In this paper, a new algorithm for the fast computation of a 2-D discrete cosine transform (DCT) is presented. It is shown that the N×N DCT, where N = 2m, can be computed using only N 1-D DCT’s and additions, in...In this paper, a new algorithm for the fast computation of a 2-D discrete cosine transform (DCT) is presented. It is shown that the N×N DCT, where N = 2m, can be computed using only N 1-D DCT’s and additions, instead of using 2N 1-D DCT’s as in the conventional row-column approach. Hence the total number of multiplications for the proposed algorithm is only half of that required for the row-column approach, and is also less than that of most of other fast algorithms, while the number of additions is almost comparable to that of others.展开更多
Secure exchange of information is the basic need of modern digital world of e-communication which is achieved either by encrypting information or by hiding information in other information called cover media. Conceali...Secure exchange of information is the basic need of modern digital world of e-communication which is achieved either by encrypting information or by hiding information in other information called cover media. Concealing information requires a well designed technique of Stegnography. This work presents a technique, variable tone variable bits (VTVB) Stegnography, to hide information in a cover image. The VTVB Stegnography hides variable data in discrete cosine transform (DCT) coefficients of the cover image. VTVB Stegnography provides variable data hiding capacity and variable distortion. Additional large data hiding this technique provide extra security due to the large key size making VTVB Stegnography technique much more immune to steganalysis. The hiding makes the existence of information imperceptible for steganalysis and the key of keeping a secret makes the recovering of information difficult for an intruder. The key size is depending on cover image and numbers of bits of discrete cosine transform (DCT) coefficients used for information embedding. This is a very flexible technique and can be used for low payload applications, e.g. watermarking to high payload applications, e.g. network Stegnography.展开更多
DHT of length p<sup>l</sup>q（p is odd and q is arbitrary） is turned into p<sup>l</sup> DHTs of length qand some additional operations, while the additional operations only involves the comput...DHT of length p<sup>l</sup>q（p is odd and q is arbitrary） is turned into p<sup>l</sup> DHTs of length qand some additional operations, while the additional operations only involves the computation ofcos-DFT and sin-DFT with length p. If the length of a DHT is p<sub>1</sub><sup>l<sub>1</sub></sup>…P<sub>N</sub><sup>l<sub>N</sub></sup>2<sup>l</sup>（P<sub>1</sub>…,P<sub>N</sub> are oddprimes）, a fast algorithm is obtained by the similar recursive technique. Therefore, the algorithmcan compute DHT of arbitrary length. The paper also Proves that operations for computingDHT of length N by the algorithm are no more than O（Nlog<sub>2</sub>N）, when the length is N=p<sup>l</sup>,operations of the algorithm are fewer than that of other known algorithms.展开更多
The Fourier transform is very important to numerous applications in science and engineering. However, its usefulness is hampered by its computational expense. In this paper, in an attempt to develop a faster method fo...The Fourier transform is very important to numerous applications in science and engineering. However, its usefulness is hampered by its computational expense. In this paper, in an attempt to develop a faster method for computing Fourier transforms, the authors present parallel implementations of two new algorithms developed for the type IV Discrete Cosine Transform （DCT-IV） which support the new interleaved fast Fourier transform method. The authors discuss the realizations of their implementations using two paradigms. The first involved commodity equipment and the Message-Passing Interface （MPI） library. The second utilized the RapidMind development platform and the Cell Broadband Engine （BE） processor. These experiments indicate that the authors＇ rotation-based algorithm is preferable to their lifting-based algorithm on the platforms tested, with increased efficiency demonstrated by their MPI implementation for large data sets. Finally, the authors outline future work by discussing an architecture-oriented method for computing DCT-IVs which promises further optimization. The results indicate a promising fresh direction in the search for efficient ways to compute Fourier transforms.展开更多
A generalized fast computational algorithm for the n -dimensional discrete cosine transform ( n- D DCT) of length N=2 m(m≥2) is presented. The developed algorithm is theoretically proved and its efficiency is evaluat...A generalized fast computational algorithm for the n -dimensional discrete cosine transform ( n- D DCT) of length N=2 m(m≥2) is presented. The developed algorithm is theoretically proved and its efficiency is evaluated. The theoretical results show that compared with the conventional method to compute the 1-D DCTs in n directions, the number of multiplications needed by this algorithm is only 1/n of that required by the conventional method; for the total number of additions, it is a bit more when N≤8 and much less when N≥16 than the coventional one. To validate the proposed algorithm, the case when n=3 is taken as an example and applied to the motion picture compression. The results show that the proposed method is superior to MPEG-2.展开更多
