Showing papers on "Discrete Hartley transform published in 2008"
TL;DR: This paper systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory.
Abstract: In this paper, we systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory. This means that instead of manipulating the transform definition, we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method. The same method yields the well-known Cooley-Tukey fast Fourier transform (FFT) as well as general radix discrete cosine and sine transform algorithms. The algebraic approach makes the derivation concise, unifies and classifies many existing algorithms, yields new variants, enables structural optimization, and naturally produces a human-readable structural algorithm representation based on the Kronecker product formalism. We show, for the first time, that the general-radix Cooley-Tukey and the lesser known Bruun algorithms are instances of the same generic algorithm. Further, we show that this generic algorithm can be instantiated for all four types of the real DFT and the DHT.
127 citations
TL;DR: It is shown that the proposed transform provides a 25% reduction in the number of arithmetic operations with a performance in image compression that is much superior to that of the SDCT and comparable to the approximated discrete cosine transform.
Abstract: An efficient 8 times 8 sparse orthogonal transform matrix is proposed for image compression by appropriately introducing some zeros in the 8 times 8 signed discrete cosine transform (SDCT) matrix. An algorithm for its fast computation is also developed. It is shown that the proposed transform provides a 25% reduction in the number of arithmetic operations with a performance in image compression that is much superior to that of the SDCT and comparable to that of the approximated discrete cosine transform.
109 citations
TL;DR: A notion of 2D discrete Radon transforms for2D discrete images, which is based on summation along lines of absolute slope less than 1, is defined, and is shown to be geometrically faithful: the summation avoids wrap-around effects.
Abstract: Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in $\mathbb{R}^{2}$ for which the transform is rapidly computable and invertible. We describe a fast algorithm using $O(N\log{N})$ operations, where $N =n^{2}$ is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764-784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.
77 citations
TL;DR: A new method for optical image encryption is introduced on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform.
56 citations
01 Jan 2008
TL;DR: The Discrete Cosine Transform (DCT) is discussed and the proposed approach is accompanied with the sample code " DCT8x8 " in the NVIDIA CUDA SDK.
Abstract: In this whitepaper the Discrete Cosine Transform (DCT) is discussed. The two-dimensional variation of the transform that operates on 8x8 blocks (DCT8x8) is widely used in image and video coding because it exhibits high signal decorrelation rates and can be easily implemented on the majority of contemporary computing architectures. The key feature of the DCT8x8 is that any pair of 8x8 blocks can be processed independently. This makes possible fully parallel implementation of DCT8x8 by definition. Most of CPU-based implementations of DCT8x8 are firmly adjusted for operating using fixed point arithmetic but still appear to be rather costly as soon as blocks are processed in the sequential order by the single ALU. Performing DCT8x8 computation on GPU using NVIDIA CUDA technology gives significant performance boost even compared to a modern CPU. The proposed approach is accompanied with the sample code " DCT8x8 " in the NVIDIA CUDA SDK.
42 citations
TL;DR: The general form of the commuting matrix is found and one can derive the complete and orthogonal eigenvector sets for offset DFTs, DCTs of types 1, 4, 5, and 8, discrete Hartley transforms oftypes 1 and 4, the Walsh transform, and the projection operation successfully.
Abstract: It is well known that some matrices (such as Dickinson-Steiglitz's matrix) can commute with the discrete Fourier transform (DFT) and that one can use them to derive the complete and orthogonal DFT eigenvector set. Recently, Candan found the general form of the DFT commuting matrix. In this paper, we further extend the previous work and find the general form of the commuting matrix for any periodic, quasi-periodic, and offset quasi-periodic operations. Using the general commuting matrix, we can derive the complete and orthogonal eigenvector sets for offset DFTs, DCTs of types 1, 4, 5, and 8, DSTs of types 1, 4, 5, and 8, discrete Hartley transforms of types 1 and 4, the Walsh transform, and the projection operation (the operation that maps a whole vector space into a subspace) successfully. Moreover, several novel ways of finding DFT eigenfunctions are also proposed. Furthermore, we also extend our theories to the continuous case, i.e., if a continuous transform is periodic, quasi-periodic, or offset quasi-periodic (such as the FT and some cyclic operations in optics), we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform.
