Showing papers on "Hartley transform published in 1998"
TL;DR: A new convolution structure for the FRFT is introduced that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
Abstract: The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
194 citations
TL;DR: The generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
Abstract: The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
108 citations
TL;DR: In this paper, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated, and the results of the eigendecomposition of the transform matrix are used to define DFRHT and DFRFT.
Abstract: This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.
105 citations
TL;DR: This work allows the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled.
Abstract: We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.
87 citations
TL;DR: In this article, the authors extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic, which makes the transform of a convolution of two functions almost equal to the product of their transform.
Abstract: In recent years the fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has been the focus of many research papers because of its application in several areas, including signal processing and optics. In this paper, we extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic. The algebraic approach requires the introduction of a new convolution operation for the fractional Fourier transform that makes the transform of a convolution of two functions almost equal to the product of their transform.
67 citations
TL;DR: The fast Hartley transform, a fast but relatively unknown computational method for frequency domain filtering of ERP/EEG data, is introduced and compared with time domain filtering.
Abstract: A general approach to time domain digital filtering is described, and examples of some filters used in EEG/ERP research are presented. Simulations are reported that evaluate the impact of the relative length of the filter weight series and the signal cycle to be filtered, the span and real-time density of the filter weights, and slow drift across the epoch being filtered. Results indicate that some filters commonly used in the EEG/ERP literature are inadequate. Frequency domain digital filtering is also briefly discussed. The fast Hartley transform, a fast but relatively unknown computational method for frequency domain filtering of ERP/EEG data, is introduced and compared with time domain filtering. Some practical recommendations are provided.
54 citations
16 Aug 1998
TL;DR: In this paper, a trigonometry for finite fields is introduced and the k-trigonometric functions over the Galois field GF(q) are defined and their main properties derived.
Abstract: A trigonometry for finite fields is introduced. In particular, the k-trigonometric functions over the Galois field GF(q) are defined and their main properties derived. This leads to the definition of the cas/sub k/(.) function over GF(q), which in turn leads to a finite field Hartley transform (FFHT). The FFHT presented here is different from an earlier version and seems to be the more natural one.
47 citations
TL;DR: A version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT are shown.
Abstract: We introduce the concept of the Fractional Wave Packet Transform(FRWPT), based on the idea of the Fractional Fourier Transform(FRFT) and Wave Packet Transform(WPT). We show a version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT.
43 citations
TL;DR: A new, fast, and effective treatment planning approach is developed to deal with a heterogeneous activity distribution and, using the 3-D FHT convolution, absorbed dose calculation time was reduced over 1000 times.
Abstract: Effective radioimmunotherapy may depend on a priori knowledge of the radiation absorbed dose distribution obtained by trace imaging activities administered to a patient before treatment. A new, fast, and effective treatment planning approach is developed to deal with a heterogeneous activity distribution. Calculation of the three-dimensional absorbed dose distribution requires convolution of a cumulated activity distribution matrix with a point-source kernel; both are represented by large matrices (64×64×64). To reduce the computation time required for these calculations, an implementation of convolution using three-dimensional (3-D) fast Hartley transform (FHT) is realized. Using the 3-D FHT convolution, absorbed dose calculation time was reduced over 1000 times. With this system, fast and accurate absorbed dose calculations are possible in radioimmunotherapy. This approach was validated in simple geometries and then was used to calculate the absorbed dose distribution for a patient’s tumor and a bone marrow sample.
41 citations
TL;DR: Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively and the integral representation of a fractional Hankel transform (FRHT) is derived from the fractional Fourier transform.
Abstract: We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.
41 citations
TL;DR: A nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case is presented and its digital and optical implementations are presented.
Abstract: Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example.
01 Jan 1998
TL;DR: This correspondence presents an improved split-radix algo- rithm that can flexibly compute the discrete Hartley transforms (DHT) of length- where is an odd integer.
