f

j

are annihilated by Δ

h

γ

,

μ

is a fixed but arbitrary complex number, and

R

μ

n

= (

R

μ

)

n

are given by

R

μ

=

μ

I +

R

0

, with

I

the identity operator and

R

0

the Euler operator.

Guangbin Ren, Helmuth R. Malonek

### An Uncertainty Principle for Operators

Hardy—s Uncertainty Principle asserts that if

f

is a function on ℝ

n

such that exp(

α

| · |

2

)

f

and exp

$$(\beta | \cdot |^2 )\hat f$$

are bounded, where

$$\alpha \beta > \tfrac{1} {4}$$

, then

f

= 0. In this paper, we prove a version of Hardy—s result for operators.

Michael G. Cowling, M. Sundari

### Uncertainty Principle for Clifford Geometric Algebras Cl n,0, n = 3 (mod 4) Based on Clifford Fourier Transform

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we define a generalized real Fourier transform on Clifford multivector-valued functions (

f

: ℝ

n

Cl

n

,0

,

n

= 3 (mod 4)). Third, we introduce a set of important properties of the Clifford Fourier transform on

Cl

n

,0

,

n

= 3 (mod 4) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving a directional uncertainty principle for

Cl

n

,0

n

= 3 (mod 4) multivector functions.

Eckhard S. M. Hitzer, Bahri Mawardi

### Orthogonal Wavelet Vectors in a Hilbert Space

In a Hilbert space, some concepts, such as orthogonal wavelet vector, multiresolution analysis(MRA), scaling vector, unitary-shift operator, are introduced, the existence of scaling vectors and orthogonal wavelet vectors are proved, and the standard forms of them are also given. Our abstract arguments give a short and brief proof of the usual existence result of orthogonal wavelet in

L

2

(ℝ).

Huai-Xin Cao, Bao-Min Yu

### Operator Frames for $$B(\mathcal{H})$$

Operator frames for the space

$$B(\mathcal{H})$$

of all bounded linear operators on a Hilbert space

$$(\mathcal{H})$$

are introduced and discussed. By introducing the concept of operator response of vectors in a Hilbert space, we establish a relationship between operator frames for

$$B(\mathcal{H})$$

and usual frames for Hilbert space

$$(\mathcal{H})$$

and show that operator frames preserve so many properties of usual frames that we can say that the concept of operator frames is a generalization of frames for Hilbert spaces. In fact, a frame for a Hilbert space or a frame of subspaces for a Hilbert space may be considered as a special case of operator frames in certain sense.

Chun-Yan Li, Huai-Xin Cao

### On the Stability of Multi-wavelet Frames

Frame plays an important role in the theory of wavelet analysis. Frame theory and stability of frames are important topics of wavelet analysis. Recently, people pay more attention to multi-wavelet frames. Among literatures, Chui [

2

], for instance, give a complete characterization of multi-wavelet frames for arbitrary dilation factor

a

> 1. There, however, is relatively less results on the stability of multi-wavelet frames. This paper devotes to the study of stability of multi-wavelet frames based on functional analysis methods. The following meaningful results are obtained: firstly, multi-wavelet frames are stable by some kinds of linear operators action; Secondly, multi-wavelet frames are stable over some kinds of perturbations conditions on ψ.

Gang Wang, ZhengXing Cheng

### Biorthogonal Wavelets Associated with Two-Dimensional Interpolatory Function

To construct biorthogonal wavelets from two-dimensional interpolatory function, a large amount of computation is involved in traditional method. In this paper, a method is developed for constructing the biorthogonal wavelets. Masks of the biorthogonal wavelets are given explicitly. Neither the Gram-Schmidt processing nor the inverse of a nonsingular polynomial matrix is needed.

