Starting from the diffraction imaging process,we have discussed the relationship between optical imaging system and fractional Fourier transform, and proposed a specific system which can form an inverse amplified imag...Starting from the diffraction imaging process,we have discussed the relationship between optical imaging system and fractional Fourier transform, and proposed a specific system which can form an inverse amplified image of input function.展开更多
The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its poten...The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.展开更多
This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained ...This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.展开更多
Two discriminant methods,partial least squares-discriminant analysis(PLS-DA)and Fisher's discriminant analysis(FDA),were combined with Fourier transform infrared imaging(FTIRI)to differentiate healthy and osteoart...Two discriminant methods,partial least squares-discriminant analysis(PLS-DA)and Fisher's discriminant analysis(FDA),were combined with Fourier transform infrared imaging(FTIRI)to differentiate healthy and osteoarthritic articular cartilage in a canine model.Osteoarthritic cartilage had been developed for up to two years after the anterior cruciate ligament(ACL)transection in one knee.Cartilage specimens were sectioned into 10μm thickness for FTIRI.A PLS-DA model was developed after spectral pre-processing.All IR spectra extracted from FTIR images were calculated by PLS-DA with the discriminant accuracy of 90%.Prior to FDA,principal component analysis(PCA)was performed to decompose the IR spectral matrix into informative princi pal component matrices.Based on the different discriminant mechanism,the discriminant accuracy(96%)of PCA-FDA with high convenience was higher than that of PLS-DA.No healthy cartilage sample was mis assigned by these two methods.The above mentioned suggested that both integrated technologies of FTIRI-PLS-DA and,especially,FTIRI-PCA-FDA could become a promising tool for the discrimination of healthy and osteoarthritic cartilage specimen as well as the diagnosis of cartilage lesion at microscopic level.The results of the study would be helpful for better understanding the pathology of osteoarthritics.展开更多
The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but ba...The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].展开更多
Texture analysis is a fundamental field in computer vision. However,it is also a particularly difficult problem for no universal mathematical model of real world textures. By extending a new application of the fractio...Texture analysis is a fundamental field in computer vision. However,it is also a particularly difficult problem for no universal mathematical model of real world textures. By extending a new application of the fractional Fourier transform( Fr FT) in the field of texture analysis,this paper proposes an Fr FT-based method for describing textures. Firstly,based on the Radon-Wigner transform,1-D directional Fr FT filters are designed to two types of texture features,i. e.,the coarseness and directionality. Then,the frequencies with maximum and median amplitudes of the Fr FT of the input signal are regarded as the output of the 1-D directional Fr FT filter. Finally,the mean and the standard deviation are used to compose of the feature vector. Compared to the WD-based method,three benefits can be achieved with the proposed Fr FT-based method,i. e.,less memory size,lower computational load,and less disturbed by the cross-terms. The proposed method has been tested on16 standard texture images. The experimental results show that the proposed method is superior to the popular Gabor filtering-based method.展开更多
文摘Starting from the diffraction imaging process,we have discussed the relationship between optical imaging system and fractional Fourier transform, and proposed a specific system which can form an inverse amplified image of input function.
基金supported by the National Natural Science Foundation of China(Grant Nos.60232010 and 60572094)the Teaching and Research Award for 0utstanding Young Teachers in Higher Education Institutions of M0E,P.R.C.the Ministerial Foundation of China(Grant No.6140445).
文摘The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.
基金Project supported by the Young People Foundation of Zhejiang Normal University, China (Grant No KYJ06Y07150)
文摘This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.
基金the National Natural Science Foundation of China for the grant of 61378087Natural Science Foundation of Jiangsu Province(BK20151478)+1 种基金Zhi-Hua Mao is grateful to the Open Funds for Graduate Innovation Lab of Nanjing University of Aeronautics and Astronautics(kfjj20150309)and Fundamental Research Funds for the Central Universities.The raw data acquisition in FTIRI was mostly carried out in the lab of Professor Yang Xia at Oakland University(Rochester,Michigan,USA).Professor Xia was supported by an NIH grant R01-AR052353 during the time of the data acquisition.
文摘Two discriminant methods,partial least squares-discriminant analysis(PLS-DA)and Fisher's discriminant analysis(FDA),were combined with Fourier transform infrared imaging(FTIRI)to differentiate healthy and osteoarthritic articular cartilage in a canine model.Osteoarthritic cartilage had been developed for up to two years after the anterior cruciate ligament(ACL)transection in one knee.Cartilage specimens were sectioned into 10μm thickness for FTIRI.A PLS-DA model was developed after spectral pre-processing.All IR spectra extracted from FTIR images were calculated by PLS-DA with the discriminant accuracy of 90%.Prior to FDA,principal component analysis(PCA)was performed to decompose the IR spectral matrix into informative princi pal component matrices.Based on the different discriminant mechanism,the discriminant accuracy(96%)of PCA-FDA with high convenience was higher than that of PLS-DA.No healthy cartilage sample was mis assigned by these two methods.The above mentioned suggested that both integrated technologies of FTIRI-PLS-DA and,especially,FTIRI-PCA-FDA could become a promising tool for the discrimination of healthy and osteoarthritic cartilage specimen as well as the diagnosis of cartilage lesion at microscopic level.The results of the study would be helpful for better understanding the pathology of osteoarthritics.
文摘The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].
基金Sponsored by the National Natural Science Foundation of China(Grant No.61003128)
文摘Texture analysis is a fundamental field in computer vision. However,it is also a particularly difficult problem for no universal mathematical model of real world textures. By extending a new application of the fractional Fourier transform( Fr FT) in the field of texture analysis,this paper proposes an Fr FT-based method for describing textures. Firstly,based on the Radon-Wigner transform,1-D directional Fr FT filters are designed to two types of texture features,i. e.,the coarseness and directionality. Then,the frequencies with maximum and median amplitudes of the Fr FT of the input signal are regarded as the output of the 1-D directional Fr FT filter. Finally,the mean and the standard deviation are used to compose of the feature vector. Compared to the WD-based method,three benefits can be achieved with the proposed Fr FT-based method,i. e.,less memory size,lower computational load,and less disturbed by the cross-terms. The proposed method has been tested on16 standard texture images. The experimental results show that the proposed method is superior to the popular Gabor filtering-based method.