For traditional JPEG image encryption,block position shuffling can achieve a better encryption effect and is resistant to non-zero counting attack.However,the numbers of non-zero coefficients in the 8×8 sub-block...For traditional JPEG image encryption,block position shuffling can achieve a better encryption effect and is resistant to non-zero counting attack.However,the numbers of non-zero coefficients in the 8×8 sub-blocks are unchanged using block position shuffle.For this defect,this paper proposes a fast attack algorithm for JPEG image encryption based on inter-block shuffle and non-zero quantization discrete cosine transformation coefficient attack.The algorithm analyzes the position mapping relationship before and after encryption of image blocks by detecting the pixel values of an image by the designed plaintext image.Then the preliminary attack result of the image blocks can be obtained from the inverse mapping relationship.Finally,the final attack result of the algorithm is generated according to the numbers of non-zero coefficients in each 8×8 block of the preliminary attack result.Every 8×8 block position is related with its number of non-zero discrete cosine transform coefficients in the designed plaintext.It is verified that the main content of the original image could be obtained without knowledge of the encryption algorithm and keys in a relatively short time.展开更多
Steganography is a technique for hiding secret messages while sending and receiving communications through a cover item.From ancient times to the present,the security of secret or vital information has always been a s...Steganography is a technique for hiding secret messages while sending and receiving communications through a cover item.From ancient times to the present,the security of secret or vital information has always been a significant problem.The development of secure communication methods that keep recipient-only data transmissions secret has always been an area of interest.Therefore,several approaches,including steganography,have been developed by researchers over time to enable safe data transit.In this review,we have discussed image steganography based on Discrete Cosine Transform(DCT)algorithm,etc.We have also discussed image steganography based on multiple hashing algorithms like the Rivest–Shamir–Adleman(RSA)method,the Blowfish technique,and the hash-least significant bit(LSB)approach.In this review,a novel method of hiding information in images has been developed with minimal variance in image bits,making our method secure and effective.A cryptography mechanism was also used in this strategy.Before encoding the data and embedding it into a carry image,this review verifies that it has been encrypted.Usually,embedded text in photos conveys crucial signals about the content.This review employs hash table encryption on the message before hiding it within the picture to provide a more secure method of data transport.If the message is ever intercepted by a third party,there are several ways to stop this operation.A second level of security process implementation involves encrypting and decrypting steganography images using different hashing algorithms.展开更多
Currently,in multimedia and image processing technologies, implementing special kinds of image manipulation operations by dealing directly with the compressed image is a work worthy to be concerned with. Theoretical a...Currently,in multimedia and image processing technologies, implementing special kinds of image manipulation operations by dealing directly with the compressed image is a work worthy to be concerned with. Theoretical analysis and experiment haVe indicated that some kinds of image processing works can be done very well by dealing with compressed image. In Ans paper, we give some efficient image manipulation operation algorithms operating on the compressed image data. These algorithms have advantages in computing complexity, storage space retirement and image quality.展开更多
For the case of atomic force microscope (AFM) automation, we extract the most valuable sub-region of a given AFM image automatically for succeeding scanning to get the higher resolution of interesting region. Two obje...For the case of atomic force microscope (AFM) automation, we extract the most valuable sub-region of a given AFM image automatically for succeeding scanning to get the higher resolution of interesting region. Two objective functions are sum- marized based on the analysis of evaluation of the information of a sub-region, and corresponding algorithm principles based on standard deviation and Discrete Cosine Transform (DCT) compression are determined from math. Algorithm realizations are analyzed and two selec...展开更多
