scholarly journals Tight Wavelet Frame Decomposition and Its Application in Image Processing

2008 ◽  
Vol 40 (2) ◽  
pp. 151-165
Author(s):  
Mahmud Yunus ◽  
Hendra Gunawan
Keyword(s):  
Image Processing ◽  
Wavelet Frame ◽  
Tight Wavelet Frame

scholarly journals Tight wavelet frames in Lebesgue and Sobolev spaces

2004 ◽  
Vol 2 (3) ◽  
pp. 227-252 ◽  
Author(s):  
L. Borup ◽  
R. Gribonval ◽  
M. Nielsen
Keyword(s):  
Sobolev Space ◽  
Jackson Inequality ◽  
Wavelet Frame ◽  
Approximation Rate ◽  
Wavelet Frames ◽  
Atomic Decompositions ◽  
Approximation Spaces ◽  
Tight Wavelet Frame ◽  
Frame Systems ◽  
Term Approximation

We study tight wavelet frame systems inLp(ℝd)and prove that such systems (under mild hypotheses) give atomic decompositions ofLp(ℝd)for1≺p≺∞. We also characterizeLp(ℝd)and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for bestm-term approximation with the systems inLp(ℝd)and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for bestm-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.

Frame multiresolution analysis of continuous piecewise linear functions

2021 ◽  
pp. 2150032
Author(s):  
S. Pitchai Murugan ◽  
G. P. Youvaraj
Keyword(s):  
Multiresolution Analysis ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Perfect Reconstruction ◽  
Wavelet Frame ◽  
Perfect Reconstruction Filter Banks ◽  
Reconstruction Filter ◽  
Dilation Factor ◽  
Tight Wavelet Frame ◽  
Frame Multiresolution Analysis

The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.

Practical design of perfect-translation-invariant real-valued discrete wavelet transform

2014 ◽  
Vol 12 (04) ◽  
pp. 1460005 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang ◽  
Takashi Imamura
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Irrational Number ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Wavelet Frame ◽  
Reconstruction Algorithms ◽  
The Real ◽  
Practical Design ◽  
Tight Wavelet Frame

The real-valued tight wavelet frame having perfect translation invariance (PTI) has already proposed. However, due to the irrational-number distances between wavelets, its calculation amount is very large. In this paper, based on the real-valued tight wavelet frame, a practical design of a real-valued discrete wavelet transform (DWT) having PTI is proposed. In this transform, all the distances between wavelets are multiples of 1/4, and its transform and inverse transform are calculated fast by decomposition and reconstruction algorithms at the sacrifice of a tight wavelet frame. However, the real-valued DWT achieves an approximate tight wavelet frame.

scholarly journals Tight wavelet frame sets in finite vector spaces

2019 ◽  
Vol 46 (1) ◽  
pp. 192-205
Author(s):  
Alex Iosevich ◽  
Chun-Kit Lai ◽  
Azita Mayeli
Keyword(s):  
Vector Spaces ◽  
Wavelet Frame ◽  
Finite Vector ◽  
Tight Wavelet Frame ◽  
Finite Vector Spaces
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