33 citations
13 Dec 2008
TL;DR: A method is introduced how to process the Discrete Fourier Transform (DFT) by a single-layer neural network with a linear transfer function to achieve a stand alone solution of neural networks without the necessity of additional computational resources.
Abstract: In this paper, a method is introduced how to process the Discrete Fourier Transform (DFT) by a single-layer neural network with a linear transfer function. By implementing the suggested solution into neuro- hardware, advantage can be taken of actual parallel processing of spectral components of different frequencies and of different coefficients of each spectral line. When computing the DFT due to input data pre-processing for a certain neural network solution, a stand alone solution of neural networks without the necessity of additional computational resources can be achieved.
29 citations
Journal Article•
29 Jun 2008-World Academy of Science, Engineering and Technology, International Journal of Computer, Electrical, Automation, Control and Information Engineering
TL;DR: This tutorial gives brief introduction to MRI physics; MRI system from instrumentation point of view; K-space signal and the process of IDFT and IFFT for One and two dimensional data.
Abstract: The use of Inverse Discrete Fourier Transform (IDFT)
implemented in the form of Inverse Fourier Transform (IFFT) is
one of the standard method of reconstructing Magnetic Resonance
Imaging (MRI) from uniformly sampled K-space data. In this tutorial,
three of the major problems associated with the use of IFFT in
MRI reconstruction are highlighted. The tutorial also gives brief
introduction to MRI physics; MRI system from instrumentation point
of view; K-space signal and the process of IDFT and IFFT for One
and two dimensional (1D and 2D) data.
23 citations
TL;DR: A new discrete fractional transform defined by two parameters (angle and fractional order) is presented that can be applied in one and two dimensional signal processing and has very important practical applications in signal and image processing.
23 citations
TL;DR: SFRHT is a real transform as Hartley transform and superior to HT in virtue of the advantage that it can also append fractional orders as additional keys for the purpose of improving the system security to some extent.
Abstract: We present a new method for image encryption on the basis of simplified fractional Hartley transform (SFRHT). SFRHT is a real transform as Hartley transform (HT) and furthermore, superior to HT in virtue of the advantage that it can also append fractional orders as additional keys for the purpose of improving the system security to some extent. With this method, one can encrypt an image with an intensity-only medium such as a photographic film or a CCD camera by spatially incoherent or coherent illumination. The optical realization is then proposed and computer simulations are also performed to verify the feasibility of this method.
16 citations
12 May 2008
TL;DR: This paper investigates the implementation of the discrete Fourier transform (DFT) in the encrypted domain, by using the homomorphic properties of the underlying cryptosystem.
Abstract: Signal processing modules working directly on the encrypted data could provide an elegant solution to application scenarios where valuable signals should be protected from a malicious processing device. In this paper, we investigate the implementation of the discrete Fourier transform (DFT) in the encrypted domain, by using the homomorphic properties of the underlying cryptosystem. Several important issues are considered for both the DFT and radix-2 fast Fourier transform, including the error analysis and the maximum size of the sequence that can be transformed.
Posted Content•
TL;DR: The discrete oscillator transform (DOT) as mentioned in this paper is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ, and it has a canonical basis of eigenvectors for the DFT.
Abstract: The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.
09 Jun 2008
TL;DR: A novel implementation of the discrete curvelet transform based on the Fast Fourier Transform that has several advantages over existing transforms, such as lower redundancy, hierarchical data structure, ease of implementation and possible extension to N dimension is proposed.