Assuring medical images protection and robustness is a compulsory necessity nowadays.In this paper,a novel technique is proposed that fuses the wavelet-induced multi-resolution decomposition of the Discrete Wavelet Tr...Assuring medical images protection and robustness is a compulsory necessity nowadays.In this paper,a novel technique is proposed that fuses the wavelet-induced multi-resolution decomposition of the Discrete Wavelet Transform(DWT)with the energy compaction of the Discrete Wavelet Transform(DCT).The multi-level Encryption-based Hybrid Fusion Technique(EbhFT)aims to achieve great advances in terms of imperceptibility and security of medical images.A DWT disintegrated sub-band of a cover image is reformed simultaneously using the DCT transform.Afterwards,a 64-bit hex key is employed to encrypt the host image as well as participate in the second key creation process to encode the watermark.Lastly,a PN-sequence key is formed along with a supplementary key in the third layer of the EbHFT.Thus,the watermarked image is generated by enclosing both keys into DWT and DCT coefficients.The fusions ability of the proposed EbHFT technique makes the best use of the distinct privileges of using both DWT and DCT methods.In order to validate the proposed technique,a standard dataset of medical images is used.Simulation results show higher performance of the visual quality(i.e.,57.65)for the watermarked forms of all types of medical images.In addition,EbHFT robustness outperforms an existing scheme tested for the same dataset in terms of Normalized Correlation(NC).Finally,extra protection for digital images from against illegal replicating and unapproved tampering using the proposed technique.展开更多
Data compression is one of the core fields of study for applications of image and video processing.The raw data to be transmitted consumes large bandwidth and requires huge storage space as a result,it is desirable to...Data compression is one of the core fields of study for applications of image and video processing.The raw data to be transmitted consumes large bandwidth and requires huge storage space as a result,it is desirable to represent the information in the data with considerably fewer bits by the mean of data compression techniques,the data must be reconstituted very similarly to the initial form.In this paper,a hybrid compression based on Discrete Cosine Transform(DCT),DiscreteWavelet Transform(DWT)is used to enhance the quality of the reconstructed image.These techniques are followed by entropy encoding such as Huffman coding to give additional compression.Huffman coding is optimal prefix code because of its implementation is more simple,faster,and easier than other codes.It needs less execution time and it is the shortest average length and the measurements for analysis are based upon Compression Ratio,Mean Square Error(MSE),and Peak Signal to Noise Ratio(PSNR).We applied a hybrid algorithm on(DWT–DCT 2×2,4×4,8×8,16×16,32×32)blocks.Finally,we show that by using a hybrid(DWT–DCT)compression technique,the PSNR is reconstructed for the image by using the proposed hybrid algorithm(DWT–DCT 8×8 block)is quite high than DCT.展开更多
The use of the multiscale generalized radial basis function(MSRBF)neural networks for image feature extraction and medical image analysis and classification is proposed for the first time in this work.The MSRBF networ...The use of the multiscale generalized radial basis function(MSRBF)neural networks for image feature extraction and medical image analysis and classification is proposed for the first time in this work.The MSRBF networks hold a simple and flexible architecture that has been successfully used in forecasting and model structure detection of input-output nonlinear systems.In this work instead,MSRBF networks are part of an integrated computer-aided diagnosis(CAD)framework for breast cancer detection,which holds three stages:an input-output model is obtained from the image,followed by a high-level image feature extraction from the model and a classification module aimed at predicting breast cancer.In the first stage,the image data is rendered into a multiple-input-single-output(MISO)system.In order to improve the characterisation,the nonlinear autoregressive with exogenous inputs(NARX)model is introduced to rearrange the available input-output data in a nonlinear way.The forward regression orthogonal least squares(FROLS)algorithm is then used to take advantage of the previous arrangement by solving the system as a model structure detection problem and finding the output layer weights of the NARX-MSRBF network.In the second stage,once the network model is available,the feature extraction takes place by stimulating the input to produce output signals to be compressed by the discrete cosine transform(DCT).In the third stage,we leverage the extracted features by using a clustering algorithm for classification to integrate a CAD system for breast cancer detection.To test the method performance,three different and well-known public image repositories were used:the mini-MIAS and the MMSD for mammography,and the BreaKHis for histopathology images.A comparison exercise was also made between different database partitions to understand the mammogram breast density effect in the performance since there are few remarks in the literature on this factor.Classification results show that the new CAD method reached an accuracy of 93.5%in mini-Mammo graphic image analysis society(mini-MIAS),93.99%in digital database for screening mammography(DDSM)and 86.7%in the BreaKHis.We found that the MSRBF networks are able to build tailored and precise image models and,combined with the DCT,to extract high-quality features from both black and white and coloured images.展开更多
