This Article 
 Bibliographic References 
 Add to: 
An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration
March 2009 (vol. 31 no. 3)
pp. 400-414
Wei Pan, Tsinghua University, Beijing
Kaihuai Qin, Tsinghua University, Beijing
Yao Chen, Tsinghua University, Beijing
a novel adaptable accurate way for calculating Polar FFT and Log-Polar FFT is developed in this paper, named Multilayer Fractional Fourier Transform (MLFFT). MLFFT is a necessary addition to the pseudo-polar FFT for the following reasons: It has lower interpolation errors in both polar and log-polar Fourier transforms; it reaches better accuracy with the nearly same computing complexity as the pseudo-polar FFT; it provides a mechanism to increase the accuracy by increasing the user-defined computing level. This paper demonstrates both MLFFT itself and its advantages theoretically and experimentally. By emphasizing applications of MLFFT in image registration with rotation and scaling, our experiments suggest two major advantages of MLFFT: 1) scaling up to 5 and arbitrary rotation angles, or scales up to 10 without rotation can be recovered by MLFFT while currently the result recovered by the state-of-the-art algorithms is the maximum scaling of 4; 2) No iteration is needed to obtain large rotation and scaling values of images by MLFFT, hence it is more efficient than the pseudopolar-based FFT methods for image registration.

[1] B. Zitova and J. Flusser, “Image Registration Methods: A Survey,” Image Vision Computing, vol. 21, no. 11, pp. 977-1000, Nov. 2003.
[2] Y. Keller, A. Averbuch, and M. Israeli, “Pseudopolar-Based Estimation of Large Translations, Rotations and Scalings in Images,” IEEE Trans. Image Processing, vol. 14, no. 1, pp. 12-22, Jan. 2005.
[3] B.S. Reddy and B.N. Chatterji, “An FFT-Based Technique for Translation, Rotation and Scale-Invariant Image Registration,” IEEE Trans. Image Processing, vol. 5, no. 8, pp. 1266-1271, Aug. 1996.
[4] A. Averbuch, R.R. Coifman, D.L. Donoho, M. Elad, and M. Israeli, “Fast and Accurate Polar Fourier Transform,” Applied and Computational Harmonic Analysis, vol. 21, pp. 145-167, 2006.
[5] S. Man and R.W. Picard, “Virtual Bellows, Constructing High Quality Still from Video,” Proc. IEEE Int'l Conf. Image Processing, Nov. 1994.
[6] S. Zokai and G. Wolberg, “Image Registration Using Log-Polar Mappings for Recovery of Large-Scale Similarity and Projective Transformations,” IEEE Trans. Image Processing, vol. 14, no. 10, Oct. 2005.
[7] H. Foroosh, J.B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Processing, vol. 11, no. 3, Mar. 2002.
[8] H. Liu, B. Guo, and Z. Feng, “Pseudo-Log-Polar Fourier Transform for Image Registration,” IEEE Signal Processing Letters, vol. 13, no. 1, Jan. 2006.
[9] C.D. Kuglin and D.C. Hines, “The Phase Correlation Image Alignment Method,” Proc. IEEE. Conf. Cybernetics Soc., pp. 163-165, Sept. 1975.
[10] H.Y. Shurn and R. Szeliski, “Construction and Refinement of Panoramic Mosaics with Global and Local Alignment,” Proc. IEEE. Conf. Computer Vision, 1998.
[11] H.S. Stone, B. Tao, and M. McGuire, “Analysis of Image Registration Noise Due to Rotationally Dependent Aliasing,”, 2008.
[12] P.N. Swarztrauber and D.H. Bailey, “The Fractional Fourier Transform and Applications,” SIAM Rev., vol. 33, no. 3, pp. 389-404, Sept. 1991.
[13] A.V. Oppenheim and R.W. Schafer, Discrete Time Signal Processing, second ed. Prentice Hall, 1999.
[14] Y. Keller, Y. Shkolnisky, and A. Averbuch, “The Angular Difference Function and Its Application to Image Registration,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 6, June 2005.
[15] M. Frigo and S.G. Johnson, “FFTW: The Fast Fourier Transform in the West,” MIT,, 2008.
[16] S. Derrode and F. Ghorbel, “Robust and Efficient Fourier-Mellin Transform Approximations for Gray-Level Image Reconstruction and Complete Invariant Description,” Computer Vision and Image Understanding, vol. 83, no. 1, pp. 57-78, July 2001.
[17] A. Averbuch, R.R. Coifman, D.L. Donoho, M. Elad, and M. Israeli, Fast and Accurate Polar Fourier Transform, journals/200430_PolarFFT_ACHA.pdf.
[18] L.R. Rabiner, R.W. Schafer, and C.M. Rader, “The Chirp-Z Transform Algorithm,” IEEE Trans. Audio Electroacoustics, vol. 17, pp. 86-92, June 1969.
[19] Y. Keller, A. Averbuch, and Y. Shkolnisky, “Algebraically Accurate Volume Registration Using Euler's Theorem and the 3D Pseudo-Polar FFT,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 795-800, 2005.
[20] F. Dufaux and J. Konrad, “Efficient, Robust, and Fast Global Motion Estimation for Video Coding,” IEEE Trans. Image Processing, vol. 9, no. 3, pp. 497-501, Mar. 2000.
[21] M. Irani and S. Peleg, “Motion Analysis for Image Enhancement: Resolution, Occlusion and Transparency,” J. Visual Comm. and Image Representation, vol. 4, no. 4, pp. 324-335, Dec. 1993.
[22] D. Steedly, C. Pal, and R. Szeliski, “Efficiently Registering Video into Panoramic Mosaics,” Proc. 10th IEEE Int'l Conf. Computer Vision, vol. 2, pp. 1300-1307, Oct. 2005.
[23] J. More, “Levenberg-Marquardt Algorithm: Implementation and Theory,” Proc. Conf. Numerical Analysis, 1967.
[24] Y. Keller, A. Avenbuch, and O. Miller, “Robust Phase Correlation,” Proc. IEEE 14th Int'l Conf. Pattern Recognition, vol. 2, pp. 740-743, Aug. 2004.
[25] G. Wolberg and S. Zokai, “Robust Image Registration Using Log-Polar Transform,” Proc. IEEE Int'l Conf. Image Processing, pp. 493-496, Sept. 2000.
[26] “The USC-SIPI Image Database,” Univ. of Southern California, http://sipi.usc.edudatabase/, 2008.
[27] D.G. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int'l J. Computer Vision, vol. 60, no. 2, pp. 91-110, Nov. 2004.
[28] M. Brown and D.G. Lowe, “Recognising Panoramas,” Proc. Ninth IEEE Int'l Conf. Computer Vision, vol. 2, pp. 1218-1225, Oct. 2003.

Index Terms:
Image Processing and Computer Vision, Computation of transforms, Pattern matching
Wei Pan, Kaihuai Qin, Yao Chen, "An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 3, pp. 400-414, March 2009, doi:10.1109/TPAMI.2008.83
Usage of this product signifies your acceptance of the Terms of Use.