Abstract: A novel implementation of the discrete curvelet transform is proposed in this work. The transform is based on the Fast Fourier Transform (FFT) and has the same order of complexity as the FFT. The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that are 2-pi periodic and form a partition of unity. The transform is named the Uniform Discrete Curvelet Transform (UDCT) because the centers of the curvelet functions at each resolution are located on a uniform grid. The forward and inverse transforms form a tight frame, in the sense that they are the exact transpose of each other. The novel discrete transform has several advantages over existing transforms, such as lower redundancy, hierarchical data structure, ease of implementation and possible extension to N dimension. Finally, we present a simple initial application of the UDCT in sparseness constraint seismic data interpolation to recover missing traces.
TL;DR: The robustness of a watermarking scheme based on the discrete fractional random transform has been investigated and demonstrated to be superior to those based onThe discrete cosine transform, discrete fractions Fourier transform, and discrete Fouriertransform.
Abstract: The robustness of a watermarking scheme based on the discrete fractional random transform has been investigated and demonstrated to be superior to those based on the discrete cosine transform, discrete fractional Fourier transform, and discrete Fourier transform The spectrum distribution of the discrete fractional random transform is random and uniform, which guarantees good robustness In addition, the discrete fractional random transform itself can serve as a secret key, and can provide high capacity while ensuring robustness in image hiding Moreover, the feature of real output of the discrete fractional random transform with a half periodicity in its eigenvalues can save storage space of image data and is convenient for storage Numerical simulations have confirmed our analysis and demonstrated the superiority of the discrete fractional random transform
TL;DR: A novel critically sampled orientation-selective orthogonal lapped transform called the lapped Hartley transform (LHT) is derived and a statistical analysis shows that the LHT and LHT-PO have better redundancy removal properties than other block or lapped transforms.
Abstract: A novel critically sampled orientation-selective orthogonal lapped transform called the lapped Hartley transform (LHT) is derived. In a first step, overlapping basis functions are generated by modulating basis functions of a 2-D block Hartley transform by a cosine wave. To achieve invertibility and orthogonality, an iterative filter is applied as prefilter in the analysis and as postfilter in the synthesis operation, respectively. Alternatively, filtering can be restricted to analysis or synthesis, ending up with a biorthogonal transform (LHT-PR, LHT-PO). A statistical analysis based on a 4000-image data base shows that the LHT and LHT-PO have better redundancy removal properties than other block or lapped transforms. Finally, image compression and noise removal examples are given, showing the advantages of the LHT especially in images containing oriented textures.
TL;DR: A vector-radix algorithm for the fast computation of a 2-D discrete Hartley transform (DHT) and a radix-3 times 3 decimation in frequency algorithm for data sequences whose length is a power of three is developed.
Abstract: In this correspondence, we propose a vector-radix algorithm for the fast computation of a 2-D discrete Hartley transform (DHT). For data sequences whose length is a power of three, a radix-3 times 3 decimation in frequency algorithm is developed. It decomposes a length-N times N DHT into nine length-(N/3) times N (N/3) DHTs. Comparison of the computational complexity with known algorithms shows that the proposed algorithm, in some cases, reduces significantly the number of arithmetic operations.
27 May 2008
TL;DR: In this paper, an alternative transform based on the correlation of the residual signal is proposed for the improvement of the H.264/AVC coding efficiency, which combines the discrete sine transform and the integer cosine transform in order to greatly compact the energy of the signal.
Abstract: An alternative transform based on the correlation of the residual signal is proposed for the improvement of the H.264/AVC coding efficiency. To make use of this alternative transform in video coding, discrete sine transform and integer sine transform are used alternately with integer cosine transform in order to greatly compact the energy of the signal when the correlation coefficients of the signal are relatively low. Therefore, the discrete sine transform and the integer sine transform are suggested to be used in conjunction with the integer cosine transform in H.264/AVC. The alternative transform that selects the optimal transform between two transforms by using rate-distortion optimization in H.264/AVC shows a coding gain compared with H.264/AVC. Based on the experimental results, the proposed method with the discrete sine transform achieves a BD-PSNR gain of up to 0.68 dB compared to JM 10.2 at relatively high bitrates.
Journal Article•
TL;DR: A new method based on Discrete Fourier Transform (DFT) that has lower computational complexity compared to the conventional approach and is mathematically equivalent to matched filtering, so there is no processing loss is proposed.