By introducing a form of reorder for multidimensional data, we propose a unified fast algo-rithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial trans-form to compute eleven t...By introducing a form of reorder for multidimensional data, we propose a unified fast algo-rithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial trans-form to compute eleven types of multidimensional discrete orthogonal transforms, which contain three types of m-dimensional discrete cosine transforms ( m-D DCTs) ,four types of m-dimensional discrete W transforms ( m-D DWTs) ( m-dimensional Hartley transform as a special case), and four types of generalized discrete Fourier transforms ( m-D GDFTs). For real input, the number of multiplications for all eleven types of the m-D discrete orthogonal transforms needed by the proposed algorithm are only 1/m times that of the commonly used corresponding row-column methods, and for complex input, it is further reduced to 1/(2m) times. The number of additions required is also reduced considerably. Furthermore, the proposed algorithm has a simple computational structure and is also easy to be im-plemented on computer, and the numerical experiments show that the computational efficiency is con-sistent with the theoretic analysis.展开更多
文摘In this paper, a new algorithm for the fast computation of a 2-D discrete cosine transform (DCT) is presented. It is shown that the N×N DCT, where N = 2m, can be computed using only N 1-D DCT’s and additions, instead of using 2N 1-D DCT’s as in the conventional row-column approach. Hence the total number of multiplications for the proposed algorithm is only half of that required for the row-column approach, and is also less than that of most of other fast algorithms, while the number of additions is almost comparable to that of others.
文摘Secure exchange of information is the basic need of modern digital world of e-communication which is achieved either by encrypting information or by hiding information in other information called cover media. Concealing information requires a well designed technique of Stegnography. This work presents a technique, variable tone variable bits (VTVB) Stegnography, to hide information in a cover image. The VTVB Stegnography hides variable data in discrete cosine transform (DCT) coefficients of the cover image. VTVB Stegnography provides variable data hiding capacity and variable distortion. Additional large data hiding this technique provide extra security due to the large key size making VTVB Stegnography technique much more immune to steganalysis. The hiding makes the existence of information imperceptible for steganalysis and the key of keeping a secret makes the recovering of information difficult for an intruder. The key size is depending on cover image and numbers of bits of discrete cosine transform (DCT) coefficients used for information embedding. This is a very flexible technique and can be used for low payload applications, e.g. watermarking to high payload applications, e.g. network Stegnography.
文摘DHT of length p<sup>l</sup>q（p is odd and q is arbitrary） is turned into p<sup>l</sup> DHTs of length qand some additional operations, while the additional operations only involves the computation ofcos-DFT and sin-DFT with length p. If the length of a DHT is p<sub>1</sub><sup>l<sub>1</sub></sup>…P<sub>N</sub><sup>l<sub>N</sub></sup>2<sup>l</sup>（P<sub>1</sub>…,P<sub>N</sub> are oddprimes）, a fast algorithm is obtained by the similar recursive technique. Therefore, the algorithmcan compute DHT of arbitrary length. The paper also Proves that operations for computingDHT of length N by the algorithm are no more than O（Nlog<sub>2</sub>N）, when the length is N=p<sup>l</sup>,operations of the algorithm are fewer than that of other known algorithms.
文摘The Fourier transform is very important to numerous applications in science and engineering. However, its usefulness is hampered by its computational expense. In this paper, in an attempt to develop a faster method for computing Fourier transforms, the authors present parallel implementations of two new algorithms developed for the type IV Discrete Cosine Transform （DCT-IV） which support the new interleaved fast Fourier transform method. The authors discuss the realizations of their implementations using two paradigms. The first involved commodity equipment and the Message-Passing Interface （MPI） library. The second utilized the RapidMind development platform and the Cell Broadband Engine （BE） processor. These experiments indicate that the authors＇ rotation-based algorithm is preferable to their lifting-based algorithm on the platforms tested, with increased efficiency demonstrated by their MPI implementation for large data sets. Finally, the authors outline future work by discussing an architecture-oriented method for computing DCT-IVs which promises further optimization. The results indicate a promising fresh direction in the search for efficient ways to compute Fourier transforms.
