DOCSLIB.ORG

The Implementation of Multidimensional Discrete Transforms for Digital Signal Processing Hong Ren Wu University of Wollongong

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Implementation of Multidimensional Discrete Transforms for Digital Signal Processing Hong Ren Wu University of Wollongong University of Wollongong Research Online University of Wollongong Thesis Collection University of Wollongong Thesis Collections 1990 The implementation of multidimensional discrete transforms for digital signal processing Hong Ren Wu University of Wollongong Recommended Citation Wu, Hong Ren, The implementation of multidimensional discrete transforms for digital signal processing, Doctor of Philosophy thesis, Department of Electrical and Computer Engineering, University of Wollongong, 1990. http://ro.uow.edu.au/theses/1353 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] THE IMPLEMENTATION OF MULTIDIMENSIONAL DISCRETE TRANSFORMS FOR DIGITAL SIGNAL PROCESSING A thesis submitted in fulfilment of the requirements for the award of the degree DOCTOR OF PHILOSOPHY from THE UNIVERSITY OF WOLLONGONG by WU, HONG REN, B.E., M.E. THE DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. FEBRUARY 1990. "Entertaining someone with fish, uou could only serve him once,', but if you teach him the, art of fishing, it vt>iCC serve fvim for a Cije time,." —cAn ancient Chinese wise man and, philosopher. 1 CONTENTS ACKNOWLEDGEMENTS vi ABSTRACT viii LIST OF ACRONYMS AND SYMBOLS x CHAPTER ONE: INTRODUCTION 1 1 -1 Introduction to Multidimensional Digital Signal Processing 1 1-2 Applications 2 1-3 History and New Achievements in Fast Signal Processing Algorithms 3 1-4 Objectives 6 1-5 Thesis Review and Contributions 8 1-6 Publications, Submitted Papers and Internal Technical Reports 11 PART I. MULTIDIMENSIONAL DISCRETE FOURIER TRANSFORMS 14 CHAPTER TWO: 1-D DISCRETE FOURIER TRANSFORM AND FAST FOURIER TRANSFORM ALGORITHMS 15 2-1 Definitions 15 2-2 Matrix Representations for 1-D Cooley-Tukey FFT Algorithms 16 2-3 Computational Considerations 23 2-4 Summary 31 ii CHAPTER THREE: 2-D DFT AND 2-D FFT ALGORITHMS 32 3-1 Introduction to 2-D Discrete Fourier Transforms 32 3-2 Definitions 37 3-3 Row-Column FFT Algorithms 38 3-4 Vector Radix FFT Algorithms 39 3-5 Matrix Representations for 2-D Vector Radix FFT Algorithms 43 3-6 Structure Theorems 46 3-7 Structural Approach via Logic Diagrams 55 3-8 2-D Vector Split-Radix FFT Algorithms 63 3-9 Comparisons of Various 2-D Vector Radix FFT Algorithms 67 3-10 Vector Radix FFT Using FDP™ A41102 69 3-11 Summary 72 CHAPTER FOUR: A PERSPECTIVE ON VECTOR RADLX FFT ALGORITHMS OF HIGHER DIMENSIONS 72 4-1 Definitions 74 4-2 Matrix Representations and Structure Theorems 75 4-3 Diagrammatical Presentations 78 4-4 Computing Power Limitations 84 Ill PART II. MULTIDIMENSIONAL DISCRETE COSINE TRANSFORMS 87 CHAPTER FIVE: INTRODUCTION TO MULTI­ DIMENSIONAL DISCRETE COSINE TRANSFORMS 88 5-1 Definitions of 1 -D DCT and Its Inverse DCT 91 5-2 Definitions of 2-D DCT and Its Inverse DCT 93 5-3 Applications of 2-D DCTs in Image Compression 95 5-4 2-D Indirect Fast DCT Algorithms 100 CHAPTER SIX: 2-D DIRECT FAST DCT ALGORITHMS 103 6-1 2-D Direct Fast DCT Algorithm Based on Lee's Method 103 6-1-1 1-D Lee's algorithm in matrix form 103 6-1-2 Derivation of 2-D fast DCT algorithm from Lee's algorithm 108 6-2 2-D Direct Fast DCT Algorithm Based on Hou's Method 114 6-2-1 1-D Hou's algorithm in matrix form 114 6-2-2 Derivation of 2-D fast DCT algorithm from Hou's algorithm 118 6-3 Comparison of Arithmetic Complexity of Various DCT Algorithms 124 6-4 Comparison of Computation Structures of 2-D Direct VR DCTs and VR FFTs 125 6-5 Summary 126 IV CHAPTER SEVEN: HARDWARE IMPLEMENTATION OF 2-D DCTS FOR REAL-TIME IMAGE CODING SYSTEMS 128 7_ l Description of Hardware Implementation of Modified 2-D Makhoul DCT Algorithm Using FDP™ A41102 129 7-2 Discussion of 2-D DCT Image Coding