Tight Wavelet Frame Decomposition and Its Application in Image Processing

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.

Download Full-text

Frame multiresolution analysis of continuous piecewise linear functions

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691321500326 ◽

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.

Download Full-text

A GPGPU accelerated compressed sensing with tight wavelet frame transform technique for MR imaging reconstruction

2012 IEEE International Conference on Imaging Systems and Techniques Proceedings ◽

10.1109/ist.2012.6295566 ◽

2012 ◽

Cited By ~ 1

Author(s):

B. Hu ◽

X. Ma ◽

M. Joyce ◽

Paul. Glover ◽

Buhary. Naleem

Keyword(s):

Mr Imaging ◽

Compressed Sensing ◽

Wavelet Frame ◽

Imaging Reconstruction ◽

Tight Wavelet Frame ◽

Wavelet Frame Transform

Download Full-text

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

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691314600054 ◽

2014 ◽

Vol 12 (04) ◽

pp. 1460005 ◽

Cited By ~ 6

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.

Download Full-text