文摘A generalized fast computational algorithm for the n -dimensional discrete cosine transform ( n- D DCT) of length N=2 m(m≥2) is presented. The developed algorithm is theoretically proved and its efficiency is evaluated. The theoretical results show that compared with the conventional method to compute the 1-D DCTs in n directions, the number of multiplications needed by this algorithm is only 1/n of that required by the conventional method; for the total number of additions, it is a bit more when N≤8 and much less when N≥16 than the coventional one. To validate the proposed algorithm, the case when n=3 is taken as an example and applied to the motion picture compression. The results show that the proposed method is superior to MPEG-2.
文摘For traditional JPEG image encryption,block position shuffling can achieve a better encryption effect and is resistant to non-zero counting attack.However,the numbers of non-zero coefficients in the 8×8 sub-blocks are unchanged using block position shuffle.For this defect,this paper proposes a fast attack algorithm for JPEG image encryption based on inter-block shuffle and non-zero quantization discrete cosine transformation coefficient attack.The algorithm analyzes the position mapping relationship before and after encryption of image blocks by detecting the pixel values of an image by the designed plaintext image.Then the preliminary attack result of the image blocks can be obtained from the inverse mapping relationship.Finally,the final attack result of the algorithm is generated according to the numbers of non-zero coefficients in each 8×8 block of the preliminary attack result.Every 8×8 block position is related with its number of non-zero discrete cosine transform coefficients in the designed plaintext.It is verified that the main content of the original image could be obtained without knowledge of the encryption algorithm and keys in a relatively short time.
文摘Steganography is a technique for hiding secret messages while sending and receiving communications through a cover item.From ancient times to the present,the security of secret or vital information has always been a significant problem.The development of secure communication methods that keep recipient-only data transmissions secret has always been an area of interest.Therefore,several approaches,including steganography,have been developed by researchers over time to enable safe data transit.In this review,we have discussed image steganography based on Discrete Cosine Transform(DCT)algorithm,etc.We have also discussed image steganography based on multiple hashing algorithms like the Rivest–Shamir–Adleman(RSA)method,the Blowfish technique,and the hash-least significant bit(LSB)approach.In this review,a novel method of hiding information in images has been developed with minimal variance in image bits,making our method secure and effective.A cryptography mechanism was also used in this strategy.Before encoding the data and embedding it into a carry image,this review verifies that it has been encrypted.Usually,embedded text in photos conveys crucial signals about the content.This review employs hash table encryption on the message before hiding it within the picture to provide a more secure method of data transport.If the message is ever intercepted by a third party,there are several ways to stop this operation.A second level of security process implementation involves encrypting and decrypting steganography images using different hashing algorithms.
文摘Currently,in multimedia and image processing technologies, implementing special kinds of image manipulation operations by dealing directly with the compressed image is a work worthy to be concerned with. Theoretical analysis and experiment haVe indicated that some kinds of image processing works can be done very well by dealing with compressed image. In Ans paper, we give some efficient image manipulation operation algorithms operating on the compressed image data. These algorithms have advantages in computing complexity, storage space retirement and image quality.
基金supported by the National High TechnologyResearch and Development Program of China (Grant No.2007AA122128)
文摘For the case of atomic force microscope (AFM) automation, we extract the most valuable sub-region of a given AFM image automatically for succeeding scanning to get the higher resolution of interesting region. Two objective functions are sum- marized based on the analysis of evaluation of the information of a sub-region, and corresponding algorithm principles based on standard deviation and Discrete Cosine Transform (DCT) compression are determined from math. Algorithm realizations are analyzed and two selec...
文摘Assuring medical images protection and robustness is a compulsory necessity nowadays.In this paper,a novel technique is proposed that fuses the wavelet-induced multi-resolution decomposition of the Discrete Wavelet Transform(DWT)with the energy compaction of the Discrete Wavelet Transform(DCT).The multi-level Encryption-based Hybrid Fusion Technique(EbhFT)aims to achieve great advances in terms of imperceptibility and security of medical images.A DWT disintegrated sub-band of a cover image is reformed simultaneously using the DCT transform.Afterwards,a 64-bit hex key is employed to encrypt the host image as well as participate in the second key creation process to encode the watermark.Lastly,a PN-sequence key is formed along with a supplementary key in the third layer of the EbHFT.Thus,the watermarked image is generated by enclosing both keys into DWT and DCT coefficients.The fusions ability of the proposed EbHFT technique makes the best use of the distinct privileges of using both DWT and DCT methods.In order to validate the proposed technique,a standard dataset of medical images is used.Simulation results show higher performance of the visual quality(i.e.,57.65)for the watermarked forms of all types of medical images.In addition,EbHFT robustness outperforms an existing scheme tested for the same dataset in terms of Normalized Correlation(NC).Finally,extra protection for digital images from against illegal replicating and unapproved tampering using the proposed technique.