Jianwei Yang, Yuan Yan Tang, Zhengxing Cheng, Xinge You

### Parameterization of Orthogonal Filter Bank with Linear Phase

For the M-channel FIR orthogonal filter bank with linear phase, a complete parameterization is obtained by applying the singular value decomposition of matrices related to the corresponding polyphase matrix. In the obtained parameterization forms, the number of the required parameters is reduced to

$$(N + 2)(\tfrac{M} {{\mathop 2\limits_2 }})$$

.

Xiaoxia Feng, Zhengxing Cheng, Zhongpeng Yang

### On Multivariate Wavelets with Trigonometric Vanishing Moments

Wavelets with trigonometric vanishing moments are studied for the first time. A practical construction algorithm of multivariate orthonormal wavelets with trigonometric vanishing moments is proposed. Based on such construction algorithm, a tight frame of

L

2

(ℝ

d

) can be obtained even at the worst case. An example of construction of bivariate orthonormal wavelets providing concrete trigonometric vanishing moments is presented.

Ying Li, Zhi-Dong Deng, Yan-Chun Liang

### Directional Wavelet Analysis with Fourier-Type Bases for Image Processing

Motivated by the fact that in natural images, there is usually a presence of local strongly oriented features such as directional textures and linear discontinuities, a representation which is both well-localised in frequency and orientation is desirable to efficiently describe those oriented features. Here we introduce a family of multiscale trigonometric bases for image processing using Fourier-type constructions, namely, the multiscale directional cosine transform and the multiscale Fourier transform. We also show that by seeking an adaptive basis locally, the proposed bases are able to capture both oriented harmonics as well as discontinuities, although the complexity of such adaptiveness varies significantly. We conducted denoising experiments with the proposed bases and the results show great promise of the proposed directional wavelet bases.

Zhen Yao, Nasir Rajpoot, Roland Wilson

### Unitary Systems and Wavelet Sets

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators de-fined in terms of translation and dilation operations. We will describe an operator-interpolation approach to wavelet theory using the local commutant of a unitary system. This is an application of the theory of operator algebras to wavelet theory. The concrete applications to wavelet theory include results obtained using specially constructed families of wavelet sets. The main section of this paper is section 5, in which we introduce the interpolation map σ induced by a pair of wavelet sets, and give an exposition of its properties and its utility in constructing new wavelets from old. The earlier sections build up to this, establishing terminology and giving examples. The main theoretical result is the Coefficient Criterion, which is described in Section 5.2.2, and which gives a matrix valued function criterion specificing precisely when a function with frequency support contained in the union of an interpolation family of wavelet sets is in fact a wavelet. This can be used to derive Meyer’s famous class of wavelets using an interpolation pair of Shannon-type wavelet sets as a starting point. Section 5.3 contains a new result on interpolation pairs of wavelet sets: a proof that every pair of sets in the generalized Journe family of wavelet sets is an interpolation pair. We will discuss some results that are due to this speaker and his former and current students. And we finish in section 6 with a discussion of some open problems on wavelets and frame-wavelets.

David R. Larson

### Clifford Analysis and the Continuous Spherical Wavelet Transform

We present a group-theoretical approach for the continuous wavelet transform on the sphere

S

n

−1

, based on the Lorentz group Spin(1,

n

) (the conformal group of the unit sphere). We introduce transformations on the sphere based on the decomposition of the group Spin(1,

n

) into the maximal compact subgroup of rotations (Spin(

n

)) and the set of Möbius transformations in ℝ

n

of the form

ϕ

a

(

x

) = (

x

a

)(1 +

ax

)

−1

, |

a

| < 1. This approach presents an advantage of allowing the full use of the whole of the conformal group Spin(1,

n

), and in such way, it is a generalization of the continuous wavelet transform defined by J. P. Antoine and P. Vandergheynst (see

[1], [2]

). We will give an account of the influence of the parameter

a

arising in the definition of dilatations / contractions on the sphere. Finally we give different representations (with different properties) for the Hilbert space

L

2

(

S

n

−1

) and the Hardy space

H

2

.

