Showing papers on "Discrete Hartley transform published in 2014"

Sparse Discrete Fractional Fourier Transform and Its Applications

Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

The Hopping Discrete Fourier Transform [sp Tips&Tricks]

Abstract: The discrete Fourier transform (DFT) produces a Fourier representation for finite-duration data sequences. In addition to its theoretical importance, the DFT plays a key role in the implementation of a variety of digital signal-?processing algorithms. Several algorithms including the fast Fourier transform (FFT) and the Goertzel algorithm have been introduced for the fast implementation of the DFT [1], [2].

Alpha-rooting method of color image enhancement by discrete quaternion Fourier transform

Abstract: This paper presents a novel method for color image enhancement based on the discrete quaternion Fourier transform. We choose the quaternion Fourier transform, because it well-suited for color image processing applications, it processes all 3 color components (R,G,B) simultaneously, it capture the inherent correlation between the components, it does not generate color artifacts or blending , finally it does not need an additional color restoration process. Also we introduce a new CEME measure to evaluate the quality of the enhanced color images. Preliminary results show that the α-rooting based on the quaternion Fourier transform enhancement method out-performs other enhancement methods such as the Fourier transform based α-rooting algorithm and the Multi scale Retinex. On top, the new method not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors. It can be used to enhance edge information and sharp features in images, as well as for enhancing even low contrast images. The proposed algorithms are simple to apply and design, which makes them very practical in image enhancement.

CORDIC-Based Unified Architectures for Computation of DCT/IDCT/DST/IDST

Abstract: In this paper, CORDIC (coordinate rotation digital computer)-based Cooley-Tukey fast Fourier transform (FFT)-like algorithms for power-of-two point discrete cosine transform/discrete sine transform/inverse discrete cosine transform/inverse discrete sine transform are proposed and their corresponding unified architectures are developed by fully reusing the unique two basic processing elements. The proposed algorithms have some distinguished advantages, such as FFT-like regular data flow, unique post-scaling factor, and arithmetic-sequence rotation angles. The developed unified architectures can compute four different transforms by simple routing the data flow according to the specific transform without feeding different transform coefficients or different transform kernels. The unfolding technique is used to overcome the problem of difficult to realize pipeline that occur in iterative CORDIC algorithms. Compared to existing unified architectures, the proposed architectures have a superior performance in terms of hardware complexity, control complexity, throughput, scalability, modularity, and pipelinability.

An algorithm for fast Hilbert transform of real functions

Abstract: A simple and accurate algorithm to evaluate the Hilbert transform of a real function is proposed using interpolations with piecewise---linear functions. An appropriate matrix representation reduces the complexity of this algorithm to the complexity of matrix-vector multiplication. Since the core matrix is an antisymmetric Toeplitz matrix, the discrete trigonometric transform can be exploited to calculate the matrix---vector multiplication with a reduction of the complexity to O(N log N), with N being the dimension of the core matrix. This algorithm has been originally envisaged for self-consistent simulations of radio-frequency wave propagation and absorption in fusion plasmas.

Image and video processing using discrete fractional transforms

Abstract: The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.

Image Watermarking in the Linear Canonical Transform Domain

Abstract: The linear canonical transform, which can be looked at the generalization of the fractional Fourier transform and the Fourier transform, has received much interest and proved to be one of the most powerful tools in fractional signal processing community. A novel watermarking method associated with the linear canonical transform is proposed in this paper. Firstly, the watermark embedding and detecting techniques are proposed and discussed based on the discrete linear canonical transform. Then the Lena image has been used to test this watermarking technique. The simulation results demonstrate that the proposed schemes are robust to several signal processing methods, including addition of Gaussian noise and resizing. Furthermore, the sensitivity of the single and double parameters of the linear canonical transform is also discussed, and the results show that the watermark cannot be detected when the parameters of the linear canonical transform used in the detection are not all the same as the parameters used in the embedding progress.

Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform

Abstract: A movie encryption scheme is proposed using a discrete multiple-parameter fractional Fourier transform and theta modulation. After being modulated by sinusoidal amplitude grating, each frame of the movie is transformed by a filtering procedure and then multiplexed into a complex signal. The complex signal is multiplied by a pixel scrambling operation and random phase mask, and then encrypted by a discrete multiple-parameter fractional Fourier transform. The movie can be retrieved by using the correct keys, such as a random phase mask, a pixel scrambling operation, the parameters in a discrete multiple-parameter fractional Fourier transform and a time sequence. Numerical simulations have been performed to demonstrate the validity and the security of the proposed method.

