Showing papers on "Discrete sine transform published in 2006"

Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations

Abstract: Preface Acknowledgements List of Acronyms 1. Discrete Cosine and Sine Transforms 2. Definitions and General Properties 3. The Karhunen-Loeve Transform and Optimal Decorrelation 4. Fast DCT/DST Algorithms 5. Integer Discrete Cosine/Sine Transforms Appendices Index

CHAPTER 1 – Discrete Cosine and Sine Transforms

Abstract: This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

The multiple-parameter discrete fractional Fourier transform

Abstract: The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.

Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices

Abstract: Based on discrete Hermite-Gaussian-like functions, a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix, which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite-Gaussian functions (HGFs). Then, new versions of DFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper

Discrete Radon transform has an exact, fast inverse and generalizes to operations other than sums along lines

Abstract: Gotz, Druckmuller, and, independently, Brady have defined a discrete Radon transform (DRT) that sums an image9s pixel values along a set of aptly chosen discrete lines, complete in slope and intercept. The transform is fast, O ( N 2 log N ) for an N × N image; it uses only addition, not multiplication or interpolation, and it admits a fast, exact algorithm for the adjoint operation, namely backprojection. This paper shows that the transform additionally has a fast, exact (although iterative) inverse. The inverse reproduces to machine accuracy the pixel-by-pixel values of the original image from its DRT, without artifacts or a finite point-spread function. Fourier or fast Fourier transform methods are not used. The inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise. Also introduced are generalizations of the DRT that combine pixel values along lines by operations other than addition. For example, there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal-to-noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude.

Fast 2-Dimensional 4 $times$ 4 Forward Integer Transform Implementation for H.264/AVC

Abstract: In this paper, the novel two-dimensional (2-D) fast algorithm for realization of 4 $times$ 4 forward integer transform in H.264 is proposed. Based on matrix operations with Kronecker product and direct sum, the efficient fast 2-D 4 $times$ 4 forward integer transform can be derived from the proposed one-dimensional fast 4 $times$ 4 forward integer transform through matrix decompositions. The proposed fast 2-D 4 $times$ 4 forward integer transform design doesn't need transpose memory for direct parallel pipelined architecture. The fast 2-D 4 $times$ 4 forward integer transform requires fewer latency delays than the state-of-the-art methods. With regular modularity, the proposed fast algorithm is suitable for VLSI implementation to achieve real-time H.264/advanced video coding (AVC) signal processing.

Why is the Linear Canonical Transform so little known

Abstract: The linear canonical transform (LCT), is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms and operators in engineering and physics such as the Fourier transform, fractional Fourier transform (FRFT), Fresnel transform (FRST), time scaling, chirping, and others. Therefore the LCT provides a unified framework for studying the behavior of many practical transforms and system responses in optics and engineering in general. From the system‐engineering point of view the LCT provides a powerful tool for design and analysis of the characteristics of optical systems. Despite this fact only few authors take advantage of the powerful and general LCT theory for analysis and design of optical systems. In this paper we review some important properties about the continuous LCT and we present some new results regarding the discretization and computation of the LCT.

An extraction method of environmental surface profile using planar end-effectors

Abstract: It is very important for robots working in unknown environment to recognize surface profile of the environment. This paper presents an extraction method of environmental surface profile based on environmental modes. Contact condition between a planar end-effector and the environment is also determined by using active motion which is named "groping motion" of the end-effector. The extraction method utilizes discrete Fourier transform (DFT) matrices as matrices transforming into environmental modes. The proposed method can be applied to a planar end-effector whose shape is an arbitrary polygon. Moreover, a compliance controller for attitude of a planar end-effector with 3 supporting points is proposed to realize stable contact motion with unknown environment. The validity of the proposed method is shown by the experimental results

Linear Canonical Transform and Its Implication

Abstract: As an emerging tool for signal processing,the linear canonical transform(LCT) proves itself to be more general and flexible than the Fourier transform as well as the fractional Fourier transform.So it can slove problems that can't be dealt with well by the latter.In this paper,the definition of LCT and some special cases are given at first,followed by its properties as listed.Besides,the discrete linear canonical transform is introduced.The implication of LCT is illustrated finally,displaying(LCT's) potentials and capabilities in the field of signal processing.

