• Calculation of the Fractional Fourier Transform
• Watermarking images with the Fractional Fourier Transform"

Paper

Matlab files

The m-file frft.m computes the fast fractional Fourier transform following the algorithm of [1]

The m-file frft2.m computes the fast fractional Fourier transform following the algorithm of [5] (see also [6] for details)

The m-file frft22d.m computes a 2D transform based on the 1D routine frft2.m.

The m-file Disfrft.m computes the discrete fractional Fourier transform following the algorithm of [2]

The m-file cdpei.m computes the discrete fractional Fourier transform following the algorithm of [3]

The m-file testfrft.m is a demonstration program for the previous four routines.

The m-file Disfrct.m computes the discrete fractional Cosine transform following the algorithm of [4]

The m-file Disfrst.m computes the discrete fractional Sine transform following the algorithm of [4]

The m-file testfrct.m test for Disfrct.m This generates figure 2 in the paper [4]

The m-file rescale.m does the rescaling preprocessing for the frft routine as described in the paper [1]

[1] H.M. Ozaktas, M.A. Kutay, and G. Bozdagi.
Digital computation of the fractional Fourier transform.
IEEE Trans. Sig. Proc., 44:2141--2150, 1996. DOI:10.1109/78.536672

[2] C. Candan, M.A. Kutay, and H.M. Ozaktas.
The discrete Fractional Fourier Transform.
IEEE Trans. Sig. Proc., 48:1329--1337, 2000 DOI:10.1109/78.839980

[3] S.-C. Pei, M.-H. Yeh, and C.-C. Tseng.
Discrete fractional Fourier-transform based on orthogonal projections.
IEEE Trans. Sig. Proc., 47(5):1335--1348, 1999. DOI:10.1109/78.757221

[4] S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198--1207, 2001 DOI:10.1109/78.923302

[5] R. Tao, G. Liang, X. Zhao.
An efficient FPGA-based implementation of fractional Fourier transform algorithm.
J. Signal Processing Systems, 60(1):47--58, 2010. DOI 10.1007/s11265-009-0401-0

[6] A. Bultheel.
A two-phase implementation of the fractional Fourier transform.
Report TW 588, Dept. Computer Science, K.U.Leuven, March 2011.

FrFT Fa(sin) for a power a= 0(0.05)2 For an animation click the image © A. Bultheel

FrFT Fa(gaussian+chirp) for a power a= 0(0.05)2 For an animation click the image © A. Bultheel

Watermarking images with the Fractional Fourier Transform

Report

Matlab files

The m-file WMdetect.m tries to detect one or more watermarks in the image

The m-file WMtest.m is a test file for the previous two routines

The zip file WMaux.zip contains some auxiliary routines and an image to run WMtest with.

Watermarked image: compressed added noise and cropped	Detection of the correct watermark 501