Discrete Fourier transform based pattern classifiers

Abstract: A technique for pattern classification using the Fourier tra nsform combined with the nearest neighbor classifier is proposed. The multidimensional fast Fourier transform ( FFT) is applied to the patterns in the data base. Then the magnitudes of the Fourier coefficients are sorted in desc ending order and the first P coefficients with largest magnitudes are selected, where P is a design parameter. These coefficients are then used in fur ther processing rather than the original patterns. When a noisy pattern is presente d for classification, the pattern’s P Fourier coefficients with largest magnitude are extracted. The coefficients are a rranged in a vector in the descending order of their magnitudes. The obtained vector is referred to as the signat ure vector of the corresponding pattern. Then the distance between the signature vector of the pattern to be cl assified and the signature vectors of the patterns in the data base are computed and the pattern to be classified is matc hed with a pattern in the data base whose signature vector is closest to the signature vector of the pattern bein g classified.

Performance Comparison of Hybrid Wavelet Transform Formed by Combination of Different Base Transforms with DCT on Image Compression

Abstract: In this paper image compression using hybrid wavelet transform is proposed. Hybrid wavelet transform matrix is formed using two component orthogonal transforms. One is base transform which contributes to global features of an image and another transform contributes to local features. Here base transform is varied to observe its effect on image quality at different compression ratios. Different transforms like Discrete Kekre Transform (DKT), Walsh, Real-DFT, Sine, Hartley and Slant transform are chosen as base transforms. They are combined with Discrete Cosine Transform (DCT) that contributes to local features of an image. Sizes of component orthogonal transforms are varied as 16-16, 32-8 and 64-4 to generate hybrid wavelet transform of size 256x256. Results of different combinations are compared and it has been observed that, DKT as a base transform combined with DCT gives better results for size 16x16 of both component transforms.

A novel method of filtration by the discrete heap transforms

Abstract: In this paper, we describe the method of filtering the frequency components of the signals and images, by using the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens transformations. The transforms are fast, because of a simple form of decomposition of their matrices, and they can be applied for signals of any length. Fast algorithms of calculation of the direct and inverse heap transforms do not depend on the length of the processed signals. Due to construction of the heap transform, if the input signal contains an additive component which is similar to the generator, this component is eliminated in the transform of this signal, while preserving the remaining components of the signal. The energy of this component is preserved in the first point, only. In particular case, when such component is the wave of a given frequency, this wave is eliminated in the heap transform. Different examples of the filtration over signals and images by the DsiHT are described and compared with the known method of the Fourier transform.

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Abstract: The discrete Fourier slice theorem is an important tool for signal processing, especially in the context of the exact reconstruction of an image from its projected views. This paper presents a digital reconstruction algorithm to recover a two dimensional (2-D) image from sets of discrete one dimensional (1-D) projected views. The proposed algorithm has the same computational complexity as the 2-D fast Fourier transform and remains robust to the addition of significant levels of noise. A mapping of discrete projections is constructed to allow aperiodic projections to be converted to projections that assume periodic image boundary conditions. Each remapped projection forms a 1-D slice of the 2-D Discrete Fourier Transform (DFT) that requires no interpolation. The discrete projection angles are selected so that the set of remapped 1-D slices exactly tile the 2-D DFT space. This permits direct and mathematically exact reconstruction of the image via the inverse DFT. The reconstructions are artefact free, except for projection inconsistencies that arise from any additive and remapped noise. We also present methods to generate compact sets of rational projection angles that exactly tile the 2-D DFT space. The improvement in noise suppression that comes with the reconstruction of larger sized images needs to be balanced against the corresponding increase in computation time.

BER performance improvement and PAPR reduction in OFDM systems based on combined DHT and μ-law companding

Abstract: High peak to average power ratio (PAPR) is a major drawback of orthogonal frequency division multiplexing (OFDM) systems. Among the various PAPR reduction methods, precoded technique and companding transform appear attractive for their simplicity and effectiveness. This paper proposes a new algorithm based on the combination of discrete Hartley transform (DHT) and μ-law companding for PAPR reduction. Theoretical analysis and numerical simulation are presented. Compared with the conventional μ-law companded OFDM system, the proposed algorithm offers an improved bit error rate (BER) and reduces the PAPR effectively for a large companding parameter.

