Showing papers on "Hartley transform published in 2000"
TL;DR: A new image encryption algorithm based on a generalized fractional Fourier transform, to which it is referred as a multifractional Fouriers transform, is proposed.
Abstract: We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.
182 citations
01 Jan 2000
TL;DR: A finite implementation of the ridgelet transform is presented that is invertible, non-redundant and achieved via fast algorithms and it is shown that this transform is orthogonal hence it allows one to use non-linear approximations for the representation of images.
Abstract: A finite implementation of the ridgelet transform is presented. The transform is invertible, non-redundant and achieved via fast algorithms. Furthermore we show that this transform is orthogonal hence it allows one to use non-linear approximations for the representation of images. Numerical results on different test images are shown. Those results conform with the theory of the ridgelet transform in the continuous domain-the obtained representation can represent efficiently images with linear singularities. Thus it indicates the potential of the proposed system as a new transform for coding of images.
76 citations
TL;DR: In this article, the authors developed the discrete fractional Hilbert transform (DFHT) and applied it to edge detection of digital images, which is a generalization of the Hilbert transform, and it presents physical interpretation in the definition.
Abstract: The Hilbert transform plays an important role in the theory and practice of signal processing. A generalization of the Hilbert transform, the fractional Hilbert transform, was recently proposed, and it presents physical interpretation in the definition. In this paper, we develop the discrete fractional Hilbert transform, and apply the proposed discrete fractional Hilbert transform to the edge detection of digital images.
59 citations
TL;DR: The integer transform introduced by Cham is modified and introduced and the complete integer transform is introduced, which requires no real number multiplication operation, no matter what the forward or inverse transform.
Abstract: The integer transform (such as the Walsh transform) is the discrete transform that all the entries of the transform matrix are integer. It is much easier to implement because the real number multiplication operations can be avoided, but the performance is usually worse. On the other hand, the noninteger transform, such as the DFT and DCT, has a good performance, but real number multiplication is required. W derive the integer transforms analogous to some popular noninteger transforms. These integer transforms retain most of the performance quality of the original transform, but the implementation is much simpler. Especially, for the two-dimensional (2-D) block transform in image/video, the saving can be huge using integer operations. In 1989, Cham had derived the integer cosine transform. Here, we will derive the integer sine, Hartley, and Fourier transforms. We also introduce the general method to derive the integer transform from some noninteger transform. Besides, the integer transform derived by Cham still requires real number multiplication for the inverse transform. We modify the integer transform introduced by Cham and introduce the complete integer transform. It requires no real number multiplication operation, no matter what the forward or inverse transform. The integer transform we derive would be more efficient than the original transform. For example, for the 8-point DFT and IDFT, there are in total four real numbers and eight fixed-point multiplication operations required, but for the forward and inverse 8-point complete integer Fourier transforms, there are totally 20 fixed-point multiplication operations required. However, for the integer transform, the implementation is simpler, and many of the properties of the original transform are kept.
42 citations
TL;DR: This study introduces several types of simplified fractional Fourier transform (SFRFT) that are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems.
Abstract: The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT’s but also their implementation. Although these SFRFT’s usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT’s in many applications.
41 citations
TL;DR: In this paper, the authors apply techniques from non-commutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups, in particular on the group of rigid-body motions in 3-space, denoted here as SE(3).
35 citations
TL;DR: A new structure for the undecimated wavelet transform (UWT) is introduced, which inherits the simplicity of the lifting scheme, such that the inverse transform is easily implemented.
Abstract: A new structure for the undecimated wavelet transform (UWT) is introduced. This structure combines the stationary wavelet transform with a lifting scheme and its design is based on a polyphase structure. The suggested structure inherits the simplicity of the lifting scheme, such that the inverse transform is easily implemented. The proposed performance of the UWT is verified on a signal denoising application.
34 citations
TL;DR: This paper outlines a fast and efficient method for the computation of the Hough transform using Fourier methods and suggests that the maxima points generated in the Radon space, corresponding to the parametrisation of straight lines, can be enhanced with a post transform convolutional filter.