Systems Using VLSI Digital Signal Processors 132 CHAPTER EIGHT: THE EFFECTS OF FINITE-WORD- LENGTH COMPUTATION FOR FAST DCT ALGORITHMS 136 8-1 Introduction 136 8-2 Simulation Design 138 8-2-1 Structure of the simulation program 138 8-2-2 Error model for the basic computation structure 140 8-2-3 DCT in infinite-word-length 141 8-2-4 Data collection 141 8-3 Simulation Results 143 8-3-1 Floating-point computation of 1-D DCTs 143 8-3-2 Floating-point computation of 2-D DCTs 147 8-4 Summary 151 CHAPTER NINE: CONCLUSIONS 152 9-1 Conclusions 152 9-2 Suggestions for Future Research 154 BIBLIOGRAPHY 156 APPENDIX A: PRELIMINARY BACKGROUND ON THE TENSOR (KRONECKER) PRODUCT AND THE LOGIC DIAGRAM 172 APPENDIX B PROOF OF STRUCTURE THEOREMS 175 APPENDIX C THE COMBINED FACTOR METHOD 176 APPENDIX D DERIVATION OF VECTOR RADIX 2-D FAST DCT BASED ON LEE'S ALGORITHM 181 APPENDIX E: ARITHMETIC COMPLEXITY OF THE VECTOR SPLIT-RADIX DIF FFT ALGORITHM 187 vi ACKNOWLEDGEMENTS The author wishes to express his deepest appreciation to his Supervisor, Dr. FJ. Paoloni, Associate Professor of the Department of Electrical and Computer Engineering, The University of Wollongong, for his guidance, support and encouragement and also for his understanding and confidence in the author throughout this research. His professional and optimistic attitude towards the research have made this research challenging, interesting, productive and enjoyable. The author wishes to thank Professor Huang, Ruji, of the Department of Industrial Automation, University of Science and Technology, Beijing (formerly Beijing University of Iron and Steel Technology), who, as his Masters' Supervisor, had a great influence on shaping the author's research skills and abilities as an independent as well as cooperative researcher. Sincere thanks are also extended to Professor B.H. Smith who introduced the author to this Institution and made this study possible in the first place. The author wishes to thank the following people for their generous help, patience and useful discussions at various stages of this program: Dr. G.W. Trott and Dr. T.S. Ng, Department of Electrical and Computer Engineering; Mr. I.C. Piper, Computer Services; Mr. J.K. Giblin, formerly with Computer Services and now with Network Technical Services, B.H.P. Steel International Group; Mr. G. Andersson, Computer Services; Dr. N. Smyth and Dr, K.G. Russell, Department of Mathematics, The University of Wollongong; Professor J.H. McClellan, School of Electrical Engineering, Georgia Institute of Technology, formerly with Schlumberger Well Services; Dr. J.D. O'Sullivan, Dr. DJ. McLean, Dr. C.E. Jacka and Mr. K.T. Hwa, Division of Radio Physics, CSIRO in Epping, New South Wales; Dr. MJ. Biggar and Dr. W.B.S. Tan, Telecom Research Laboratories (Australia); Professor K.R. Rao, Department of Electrical Engineering, The University of Texas at Arlington; Professor M. Vetterli, Department of Electrical Engineering, Columbia University; Mr. P. Single, Austek Microsystems (Australia); Dr. M.A. Magdy, Mr. J.F. Chicharo and Mrs. C. Quinn, Department of vii Electrical and Computer Engineering, The University of Wollongong; Mr. P.J. Costigan and all technical staff in the Department. The author is deeply grateful to his friend and English teacher Mrs. B. S. Perry for her generous help and professional assistance in the author's understanding of English and Australian culture, and her and her husband's, Mr. E.J.W. Perry, understanding, friendship and encouragement which made the author's stay in Wollongong worthwhile, most pleasant and enjoyable. The assistance from Miss. M.J. Fryer, of the Department of Electrical and Computer Engineering, The University of Wollongong, throughout this research and particularly in reading the final manuscript of this thesis is warmly appreciated. Financial support received from the Department of Electrical and Computer Engineering and the Committee