Paula Cerejeiras, Milton Ferreira, Uwe Kähler

### Clifford-Jacobi Polynomials and the Associated Continuous Wavelet Transform in Euclidean Space

Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are presented within the framework of Clifford analysis. These multi-dimensional wavelets are constructed by taking the Clifford-monogenic extension to ℝ

m

+1

of specific functions in ℝ

m

generalizing the traditional Jacobi weights. The notion of Clifford-monogenic function is a direct higher dimensional generalization of that of holomorphic function in the complex plane. Moreover, crucial to this construction is the orthogonal decomposition of the space of square integrable functions into the Hardy space

H

2

(ℝ

m

) and its orthogonal complement. In this way a nice relationship is established between the theory of the Clifford Continuous Wavelet Transform on the one hand, and the theory of Hardy spaces on the other hand. Furthermore, also new multi-dimensional polynomials, the so-called Clifford-Jacobi polynomials, are obtained.

Fred Brackx, Nele De Schepper, Frank Sommen

### Wavelet Leaders in Multifractal Analysis

The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Hölder singularities. The properties of this new multifractal formalism and of the alternative weak scaling exponent multifractal formalism are investigated. Examples based on known synthetic multifractal processes are illustrating its numerical implementation and abilities.

Stéphane Jaffard, Bruno Lashermes, Patrice Abry

### Application of Fast Wavelet Transformation in Parametric System Identification

This work deals with discrete embedding of system operators in identification models on basis of Fast Wavelet Transformation (FWT). In particular for FWT-models of linear dynamic systems the missing variables can be calculated with the help of connection coefficients. The application of connection coefficients provides the direct projection of the system operators into the corresponding wavelet space. Here a class of operators is introduced, which satisfies certain permutability relations with respect to dilations and translations. This class contains especially derivation and integration operators and some special convolution operators, like the Hilbert-transform. Such a definition allows the systematic determination of generalized connection coefficients. It gives so the possibility to realize identification procedures for different models and their implementations in a unified pattern. The method can be used for all biorthogonal wavelet systems whose synthesis functions are in the domain of the system operators.

Klaus Markwardt

### Image Denoising by a Novel Digital Curvelet Reconstruction Algorithm

For an anisotropic image, wavelets lose their effects on singularity detection because discontinuities across edges are spatially distributed. Based on the idea of curvelet, a new digital curvelet reconstruction algorithm is proposed. Our algorithm provides sparser representations and keeps low computational complexity. When applying it to the image denoising, much better results than the original algorithm are obtained.

Jian Bai, Xiang-Chu Feng

### Condition Number for Under-Determined Toeplitz Systems

In this note, we prove that the structured condition number for Toeplitz under-determined systems with full row rank is not better than unstructured condition number in probability sense.

Huaian Diao, Yimin Wei

### Powell-Sabin Spline Prewavelets on the Hexagonal Lattice

In this paper we give an explicit construction of compactly supported prewavelets on differentiable, twodimensional, piecewise polynomial quadratic finite element spaces of

L

2

(ℝ

2

), sampled on the hexagonal grid. The obtained prewavelet basis is stable in the Sobolev spaces

$$\mathcal{H}^s$$

for

$$|s| < \tfrac{5} {2}$$

. In particular, the prewavelet basis is generated by one single function vector

ψ

consisting of three generating functions

ψ

1

,

ψ

2

,

ψ

3

that are globally invariant by a rotation of 2

π

/3.

### Time-Frequency Aspects of Nonlinear Fourier Atoms

In the standard Fourier analysis one uses the linear Fourier atoms

e

int

:

n

∈ ℤ. With only the linear phases

nt

Fourier analysis can not expose the essence of time-varying frequencies of nonlinear and non-stationary signals. In this note we study time-frequency properties of a new family of atoms

e

inθa

(t)

:

n

∈ ℤ, non-linear Fourier atoms, where

a

is any but fixed complex number with |a| < 1, and

d

θ

a

(

t

) a harmonic measure on the unit circle parameterized by

t

. The nonlinear Fourier atoms

e

inθ

a

(t)

:

n

∈ ℤ were first noted in [

12

] with some examples and theoretically studied in [

8

]. In this note we show that the real parts cos

θ

a

(

t

), |a| < 1, form a family of intrinsic mode functions introduced in the HHT theory [

5

]. We prove that for a fixed a the set

e

inθa (t)

:

n

∈ ℤ, constitutes a Riesz basis in the space

L

2

([0, 2π]). Some miscellaneous results including Shannon type sampling theorems are obtained.