Unified Systolic-Like Architecture for DCT and DST Using Distributed Arithmetic

Abstract: A common computing-core representation of the discrete cosine transform and discrete sine transform is derived and a reduced-complexity algorithm is developed for computation of the proposed computing-core. A parallel architecture based on the principle of distributed arithmetic is designed further for the computation of these transforms using the common-core algorithm. The proposed scheme not only leads to a systolic-like regular and modular hardware for computing these transforms, but also offers significant improvement in area-time efficiency over the existing structures. The structure proposed here is devoid of complicated input/output mapping and does not involve any complex control. Unlike the convolution-based structures, it does not restrict the transform length to be a prime or multiple of prime and can be utilized as a reusable core for cost-effective, memory-efficient, high-throughput implementation of either of these transforms

On OFDM and single-carrier frequency-domain systems based on trigonometric transforms

Abstract: Orthogonal-frequency-division-multiplex (OFDM) and single-carrier (SC) frequency domain structures are simple equalization schemes that make use of the discrete Fourier transform (DFT) diagonalization properties. This letter approaches these problems within a well-known matrix algebra that allows for straightforward extensions of recent results on real trigonometric transforms, namely, the discrete cosine (DCT), sine (DST), as well as new Hartley transform (DHT) schemes. They are especially useful for real modulations and can outperform the corresponding DFT-based schemes when the channel memory is smaller than the introduced redundancy. We further look into efficient DHT schemes yielding the same throughput as the DFT counterparts and provide simulations assuming perfect channel state information (CSI)

A method for computation of the discrete Fourier transform over a finite field

Abstract: The discrete Fourier transform over a finite field finds applications in algebraic coding theory. The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circulant matrix.

Towards an Inverse Constant Q Transform

Abstract: The Constant Q transform has found use in the analysis of musical signals due to its logarithmic frequency resolution. Unfortunately, a considerable drawback of the Constant Q transform is that there is no inverse transform. Here we show it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier Transform domain. This inverse is obtained through the use of `0 and `1 minimisation approaches to project the signal from the constant Q domain back to the Discrete Fourier Transform domain. Once the signal has been projected back to the Discrete Fourier Transform domain, the signal can be recovered by performing an inverse Discrete Fourier Transform. 1. THE CONSTANT Q TRANSFORM The Constant Q transform (CQT) was derived by Brown as a means of creating a log-frequency resolution spectrogram [1]. This has considerable advantages for the analysis of musical signals, as the frequency resolution can be set to match that of the equal tempered scale used in western music, where the frequencies are geometrically spaced, as opposed to the linear spacing that occurs in the discrete Fourier transform (DFT). The frequency components of the CQT have a constant ratio of center frequency to resolution, as opposed to the constant frequency difference and constant resolution of the DFT. This constant ratio results in a constant pattern for the spectral components making up notes played on a given instrument, and this has been used to attempt sound source separation of pitched instruments from both single channel and multi-channel mixtures of instruments[2],[3]. Given an inital minimum frequency f0 for the CQT, the center frequencies for each band can be obtained from: fk = f02 k b (k = 0, 1, ...) (1) where b is the number of bins per octave. The fixed ratio of center frequency to bandwidth is then given by Q = ( 2 1 b − 1 )−1 (2) The desired bandwidth of each frequency band is FitzGerald et al. Towards an ICQT then obtained by choosing a window of length

Modified Discrete Radon Transforms and Their Application to Rotation-Invariant Image Analysis

Abstract: This paper presents two novel transforms based on the discrete Radon transform. The proposed transforms smartly solve two inherent problems of the Radon transform in rotation estimation in digital images, i.e., direction-dependency and nonhomogeneity, that come from the different numbers of pixels projected on a line for different directions and/or coordinates of a direction. While the first transform considers the sample mean operator on the same sets of pixels for a direction instead of summation in the discrete Radon transform, the second transform uses the mean operator on sets of pixels with the equal number of elements. In order to show the efficiency of the proposed transforms, we apply them on image collections from the Brodatz album for estimating the directional information. Experimental results show a significant increase in correct estimation as well as in the processing time compared to the conventional Radon transform.

A new design of carrier interferometry OFDM with FFT as spreading codes

Abstract: In this paper, we propose a new design of efficient carrier interferometry orthogonal frequency division multiplexing (CI/OFDM) by using fast Fourier transform (FFT) as spreading codes. First, we propose the use of both positive and negative frequency of the IFFT rather than only using the positive part as in conventional method. It results new CI/OFDM waveforms which only requires a half of FFT points. Second, we replace CI spreading with FFT for achieving lower complexity because complexity of CI is similar to that of discrete Fourier transform (DFT). Our results confirm that the proposed design is capable of achieving high efficiency and low complexity while providing high performance.

Tempered distributional Fourier sine (cosine) transform

Abstract: Certain distributional and tempered distributional Fourier sine (cosine) transformations are discussed on certain testing function spaces.

