Showing papers on "Hartley transform published in 1999"

Introduction to the Fractional Fourier Transform and Its Applications

Abstract: Publisher Summary This chapter is an introduction to the fractional Fourier transform and its applications. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a . Mathematically, the a th order fractional Fourier transform is the a th power of the Fourier transform operator. The a = 1st order fractional transform is the ordinary Fourier transform. In essence, the a th order fractional Fourier transform interpolates between a function f(u) and its Fourier transform F(μ) . The 0th order transform is simply the function itself, whereas the 1st order transform is its Fourier transform. The 0.5th transform is something in between, such that the same operation that takes us from the original function to its 0.5 th transform will take us from its 0.5th transform to its ordinary Fourier transform. More generally, index additivity is satisfied: The a 2 th transform of the a 1 th transform is equal to the ( a 2 + a 1 )th transform. The –1th transform is the inverse Fourier transform, and the – a th transform is the inverse of the a th transform.

Transform domain analysis of DES

Abstract: The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input. From this point of view, the tools for pseudo-random sequence analysis are applied to the S-boxes in DES. The properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamard transform, and the Avalanche transform are investigated. Two important results about the S-boxes of DES are found. The first result is that nearly two-thirds of the total 32 functions from GF (2/sup 6/) to GF(2) which are associated with the eight S-boxes of DES have the maximal linear span G3, and the other one-third have linear span greater than or equal to 57. The second result is that for all S-boxes, the distances of the S-boxes approximated by monomial functions has the same distribution as for the S-boxes approximated by linear functions. Some new criteria for the design of permutation functions for use in block cipher algorithms are discussed.

Integral transforms for engineers

Abstract: Special Functions Fourier Integrals and Fourier Transforms Applications Involving Fourier Transforms The Laplace Transformation Applications Involving Laplace Transforms The Melling Transform The Hankel Transform Finite Transforms Discrete Transforms Appendix A - Review of Complex Variables Appendix B - Table of Fourier Transforms Appendix C - Table of Laplace Transforms.

On the choice of transforms for data hiding in compressed video

Abstract: We present an information-theoretic approach to obtain an estimate of the number of bits that can be hidden in compressed image sequences. We show how addition of the message signal in a suitable transform domain rather than the spatial domain can significantly increase the data hiding capacity. We compare the data hiding capacities achievable with different block transforms and show that the choice of the transform should depend on the robustness needed. While it is better to choose transforms with good energy compaction property (like DCT, wavelet etc.) when the robustness required is low, transforms with poorer energy compaction property (like the Hadamard or Hartley transform) are preferable choices for higher robustness requirements.

The Funk transform as a Penrose transform

Abstract: The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.

Theorems on L 2 -transform and its applications

Abstract: In the present paper the author proves Parseval-Goldstein-type theorems involving a Laplace-type integral tranform, the Widder transform and the K-transform. The theorem is then shown to yield a number of new identities involving several well-known integral transforms. Using the theorems and its corollaries, a number of interesting infinite integrals of elementary and special functions are presented. Some illustrative examples are also given.

Fourier transform based techniques in efficient retrieval of similar time sequences

Abstract: Fourier-Transform Based Techniques in Efficient Retrieval of Similar Tirne Sequences Davood Rafiei Doctor of Philosophy Graduate Department of Computer Science University of Toronto 1999 The idea of posing queries in terms of similarity of objects, rather than equality or inequality, is of growing importance in new database applications, such as data mining or data warehousing. In this dissertation, the notion of similarity is defined in terms of a distance function and a set of linear transformations. This turns out to be a proper notion of similarity for time series data since it can eliminate seasonal effects and shortterm fluctuations before aligning them. The focus of this dissertation is on efficiently processing s i rn i l~ i ty queries on time series data. The dissertation first describes an important property of real-valued time sequences in the frequency domain, i.e. symmetry, and presents an algorithm that uses this property to improve the performance of a multidimensional index built on a sequence data set by more than a factor of two. This improvement is confirmed both analytically and

Lapped directional transform: a new transform for spectral image analysis

Abstract: We propose a new real-valued lapped transform for 2D-signal and image processing Lapped transforms are particularly useful in block-based processing, since their intrinsically overlapping basis functions reduce or prevent block artifacts Our transform is derived from the modulated lapped transform (MLT), which, as a real-valued and separable transform like the discrete cosine transform, does not allow to unambiguously identify oriented structures from modulus spectra This is in marked contrast to the (complex-valued) discrete Fourier transform (DFT) The new lapped transform is real-valued, and at the same time allows unambiguous detection of spatial orientation Furthermore, a fast algorithm for this transform exists As an application example, we investigate the transform's performance in spectral approaches to image restoration and enhancement in comparison to the DFT

Optimized Fast Algorithms for the Quaternionic Fourier Transform

Abstract: In this article, we deal with fast algorithms for the quaternionic Fourier transform (QFT). Our aim is to give a guideline for choosing algorithms in practical cases. Hence, we are not only interested in the theoretic complexity but in the real execution time of the implementation of an algorithm. This includes floating point multiplications, additions, index computations and the memory accesses. We mainly consider two cases: the QFT of a real signal and the QFT of a quaternionic signal. For both cases it follows that the row-column method yields very fast algorithms. Additionally, these algorithms are easy to implement since one can fall back on standard algorithms for the fast Fourier transform and the fast Hartley transform. The latter is the optimal choice for real signals since there is no redundancy in the transform. We take advantage of the fact that each complete transform can be converted into another complete transform. In the case of the complex Fourier transform, the Hartley transform, and the QFT, the conversions are of low complexity. Hence, the QFT of a real signal is optimally calculated using the Hartley transform.