of Post-Graduate Study, The University of Wollongong, by means of the Departmental Teaching Fellowship and the Post-Graduate Research Scholarship respectively, which made this research possible, is sincerely acknowledged. Financial supports from Australian Telecommunication and Electronics Research Board and from Telecom Research Laboratories (Australia) through R&D contract No.7066 are also acknowledged. Finally, the author wishes to express his deepest gratitude to Mei Mei, his best friend, colleague and wife, without whose patience, understanding, appreciation and continuous support, encouragement and inspiration, this work would not have been accomplished. The continuous support and understanding from his parents, from whom he has been separated for the cause is also greatly appreciated. vm ABSTRACT A structural approach to the construction of multidimensional vector radix fast Discrete Fourier Transforms (DFTs) and fast direct Discrete Cosine Transforms (DCTs) is presented in this thesis. The approach features the use of matrix representation of one- dimensional (1-D) and two-dimensional (2-D) FFT and fast DCT algorithms along with the tensor product, and the use of logic diagram and rules for modifications. In the first pan of the thesis, the structural approach is applied to construct 2-D Decimation-In-Time (DIT), Decimation-In-Frequency (DIF) and mixed (DIT & DIF) vector radix FFT algorithms from corresponding 1-D FFT algorithms by the Cooley- Tukey method. The results are summarized in theorems as well as examples using logic diagrams. It has been shown that the logic diagram (or signal flow graph) as well as being a form of representation and interpretation of fast algorithm equations, is a stand­ alone engineering tool for the construction of fast algorithms. The concept of "vector signal processing" is adapted into the logic diagram representation which reveals the structural features of multidimensional vector radix FFTs and explains the relationships and differences between the row-column FFT, the vector radix FFT reported previously and the approach presented in this thesis. The introduction of the structural approach makes the formulation of a multidimensional vector radix FFT algorithm of high radix and dimension easy to evaluate and implement by both software and hardware.
Load more

Recommended publications
  • 7 Image Processing: Discrete Images
    7 Image Processing: Discrete Images In the previous chapter we explored linear, shift-invariant systems in the continuous two-dimensional domain. In practice, we deal with images that are both limited in extent and sampled at discrete points. The results developed so far have to be specialized, extended, and modified to be useful in this domain. Also, a few new aspects appear that must be treated carefully. The sampling theorem tells us under what circumstances a discrete set of samples can accurately represent a continuous image. We also learn what happens when the conditions for the application of this result are not met. This has significant implications for the design of imaging systems. Methods requiring transformation to the frequency domain have be- come popular, in part because of algorithms that permit the rapid compu- tation of the discrete Fourier transform. Care has to be taken, however, since these methods assume that the signal is periodic. We discuss how this requirement can be met and what happens when the assumption does not apply. 7.1 Finite Image Size In practice, images are always of finite size. Consider a rectangular image 7.1 Finite Image Size 145 of width W and height H. Then the integrals in the Fourier transform no longer need to be taken to infinity: H/2 W/2 F (u, v)= f(x, y)e−i(ux+vy) dx dy. −H/2 −W/2 Curiously, we do not need to know F (u, v) for all frequencies in order to reconstruct f(x, y). Knowing that f(x, y)=0for|x| >W/2 and |y| >H/2 provides a strong constraint.