Abstract: The Hough transform is a useful technique in the detection of straight lines and curves in an image. Due to the mathematical similarity of the Hough transform and the forward Radon transform, the Hough transform can be computed using the Radon transform which, in turn, can be evaluated using the central slice theorem. This involves a two-dimensional Fourier transform, an x-y to r-? mapping and a ID Fourier transform. This can be implemented in specialized hardware to take advantage of the computational savings of the fast Fourier transform. In this paper, we outline a fast and efficient method for the computation of the Hough transform using Fourier methods. The maxima points generated in the Radon space, corresponding to the parametrisation of straight lines, can be enhanced with a post transform convolutional filter. This can be applied as a ID filtering operation on the resampled data whilst in the Fourier space, so further speeding the computation. Additionally, any edge enhancement or smoothing operations on the input function can be combined into the filter and applied as a net filter function.
32 citations
05 Jun 2000
TL;DR: This paper presents a real-valued discrete multicarrier modulation approach that is based on the use of the discrete Hartley transform and its inverse (IDHT) to perform the modulation and demodulation operations.
Abstract: This paper presents a real-valued discrete multicarrier modulation approach that is based on the use of the discrete Hartley transform (DHT) and its inverse (IDHT) to perform the modulation and demodulation operations. Since the DHT and IDHT definitions are identical, we can use the same hardware or program to implement the modulator and demodulator of the proposed multicarrier method. As compared to the complex-valued discrete Fourier transform based multicarrier modulation method, the proposed one achieves the same transmission performance with reduced computational complexity and implementation cost.
32 citations
TL;DR: New odd-factor algorithms are derived to support transforms whose sequence length contains multiple odd factors and have a regular computational structure, achieve a substantial reduction of computational complexity, and support a wider range of choices on the sequence length.
Abstract: This paper presents fast algorithms for type-II, type-III, and type-IV generalized discrete Hartley transform. In particular, new odd-factor algorithms are derived to support transforms whose sequence length contains multiple odd factors. By jointly using the odd-factor and radix-2 algorithms, fast computation for arbitrarily composite sequence length can be achieved. Compared to other reported algorithms, the proposed ones have a regular computational structure, achieve a substantial reduction of computational complexity, and support a wider range of choices on the sequence length.
10 Sep 2000
TL;DR: The generalized lapped biorthogonal transform embedded inverse discrete cosine transform (ge-IDCT) with nonlinear weighting in the embedded transform domain can reconstruct the signal with alleviated blockishness.
Abstract: This paper presents the generalized lapped biorthogonal transform embedded inverse discrete cosine transform (ge-IDCT) as an alternative to the IDCT. The ge-IDCT with nonlinear weighting in the embedded transform domain can reconstruct the signal with alleviated blockishness. Additional complexity, imposed by the replacement, is trivial thanks to an efficient lattice structure. The proposed ge-IDCT is applied in the JPEG still image compression standard to demonstrate its validity.
TL;DR: It is shown that, for positive definite Toeplitz matrices, the relative error of the computed vector can be estimated by a multiple of the condition number of the matrix.
Abstract: Representations for inverses of real symmetric Toeplitz matrices involving discrete Hartley transformations are presented which can be used for fast matrix-vector multiplication. In this way, multiplication of a column vector by an inverse real symmetric Toeplitz matrix can be carried out with the help of six Hartley transformations plus two for preprocessing. Besides complexity, stability issues will also be discussed. In particular, it is shown that, for positive definite Toeplitz matrices, the relative error of the computed vector can be estimated by a multiple of the condition number of the matrix.
TL;DR: A new real-valued lapped transform for 2D-signal and image processing that allows unambiguous detection of spatial orientation from spectral energy and compares it to the complex-valued discrete Fourier transform.
TL;DR: In this article, a column vector can be multiplied by the inverse of a Toeplitz-plus-Hankel matrix with the help of 6 Hartley transforms plus O(n) operations.
Abstract: Representations for inverses of Toeplitz-plus-Hankel matrices and more general Bezoutians involving only discrete Hartley transforms and diagonal matrices are presented. Using these representations a column vector can be multiplied by the inverse of a Toeplitz-plus-Hankel matrix with the help of only 6 Hartley transforms plus O(n) operations. This complexity estimate is significantly better than previous ones.
TL;DR: In this paper, Hartley transform analysis of the Bouguer gravity values in the Shamakhy-Gobustan and Absheron regions suggests that a gravity field may be separated into long and short-wavelength components.
TL;DR: It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform.
