Showing papers on "Discrete Hartley transform published in 1998"

Discrete fractional Hartley and Fourier transforms

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.

Efficient moving-window DFT algorithms

Abstract: The authors deal with the real-valued, moving-window discrete Fourier transform. After reviewing the basic recursive versions appearing in the literature, additional recursive equations are presented. Then, these equations are combined so that nonrecursive expressions involving only consecutive discrete Fourier transform (DFT) sine components are obtained for both the DFT cosine component and squared harmonic amplitude. The computational complexity of this new scheme is finally studied and compared to that of existing methods, showing that, in most practical situations, a reduction in the operation count is achieved.

The quick Fourier transform: an FFT based on symmetries

Abstract: This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm called the quick Fourier transform (QFT) that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. By further application of the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(NlogN) algorithm, with computational complexities comparable to the Cooley-Tukey algorithm. We show that the power-of-two QFT can be implemented in terms of discrete sine and cosine transforms. The algorithm can be easily modified to compute the DFT with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real.

Analysis of linear time-varying systems via Hartley series

Abstract: A method for finding approximated solutions of linear time-varying systems via Hartley series is proposed. The most relevant properties of the Hartley series are presented first. The operational matrix of integration together with the product and coefficient matrices are developed. They are used to transform dynamical equations of linear, time varying systems to a set of linear algebraic equations. Practical applications relating to the operation of electric networks are given.

The problem of defining the Fourier transform of a colour image

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.

Method and apparatus for efficient computation of discrete fourier transform (dft) and inverse discrete fourier transform (idft)

Abstract: The present invention significantly reduces the number of complex computations that must be performed in computing the discrete Fourier transform (DFT) and inverse DFT (IDFT) operations. In particular, the DFT and IDFT operations are computed using the same computing device. The computation operations are substantially identical for both operations with the exception that for the IDFT operation, the data are complex conjugated before and after processing. Using the same computing device/operations, both DFT and IDFT computations are optimized for maximum efficiency. A common transform process is selectively connected to first and second data processing paths. A DFT operation is performed on an N-point sequence on the first data processing path, and an IDFT operation is performed on an N-point sequence on the second data processing path using the same N-point fast Fourier transform (FFT).

Method and apparatus for fast image compression

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.

Recursive formulation of short-time discrete trigonometric transforms

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.

Discrete fractional Hilbert transform

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.

A new definition of continuous fractional Hartley transform

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.

Efficiency of analytical transforms for image compression

Abstract: Digital image transforms, for decorrelating image pixels, have received wide spread interest in the literature. Analytical transforms play a great role in the gray level decorrelation and energy compaction of the image and hardly affect the performance of the image compression technique. This paper provides a comparison of the bit rate reduction capability and signal to noise ratio among five transforms; namely, Karhunen-Loeve transform (KLT), discrete cosine transform (DCT), discrete Hartley transform (DHT), discrete Gabor transform (DGT), and the discrete wavelet transform (DWT), where they have shown the most promise in image compression coding systems. The coefficients of the transform are truncated with a specified threshold and the bit rate is computed after Huffman coding. The image is reconstructed from the truncated version of the coefficient matrix. Peak signal to noise ratio and compression ratio are considered to evaluate the efficiency of the analytical transform. The error image between the original and the reconstructed image is computed to follow the error distribution over the image.

The Hartley transform in a finite field

Abstract: In this paper the k-trigonometric functions over the Galois field GF(q) are introduced 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. The main properties of this new discrete transform are presented and areas for possible applications are mentioned, including digital signal processing and digital multiplexing.

Computation of 2D discrete Hartley transform

Abstract: Based on a decimation-in-time decomposition, a fast split-radix algorithm for the 2D discrete Hartley transform is presented. Compared to other reported algorithms, the proposed algorithm achieves substantial savings on the number of operations and provides a wider choice of transform sizes.

A New Fast Discrete Fourier Transform

Abstract: This paper presents a new fast Discrete Fourier Transform (DFT) algorithm. By rewriting the DFT, a new algorithm is obtained that uses 2n?2(3n?13)+4n?2 real multiplications and 2n?2(7n?29)+6n+2 real additions for a real data N=2n point DFT, comparable to the number of operations in the Split-Radix method, but with slightly fewer multiply and add operations in total. Because of the organization of multiplications as plane rotations in this DFT algorithm, it is possible to apply a pipelined CORDIC algorithm in a hardware implementation of a long-point DFT, e.g., at a 100 MHz input rate, a 1024-point transform can be realized with a 200 MHz clocking of a single CORDIC pipeline.