Abstract: —Multicarrier Phase-Coded signals have been recently introduced to achieve high range resolution in radar systems. As in single carrier phase coded radars, the conventional method for compression of these signals is based on using matched filter or direct computation of autocorrelation function. In this paper we propose a new method based on Discrete Fourier Transform (DFT) that has lower computational complexity compared to the conventional approach. It has been proved that the proposed method is mathematically equivalent to matched filtering, so there is no processing loss. Also the effect of sampling frequency on compression loss has been investigated and for the oversampled matched filter of MCPC signals, a computational efficient algorithm based on polyphase implementation has been proposed.
01 Jul 2008
TL;DR: The Discrete Pulse Transform of images is defined by using a new class of LULU operators on multidimensional arrays that generalizes the DPT of sequences and replicates its essential properties, e.g. total variation preservation.
Abstract: The Discrete Pulse Transform (DPT) of images is defined by using a new class of LULU operators on multidimensional arrays. This transform generalizes the DPT of sequences and replicates its essential properties, e.g. total variation preservation. Furthermore, the discrete pulses in the transform capture the contrast in the original image on the boundary of their supports. Since images are perceived via the contrast between neighbour pixels, the DPT may be a convenient new tool for image analysis.
TL;DR: The conditions which provide a correct harmonic analysis are emphasized and analytically justified in this paper and a virtual instrument for the harmonics analysis has been realised in LabVIEW by the authors.
Abstract: 54 Abstract—The importance of the harmonic analysis and its frequent utilization in different applications impose a complete accuracy. The conditions which provide a correct harmonic analysis are emphasized and analytically justified in this paper. By using an efficient algorithm, a virtual instrument for the harmonics analysis has been realised in LabVIEW by the authors. This program was a very useful and efficient tool in order to emphasize, from the practical point of view, all the errors analytically outlined. The translation of some superior harmonics to inferior order ones represents a possible and harmful phenomenon pointed out as well.
TL;DR: Comparison with recently proposed circular convolution technique in DCT II domain shows that the proposed new method is computationally more efficient.
Abstract: We derive expressions for convolution multiplication properties of discrete cosine transform II (DCT II) starting from equivalent discrete Fourier transform (DFT) representations. Using these expressions, a method for implementing linear filtering through block convolution in the DCT II domain is presented. For the case of nonsymmetric impulse response, additional discrete sine transform II (DST II) is required for implementing the filter in DCT II domain, where as for a symmetric impulse response, the additional transform is not required. Comparison with recently proposed circular convolution technique in DCT II domain shows that the proposed new method is computationally more efficient.
22 Jun 2008
TL;DR: Fiber or planar lightwave circuit architectures are given to implement the discrete Fourier transform and the discrete Hartley transform directly in the optical domain, and a recursive approach for the decimation-in-frequency algorithm is presented.
Abstract: We give fiber or planar lightwave circuit (PLC) architectures to implement the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) directly in the optical domain. In both cases, we present a recursive approach for the decimation-in-frequency algorithm, considering parallel and serial input configurations. We also describe PLC devices for high-speed optical filtering and data compression, based on discrete wavelet transforms (DWT) and discrete cosine transform (DCT).
TL;DR: The VC-FFT therefore comprehensively reduces the complexity of the algorithm as compared with other current multidimensional DFT algorithms.
08 Jun 2008
TL;DR: A new unified multirate DSP model is derived that allows the derivation of new closed form DNA spectrum expressions for the above trigonometric transforms and suggests an efficient way to improve the detection of repeats by digital filtering.
Abstract: The detection of many forms of periodicities in DNA sequences has been an active area of research in recent years. Most of the signal processing based methods have primarily focussed on using the short-time discrete Fourier transform (ST-DFT) as the key tool in identifying such repeat sequences. In this paper, we propose to use different fast discrete transforms such as the discrete cosine transform (DCT), the discrete sine transform (DST), and the discrete Hartley transform (DHT), to locate these patterns. In specific, we derive a new unified multirate DSP model that i) allows the derivation of new closed form DNA spectrum expressions for the above trigonometric transforms, ii) includes the DFT model as a special case, and iii) suggests an efficient way to improve the detection of repeats by digital filtering.