Qiuhui Chen, Luoqing Li, Tao Qian

### Mono-components for Signal Decomposition

In relation to the study of instantaneous frequency, HHT and the EMD algorithm in signal analysis people have been trying to find solutions of the eigenfunction problem: Find

f

(

t

) = ρ(

t

)

e

iθ(t)

such that

Hf

= −

if

, ρ(

t

) ≥ 0 and

θ

′(

t

) ≥ 0, a.e., where

Hf

is Hilbert transform of

f

. This article serves as a survey on some recent studies, and presents some new results as well. In the survey part we first review the systematic study on the unimodular case, and then give a detailed account on a fundamental class of non-unimodular solutions, called H-atoms, in terms of starlike functions in one complex variable. As new result we construct certain circular monocomponents that do not fall into the category of H-atoms but of the form ρ(

t

)

e

iθa (t)

, where ρ(

t

) ≥ 0, and

e

iθa (t)

is some Fourier atom, as well as those of the form ρ(

s

)

e

iφa (s)

, where

e

iφa (s)

is one on the line induced from some Fourier atom under Cayley transform.

Tao Qian

### Signal-Adaptive Aeroelastic Flight Data Analysis with HHT

This paper investigates the utility of the Hilbert-Huang transform for the analysis of aeroelastic flight data. The recently-developed Hilbert-Huang algorithm addresses the limitations of the classical Hilbert transform through a process known as empirical mode decomposition. Using this approach, the data is filtered into a series of intrinsic mode functions, each of which admits a well-behaved Hilbert transform. In this manner, the Hilbert-Huang algorithm affords time-frequency analysis of a large class of signals. The purpose of this paper is to demonstrate the potential applications of the Hilbert-Huang algorithm for the analysis of aeroelastic systems. Applications for correlations between system input and output, and amongst output sensors, are discussed to characterize the time-varying amplitude and frequency correlations present in the various components of multiple data channels. Examples are given using aeroelastic flight test data from the F/A-18 Active Aeroelastic Wing aircraft and Aerostructures Test Wing.

Martin J. Brenner, Sunil L. Kukreja, Richard J. Prazenica

### An Adaptive Data Analysis Method for Nonlinear and Nonstationary Time Series: The Empirical Mode Decomposition and Hilbert Spectral Analysis

An adaptive data analysis method, the Empirical Mode Decomposition and Hilbert Spectral Analysis, is introduced and reviewed briefly. The salient properties of the method is emphasized in this review; namely, physical meaningful adaptive basis, instantaneous frequency, and using intra-wave frequency modulation to represent nonlinear waveform distortion. This method can perform and enhance most of the traditional data analysis task such as filtering, regression, and spectral analysis adaptively. Also presented are the mathematical problems associated with the new method. It is hope that this presentation will entice the interest of the mathematical community to examine this empirically based method and inject mathematical rigor into the new approach.

Norden E. Huang

### Transfer Colors from CVHD to MRI Based on Wavelets Transform

A new algorithm based on wavelet transform to transfer colors from image of Chinese Virtual Human Data (CVHD) to Magnetic Resonance Images (MRI) has been proposed and implemented. The algorithm firstly extracts the primary components from both CVHD and MR images by waveletbased multi-resolution analysis, then mapping colors from CVHD to MRI between these primary components with the similar characters. Finally these colors in MRI will be transferred to all pixels of the MRI according to their characters. Several experiment results have been reported, which have confirmed the effectiveness of this new color-transferring scheme.