    [Show full text]
  • Arxiv:1801.05832V2 [Cs.DS]
    Efficient Computation of the 8-point DCT via Summation by Parts D. F. G. Coelho∗ R. J. Cintra† V. S. Dimitrov‡ Abstract This paper introduces a new fast algorithm for the 8-point discrete cosine transform (DCT) based on the summation-by-parts formula. The proposed method converts the DCT matrix into an alternative transformation matrix that can be decomposed into sparse matrices of low multiplicative complexity. The method is capable of scaled and exact DCT computation and its associated fast algorithm achieves the theoretical minimal multiplicative complexity for the 8-point DCT. Depending on the nature of the input signal simplifications can be introduced and the overall complexity of the proposed algorithm can be further reduced. Several types of input signal are analyzed: arbitrary, null mean, accumulated, and null mean/accumulated signal. The proposed tool has potential application in harmonic detection, image enhancement, and feature extraction, where input signal DC level is discarded and/or the signal is required to be integrated. Keywords DCT, Fast Algorithms, Image Processing 1 Introduction Discrete transforms play a central role in signal processing. Noteworthy methods include trigonometric transforms— such as the discrete Fourier transform (DFT) [1], discrete Hartley transform (DHT) [1], discrete cosine transform (DCT) [2], and discrete sine transform (DST) [2]—as well as the Haar and Walsh-Hadamard transforms [3]. Among these methods, the DCT has been applied in several practical contexts: noise reduction [4], watermarking methods [5], image/video compression techniques [2], and harmonic detection [2], to cite a few. In fact, when processing signals modeled as a stationary Markov-1 type random process, the DCT behaves as the asymptotic case of the optimal Karhunen–Lo`eve transform in terms of data decorrelation [2].
    [Show full text]
  • Comparison of FFT, DCT, DWT, WHT Compression Techniques on Electrocardiogram & Photoplethysmography Signals
    Special Issue of International Journal of Computer Applications (0975 – 8887) International Conference on Computing, Communication and Sensor Network (CCSN) 2012 Comparison of FFT, DCT, DWT, WHT Compression Techniques on Electrocardiogram & Photoplethysmography Signals Anamitra Bardhan Roy Debasmita Dey Devmalya Banerjee Programmer Analyst Trainee, B.Tech. Student, Dept. of CSE, JIS Asst. Professor, Dept. of EIE Cognizant Technology Solution, College of Engineering College of Engineering, WB, India Kolkata, WB, India Kolkata, WB, India, Bidisha Mohanty B.Tech. Student, Dept. of ECE, JIS College of Engineering, WB, India. ABSTRACT media requires high security and authentication [1]. This Compression technique plays an important role indiagnosis, concept of telemedicine basically and foremost needs image prognosis and analysis of ischemic heart diseases.It is also compression as a vital part of image processing techniques to preferable for its fast data sending capability in the field of manage with the data capacity to transfer within a short and telemedicine.Various techniques have been proposed over stipulated time interval. The therapeutic remedies are derived years for compression. Among those Discrete Cosine from the minutiae’s of the biomedical signals shared Transformation(DCT), Discrete Wavelet worldwide through telelinking. This communication of Transformation(DWT), Fast Fourier Transformation(FFT) exchanging medical data through wireless media requires high andWalsh Hadamard Transformation(WHT) are mostly level of compression techniques to cope up with the restricted used.In this paper a comparative study of FFT, DCT, DWT, data transfer capacity and rate. Compression of signals and and WHT is proposed using ECG and PPG signal, images can cause distortion in them. As the medical signals whichshows a certain relation between them as discussed in and images contains and relays information required for previous papers[1].