Hierarchical finite state vector quantization for MRI and CT image compression

Abstract: Magnetic resonance imaging (MRI) and computer tomography (CT) are very important techniques that are used in disease diagnosis in medicine. In an average sized hospital, many tera-bytes of digital imaging data (MRI and CT) are generated every year, almost all of which has to be kept and archived. The compression of medical images is currently performed by using different algorithms. The most common compression technique is vector quantization. Interframe coding, the discrete Hartley transform, mixed transform, entropy-coded DPCM and JPEG algorithm are also used. Hierarchical finite-state vector quantization (HFSVQ), which is the improved version of vector quantization, has been introduced as a compression technique with high compression ratios for video images. Although the HFSVQ algorithm is the most efficient compression technique according to its compression ratios, we show that it has also a good performance for brain tomography and magnetic resonance images.

A new algorithm for the 2-D discrete cosine transform

Abstract: A simple relation between the two-dimensional discrete cosine transform (2D DCT) and two-dimensional discrete W transform (2D DWT) is found for multidimensional signal processing. Using the relation, the 2D type-IV DCT (DCT-IV) can be mapped to a number of one-dimensional type-III DWT (DWT-III) by Ma's (1989) real-valued FFT algorithm and the Wang (1992) mapping. Thus, a fast algorithm for 2D DCT-IV based on 1D DWT-III can be obtained. Computation efficiency of the proposed algorithm is higher than the existing algorithms for 2D DCT-IV.

Computationally efficient inverse discrete cosine transform method and apparatus

Abstract: A method and apparatus for efficiently computing an Inverse Discrete Cosine Transform (IDCT).

Discrete Fourier Transform

Abstract: This chapter discusses the Fourier representations for finite-length time sequences, referred to as the discrete Fourier transform (DFT), which is not a continuous-frequency function but a discrete-frequency sequence obtained by sampling the discrete-time Fourier transform of the sequence with an equally spaced frequency interval. It shows some relationships between DFT and the Fourier series representation of periodic sequences, some properties of DFT, efficient algorithms for computing DFT, and applications of DFT to the convolution computations.. The chapter also shows the properties of the DFT, including the circular shift and the circular convolution properties. The chapter introduces an efficient algorithm for computing DFTs, which is called the fast Fourier transform (FFT).. It also proves that the computing time required for linear convolutions in the frequency domain is less than that which is necessary for the direct computation of the linear convolution in the time domain. Finally, the chapter discusses practical methods for computing linear convolutions where an indefinitely long sequence is convolved with a finite-length sequence.

Discrete transforms with Gaussian periods of cyclotomic fields as basis for set functions

Abstract: In the paper algebraic foundations of fast algorithms like Rader's for discrete orthogonal transforms are analyzed. It is shown that the connection between discrete Fourier transform of length p (p is a prime) and cyclic convolution of length (p - 1) is defined by cyclic structure of Galois group of some cyclotomic field. A class of discrete orthogonal transforms with fast algorithms like Rader- Vinograd is introduced.

Fast edge-preserving image interpolation algorithm with an arbitrary aspect ratio

Abstract: Conventional interpolation techniques, including pixel replication, bilinear interpolation and spline based methods have been popularly used in commercial applications. It is often desired that the interpolation process is able to accurately and fastly increase the spatial resolution of an image at an arbitrary aspect ratio and with sharp edges. These do not generally happen to conventional algorithms, which will consume a great effort to interpolate an image at an arbitrary ratio and tend to blur edges or introduce blocking artifacts. In this correspondence, we propose an interpolation system that is more general than the existing ones. The method is based on the fact that the Hartley transform (HT) at an arbitrary frequency can be expressed as a weighted sum of its Discrete Hartley transform (DHT) coefficients. These weights can be suitably approximated so that the HT is very nearly the sum of (1) a few dominant terms of the sum of the DHT coefficients, and (2) the DHT of a new sequence obtained by multiplying the original sequence with a saw-tooth function. If we take the inverse discrete Hartley transform (IDHT) of an image; then by using the algorithm described above, the spatial sample at an arbitrary location can be fastly computed by the fast Hartley transform (FHT) algorithm. In addition to retaining the computational efficiency of the FHT algorithm, experimental results have revealed that the proposed system preserves the sharp edge of the original image.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.