Xiaolin Tian, Xueke Li, Yankui Sun, Zesheng Tang

### Medical Image Fusion by Multi-resolution Analysis of Wavelets Transform

A novel algorithm for the multimodalities medical images fusion based on wavelet transform has been proposed and implemented. The autoadaptive weighted coefficients have been calculated recursively to maximize the mutual information between the source image and the result image. Adopting multi-resolution analysis of wavelet transform, we achieved the MRI and CT image fusion. In addition, the new algorithm has been extended to MRI and color image fusion. The experiment results demonstrate that the new algorithm with wavelet transform have better fusion results compared with other mutual information fusion schemes without wavelet transform.

Xueke Li, Xiaolin Tian, Yankui Sun, Zesheng Tang

### Salient Building Detection from a Single Nature Image via Wavelet Decomposition

We describes how wavelet decomposition can be used in detecting salient building from a single nature image. Firstly, we use Haar wavelet decomposition to obtain the enhanced image which is the sum of the square of LH sub-image and HL sub-image. Secondly, we separate the candidates of building from the background based on projection profile. Finally, we discriminate the building by Principle Component Analysis(PCA) in RGB color space. The proposed approach has been tested on many real-world images with promising results.

Yanyun Qu, Cuihua Li, Nanning Zheng, Zejian Yuan, Congying Ye

### SAR Images Despeckling via Bayesian Fuzzy Shrinkage Based on Stationary Wavelet Transform

An efficient despeckling method is proposed based on stationary wavelet transform (SWT) for synthetic aperture radar (SAR) images. The statistical model of wavelet coefficients is analyzed and its performance is modeled with a mixture density of two zero-mean Gaussian distributions. A fuzzy shrinkage factor is derived based on the minimum mean error (MMSE) criteria with bayesian estimation. In this case, the ideas of region division and fuzzy shrinkage are adopted according to the interscale dependencies among wavelet coefficients. The noise-free wavelet coefficients are estimated accurately. Experimental results show that our method outperforms the refined Lee filterwavelet soft thresholding shrinkage and SWT shrinkage algorithms in terms of smoothing effects and edges preservation.

Yan Wu, Xia Wang, Guisheng Liao

### Super-Resolution Reconstruction Using Haar Wavelet Estimation

High resolution image reconstruction refers to the reconstruction of a high resolution image from a set of shifted, blurred low resolution images. Many methods have been developed, and most of them are iterative methods. In this paper, we present a direct method to obtain the reconstruction. Our method takes advantages of the properties of Haar wavelet transform of the high resolution image and its relationship with the low resolution images. Thus the coefficients of the Haar wavelet transform of the high resolution image can be estimated from the low resolution images. Our method is very simple to implement and is very efficient. Experiments show that it is robust to boundary conditions and superior to the least - squares method especially in the low - noise case.

C. S. Tong, K. T. Leung

### The Design of Hilbert Transform Pairs in Dual-Tree Complex Wavelet Transform

An approach for designing biorthogonal dual- tree complex wavelet transform filters is proposed, where the two related wavelets form an approximate Hilbert transform pair. Different from the existing design techniques, the two wavelet filter banks obtained here are both of linear phase. By adjusting the parameters, wavelet filters with rational coefficients may be achieved. The designed examples show that the lengths of wavelet filters may be effectively shorted while efficient approximation to Hilbert transform pairs is still kept. The validity of the proposed design scheme is exhibited through an application to dual-tree complex wavelet for iris image enhancement.

Fengxia Yan, Lizhi Cheng, Hongxia Wang

### Supervised Learning Using Characteristic Generalized Gaussian Density and Its Application to Chinese Materia Medica Identification

This paper presents the estimation of the characteristic generalized Gaussian density (CGGD) given a set of known GGD distributions based on some optimization techniques, and its application to the Chinese Materia Medica identification. The CGGD parameters are estimated by minimizing the distance between the CGGD distribution and known GGD distributions. Our experimental results show that the proposed signature based on the CGGD together with the use of Kullback-Leibler distance outperforms the traditional wavelet-based energy signature. The recognition rate for the proposed method is higher than the energy signature by at least 10% to around 60% – 70%. Nevertheless, the extraction of CGGD estimators still retains comparable level of computational complexity. In general, our proposed method is very competitive compared with many other existing Chinese Materia Medica classification methods.