Abstract: This correspondence presents an improved split-radix algo- rithm that can flexibly compute the discrete Hartley transforms (DHT) of length- where is an odd integer. Comparisons with previously reported algorithms show that savings on the number of arithmetic operations can be made. Furthermore, a wider range of choices on different DHT lengths is naturally provided.
TL;DR: In this article, a class of Fourier-related functions such as the fractional Fourier transform, the Hartley and Hartley transform, Mellin and Mellin transform, and fractional Mellin Transform are implemented in the time domain.
04 Oct 1998
TL;DR: A discrete two-dimensional Fourier transform based on quaternion (or hypercomplex) numbers allows colour images to be transformed as a whole, rather than as colour-separated components.
Abstract: A discrete two-dimensional Fourier transform based on quaternion (or hypercomplex) numbers allows colour images to be transformed as a whole, rather than as colour-separated components. The transform is reviewed and its basis functions presented with example images.
Patent•
12 Jun 1998
TL;DR: In this article, the authors employ a texture filter in a graphics processor to perform a transform such as a Fast Fourier Transform (FFT), which can include an array of linear interpolators.
Abstract: The method and apparatus employ a texture filter in a graphics processor to perform a transform such as, for example, a Fast Fourier Transform. The texturizer can include an array of linear interpolators. The architecture reduces the computational complexity of the transform processes.
TL;DR: In this article, a spectral analysis of gravity anomalies due to slab like structures with linearly varying density using the Hartley transform, a real valued replacement for the well known complex Fourier transform which is conventionally used in such an analysis, is presented.
01 Jan 1998
TL;DR: This chapter begins with an introduction to basic Fourier principles and the notation used, and follows in succeeding chapters with specific applications in the various areas in biomedical engineering.
Abstract: We begin this chapter with an introduction to basic Fourier principles and the notation used, and follow in succeeding chapters with specific applications in the various areas in biomedical engineering. For the nonmathematical readers, we first introduce the basic concepts of sine and cosine waves, their representation in terms of complex numbers, and their role in Fourier transforms.
ENEA1
TL;DR: In this paper, the concept of the fractional Fourier transform is framed within the context of quantum evolution operators, which yields an extension of the above concept and greatly simplifies the underlying operational algebra.
Abstract: The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It t is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.
Patent•
AT&T1
TL;DR: An image compression scheme uses a reversible transform such as the Discrete Hartley Transform (DHT) to efficiently compress and expand image data for storage and retrieval of images in a digital format as mentioned in this paper.
Abstract: An image compression scheme uses a reversible transform, such as the Discrete Hartley Transform, to efficiently compress and expand image data for storage and retrieval of images in a digital format. The image data is divided into one or more image sets, each image set representing a rectangular array of pixel data from the image. Each image set is transformed using a reversible transform, such as the Hartley transform, into a set of coefficients which are then quantized and encoded using an entropy coder. The resultant coded data sets for each of the compressed image sets are then stored for subsequent expansion. Expansion of the stored data back into the image is essentially the reverse of the compression scheme.
TL;DR: In this paper, second-order recursive expressions for the DCT, DST, and DHT, intended for real-valued windowed sequences, are presented.
Abstract: Recursive formulations of the moving-window discrete Fourier transform (DFT) are well known. However, recursive versions of other useful discrete transforms, like the moving-window discrete cosine transform (DCT), discrete sine transform (DST), or discrete Hartley transform (DHT), have not been developed so far. In this paper, second-order recursive expressions for the DCT, DST, and DHT, intended for real-valued windowed sequences, are presented.
TL;DR: A novel interpretation of the 8-point discrete Hartley transform (DHT) as a new edge operator in the frequency domain is introduced and application of the 3 x 3 DHT masks to edge detection of a two-dimensional image is shown.
31 May 1998
TL;DR: In this article, a discrete fractional Hilbert transform (DFHT) was proposed for edge detection applications, which is a generalization of the Hilbert transform, and it presents a physical interpretation in the definition.