    [Show full text]
  • Discrete Cosine Transform Over Finite Prime Fields
    THE DISCRETE COSINE TRANSFORM OVER PRIME FINITE FIELDS M.M. Campello de Souza, H.M. de Oliveira, R.M Campello de Souza, M. M. Vasconcelos Federal University of Pernambuco - UFPE, Digital Signal Processing Group C.P. 7800, 50711-970, Recife - PE, Brazil E-mail: {hmo, marciam, ricardo}@ufpe.br, [email protected] Abstract - This paper examines finite field the FFFT have been found, not only in the fields trigonometry as a tool to construct of digital signal and image processing [3-5], but trigonometric digital transforms. In particular, also in different contexts such as error control by using properties of the k-cosine function coding and cryptography [6-7]. over GF(p), the Finite Field Discrete Cosine The Finite Field Hartley Transform, which is a Transform (FFDCT) is introduced. The digital version of the Discrete Hartley Transform, FFDCT pair in GF(p) is defined, having has been recently introduced [8]. Its applications blocklengths that are divisors of (p+1)/2. A include the design of digital multiplex systems, special case is the Mersenne FFDCT, defined multiple access systems and multilevel spread when p is a Mersenne prime. In this instance spectrum digital sequences [9-12]. blocklengths that are powers of two are Among the discrete transforms, the DCT is possible and radix-2 fast algorithms can be specially attractive for image processing, because used to compute the transform. it minimises the blocking artefact that results when the boundaries between subimages become Key-words: Finite Field Transforms, Mersenne visible. It is known that the information packing primes, DCT. ability of the DCT is superior to that of the DFT and other transforms.
    [Show full text]
  • Wavelets for Feature Detection; Theoretical Background
    Wavelets for Feature Detection; Theoretical background M. van Berkel CST 2010.009 Literature study Coach: Ir. G. Witvoet Supervisors: Dr. ir. P.W.J.M. Nuij Prof. dr. ir. M. Steinbuch Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group Eindhoven, March 2010 Contents Contents 4 1 Introduction5 2 Literature overview7 3 Decomposition of signals and the Fourier Transform9 3.1 Decomposition of signals...............................9 3.2 Approximation of signals............................... 11 3.3 The Fourier Transform................................ 12 3.4 The Short Time Fourier Transform......................... 13 3.5 Conclusions and considerations........................... 15 4 Introduction to Wavelets 17 4.1 Types of Wavelet Transforms............................ 17 4.2 The Continuous Wavelet Transform........................ 18 4.3 Morlet example of a Continuous Wavelet Transform............... 20 4.4 Mexican hat example of a Continuous Wavelet Transform............ 21 4.5 Conclusions and considerations........................... 22 5 The Discrete Wavelet Transform and Filter Banks 23 5.1 Multiresolution.................................... 23 5.2 Scaling function φ .................................. 24 5.3 Wavelets in the Discrete Wavelet Transform................... 26 5.4 DWT applied on a Haar wavelet.......................... 28 5.5 Filter Banks...................................... 30 5.6 Downsampling and Upsampling........................... 32 5.7 Synthesis and Perfect reconstruction.......................