Hybrid Encryption-Compression Scheme Based on Multiple Parameter Discrete Fractional Fourier Transform with Eigen Vector Decomposition Algorithm

Abstract: Encryption along with compression is the process used to secure any multimedia content processing with minimum data storage and transmission. The transforms plays vital role for optimizing any encryptioncompression systems. Earlier the original information in the existing security system based on the fractional Fourier transform (FRFT) is protected by only a certain order of FRFT. In this article, a novel method for encryption-compression scheme based on multiple parameters of discrete fractional Fourier transform (DFRFT) with random phase matrices is proposed. The multiple-parameter discrete fractional Fourier transform (MPDFRFT) possesses all the desired properties of discrete fractional Fourier transform. The MPDFRFT converts to the DFRFT when all of its order parameters are the same. We exploit the properties of multipleparameter DFRFT and propose a novel encryptioncompression scheme using the double random phase in the MPDFRFT domain for encryption and compression data. The proposed scheme with MPDFRFT significantly enhances the data security along with image quality of decompressed image compared to DFRFT and FRFT and it shows consistent performance with different images. The numerical simulations demonstrate the validity and efficiency of this scheme based on Peak signal to noise ratio (PSNR), Compression ratio (CR) and the robustness of the schemes against bruit force attack is examined.

Conjugate Symmetric Discrete Orthogonal Transform

Abstract: A new conjugate symmetric discrete orthogonal transform (CS-DOT)-generating method is proposed. The spectra of the CS-DOT for real input signals are conjugate symmetric so that we only need half memory size to store data. Meanwhile, the proposed CS-DOT also has a radix-2 fast algorithm so that it is suitable for hardware implementation. The CS-DOT generalized the existing transforms such that the discrete Fourier transform (DFT) and the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) are special cases of the CS-DOT. The CS-DOT-generating method is more systematic and generalized than that of the original CS-SCHT. We can use the same implementation structure but only adjust the twiddle factors to construct the CS-SCHT and DFT so that it is easy to switch the behaviors between these transforms.

Fourier Transform: Applications

Abstract: The Fourier transform is very useful in solving a variety of linear constant coefficient ordinary and partial differential equations describing processes which take place over an infinite interval, −∞ < x < ∞. We will provide a number of examples of this sort of application in the present section. Our first example involves a simple, time independent, equilibrium process. Example 1 We consider a stretched string, or cord, with small transverse displacement y(x), subject to an external transverse force f (x) and a transverse restoring force −κ y(x), maintained at tension τ > 0 over the interval −∞ < x < ∞ and constrained so that lim |x| → ∞ y(x) = 0. It can then be shown that y(x) satisfies τ d 2 y dx 2 − κ y(x) + f (x) = 0. Taking a 2 = κ/τ and applying the Fourier transform to both sides of this equation, using the differentiation property (twice) we have − ξ 2 + a 2 ˆ y(ξ) + 1 τ ˆ f (ξ) = 0 ⇒ ˆ y(ξ) = 1 τ ˆ f (ξ) ξ 2 + a 2. Using the convolution property of the Fourier transform we obtain y(x) = 1 τ ∞ −∞ F −1 1 ξ 2 + a 2 (r) f (x − r) dr as the solution. To make any further progress on this we need to find the function K(x) such thatˆK(ξ) ≡ (F K) (ξ) = 1 ξ 2 +a 2. It turns out that this function is K(x) =        e −ax 2a , x > 0, e ax 2a , x < 0.

Digital image sharpening using Riesz fractional order derivative and discrete hartley transform

Abstract: In this paper, a digital image sharpening method using Riesz fractional order derivative (RFOD) and discrete Hartley transform (DHT) is presented. First, the definition of Riesz fractional order derivative is reviewed briefly. Then, the DHT interpolation method is described. Next, the RFOD, DHT interpolation and Mach band effect are used to construct a digital image sharpening algorithm. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed digital image sharpening approach.