文摘Data compression is one of the core fields of study for applications of image and video processing.The raw data to be transmitted consumes large bandwidth and requires huge storage space as a result,it is desirable to represent the information in the data with considerably fewer bits by the mean of data compression techniques,the data must be reconstituted very similarly to the initial form.In this paper,a hybrid compression based on Discrete Cosine Transform(DCT),DiscreteWavelet Transform(DWT)is used to enhance the quality of the reconstructed image.These techniques are followed by entropy encoding such as Huffman coding to give additional compression.Huffman coding is optimal prefix code because of its implementation is more simple,faster,and easier than other codes.It needs less execution time and it is the shortest average length and the measurements for analysis are based upon Compression Ratio,Mean Square Error(MSE),and Peak Signal to Noise Ratio(PSNR).We applied a hybrid algorithm on(DWT–DCT 2×2,4×4,8×8,16×16,32×32)blocks.Finally,we show that by using a hybrid(DWT–DCT)compression technique,the PSNR is reconstructed for the image by using the proposed hybrid algorithm(DWT–DCT 8×8 block)is quite high than DCT.
基金the financial support to Carlos Beltran-Perez from the Mexican National Council of Science and Technology (CONACYT)part of the work was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/I011056/1 and platform grant EP/H00453X/1
文摘The use of the multiscale generalized radial basis function(MSRBF)neural networks for image feature extraction and medical image analysis and classification is proposed for the first time in this work.The MSRBF networks hold a simple and flexible architecture that has been successfully used in forecasting and model structure detection of input-output nonlinear systems.In this work instead,MSRBF networks are part of an integrated computer-aided diagnosis(CAD)framework for breast cancer detection,which holds three stages:an input-output model is obtained from the image,followed by a high-level image feature extraction from the model and a classification module aimed at predicting breast cancer.In the first stage,the image data is rendered into a multiple-input-single-output(MISO)system.In order to improve the characterisation,the nonlinear autoregressive with exogenous inputs(NARX)model is introduced to rearrange the available input-output data in a nonlinear way.The forward regression orthogonal least squares(FROLS)algorithm is then used to take advantage of the previous arrangement by solving the system as a model structure detection problem and finding the output layer weights of the NARX-MSRBF network.In the second stage,once the network model is available,the feature extraction takes place by stimulating the input to produce output signals to be compressed by the discrete cosine transform(DCT).In the third stage,we leverage the extracted features by using a clustering algorithm for classification to integrate a CAD system for breast cancer detection.To test the method performance,three different and well-known public image repositories were used:the mini-MIAS and the MMSD for mammography,and the BreaKHis for histopathology images.A comparison exercise was also made between different database partitions to understand the mammogram breast density effect in the performance since there are few remarks in the literature on this factor.Classification results show that the new CAD method reached an accuracy of 93.5%in mini-Mammo graphic image analysis society(mini-MIAS),93.99%in digital database for screening mammography(DDSM)and 86.7%in the BreaKHis.We found that the MSRBF networks are able to build tailored and precise image models and,combined with the DCT,to extract high-quality features from both black and white and coloured images.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 69974041).
文摘By introducing a form of reorder for multidimensional data, we propose a unified fast algo-rithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial trans-form to compute eleven types of multidimensional discrete orthogonal transforms, which contain three types of m-dimensional discrete cosine transforms ( m-D DCTs) ,four types of m-dimensional discrete W transforms ( m-D DWTs) ( m-dimensional Hartley transform as a special case), and four types of generalized discrete Fourier transforms ( m-D GDFTs). For real input, the number of multiplications for all eleven types of the m-D discrete orthogonal transforms needed by the proposed algorithm are only 1/m times that of the commonly used corresponding row-column methods, and for complex input, it is further reduced to 1/(2m) times. The number of additions required is also reduced considerably. Furthermore, the proposed algorithm has a simple computational structure and is also easy to be im-plemented on computer, and the numerical experiments show that the computational efficiency is con-sistent with the theoretic analysis.