    [Show full text]
  • Digital Image Processing Question & Answers GRIET/ECE 1. Define
    for more :- http://www.UandiStar.org Digital Image Processing Question & Answers 1. Define Fourier Transform and its inverse. Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is defined by the equation Where j = √-1 Given F(u), f(x) can be obtained by using the inverse Fourier transform The Fourier transform exists if f(x) is continuous and integrable and F(u) is integrable. The Fourier transform of a real function, is generally complex, F(u) = R(u) + jI(u) Where R(u) and I(u) are the real and imiginary components of F(u). F(u) can be expressed in exponential form as F(u) = │F(u)│ejØ(u) where │F(u)│ = [R2(u) + I2(u)]1/2 and Ø (u, v) = tan-1[ I (u, v)/R (u, v) ] The magnitude function |F (u)| is called the Fourier Spectrum of f(x) and Φ(u) its phase angle. The variable u appearing in the Fourier transform is called the frequency variable. GRIET/ECE 1 100% free SMS:- ON<space>UandiStar to 9870807070 for JNTU, Job Alerts, Tech News , GK News directly to ur Mobile for more :- http://www.UandiStar.org Digital Image Processing Question & Answers Fig 1 A simple function and its Fourier spectrum The Fourier transform can be easily extended to a function f(x, y) of two variables. If f(x, y) is continuous and integrable and F(u,v) is integrable, following Fourier transform pair exists and \ Where u, v are the frequency variables The Fourier spectrum, phase, are │F(u, v)│ = [R2(u, v) + I2(u, v )]1/2 Ø(u, v) = tan-1[ I(u, v)/R(u, v) ] 2.
    [Show full text]
  • Performance Analysis of Lossless and Lossy Image Compression Techniques for Human Object
    International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 15 (2018) pp. 11715-11723 © Research India Publications. http://www.ripublication.com Performance Analysis of Lossless and Lossy Image Compression Techniques for Human Object 1S.Gomathi, 2T.Santhanam 1Research Scholar, Manonmaniam Sundaranar University, Tirunelveli and Assistant Professor, Department of Computer Science, S.D.N.B Vaishnav College for women, Chennai, Tamil Nadu, India. 2 Associate Professor, PG&Research Department of Computer Science and Applications, DG Vaishnav College, Chennai, India. Abstract techniques. They are Chroma sampling, Transform coding and Fractal compression. In this work use a lossless compression Image compression is an important step in image transmission and decompression through a technique called Run-length and storage. In order to have efficient utilization of disk space coding is compared with Huffman coding (i.e. Huffman and transmission rate, images need to be compressed. The two encoding and decoding). And also lossy compression technique standard compression technique are lossy and lossless image DCT (Discrete Cosine Transform) and FFT (Fast Fourier compression. Though there are many compression technique Transform) are compared with Haar wavelet transform. already available, a technique which is faster, memory efficient and simple surely suits the requirements of the user. In this Run-length encoding (RLE) [2] is a data compression work, the lossless method of image compression and algorithm that is supported by most bitmap file formats, such decompression using a simple coding technique called Run- as TIFF, BMP, and PCX. RLE is suited for compressing any length coding is compared with Huffman coding and lossy type of data regardless of its information content, but the compression technique using DCT (Discrete Cosine content of the data will affect the compression ratio achieved Transform) and FFT(Fast Fourier Transform) are compared by RLE.
    [Show full text]
  • Fourier and Hartley Codes
    Fourier Codes and Hartley Codes H.M. de Oliveira, C.M.F. Barros, R.M. Campello de Souza [15]. Discrete transforms have been successfully applied in Abstract—Real-valued block codes are introduced, which are derived from Discrete Fourier Transforms (DFT) and Discrete error control coding schemes, both in the design of new codes Hartley Transforms (DHT). These algebraic structures are built and decoding algorithms [16], [17], [18], [19]. from the eigensequences of the transforms. Generator and parity Some cryptographic systems have been devised that exploits check matrices were computed for codes up to block length N=24. discrete transforms [20]. The discrete multitone (DMT) They can be viewed as lattices codes so the main parameters systems are essentially based on DFTs [21]. A related type of (dimension, minimal norm, area of the Voronoi region, density, modulation, the orthogonal frequency division multiplex and centre density) are computed. Particularly, Hamming-Hartley and Golay-Hartley block codes are presented. (OFDM multicarrier systems), has been effectively applied in These codes may possibly help an efficient computation of a digital broadcast and wireless channel communication [22], DHT/DFT. [23], [24]. They are also a very efficient tool for spectral Index Terms—-discrete transforms, DFT, Hartley-DHT, real monitoring, therefore are extensively used in spectral managing block codes, eigensequences, lattices. [25], [26]. Eigenfunctions of discrete transforms have been one of the focus of several studies [27], [28], [29]. This analysis recently I. INTRODUCTION derived new multi-user systems [30]. This paper links the iscrete transforms over finite or infinite fields have long eigenvalues of discrete transforms [31] with the design of block Dbeen used in the Telecommunication field to achieve codes defined over the field of real numbers [32], [33].