Abstract: In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.
TL;DR: In this paper, it was shown that for continuous functions, the Hilbert transform and Cauchy transform have a similar divergence behavior as for absolute integrable functions, and that for each set of the set E with Lebesgue measure zero, there exists a continuous function such that the Hilbert transformation and the cauchy transformation of this function are infinite for all points in E.
Abstract: In this paper we investigate the behavior of the Hilbert transform and the Cauchy transform. It is well known, that for absolut integrable functions the Hilbert transform and the Cauchy transform is finite almost everywhere. In this paper it is shown, that for each set \( E\subset [-\pi ,\pi ) \) with Lebesgue measure zero there exists a continuous function such that the Hilbert transform and the Cauchy transform of this function is infinite for all points of the set E. So for continuous functions the Hilbert transform and the Cauchy transform have a similar divergence behavior as for absolute integrable functions.
TL;DR: In this article, the wavelet transform (WT) is used for data analysis and signal processing in the context of chromatographic data analysis, which is used to solve the problems of noise suppression, signal enhancement, peak detection, resolution enhancement, and multivariate signal resolution.
Abstract: This chapter describes application of wavelet transform in processing chromatographic data. Chromatography is used widely in analytical chemistry for the separation of compounds in sample mixtures. By adopting different chemical and physical properties, various chromatographic techniques and instruments are developed for chemical analysis. Such techniques include paper chromatography, thin layer chromatography (TLC), gas chromatography (GC), and many more. There was a tendency to combine different analytical techniques or instruments with chromatography for separation and characterization. In chromatographic data analysis and signal processing, analytical chemists always face problems such as noise suppression, signal enhancement,peak detection, resolution enhancement, and multivariate signal resolution. Various chemometric methods are proposed for tackling these problems. Transformation techniques, such as the Fourier transform, Laplace transform, and Hartley transform are utilized in chromatography for data processing. Recently, the new mathematical technique wavelet transform (WT) is introduced to help solve various problems.
05 Jun 2000
TL;DR: The aim of this paper is to derive the 3-D vector radix for the3-D discrete Hartley transform and the arithmetic operations of this algorithm are compared to similar algorithms using the row-column approach.
Abstract: The application of multidimensional fast transforms to solve problems in image processing, motion analysis and multidimensional signal processing is growing The discrete Hartley transform (DHT) is one of the new tools used in many applications including signal and image processing, digital filters, communication etc This transform is closely related to the discrete Fourier transform, but it is a real-to-real transform and it has the same inverse Many fast algorithms have been developed for the calculation of one-dimensional DHT These algorithms are then used for the calculation of multidimensional Hartley transform through an intermediate transform using the row-column approach However proper multidimensional algorithms can be more efficient and need to be developed It is the aim of this paper to derive the 3-D vector radix for the 3-D discrete Hartley transform The arithmetic operations of this algorithm are compared to similar algorithms using the row-column approach
28 May 2000
TL;DR: This paper introduces the integer Fourier transform (ITFT), which is approximated to the DFT, but all the entries in the transform matrix are integer numbers, and it only requires fixed-point multiplication, and the implementation can be much simplified, especially for VLSI.
Abstract: DFT has good quality of performance and fast algorithms. But when we implement the DFT, we require the floating-point multiplication. In this paper, we introduce the integer Fourier transform (ITFT). ITFT is approximated to the DFT, but all the entries in the transform matrix are integer numbers. So it only requires fixed-point multiplication, and the implementation can be much simplified, especially for VLSI. This new transform will work similarly to the original DFT, for example, the transform results are similar and the shifting-invariant property is also preserved for ITFT. We also introduce the general method to derive the integer transform. By this approach, we can derive many types of integer transforms (such as integer cosine, sine, and Hartley transforms).
Proceedings Article•
01 Jan 2000TL;DR: An easily comprehensible and efficient implementation of the fast HT and its multi-dimensional extension is demonstrated by adapting this algorithm to volume rendering by the projection-slice theorem and by the use for filter analysis in frequency domain to demonstrate the importance of the HT in this application area.
