Inpainting of multiple blind motion-blurred images based on multi-scale tight wavelet frame

2017 ◽  
Author(s):  
Cao Chunhong ◽  
Duan Wei ◽  
Wang Minmin ◽  
Hu Kai
Keyword(s):  
Wavelet Frame ◽  
Multi Scale ◽  
Tight Wavelet Frame ◽  
Blurred Images

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.

scholarly journals Nonconvex regularization for blurred images with Cauchy noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiao Ai ◽  
Guoxi Ni ◽  
Tieyong Zeng
Keyword(s):  
Minimization Problem ◽  
Variational Approach ◽  
Numerical Experiments ◽  
Alternating Direction Method ◽  
Wavelet Frame ◽  
Alternating Direction ◽  
Nonconvex Regularization ◽  
Blurred Images ◽  
Different Levels

<p style='text-indent:20px;'>In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of <inline-formula><tex-math id="M1">\begin{document}$ l_1 $\end{document}</tex-math></inline-formula>-norm and <inline-formula><tex-math id="M2">\begin{document}$ l_2 $\end{document}</tex-math></inline-formula>-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.</p>

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