Discrete and continuous sine transform generalized to semisimple Lie groups of rank two

Abstract: A new continuous group transform, together with its discretization over a lattice of any density and admissible symmetry, is defined and described for the four compact semisimple Lie groups of rank 2. In the case of rank 1, the discrete version is the sine transform. Properties of the expansion functions of the transform, called S-functions, are studied. Digital data processing is our motivating application.

On the generalized convolution with a weight function for the Fourier sine and cosine transforms

Abstract: The generalized convolution with a weight function for the Fourier sine and cosine transforms is introduced. Its properties and applications to solving system of integral equations are considered.

A full-wave analysis of a coaxial waveguide slot bridge using the Fourier transform technique

Abstract: Electromagnetic wave scattering from a coaxial waveguide slot bridge is theoretically investigated. The Fourier transform/series technique is utilized to constitute a set of simultaneous equations for the discrete modal coefficients. The residue calculus is applied to transform the scattered-field integral representations into fast-converging series forms. Numerical computations illustrate the behavior of scattering in terms of the slot geometry, the incident mode, and the operating frequency.

Noisy Speech Enhancement Based on Discrete Sine Transform

Abstract: When the speech signal and the random noise are transformed with discrete sine transform (DST), the discrete sine transform coefficients at the low frequency regions are predominantly speech, and at the high frequency regions are predominantly noise. According to this, a new speech enhancement method based discrete sine transform is proposed in this paper. The level of the trace of the signal present in the high frequency region of DST coefficients is estimated to obtain the threshold parameter, and after the signal-to-noise ratio estimation and speech present probability is calculated, an improved method is adopted. The experiment results show that it is an effective speech enhanced method.

3-D Discrete Analytical Ridgelet Transform

Abstract: In this paper, we propose an implementation of the 3-D Ridgelet transform: the 3-D discrete analytical Ridgelet transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and color video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient

On the DCT Coefficient Distributions

Abstract: Explicit expressions are derived for the actual discrete cosine transform (DCT) coefficient distribution. Associated estimation procedures are also derived for ready use in practical applications

Robust DFT with high breakdown point for complex-valued impulse noise environment

Abstract: Modification of the robust discrete Fourier transform (DFT) is proposed in order to achieve a high breakdown point for signals corrupted by complex-valued impulse noise with independent real and imaginary parts. Obtained results are compared with existing robust DFT forms. In addition, an adaptive procedure for selection of the modified robust DFT form is developed.

Continuous extension of discrete transform for data processing

Abstract: A method for interpolating images using continuous extension functions of inverse discrete transformation is described. The method comprises defining at least one block of an image dataset and calculating and storing forward discrete orbit function transform coefficients representing the at least one block. The method involves defining a first and second different spatial resolution and calculating a first processed image dataset using at least one continuous extension of discrete orbit function transform representing said at least one block using the stored discrete orbit function transform coefficients and the first different spatial resolution. A second processed image dataset is calculated using at least one continuous extension of discrete orbit function transform representing said at least one block using the stored discrete orbit function transform coefficients and the second different spatial resolution.

multivariate fourier transform methods over simplex and super-simplex domains

Abstract: In this paper we propose the well-known Fourier method on some non-tensor product domains in R d , including simplex and so-called super-simplex which consists of (d + 1)! simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallel hexagon and a parallel quadrilateral dodecahedron, respectively. We have extended most of concepts and results of the traditional Fourier methods on multivariate cases, such as Fourier basis system, Fourier series, discrete Fourier transform (DFT) and its fast algorithm (FFT) on the super-simplex, as well as generalized sine and cosine transforms (DST, DCT) and related fast algorithms over a simplex. The relationship between the basic orthogonal system and eigen-functions of a Laplacian-like operator over these domains is explored.

A New Method for Spectral Analysis of the Potential Field and Conversion of Derivatives of Gravity Anomalies: Cosine Transform

Abstract: In order to improve the accuracy of derivative conversion of gravity anomalies and reflect anomaly characteristics of geologic bodies effectively, we propose a new method of calculating anomaly derivatives using cosine transform. Two theorems are put forward and proved and the common expression of the cosine transform spectrum of the gravity potential field and formula to calculate derivatives of gravity anomalies are deduced. So a theory of cosine-transform-spectra of the potential-field is established. In model experiments, we find that the deviation of the first derivative calculated by Fourier transform is very large comparing with the theoretical derivative, but the fitting effect of the anomaly derivative calculated by cosine transform is excellent. The calculation accuracy of data is all very high except that errors of several data on the boundary are large because of the residual Gibbus effect induced by finite truncation of gravity anomalies. Errors are –0.09%~5%.