A Novel VLSI DHT Algorithm for A Highly Modular And Parallel Architecture

Abstract: A new very large scale integration (VLSI) algorithm for a 2N-length discrete Hartley transform (DHT) that can be efficiently implemented on a highly modular and parallel VLSI architecture having a regular structure is presented. The DHT algorithm can be efficiently split on several parallel parts that can be executed concurrently. Moreover, the proposed algorithms' well suited for the sub expression sharing technique that can be used to significantly reduce the hardware complexity of the highly parallel VLSI implementation. Using the advantages of the proposed algorithm and the fact that we can efficiently share the multipliers with the same constant, the number of the multipliers has been significantly reduced such that the number of multipliers is very small comparing with that of the existing algorithms. Moreover, the multipliers with a constant can be efficiently implemented in VLSI. Image compression is largely possible by exploiting various kinds of redundancies which are typically present in an image. The extent of redundancies may vary from image to image. Image compression algorithms aim to remove redundancy in data in a way which makes image reconstruction possible. This basically means that image compression algorithms try to exploit redundancies in the data, they calculate which data needs to be kept in order to reconstruct the original image and therefore which data can be ‗thrown away. Image compression and coding techniques explore three types of redundancies: coding redundancy, inter pixel (spatial) redundancy, and psycho visual redundancy. Coding redundancy consists in using variable-length code words selected as to match the statistics of the image itself or a processed version of its pixel values. This type of coding is always reversible and usually implemented using look-up tables (LUTs).

Multiple-parameter real discrete fractional Fourier and Hartley transforms

Abstract: In this paper, two new real fractional transforms with many parameters are constructed. They are the real discrete fractional Fourier transform (RDFRFT) and the real discrete fractional Hartley transform (RDFRHT). The eigenvectors of these two new transforms are all random, and they both have only two distinct eigenvalues: 1 or -1. Real eigenvectors of both two transforms are constructed from random DFT-commuting matrices. We also propose an alternative definition of RDFRHT based on a diagonal-like matrix. All of the proposed new transforms have required good properties to be fractional transforms. Finally, since outputs of proposed new transforms are random, they can be applied in image encryptions.

Implementation and evaluation of the Vandermonde transform

Abstract: The Vandermonde transform was recently presented as a time-frequency transform which, in difference to the discrete Fourier transform, also decorrelates the signal. Although the approximate or asymptotic decorrelation provided by Fourier is sufficient in many cases, its performance is inadequate in applications which employ short windows. The Vandermonde transform will therefore be useful in speech and audio processing applications, which have to use short analysis windows because the input signal varies rapidly over time. Such applications are often used on mobile devices with limited computational capacity, whereby efficient computations are of paramount importance. Implementation of the Vandermonde transform has, however, turned out to be a considerable effort: it requires advanced numerical tools whose performance is optimized for complexity and accuracy. This contribution provides a baseline solution to this task including a performance evaluation.

On the Hartley – Fourier sine generalized convolution

Abstract: In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H1,H2) and the Fourier sine (Fs) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities: We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.

Novel Analytical Approach of Non Conventional Mapping Scheme with Discrete Hartley Transform in OFDM System

Abstract: The system performance has been analyzed for π/4 DQPSK mapping scheme, which is differential in nature and hence adding additional advantage. Performance evaluation with random data as well as some images has been taken. Channel modeling has been performed in multipath fading environment. For elaboration of the concept mathematical modeling has been implemented using computer simulation. In this paper, an attempt is made to know the capabilities of DHT-OFDM with non conventional mapping technique π/4 DQPSK.

Phase-only correlation function by means of Hartley Transform

Abstract: In image processing or pattern recognition, Fourier Transform is widely used for frequency-domain analysis. In particular, the Phase Only Correlation (POC) method demonstrates high robustness and accuracy in the pattern matching and the image registration. However, there is a disadvantage in required memory machine because of the calculation of 2D-FFT. In this case, Hartley transform can be a very good substitute for more commonly used Fourier transform when the real input data are concerned. The Hartley transform is similar to the Fourier transform, but it is free from the need to process complex numbers. It also has some distinctive features that make it an interesting choice when a greater efficiency in memory requirements is needed. In this paper we show the correspondence between the Phase-Only Correlation (POC) function obtained by means of FFT and by FHT.

