Image and video processing using discrete fractional transforms

The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.

[1]  John T. Sheridan,et al.  Image encryption and the fractional Fourier transform , 2003, SPIE OPTO-Ireland.

[2]  N. Wiener.  Hermitian Polynomials and Fourier Analysis , 1929 .

[3]  James H. McClellan,et al.  The DRFT-a rotation in time-frequency space , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[4]  K. W. Cattermole.  The Fourier Transform and its Applications , 1965 .

[5]  V. Namias.  The Fractional Order Fourier Transform and its Application to Quantum Mechanics , 1980 .

[6]  Ioannis Pitas,et al.  Digital watermarking in the fractional Fourier transformation domain , 2001, J. Netw. Comput. Appl..

[7]  Chaur-Chin Chen.  On the selection of image compression algorithms , 1998, Proceedings. Fourteenth International Conference on Pattern Recognition (Cat. No.98EX170).

[8]  Maher A. Sid-Ahmed,et al.  Image processing, theory, algorithms and architectures , 1995 .

[9]  Omar Zakaria,et al.  An Overview of Video Encryption Techniques , 2009 .

[10]  H. Kober.  WURZELN AUS DER HANKEL-, FOURIER-UND AUS ANDEREN STETIGEN TRANSFORMATIONEN , 1939 .

[11]  James H. McClellan,et al.  The discrete rotational Fourier transform , 1996, IEEE Trans. Signal Process..

[12]  Anil K. Jain.  Fundamentals of Digital Image Processing , 2018, Control of Color Imaging Systems.

[13]  C. Vijaya,et al.  Signal compression using discrete fractional Fourier transform and set partitioning in hierarchical tree , 2006, Signal Process..

[14]  M. J. Narasimha,et al.  On the Computation of the Discrete Cosine Transform , 1978, IEEE Trans. Commun..

[15]  Anja Walter.  Still Image And Video Compression With Matlab , 2016 .

[16]  Juan G. Vargas-Rubio,et al.  On the multiangle centered discrete fractional Fourier transform , 2005, IEEE Signal Processing Letters.

[17]  Adhemar Bultheel,et al.  A shattered survey of the Fractional Fourier Transform , 2002 .

[18]  Narendra Singh,et al.  Optical image encryption using fractional Fourier transform and chaos , 2008 .

[19]  Wilhelm Burger,et al.  Digital Image Processing - An Algorithmic Introduction using Java , 2008, Texts in Computer Science.

[20]  Tsuhan Chen,et al.  Multimedia Systems, Standards and Networks , 2000 .

[21]  S. Pei,et al.  Improved discrete fractional Fourier transform. , 1997, Optics letters.

[22]  Soo-Chang Pei,et al.  A novel method for discrete fractional Fourier transform computation , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[23]  M. Fatih Erden,et al.  THE DISCRETE FRACTIONAL COSINE TRANSFORM , .

[24]  Mihir Bellare,et al.  A concrete security treatment of symmetric encryption , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[25]  Lu Wang,et al.  A Novel Split-Radix Fast Algorithm for 2-D Discrete Hartley Transform , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[26]  Imam Samil Yetik,et al.  Continuous and discrete fractional Fourier domain decomposition , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[27]  Alan C. Bovik,et al.  The Essential Guide to Image Processing , 2009, J. Electronic Imaging.

[28]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

[29]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[30]  A. P. Prudnikov,et al.  Integrals and series of elementary functions , 1981 .

[31]  Wade Trappe,et al.  Introduction to Cryptography with Coding Theory , 2002 .

[32]  Soo-Chang Pei,et al.  A method for the discrete fractional Fourier transform computation , 2003, IEEE Trans. Signal Process..

[33]  Soo-Chang Pei,et al.  The discrete fractional cosine and sine transforms , 2001, IEEE Trans. Signal Process..

[34]  Ran Tao,et al.  Research progress of the fractional Fourier transform in signal processing , 2006, Science in China Series F.

[35]  E. Condon,et al.  Immersion of the Fourier Transform in a Continuous Group of Functional Transformations. , 1937, Proceedings of the National Academy of Sciences of the United States of America.

[36]  Li Yu,et al.  Optical image encryption by cascaded fractional Fourier transforms with random phase filtering , 2001 .

[37]  Luís B. Almeida,et al.  The fractional Fourier transform and time-frequency representations , 1994, IEEE Trans. Signal Process..

[38]  Neeru Jindal,et al.  Image Encryption Using Discrete Fractional Transforms , 2010, 2010 International Conference on Advances in Recent Technologies in Communication and Computing.

[39]  Ibrahim Sogukpinar,et al.  Analysis and Comparison of Image Encryption Algorithms , 2007 .

[40]  Rafael C. González,et al.  Digital image processing, 3rd Edition , 2008 .

[41]  N. Ahmed,et al.  Discrete Cosine Transform , 1996 .

[42]  G. Unnikrishnan,et al.  Optical encryption by double-random phase encoding in the fractional Fourier domain. , 2000, Optics letters.

[43]  Zeev Zalevsky,et al.  Some important fractional transformations for signal processing , 1996 .

[44]  A. Lohmann.  Image rotation, Wigner rotation, and the fractional Fourier transform , 1993 .

[45]  F. H. Kerr,et al.  On Namias's fractional Fourier transforms , 1987 .

[46]  Ran Tao,et al.  Double image encryption based on random phase encoding in the fractional Fourier domain. , 2007, Optics express.

[47]  Chien-Cheng Tseng,et al.  Discrete fractional Fourier transform based on orthogonal projections , 1999, IEEE Trans. Signal Process..

[48]  Pat Yip,et al.  Sine and Cosine Transforms , 2000 .

[49]  Klara Nahrstedt,et al.  Comparison of MPEG encryption algorithms , 1998, Comput. Graph..

[50]  A. Bultheel,et al.  Computation of the fractional Fourier transform , 2004 .

[51]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[52]  Nikolaos G. Bourbakis,et al.  Image and video encryption using SCAN patterns , 2004, Pattern Recognit..

[53]  Lotfi Senhadji,et al.  Direct Computation of Type-II Discrete Hartley Transform , 2007, IEEE Signal Processing Letters.

[54]  Arun N. Netravali,et al.  Digital Pictures: Representation and Compression , 1988 .

[55]  S. Pei,et al.  Discrete fractional Fourier transform , 1996, 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96.

[56]  K. Wolf,et al.  Continuous vs. discrete fractional Fourier transforms , 1999 .

[57]  Chandra Shakher,et al.  Image encryption and decryption using fractional Fourier transform and radial Hilbert transform , 2008 .

[58]  K. M. M. Prabhu,et al.  The fractional Fourier transform: theory, implementation and error analysis , 2003, Microprocess. Microsystems.

[59]  Kulbir Singh,et al.  Performance of Discrete Fractional Fourier TransformClasses in Signal Processing Applications , 2006 .

[60]  Zhengjun Liu,et al.  Double image encryption based on iterative fractional Fourier transform , 2007 .