Showing papers on "Discrete sine transform published in 1999"

Efficient time series matching by wavelets

Abstract: Time series stored as feature vectors can be indexed by multidimensional index trees like R-Trees for fast retrieval. Due to the dimensionality curse problem, transformations are applied to time series to reduce the number of dimensions of the feature vectors. Different transformations like Discrete Fourier Transform (DFT) Discrete Wavelet Transform (DWT), Karhunen-Loeve (KL) transform or Singular Value Decomposition (SVD) can be applied. While the use of DFT and K-L transform or SVD have been studied on the literature, to our knowledge, there is no in-depth study on the application of DWT. In this paper we propose to use Haar Wavelet Transform for time series indexing. The major contributions are: (1) we show that Euclidean distance is preserved in the Haar transformed domain and no false dismissal will occur, (2) we show that Haar transform can outperform DFT through experiments, (3) a new similarity model is suggested to accommodate vertical shift of time series, and (4) a two-phase method is proposed for efficient n-nearest neighbor query in time series databases.

Discrete fractional Fourier transform based on orthogonal projections

Abstract: The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.

The discrete fractional Fourier transform

Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.

Discrete cosine transform based regularized high-resolution image reconstruction algorithm

Abstract: While high-resolution images are required for various applica- tions, aliased low-resolution images are only available due to the physi- cal limitations of sensors. We propose an algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique. The conventional approaches are based on the discrete Fourier transform (DFT) since the aliasing effect is easily analyzed in the frequency domain. However, the useful solution may not be available in many cases, i.e., the underdeter- mined cases or the insufficient subpixel information cases. To compen- sate for such ill-posedness, the generalized regularization is adopted in the spatial domain. Furthermore, the usage of the discrete cosine trans- form (DCT) instead of the DFT leads to a computationally efficient recon- struction algorithm. The validity of the proposed algorithm is both theo- retically and experimentally demonstrated. It is also shown that the artifact caused by inaccurate motion information is reduced by regular- ization. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)00508-5)

Motion field modeling and estimation using motion transformation

Abstract: This paper presents the concept of using a motion transform for finding the motion field between two images. A motion transform is a representation for modeling the motion field in the transform domain. Compared to other parametric motion models, e.g., affine, projective, etc., a motion transform offers a considerable advantage by its capability to model any motion field, including one with motion discontinuities. It also offers the flexibility of dynamically choosing the significant time-frequency components used to model the underlying motion. Simulation results on motion estimation using the DCT/DFT for motion modeling are presented. These results are comparable to the results from a wavelet-based approach.

Numerical solution of the identification problem for the attenuated Radon transform

Abstract: The attenuated Radon transform serves as a mathematical tool for single-photon emission computerized tomography (SPECT). The identification problem for the attenuated Radon transform is to find the attenuation coefficient, which is a parameter of the transform, from the values of the transform alone. Previous attempts to solve this problem used range theorems for the continuous attenuated/exponential Radon transform. We consider a matrix representation of the transform and formulate the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthogonal complement of the column space of the matrix. The singular value decomposition is applied to compute the orthogonal projector and its Frechet derivative. The numerical algorithm suggested is based on the Newton method with the Tikhonov regularization. Results of numerical experiments and inversion of the measured SPECT data are considered.

Fast DCT domain filtering using the DCT and the DST

Abstract: A method for efficient spatial domain filtering, directly in the discrete cosine transform (DCT) domain, is developed and proposed. It consists of using the discrete sine transform (DST) and the DCT for transform-domain processing on the in JPEG basis of the previously derived convolution-multiplication properties of discrete trigonometric transforms. The proposed scheme requires neither zero padding of the input data nor kernel symmetry. It is demonstrated that, in typical applications, the proposed algorithm is significantly more efficient than the conventional filtered spatial domain and earlier proposed DCT domain methods. The proposed method is applicable to any DCT-based image compression standard, such as JPEG, MPEG, and H.261.

