In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.
Published in | Applied and Computational Mathematics (Volume 7, Issue 3) |
DOI | 10.11648/j.acm.20180703.21 |
Page(s) | 155-160 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2018. Published by Science Publishing Group |
Frame Multiresolution Analysis, Polyphase Decomposition, Unitary Matrix Extension
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APA Style
Zhaofeng Li, Yuanyuan Zhang. (2018). An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Applied and Computational Mathematics, 7(3), 155-160. https://doi.org/10.11648/j.acm.20180703.21
ACS Style
Zhaofeng Li; Yuanyuan Zhang. An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Appl. Comput. Math. 2018, 7(3), 155-160. doi: 10.11648/j.acm.20180703.21
AMA Style
Zhaofeng Li, Yuanyuan Zhang. An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA. Appl Comput Math. 2018;7(3):155-160. doi: 10.11648/j.acm.20180703.21
@article{10.11648/j.acm.20180703.21, author = {Zhaofeng Li and Yuanyuan Zhang}, title = {An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA}, journal = {Applied and Computational Mathematics}, volume = {7}, number = {3}, pages = {155-160}, doi = {10.11648/j.acm.20180703.21}, url = {https://doi.org/10.11648/j.acm.20180703.21}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180703.21}, abstract = {In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.}, year = {2018} }
TY - JOUR T1 - An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA AU - Zhaofeng Li AU - Yuanyuan Zhang Y1 - 2018/08/13 PY - 2018 N1 - https://doi.org/10.11648/j.acm.20180703.21 DO - 10.11648/j.acm.20180703.21 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 155 EP - 160 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20180703.21 AB - In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M>2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm. VL - 7 IS - 3 ER -