Multidimensional polynomial transform algorithm for multidimensional discrete W transform

Abstract: The multidimensional (MD) polynomial transform is used to convert the MD W transform (MDDWT) into a series of one-dimensional (1-D) W transforms (DWTs). Thus, a new polynomial transform algorithm for the MDDWT is obtained. The algorithm needs no operations on complex data. The number of multiplications for computing an r-dimensional DWT is only 1 times that of the commonly used row-column method. The number of additions is also reduced considerably.

Fourier transform of spherical Laguerre Gaussian functions and its application in molecular integrals

Abstract: The Fourier transform of the spherical Laguerre Gaussian-type function (LGTF), L(αr2)rlYlm(r)e, was derived. Applying the Fourier transform convolution theorem, the basic two-center integrals over the general two-electron irregular solid harmonic operator, YLM(r12)/r (which becomes Coulomb repulsion, spin–other-orbit interaction or spin–spin interaction when L=0, 1, or 2, respectively) as well as the overlap were evaluated analytically. These basic integral results generate the two-electron integrals of the Coulomb type, hybrid type, and exchange type as well as that of three- and four-center. The formulas obtained, which are general for electronic wave functions of unrestricted quantum numbers n, l, and m, are expressed explicitly in terms of nuclear spherical LGTFs of internuclear geometrical variables. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 265–273, 1999

Numerical inversion of 2-D Laplace transforms by fast Hartley transform computations ☆

Abstract: Laplace transforms in two variables are useful in the solution of partial differential equations. However, their analytic inverses are often difficult to obtain. In this paper, the numerical method based on evaluating the Bromwich integral and using the computation algorithm of fast Fourier transform (FFT) is extended to the inversion of two-dimensional (2-D) Laplace transforms. A new inversion formula is devised so that the values of the inverted 2-variable function on the rectangle grids can be efficiently obtained via multiple sets of 2-D fast Hartley transform (FHT) computations. The proposed FHT-based inversion algorithm is thus suitable for the computation in a multiprocessor environment.

Hartley transform in compression of medical ultrasonic images

Abstract: In this paper the Hartley transform has been used for both compression and filtering of medical ultrasonic images. The Hartley transform shares some features of the Fourier transform and, most importantly, there exists a computationally effective butterfly algorithm of the transform. Compression relies on filtering out higher harmonics of the forward Hartley transform and saving the result rather as the image than the coefficients of the transform. In this case the images' size is reduced 16 times without significant loss of valuable medical information.

FHT algorithm for length N=q.2/sup m/

Abstract: Of late, the discrete Hartley transform (DHT) has become an important real-valued transform Many fast algorithms for computing the DHT of sequence length N=2/sup m/ have been reported Fast computation of the DHT of length N=q2/sup m/, where q is an odd integer, is proposed The key feature of the algorithm is its flexibility in the choice of sequence length N, where N need not necessarily be a power of 2, while giving rise to a substantial reduction in computational complexity when compared to other algorithms

A novel multiwavelet-based integer transform for lossless image coding

Abstract: Integer Haar wavelet transform or S-transform is used as the basic building block for many exiting integer wavelet transform. As an alternative, a new integer multiwavelet transform and its associated integer prefilter are designed based on box-and-slope multi-scaling system. Both the transform and prefilter can be implemented with simple integer Haar transform requiring only addition and bit shift operations. Since the new integer transform is an approximation to nontruncated transform with higher vanishing moment than that of Haar transform, better approximation accuracy is expected and verified experimentally. The transform is successfully applied to lossless image coding with results outperforming that of lossless JPEG and S-transform.

Computing discrete Hartley transform using algebraic integers

Abstract: An algorithm for computing the discrete Hartley transform is presented that is based on the algebraic integers encoding scheme. With the aid of this scheme, an error-free representation of the cos function becomes possible. In addition, for further complexity reduction an approximation scheme is proposed. Finally, for the implementation of the algorithm a fully pipelined systolic architecture with O(N) throughput is proposed.

Applications of the fractional Fourier transform to filtering, estimation and restoration.

Abstract: The fractional Fourier transform is more general and exible than the ordinary Fourier transform, but its optical and digital implementation is just as e cient. This underlies its potential for generalizations and improvements in every area of digital and optical signal processing. Here we consider applications of the transform to ltering, estimation and restoration. We see that the use of fractional Fourier transform based ltering con gurations allow one to exibly trade o between cost and accuracy in these applications.