Abstract: The Hilbert transform plays an important role in signal processing. A generalization of the Hilbert transform, the fractional Hilbert transform, was recently proposed, and it presents a physical interpretation in the definition. In this paper, we develop the discrete fractional Hilbert transform, and apply the proposed transform to edge detection applications.
TL;DR: In this paper, the authors invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operator, α>−1/2.
Abstract: We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL
α, α>−1/2. Next, we use the connection between radial classical wavelets onR
n
and generalized wavelets associated with the Bessel operatorL(
n−2)/2 to derive new inversion formulas for the Radon transform onR
n
,n≥2.
12 May 1998
TL;DR: The results of general theory are used to derive the definitions of the fractional Fourier transform and fractional Hartley transform which satisfy the boundary conditions and additive property simultaneously.
Abstract: This paper is concerned with the definition of the continuous fractional Hartley transform. First, a general theory of the linear fractional transform is presented to provide a systematic procedure to define the fractional version of any well-known linear transforms. Then, the results of general theory are used to derive the definitions of the fractional Fourier transform (FRFT) and fractional Hartley transform (FRHT) which satisfy the boundary conditions and additive property simultaneously. Next, an important relationship between FRFT and FRHT is described. Finally, a numerical example is illustrated to demonstrate the transform results of the delta function of FRHT.
TL;DR: In this paper, the self-imaging phenomenon in a fractional Fourier transform optical system is described in the framework of self-fractional Fouriers functions and the main properties of these functions are investigated.
TL;DR: Several windowed fractional transforms are proposed and their characters are envisaged in detail, which can provide information of a signal in both the spatial and the fractional spectral domains simultaneously.
Patent•
15 Oct 1998
TL;DR: In this article, the problem of decoding an image signal that is compression-coded by adopting wavelet transform for the transform system with resolution of an optional rational number is solved. But the problem is not addressed in this paper.
Abstract: PROBLEM TO BE SOLVED: To decode an image signal that is compression-coded by adopting wavelet transform for the transform system with resolution of an optional rational number. SOLUTION: The decoder is provided with an entropy decoding section 1 that applies entropy decoding to a coded bit stream 100, an inverse quantization section 2 that applies inverse quantization to a quantization coefficient 101 to output a transform coefficient 102, a transform coefficient inverse scanning section 3 that scans the transform coefficient 102 by a prescribed method to rearrange the transform coefficients, and a wavelet inverse transform section 4 that applies inverse transform to the transform coefficient 108 to provide a decoded image 104. The wavelet inverse transform section 4 has a band limit means of the transform coefficient in response to a resolution transform magnification and configures an up-sampler, a down-sampler, and a composite filter adaptively according to the prescribed resolution transform magnification.
TL;DR: From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions as discussed by the authors, with mere...
Abstract: From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions. With mere ...
TL;DR: A simple technique for multichannel image addition and subtraction has been proposed using a basic joint-transform correlator architecture that can be modified to measure small relative displacement between objects, and to implement Hartley transform.
TL;DR: In this article, it was shown that if f is a polynomial of even degree, then the Fourier transform F(e−f )(ξ) can be estimated by e− f∗(Ξ) where f ∗(x) is the Legendre transform of f defined by supx(xξ − f(x)).
Abstract: We will prove that if f is a polynomial of even degree then the Fourier transform F(e−f )(ξ) can be estimated by e− f∗(ξ) where f∗(ξ) is the Legendre transform of f defined by f∗(ξ) = supx(xξ − f(x)). This result was previously proved by H. Kang [K] for a case of a convex polynomial which is a finite sum of monomials of even order with positive coefficients. Our result is the most general one for the polynomial f(x) since the convexity condition is not imposed and e−f(x) belongs to the space L1 if and only if f(x) is a polynomial of even degree with the coefficient of the highest degree a2m > 0. Also, we will make a more precise estimate of constants.