Abstract: The Fast Hartley Transform (FHT), a discrete version of the Hartley Transform (HT), has been studied in various papers and shown to be faster and more convenient to implement and handle than the corresponding Fast Fourier Transform (FFT). As the HT is not as nicely separable as the FT, a multidimensional version of the HT needs to perform a final correction step to convert the result of separate HTs for each dimension into the final multi-dimensional transform. Although there exist algorithms for two and three dimensions, no generalization to arbitrary dimensions can be found in the literature. We demonstrate an easily comprehensible and efficient implementation of the fast HT and its multi-dimensional extension. By adapting this algorithm to volume rendering by the projection-slice theorem and by the use for filter analysis in frequency domain we further demonstrate the importance of the HT in this application area.
28 May 2000
TL;DR: These networks realize the data reordering found in constant geometry radix-2/sup r/ algorithms, which exist, e.g., for discrete Fourier, sine, cosine, and Hartley transforms.
Abstract: In parallel architectures for discrete trigonometric transforms, the number of processing elements is typically dependent on the transform size. Scalable architectures can be constructed with a partial column approach where the computation is performed iteratively with less number of processing elements. This approach results in a need for complex data reordering for realizing the interconnections between the processing columns. In this paper, such interconnection networks performing temporal and spatial reordering are proposed. These networks realize the data reordering found in constant geometry radix-2/sup r/ algorithms, which exist, e.g., for discrete Fourier, sine, cosine, and Hartley transforms. A general decomposition of stride by 2/sup r/ permutation is shown with corresponding network implementations. Furthermore, modifications to support mixed-size and 2-D transforms are discussed.
TL;DR: Based on the delay theorem of Laplace transform, a new algorithm for the G-S transform is proposed that can reduce the computation quantity by orders of magnitude, and therefore increase markedly the speed of computation for TEM responses with dense samples.
Abstract: Calculations for the transient electromagnetic (TEM) method are frequently performed by using the Gaver-Stehfest algorithm to compute the inverse Laplace transform (G-S transform). It is a pure real number operation and only needs implementing the calculation for the values of a small number of Laplace transform variable s (usually, 12s values for one sample time). So it is a rather rapid algorithm. However, when we need to calculate TEM responses for much sampling time, the computation quantity of this algorithm will be quite large. Based on the delay theorem of Laplace transform, a new algorithm for the G-S transform is proposed in this paper. This method can reduce the computation quantity by orders of magnitude, and therefore increase markedly the speed of computation for TEM responses with dense samples.
05 Jun 2000
TL;DR: New hardware-oriented matrix formulations and their corresponding VLSI architectures are proposed for both radix-2 andRadix-4 fast algorithms of a variety of discrete trigonometric transforms, leading to hardware-efficient architectures requiring much fewer processing elements compared to other previously proposed unified DXTs of the same throughput rate.
Abstract: New hardware-oriented matrix formulations and their corresponding VLSI architectures are proposed for both radix-2 and radix-4 fast algorithms of a variety of discrete trigonometric transforms (DXTs), including the discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST), and discrete Hartley transform (DHT). All the DXTs have the same complex kernel operation consisting of products of diagonal matrices and band matrices with equally spaced nonzero diagonals. New linear arrays based on the systolic mapping of the common kernel operation are designed which lead to hardware-efficient architectures requiring much fewer processing elements compared to other previously proposed unified DXT architectures of the same throughput rate.
Journal Article•
TL;DR: In this article, the Collins diffraction integral formula for description through any axial symmetrical optical Systems in spatial-domain could be written in terms of the fractional Fourier transform in the Cartesian coordinates, or could be described in the cylindrical coordinates.
05 Jun 2000
TL;DR: The eigenfunctions of the affine Fourier transform are just the light distributions that will cause the self-imaging phenomena for some optical systems and are used to discuss theSelf-imaged phenomena.
Abstract: The affine Fourier transform (AFT) also called as the canonical transform. It generalizes the fractional Fourier transform (FRFT), Fresnel transform, scaling operation, etc., and is a very useful tool for signal processing. We derive the eigenfunctions of the AFT. The eigenfunctions seems hard to be derived, but since the AFT can be represented by the time-frequency matrix (TF matrix), we can use the matrix operations to derive its eigenfunctions. Then, because many optical systems can be represented as a special case of the AFT, the eigenfunctions of the AFT are just the light distributions that will cause the self-imaging phenomena for some optical systems. We use the eigenfunctions we derive to discuss the self-imaging phenomena.