Fast and exact 2D image reconstruction by means of Chebyshev decomposition and backprojection

Abstract: A new algorithm for the reconstruction of two-dimensional (2D) images from projections is described. The algorithm is based on the decomposition of the projections into Chebyshev polynomials of the second kind, which are the ideal basis functions for this application. The Chebyshev decomposition is done via the fast discrete sine transform. A discrete reconstruction filter is applied that corresponds to the ramp filter used in standard filtered backprojection (FBP) reconstruction. In contrast to FBP, the filter is applied to the Chebyshev coefficients and not to the Fourier coefficients of the projections. Then the reconstructed image is simply obtained by means of backprojection. Consequently, the method can be considered as a Chebyshev-domain filtered backprojection (CD-FBP). The total calculation time is dominated by the backprojection step only and is comparable to FBP. The merits of CD-FBP as compared with standard FBP are that: (a) The result is exact if the 2D function to be reconstructed can be decomposed into polynomials of finite degree, and if the sampling is adequate. Otherwise a polynomial approximation results. (b) The algorithm is inherently discrete. (c) It is particularly well suited for reconstructions from projections with non-equidistant samples that occur for instance in 2D PET (positron emission tomography) imaging and in a special form of fan beam scanning. Examples of applications comprise reconstructions of the Shepp and Logan head phantom in various sampling geometries, and a real PET test object. In the PET example an increased resolution is observed in comparison with standard FBP.

A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Lagrange multiplier method

Abstract: This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several 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.

A refined fast 2-D discrete cosine transform algorithm

Abstract: An index permutation-based fast two-dimensional discrete cosine transform (2-D DCT) algorithm is presented. It is shown that the N/spl times/N 2-D DCT, where N=2/sup m/, can be computed using only N 1-D DCTs and some post additions.

Direct transform to transform computation

Abstract: An efficient direct method for the computation of a length-N discrete cosine transform (DCT) given two adjacent length-(N/2) DCT coefficients, is presented. The computational complexity of the proposed method is lower than the traditional approach for lengths N>8. Savings of N memory locations and 2N data transfers are also achieved.

Discrete fractional Hadamard transform

Abstract: Hadamard transform is an important tool in discrete signal processing. In this paper, we define the discrete fractional Hadamard transform which is a generalized one. The development of discrete fractional Hadamard is based upon the same spirit as that of the discrete fractional Fourier transform.

Inverse discrete cosine transformer in an MPEG decoder

Abstract: An invention about inverse discrete cosine transformer of MPEG decoder is disclosed. By using the symmetry of N×N IDCT kernel matrix, the invention reduces the number of multipliers to N/4, the number of accumulators to N/2 in IDCT block without loss of decoding speed. This invention include memory parts, N/4 multipliers, M/2 accumulators and transposing means. Memory parts store absolute values of kernel matrix of inverse discrete cosine transform. N/4 multipliers receive elements of discrete cosine transform coefficient matrix or of transpose matrix of one-dimensional inverse discrete cosine transform coefficient matrix, as their multiplicand input, and elements of kernel matrix of inverse discrete cosine transform as their multiplier input. N/2 accumulators accumulate data outputted from multiplier. Transposing means transpose data outputted from accumulator and output one-dimensional inverse discrete cosine transform coefficient matrix or two-dimensional inverse discrete cosine transform coefficient matrix. The effects of this invention is the reduction of hardware size due to reducing the number of multipliers and accumulators. Also, in spite of the reduction of hardware size, this invention satisfies the resolution and operational speed requirements of MP@ML and CCIR 601 of MPEG2.

On fast algorithms for computing the inverse modified discrete cosine transform

Abstract: Two new fast algorithms for computing the inverse modified discrete cosine transform (IMDCT) as used in the adaptive spectral entropy coder (ASPEC) (Brandenburg et al. 1991) are proposed. A fixed-point error analysis is presented to determine the number of significant bits required for fixed-point implementations.

The Discrete Fourier Transform in Coding and Cryptography

Abstract: | Some applications of the Discrete Fourier Transform (DFT) in coding and in cryptography are described. The DFT over general commutative rings is introduced and the condition for its existence given. Blahut's Theorem, which relates the DFT to linear complexity, is shown to hold unchanged in general commutative rings. I. The (Usual) Discrete Fourier Transform Let be a primitive N th root of unity in a eld F , i.e., N = 1 but i 6= 1 for 1 i < N . The (usual) Discrete Fourier Transform (DFT) of length N generated by is the mapping DFT ( ) from F N to F de ned by B =DFT (b) in the manner

Matrix description of near-field diffraction and the fractional Fourier transform

Abstract: We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with N×N pixels, in an area d×d, it is necessary to evaluate the Fresnel diffraction pattern at distances z⩾d2/λN. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ϵ. If the phase matrix is evaluated at ϵ=1, we find the discrete Fourier transform matrix.