    [Show full text]
  • Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
    Article Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules Natalie Baddour 1,* 1 Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, , Ottawa, Canada * Correspondence: [email protected]; Tel.: (+1(613)5625800 X2324; Received: 9 July 2019; Accepted: 26 July 2019; Published: 2 August 2019 Abstract: The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two- part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT. Keywords: Fourier Theory, DFT in polar coordinates, polar coordinates, multidimensional DFT, discrete Hankel Transform, discrete Fourier Transform, Orthogonality. 1. Introduction The Fourier transform (FT) in continuous and discrete forms has seen much application in various disciplines [1]. It easily expands to multiple dimensions, with all the same rules of the one- dimensional (1D) case carrying into the multiple dimensions.
    [Show full text]
  • Quantum Spectral Analysis: Frequency in Time, with Applications to Signal and Image Processing Mario Mastriani
    Quantum spectral analysis: frequency in time, with applications to signal and image processing Mario Mastriani To cite this version: Mario Mastriani. Quantum spectral analysis: frequency in time, with applications to signal and image processing. 2017. hal-01654125v2 HAL Id: hal-01654125 https://hal.inria.fr/hal-01654125v2 Preprint submitted on 22 Jan 2018 (v2), last revised 26 Dec 2018 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum spectral analysis: frequency in time, with applications to signal and image processing. Mario Mastriani Quantum Communications Division, Merx Communications LLC, 2875 NE 191 st, suite 801, Aventura, FL 33180, USA [email protected] Abstract A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it’s based on Schrödinger equation, which is a partial differential equation that describes how the quantum state of a non-relativistic physical system changes with time. In classic world is named frequency in time (FIT), which is presented here in opposition and as a complement of traditional spectral analysis frequency-dependent based on Fourier theory. Besides, FIT is a metric, which assesses the impact of the flanks of a signal on its frequency spectrum, which is not taken into account by Fourier theory and even less in real time.
    [Show full text]
  • Face Recognition Using the Discrete Cosine Transform
    International Journal of Computer Vision 43(3), 167–188, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Face Recognition Using the Discrete Cosine Transform ZIAD M. HAFED AND MARTIN D. LEVINE Center for Intelligent Machines, McGill University, 3480 University Street, Montreal, Canada H3A 2A7 [email protected] [email protected] Abstract. An accurate and robust face recognition system was developed and tested. This system exploits the feature extraction capabilities of the discrete cosine transform (DCT) and invokes certain normalization techniques that increase its robustness to variations in facial geometry and illumination. The method was tested on a variety of available face databases, including one collected at McGill University. The system was shown to perform very well when compared to other approaches. Keywords: Face recognition, discrete cosine transform, Karhunen-Loeve transform, geometric normalization, illumination normalization, feature extraction, data compression 1. Introduction This paper discusses a new computational approach to face recognition that, when combined with proper Face recognition by humans is a high level visual face localization techniques, has proved to be very task for which it has been extremely difficult to con- efficacious. struct detailed neurophysiological and psychophysi- This section begins with a survey of the face recog- cal models. This is because faces are complex natu- nition research performed to date. The proposed ap- ral stimuli that differ dramatically from the artificially proach is then presented along with its objectives and constructed data often used in both human and com- the motivations for choosing it. The section concludes puter vision research. Thus, developing a computa- with an overview of the structure of the paper.
    [Show full text]
  • Difference Between DFT and DTFT
    Difference between DFT and DTFT Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation. Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence. The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications.[1] In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function[2]).
    [Show full text]