S. K. Choy, C. S. Tong

### A Novel Algorithm of Singular Points Detection for Fingerprint Images

It is very important to effectively detect singularities (core and delta)for fingerprint matching, fingerprint classification and orientation flow modeling. In this paper, based on multilevel partitions in a fingerprint image, we present a new method of singularity detection to improve the accuracy and reliability of the singularities. Firstly, based on the information of the orientation field, with the Poincaré index method, we detect singularities which are estimated by different block sizes and various methods of orientation field estimation (smoothing or no smoothing). Secondly, based on the corresponding relationship between the singularities detected by multilevel block sizes and by different methods of orientation field estimation, we extract the singularities precisely and reliably. Finally, an experiment is done in the NJU-2000 fingerprint database that has 2500 fingerprints. The result shows that the method performs well and it is robust to poor quality images.

Taizhe Tan, Jiwu Huang

We present a new receiver scheme, termed as

, for doubly-selective channels to combat the annoying Doppler effect. The key point is to convert the Doppler effect to Doppler diversity, taking advantage of the diversity technique to improve system performance. To this end, a new framework based on multiresolution analysis (MRA) is established. In this framework, we find that RAKE receiver which can only combat the multipath fading, is a special case of

which can exploit joint multipath and Doppler fading. Theoretical analysis and experimental simulation show that

can greatly enhance system performance.

Guangjian Shi, Silong Peng

### Face Retrieval with Relevance Feedback Using Lifting Wavelets Features

By using support vector machine (SVM), this paper presents a novel face retrieval scheme in face database based on lifting wavelets features. The relevance feedback mechanism is also performed. The scheme can be described in three stages as follows. First, lifting wavelets decomposition technique is employed because it not only can extract the optimal intrinsic features for representing a face image, but also can accelerate the speed of the wavelets transform. Second, Linear Discriminant Analysis (LDA) is adopted to reduce the feature dimensionality and enhance the class discriminability. Third, relevance feedback using SVM is applied to learn on user’s feedback to refine the retrieval performance. The experimental evaluation has been conducted on ORL dataset in which the results show that our proposed approach is effective and promising.

Chon Fong Wong, Jianke Zhu, Mang I Vai, Peng Un Mak, Weikou Ye

### High-Resolution Image Reconstruction Using Wavelet Lifting Scheme

High-resolution image reconstruction refers to reconstruction of high-resolution images from multiple low-resolution, shifted, blurred samples of a true image. By expressing the true image as a square integrable function, Point Spread Function (PSF) can be used to construct biorthogonal wavelet filters directly and some algorithms for high-resolution image reconstruction were proposed based on the filters. However, the filters are the piecewise linear spline and corresponding primal and dual wavelet functions are all one vanishing moments. In order to improve the quality of reconstructed high-resolution images, we propose a method in this paper which can increase the numbers of vanishing moments of the wavelet functions so as to improve the performance of the biorthogonal filters using wavelet lifting scheme. Experiment results show that the method can improve the quality of reconstructed high-resolution images effectively. Also, we derive a fast algorithm that can reconstruct high-resolution images efficiently when blurring matrix is block-circulant-circulant-block (BCCB) matrix or Toeplitze-plus-Hankel system with Toeplitze-plus-Hankel block (THTH) matrix.