20 Jun 2008
TL;DR: Comparisons of the DVT with slant transform, slantlet transform and discrete cosine transform in image transform indicate that the linear DVT has obvious advantages in the image data compression.
Abstract: Based on a class of complete orthogonal function system on L2[0,1] called as the V-system, a new kind of orthogonal discrete transform called DVT is proposed, and the application of DVT in image transform is studied. The comparisons of the DVT with slant transform, slantlet transform and discrete cosine transform (DCT) in image transform are given. Slant transform is an effective mathematical method in signal processing, and has a good effect on the images which represent variations with gradual brightness; slantlet transform further improves slant transform. Slant transform, slantlet transform and linear DVT are corresponded to discrete form of a class of linear functions respectively. They are equivalent to each other. The comparison results indicate that the linear DVT has obvious advantages in the image data compression.
TL;DR: The three sliding discrete fractional transforms are compared with sliding discrete Fourier transform (SDFT) in terms of SNR and the SDFRST performs better in comparison with SDFSDFRFT and SDFT.
TL;DR: The idea of employing the discrete Fourier transform casted as the representative cell method for the solution of multiscale problems is illustrated in this article, where its application in combination with analytical (structural mechanics methods, Wiener-Hopf method, integral transform methods) and numerical (finite element method, higher-order theory) methods is demonstrated.
Abstract: The idea of employing the discrete Fourier transform casted as the representative cell method for the solution of multiscale problems is illustrated. Its application in combination with analytical (structural mechanics methods, Wiener-Hopf method, integral transform methods) and numerical (finite element method, higher-order theory) methods is demonstrated. Both cases of 1-D and 2-D translational symmetry are addressed. In particular, the problems for layered, cellular, and perforated materials with and without flaws (cracks) are considered. The method is shown to be a convenient and universal analysis tool. Its numerical efficiency allowed us to solve optimization problems characterized by multiple reanalysis.
12 May 2008
TL;DR: A large class of alternatives to the DFT is identified using the theory of polynomial algebras, each of which has its own signal extension and own notion of convolution and has Vandermonde structure, which enables their fast computation.
Abstract: It is well-known that the discrete Fourier transform (DFT) of a finite length discrete-time signal samples the discrete-time Fourier transform (DTFT) of the same signal at equidistant points on the unit circle. Hence, as the signal length goes to infinity, the DFT approaches the DTFT. Associated with the DFT are circular convolution and a periodic signal extension. In this paper we identify a large class of alternatives to the DFT using the theory of polynomial algebras. Each of these transforms approaches the DTFT just as the DFT does, but has its own signal extension and own notion of convolution. Further, these transforms have Vandermonde structure, which enables their fast computation. We provide a few experimental examples that confirm our theoretical results.
TL;DR: Property of the linear canonical transform are examined analytically, including how the support and bandwidth of the signal are related to transform parameters, a review of sampling issues and some new proposals in this area.
Abstract: The linear canonical transform (also known as the quadratic phase integral and the special affine Fourier transform, among others) is an important tool for the modeling of quadratic phase systems for coherent optical signal processing, as it is a generalization of a number of important and widely used transforms such as the Fresnel transform, the Fourier transform and the fractional Fourier transform. We consider properties of the linear canonical transform which are important for numerical approximation of the integral transform, and thus for simulation of the related paraxial optical systems. Some of these properties have been previously developed in the literature, but are analyzed here in the context of linear canonical transform simulations, others are developed here for the first time. We examine these properties analytically, including how the support and bandwidth of the signal are related to transform parameters, a review of sampling issues and some new proposals in this area. Finally, we examine the effect of the linear canonical transform on the sparsity of signals, which is useful for efficient transmission or storage or to aid certain signal processing tools such as blind source separation.