Implementation of the discrete cosine transform and its inverse by recursive structures

Abstract: This paper discusses the recursive implementation of the discrete cosine transform (DCT) and its inverse (IDCT). The transform is constructed by using recursive filter structure to generate the transform kernel values. We first derive two trigonometric equations, which can be represented as the Chebyshev polynomial. Then we demonstrate that general length of the DCT and IDCT can be efficiently implemented by using the regressive structure derived from the recursive formulae. The computational complexity of each data throughput in these architectures is less than that in the conventional ones by as many as 50%. The proposed architectures are regular and suitable for parallel VLSI implementation.

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

Methods and systems for performing short integer chen IDCT algorithm with fused multiply/add

Abstract: Methods and apparatus for performing a fast two-dimensional inverse discrete cosine transform (IDCT) in a media processor are disclosed. A processor receives discrete cosine transform data and combines, in a first stage, the discrete cosine transform data with a first set of constants. In a media processor with a partitioned SIMD architecture, the discrete cosine transform data and first set of constants may be combined, for example, by multiplying the at least one input component with a first set of constants using a complex multiplication instruction. The output is transposed in a second stage and combined with constants in a third stage to obtain the pixel information of an image.

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.

Computationally efficient methods for analysis and synthesis of real signals using FFT and IFFT

Abstract: The discrete Fourier transform (DFT) and the inverse discrete Fourier transform (IDFT) are used in a wide variety of signal processing applications. Even with the increased speed of modern processors, there is an ongoing need to further develop more efficient methods for computing DFT and IDFT, with a particular effort to reduce the number of complex multiplications. The properties of certain complex sequences are extraordinarily useful in the sense that they lead to data manipulation schemes that result in the sequences to which traditional but much shorter fast Fourier transform (FFT) algorithms may be applied. This is achieved by exploiting a certain regularity in the complex data. The index-reversed complex conjugate sequence and the mirror symmetric complex conjugate sequence were defined. A significant reduction in the number of complex computations is achieved if a sequence in either domain exhibits such symmetry.

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.

Unified Fully-Pipelined VLSI Implementations of the One-and Two-Dimensional Real Discrete Trigonometric Transforms

Abstract: This paper presents unified VLSI architectures which can efficiently realize some widespread one-dimensional (1D) and two-dimensional (2-D) real discrete trigonometric transforms, including the discrete Hartley transform (DHT), discrete sine transform (DST), and discrete cosine transform (DCT). First, succinct and unrestrictive Clenshaw’s recurrence formula along with the inherent symmetry of the trigonometric functions are adequately employed to render efficient recurrences for computing these 1-D RDTT. By utilizing an appropriate row-column decomposition approach, the same set of recurrences can also be used to compute both of the row transform and column transform of the 2-D RDTT. Array architectures, basing on the developed recurrences, are then introduced to implement these 1-D and 2-D RDTT. Both architectures provide substantial hardware savings as compared with previous works. In addition, they are not only applicable to the 1-D and 2-D RDTT of arbitrary size, but they can also be easily adapted to compute all aforementioned RDTT with only minor modifications. A complete set of input/output (I/O) buffers along with a bidirectional circular shift matrix are addressed as well to enable the architectures to operate in a fullypipelined manner and to rectify the transformed results in a natural order. Moreover, the resulting architectures are both highly regular, modular, and locally-connected, thus being amenable to VLSI implementations. key words: discrete trigonometric transforms, VLSI array processors, Clenshaw’s recurrence, pipelining, digital signal processing

A smart method makes DFT more precise for power system frequency estimation

Abstract: A precise digital algorithm based on discrete Fourier transforms (DFT) to estimate the frequency of a sinusoid with harmonics in real-time is proposed. This algorithm that we called the smart discrete Fourier transform (SDFT) smartly avoids the errors that arise when frequency deviates from the fundamental frequency, and keeps all the advantages of the DFT, e.g., immune to harmonics of fundamental frequency, easily obtaining the parameters of amplitude and phase, and even the recursive computing can be used in SDFT. These make the SDFT more accurate than conventional DFT based techniques. In addition, this method is recursive and very easy to implement, so it is very suitable for use in real-time. We offer the simulation results compared with conventional DFT method and second-order Prony method to validate the claimed benefits of SDFT.

New fast algorithms of multidimensional Fourier and Radon discrete transforms

Abstract: This paper describes a fast new n-D discrete Radon transform (DRT) and a fast exact inversion algorithm for it, without interpolating from polar to Cartesian coordinates of using the backprojection operator. The new approach is based on the fast Nussbaumer's (1982) polynomial transform (NPT).