Shengwei Pei, Haiyan Feng, Minghui Du

### Mulitiresolution Spatial Data Compression Using Lifting Scheme

In many applications referring to terrain visualization, there is need to visualize terrains at different levels of detail. The terrain should be visualized at different levels for different parts; for example, a region of high interest should be in a higher resolution than a region of low or no interest. The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. In this paper, it is extended to the GIS data compression. A newly developed data compression approach to approximate the land surface with a series of non-overlapping triangle s has been presented. Over the years the TIN data representation has become a case in point for many researchers due its large data size. Compression of TIN is needed for efficient management of large data and good surface visualization. T his approach covers following steps: First, by using a Delaunay triangulation, an efficient algorithm is developed to generate TIN, which forms the terrain from an arbitrary set of data. A new interpolation wavelet filter for TIN has been applied in two steps, namely splitting and elevation. In the splitting step, a triangle has been divided into several sub-triangles and the elevation step has been used to ‘modify’ the point values (point coordinates for geometry) after the splitting. Then, this data set is compressed at the desired locations by using second generation wavelets. The quality of geographical surface representation after using proposed technique is compared with the original terrain. The results show that this method can be used for significant reduction of data set.

B. Pradhan, K. Sandeep, Shattri Mansor, Abdul Rahman Ramli, Abdul Rashid B. Mohamed Sharif

### Ridgelet Transform as a Feature Extraction Method in Remote Sensing Image Recognition

Using ridgelet transform to do the feature extraction, and RBFNN to do the recognition and classification, a remote sensing image recognition method is put forward in this paper. We do mathematical implementation and experimental investigation of ridgelet transform to analyze its characteristic and show its performance. Since ridgelet transform outperforms wavelet transform in extracting the linear features of objects, the proposed method has higher efficiency than that of wavelets. The simulation in remote sensing image shows its feasibility..

Yuanyuan Ren, Shuang Wang, Shuyuan Yang, Licheng Jiao

### Analysis of Frequency Spectrum for Geometric Modeling in Digital Geometry

In order to explore the properties of frequency spectrum for geometric modeling, a complete orthogonal piecewise k -degree polynomials in

L

2

[0,1], so-called U-system, is introduced. The expansion in U-series has advantageous properties for approximations in both quadratic norm and uniform. Using U-system with finite items, it can be realized to exactly represent geometric modeling. This paper analyzes the properties of frequency spectrum for geometric modeling in theory and gives some interesting results in geometric transform. By comparing U-system with Fourier system, experiments indicate that U-system is more suitable for analyzing frequency spectrum for geometric modeling than Fourier system is.

Zhanchuan Cai, Hui Ma, Wei Sun, Dongxu Qi

### Detection of Spindles in Sleep EEGs Using a Novel Algorithm Based on the Hilbert-Huang Transform

A novel approach for detecting spindles from sleep EEGs (electroencephalograph) automatically is presented in this paper. Empirical mode decomposition (EMD) is employed to decompose a sleep EEG, which are usually typical nonlinear and non-stationary data, into a finite number of intrinsic mode functions (IMF). Based on these IMFs, the Hilbert spectrum of the EEG can be calculated easily and provides a high resolution time-frequency presentation. An algorithm is developed to detect spindles from a sleep EEG accurately, experiments of which show encouraging detection results.

Zhihua Yang, Lihua Yang, Dongxu Qi

### A Wavelet-Domain Hidden Markov Tree Model with Localized Parameters for Image Denoising

Wavelet-domain hidden Markov tree models have been popularly used in many fields. The hidden Markov Tree (HMT) model provides a natural framework for exploiting the statistics of the wavelet coefficients. However, the training process of the model parameters is computationally expensive. In this paper, we propose a HMT model with localized parameters which has a fast parameter estimation algorithm with no complex training process. Firstly, Wold decomposition is used to reduce the influence on the estimation of image noise variance due to texture. Secondly, coefficients in each subband are classified into two classes based on spatially adaptive thresholds. Thirdly, parameters of different class are estimated using the local statistics. Finally, the posterior state probability is estimated with an up-down step like the traditional HMT model. We apply this model to image denoising and compare it with other models for several test images to demonstrate its competitive performance.

Minghui Yang, Zhiyun Xiao, Silong Peng
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