Lifting scheme: a new philosophy in biorthogonal wavelet constructions

1995 ◽  
Author(s):  
Wim Sweldens
Keyword(s):  
Lifting Scheme ◽  
Biorthogonal Wavelet

STABILITY OF BIORTHOGONAL WAVELET BASES

2003 ◽  
Vol 01 (01) ◽  
pp. 75-92
Author(s):  
PAUL F. CURRAN ◽  
GARY McDARBY
Keyword(s):  
Linear Operator ◽  
Lifting Scheme ◽  
Simple Eigenvalue ◽  
Single Parameter ◽  
Multiple Eigenvalue ◽  
Scaling Functions ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases ◽  
Compactly Supported

We investigate the lifting scheme as a method for constructing compactly supported biorthogonal scaling functions and wavelets. A well-known issue arising with the use of this scheme is that the resulting functions are only formally biorthogonal. It is not guaranteed that the new wavelet bases actually exist in an acceptable sense. To verify that these bases do exist one must test an associated linear operator to ensure that it has a simple eigenvalue at one and that all its remaining eigenvalues have modulus less than one, a task which becomes numerically intensive if undertaken repeatedly. We simplify this verification procedure in two ways: (i) we show that one need only test an identifiable half of the eigenvalues of the operator, (ii) we show that when the operator depends upon a single parameter, the test first fails for values of that parameter at which the eigenvalue at one becomes a multiple eigenvalue. We propose that this new verification procedure comprises a first step towards determining simple conditions, supplementary to the lifting scheme, to ensure existence of the new wavelets generated by the scheme and develop an algorithm to this effect.

scholarly journals DESIGN OF BIORTHOGONAL WAVELET FILTERS USING DYADIC LIFTING SCHEME

10.5109/12595 ◽  
2005 ◽  
Vol 37 ◽  
pp. 123-136 ◽  
Author(s):  
Turghunjan Abdukirim ◽  
Koichi Niijima ◽  
Shigeru Takano
Keyword(s):  
Lifting Scheme ◽  
Biorthogonal Wavelet ◽  
Wavelet Filters

CONSTRUCTION AND APPLICATIONS OF BIORTHOGONAL TERNARY LOOP SUBDIVISION WAVELETS

2011 ◽  
Vol 09 (04) ◽  
pp. 531-548
Author(s):  
YAOHONG XUE ◽  
XUEZHANG LIANG ◽  
QIANG LI
Keyword(s):  
Multiresolution Analysis ◽  
Numerical Experiments ◽  
Lifting Scheme ◽  
3D Graphics ◽  
Biorthogonal Wavelet ◽  
Progressive Transmission ◽  
Subdivision Surface ◽  
Initial Mesh ◽  
Sufficient Stability ◽  
The Given

In this paper, we present an efficient biorthogonal wavelet construction for the ternary Loop subdivision based on the lifting scheme, and give the corresponding decomposition and reconstruction formulas. We apply the proposed algorithm to the progressive transmission and the denoising of 3D graphics, and show its sufficient stability and good performance by some numerical experiments. Since the usual input mesh does not possess the property of subdivision connectivity, while the multiresolution analysis based on subdivision surface requires the processed initial mesh to be semi-regular, we further propose a new rapid algorithm of constructing the initial mesh which not only possesses ternary subdivision connectivity but also approximates the given input mesh.

A Method of Fabric Defects Detection Based on Biorthogonal Wavelet Lifting Scheme

2012 ◽  
Vol 485 ◽  
pp. 266-270
Author(s):  
Shu Yue Chen ◽  
Ming Qian Xu
Keyword(s):  
Lifting Scheme ◽  
Biorthogonal Wavelet ◽  
Fabric Defect Detection ◽  
Fabric Defect ◽  
Mallat Algorithm ◽  
Image Characteristic ◽  
Space Symmetry ◽  
Characteristic Features ◽  
A New Technique ◽  
Wavelet Lifting

A new technique to inspect fabric defects based on biorthogonal wavelet lifting scheme is presented. In order to get fabric defect detection of rapidity and accuracy, the method is applied to constructs biorthogonal wavelet by lifting scheme and extracts the image characteristic features. The results show that its computing speed is increased by an average of 47% compared with the Mallat algorithm. And it overcomes the lack of space symmetry such as the orthogonal wavelet. The experimental results confirm that it can efficiently inspect four kinds of common fabric defects, warp-lacking, weft-lacking, oil stains, and holes with higher inspection speed.

Multiscale fusion of multimodal medical images using lifting scheme based biorthogonal wavelet transform

Optik ◽  
2019 ◽  
Vol 182 ◽  
pp. 995-1014 ◽  
Author(s):  
Om Prakash ◽  
Chang Min Park ◽  
Ashish Khare ◽  
Moongu Jeon ◽  
Jeonghwan Gwak
Keyword(s):  
Wavelet Transform ◽  
Medical Images ◽  
Lifting Scheme ◽  
Biorthogonal Wavelet ◽  
Biorthogonal Wavelet Transform

scholarly journals Application of wavelets and conformal reflections to finding optimal scheme of fiber placement at 3d printing constructions from composition materials

2021 ◽  
Vol 244 ◽  
pp. 05004
Author(s):  
Yuri Bityukov ◽  
Yuri Deniskin ◽  
Galina Deniskina ◽  
Irina Pocebneva
Keyword(s):  
3D Printing ◽  
Strain State ◽  
Computational Algorithm ◽  
Local Coordinate System ◽  
Lifting Scheme ◽  
Multiple Solution ◽  
Biorthogonal Wavelet ◽  
Fiber Placement ◽  
Analytical Functions ◽  
Composition Materials

The article is devoted to finding the optimal schemes of fiber placement at the production of constructions, reinforced with continuous fibers by 3D printing method. As the optimization of the objective function one of the criteria for the destruction of the composite was chosen. For the process acceleration of multiple solution of the system of partial differential equations describing the stress-strain state of the structure, a computational algorithm based on wavelets built through subdivision schemes is proposed. To set the local coordinate system, it is proposed to use analytical functions, which will be constructed using the well-known Dini and Cisotti formulas, just by specifying the direction of laying the fiber at the product boundary. The article also presents a lifting scheme (lifting scheme) allowing to construct biorthogonal wavelet systems with specified properties using some initial biorthogonal wavelet systems with filters.

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