Image encryption using various transforms-a brief comparative analysis

Abstract: A comparative analysis for the gray scale image encryption technique is presented in this paper and more focus on security management of the bulk data (i.e. image) transfer in the secure way. This will provide authentication for user, ethical code, accuracy and safety of images which is travelling over internet while an image based data requires more effort during encryption & decryption. The most proposed architecture for encryption and decryption is developed with the same objective of an image using suitable user defined key. There are various methods by which image can be encrypted and decrypted to ensure the security. But here in this paper image encryption and decryption is performed using two major transforms i.e. Discrete Fractional Fourier Transform and Discrete Linear Canonical Transform.

Method for reducing peak-to-average ratio of asymmetric truncated orthogonal frequency division multiplexing signal

Abstract: The invention relates to a method for reducing a peak-to-average ratio by a DHT (Discrete Hartley Transform) precode, applicable to an ACO-OFDM system based on DHT, and belongs to the technical field of photo-communication. The DHT precode is introduced into the ACO-OFDM system based on DHT, signals do not loss; as the DHT algorithm is low in complexity, and PAPR of the system can be greatly reduced by only slightly adding implementation complexity of the system. The method is applicable to the systems with strict requirements on cost and implementation complexity, such as passive optical networks, indoor optical wireless communication, interconnection among data centers and the like.

The Plancherel, Titchmarsh and convolution theorems for the half-Hartley transform

Abstract: The generalized Parseval equality for the Mellin transform is employed to prove the Plancherel-type theorem in L2 with the respective inverse operator related to the Hartley transform on the nonnegative half-axis (the half-Hartley transform) Moreover, involving the convolution method, which is based on the double Mellin–Barnes integrals, the corresponding convolution and Titchmarsh's theorems for the half-Hartley transform are established As an application, we consider solvability conditions for a homogeneous integral equation of the second kind involving the Hartley kernel

2-D Discrete Wavelet Transform Using GPU

Abstract: With the wide spread of the discrete wavelet transform, the need for its efficient implementation becomes increasingly important. This work presents an improved version of an algorithm suitable to compute the 2-D discrete wavelet transform on GPU. Depending on the GPU platform, it is suitable to split the 2-D transform computation into separated horizontal and vertical passes. Considering the horizontal passes, we have examined and chosen the best performing method among the already known ones. Furthermore, we have adapted this method for an existing algorithm computing the vertical transform pass. This step helps to reduce several synchronizations and arithmetic operations in the utilized computation scheme. For large data, the proposed vertical method achieves speed-up about 30% compared to the current state of the art methods. In contrast to previously published works, the presented approach is built on the OpenCL parallel programming framework.

A fast modified signed Discrete Cosine Transform for image compression

Abstract: The Discrete Cosine Transform (DCT) is widely used in image compression for its high power compaction property. The Signed DCT (SDCT) and its modifications approximate the DCT and proceed faster. This paper introduces an efficient and low complexity 8 point transform. The proposed algorithm is derived by applying the signum function operator to an existing SDCT modification transform with good power compaction capabilities. Consequently, the elements of the proposed transform are only zeroes and ones. No multiplications or shift operations are required. The introduced transform keeps the high power compaction capabilities of its originating transform and, in the same time, provides a saving in computational complexity. A flow diagram is provided for the fast implementation of the transform. Only 17 additions are required for both forward and backward transformations. Simulation experiments are provided to justify the efficiency and improved performance of the proposed transform in image compression compared to other transforms.

Fast computation of the multidimensional discrete Fourier transform and discrete backward Fourier transform on sparse grids

Abstract: We propose a fast discrete Fourier transform for a given data set which may be generated from sampling a function of d-variables on a sparse grid and a fast discrete backward Fourier transform on a hyperbolic cross index set. Computation of these transforms can be formulated as evaluation of dimension-reducible sums on sparse grids. We introduce a fast algorithm for evaluating such sums and prove that the total number of operations needed in the algorithm is O(n log n), where n is the number of components along each coordinate direction of the data set. We then use it to develop fast algorithms for computing the discrete Fourier transform on the sparse grid and the discrete backward Fourier transform on the hyperbolic cross index set. We also show that if the given data set is sampled from a function having regularity of order s, then its discrete Fourier transform has the optimal approximation order O(n−s). Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed algorithms.