Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet framesψ={ψ1,ψ2}are derived.Firstly,a necessary and sufficient condition for constructing the conjugate symmet...Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet framesψ={ψ1,ψ2}are derived.Firstly,a necessary and sufficient condition for constructing the conjugate symmetric complex tight wavelet frames is established.Secondly,based on a given conjugate symmetric low pass filter,a description of a family of complex wavelet frame solutions is provided when the low pass filter is of even length.When one wavelet is conjugate symmetric and the other is conjugate antisymmetric,the two wavelet filters can be obtained by matching the roots of associated polynomials.Finally,two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames which have some vanishing moments.展开更多
From the inequality |P(z)|2 + |P(-z)|2 ≤1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows th...From the inequality |P(z)|2 + |P(-z)|2 ≤1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and high-pass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them.展开更多
We investigate the construction of two-direction tight wavelet frames First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple con...We investigate the construction of two-direction tight wavelet frames First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.展开更多
We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with su...We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete characterization of the approximation spaces is derived.展开更多
In quantitative susceptibility mapping(QSM),the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in...In quantitative susceptibility mapping(QSM),the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in the measured local field data.Even though the sparsity based first generation harmonic incompatibility removal(1GHIRE)model has achieved the performance gain over the traditional approaches,the 1GHIRE model has to be further improved as there is a basis mismatch underlying in numerically solving Poisson’s equation for the background removal.In this paper,we propose the second generation harmonic incompatibility removal(2GHIRE)model to reduce a basis mismatch,inspired by the balanced approach in the tight frame based image restoration.Experimental results shows the superiority of the proposed 2GHIRE model both in the restoration qualities and the computational efficiency.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.10631080,Grant No.11126291)Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University,the Scientific Research Foundation of Nanjing University of Information Science and Technology(Grant No.2012X057).
文摘Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet framesψ={ψ1,ψ2}are derived.Firstly,a necessary and sufficient condition for constructing the conjugate symmetric complex tight wavelet frames is established.Secondly,based on a given conjugate symmetric low pass filter,a description of a family of complex wavelet frame solutions is provided when the low pass filter is of even length.When one wavelet is conjugate symmetric and the other is conjugate antisymmetric,the two wavelet filters can be obtained by matching the roots of associated polynomials.Finally,two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames which have some vanishing moments.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.90104004 and 69735020)973 Project of China(Grant No.G1999075105).
文摘From the inequality |P(z)|2 + |P(-z)|2 ≤1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and high-pass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11071152）, the Natural Science Foundation of Guangdong Province （Grant No. $2011010004511） Education Department of Henan Province and the Science and Technology Research of （Grant No. 14B520045）.
文摘We investigate the construction of two-direction tight wavelet frames First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.
基金This work is in part supported by the Danish Technical Science Foundation, Grant no. 9701481.
文摘We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete characterization of the approximation spaces is derived.
基金The research of the first author is supported in part by the NSFC Youth Program 11901338The research of the second author is supported by the Hong Kong Research Grant Council(HKRGC)GRF 16306317 and 16309219+2 种基金The research of the third author is supported by the NSFC Youth Program 11901436 and the Fundamental Research Program of Science and Technology Commission of Shanghai Municipality(20JC1413500)The research of the fourth author is supported by the NSFC grant 11831002The research of the fifth author is supported by the National Natural Science Foundation of China Youth Program grant 11801088 and the Shanghai Sailing Program(18YF1401600).
文摘In quantitative susceptibility mapping(QSM),the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in the measured local field data.Even though the sparsity based first generation harmonic incompatibility removal(1GHIRE)model has achieved the performance gain over the traditional approaches,the 1GHIRE model has to be further improved as there is a basis mismatch underlying in numerically solving Poisson’s equation for the background removal.In this paper,we propose the second generation harmonic incompatibility removal(2GHIRE)model to reduce a basis mismatch,inspired by the balanced approach in the tight frame based image restoration.Experimental results shows the superiority of the proposed 2GHIRE model both in the restoration qualities and the computational efficiency.