Prime-factor algorithms for generalised discrete Hartley transform

Abstract: A prime factor fast algorithm for the type-II generalised discrete Hartley transform is presented. In addition to reducing the number of arithmetic operations and achieving a regular computational structure, a simple index mapping method is proposed to minimise the overall implementation complexity.

Generalized scale transforms

Abstract: We are presenting a new class of transforms which facilitates the processing of signals that are nonlinearly stretched or compressed in time. We refer to nonlinear stretching and compression as warping. While the magnitude of the Fourier transform is invariant under time shift operations, and the magnitude of the scale transform is invariant under (linear) scaling operations, the new class of transforms is magnitude invariant under warping operations. The new class contains the Fourier transform and the scale transform as special cases. Important theorems, like the convolution theorem for Fourier transforms, are generalized into theorems that apply to arbitrary members of the transform class. Cohen's class of time-frequency distributions is generalized to joint representations in time and arbitrary warping variables. Special attention is paid to a modification of the new class of transforms that maps an arbitrary time-frequency contour into an impulse in the transforms that maps an arbitrary time-frequency contour into an impulse in the transform domain. A chirp transform is derived as an example.

Determination of error values for decoding hermitian codes with the inverse affine fourier transform

Abstract: With the knowledge of the syndromes Sa,b, 0 ≤ a, b ≤ q − 2, the exact error values cannot be determined by using the conventional (q − 1)2-point discrete Fourier transform in the decoding of a plane algebraic-geometric code over GF (q). In this letter, the inverse q-point 1-dimensional and q2-point 2dimensional affine Fourier transform over GF (q) are presented to be used to retrieve the actual error values, but it requires much computation efforts. For saving computation complexity, a modification of the affine Fourier transform is derived by using the property of the rational points of the plane Hermitian curve. The modified transform, which has almost the same computation complexity of the conventional discrete Fourier transform, requires the knowledge of syndromes Sa,b, 0 ≤ a, b ≤ q − 2, and three more extended syndromes Sq−1,q−1, S0,q−1, Sq−1,0. key words: algebraic-geometric code, Hermitian code, affine Fourier transform, error value

Edge detection based on the multiresolution Fourier transform

Abstract: In this paper, an edge detection technique is proposed by using the multiresolution Fourier transform (MFT) to analyze the local properties in the spatial frequency domain. Five major steps are adopted to implement the detection of edges. First, the Laplacian pyramid method is used to create a high-pass filtered image. Secondly, the Multiresolution Fourier Transform (MFT) is applied to divide the high-pass filtered image into blocks and transform each of the blocks into spatial frequency domain. Thirdly, single-feature and non-single-feature blocks are differentiated. Subsequently, the blocks containing single feature are then subject to a process for estimating the orientation and the centroid of the feature in order to locate it. Finally, the accuracy of the estimated centroid of the local feature is checked. Once all the blocks are analyzed at a resolution level, the overall procedure is repeated at the next resolution level and the blocks with their father block being classified as non-single-feature or being rejected in the accuracy check stage at the previous level are analyzed. The algorithm stops when a specific level is reached.

Invariant Spaces and Fast Transforms

Abstract: In this paper a new approach to the convolution based on the linear representation of the dihedral group is presented. In the decomposition of this representation, the Fourier operator appears. Some useful properties of the Fourier operator are summarized. Its projectors onto its eigenspaces are expressed with the Hartley operator. An orthogonal basis of the invariant spaces of the dihedral group is defined. A generalized method for analyzing and constructing fast transforms is proposed.

Image coding using vector-embedded Karhunen-Loeve transform

Abstract: In this paper, the theory and the design of a new class of orthogonal transforms are presented. The novel transform is derived from a correlation matrix in which an arbitrary orthonormal system is embedded. By embedding an orthonormal system designed empirically, we obtain the transform that can not only represent intuitive features but also possesses statistical property like the KLT. Our main motivation is the application in block-based adaptive transforms coding. We show a design example of the transform, which adapts orientational features such as edges and lines. Using this transform, we perform orientation adaptive coding. In experimental results, it is shown that image coding using the transform is effective in rate-distortion criterion and subjective quality.

Radon-Wigner transform for optical field analysis

Abstract: The Radon-Wigner transform, associated with the intensity distribution in the fractional Fourier transform system, is used for the analysis of complex structures of coherent as well as partially coherent optical fields. The application of the Radon-Wigner transform to the analysis of fractal fields is presented.

2-D affine generalized fractional Fourier transform

Abstract: The 2-D Fourier transform has been generalized into the 2-D separable fractional Fourier transform (replaces the 1-D Fourier transform by the l-D fractional Fourier transform for each variable) and the 2-D separable canonical transform (further replaces the fractional Fourier transform by canonical transform) of Sahin, Ozaktas and Mendlovic (see Appl. Opt., vol.37, no.11, p.2130-41, 1998). It also has been generalized into the 2-D unseparable fractional Fourier transform with 4 parameters of Sahin et al. (see Appl. Opt., vol.37, no.23, p.5444-53, 1998). In this paper, we introduce the 2-D affine generalized fractional Fourier transform (AGFFT). These 2-D transforms has been further generalized. We show it can deal with many problems that can not be dealt with by these 2-D transforms and extend their utility.