This Article 
 Bibliographic References 
 Add to: 
Space-Variant Fourier Analysis: The Exponential Chirp Transform
October 1997 (vol. 19 no. 10)
pp. 1080-1089

Abstract—Space-variant, or foveating, vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular space-variant map, the log-polar map, which approximates the primate visual map, and which has been applied in machine vision by a number of investigators during the past two decades. Associated with the log-polar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform. We then show that a log-polar coordinate transform in frequency (similar to the Mellin-Transform) provides a fast exponential chirp transform. This provides size and rotation, in addition to shift, invariant properties in the transformed space. Finally, we demonstrate the use of the fast exponential chirp algorithm on a database of images in a template matching task, and also demonstrate its uses for spatial filtering. Given the general lack of algorithms in space-variant image processing, we expect that the fast exponential chirp transform will provide a fundamental tool for applications in this area.

[1] E.L. Schwartz, "Computational Studies of the Spatial Architecture of Primate Visual Cortex: Columns, Maps, and Protomaps," Primary Visual Cortex in Primates, A. Peters and K. Rocklund, eds., vol. 10, Cerebral Cortex. Plenum Press, 1994.
[2] C.F. Weiman and G. Chaikin, "Logarithmic Spiral Grids for Image-Processing and Display," Computer Graphics and Image Processing, vol. 11, pp. 197-226, 1979.
[3] D. Asselin and H.H. Arsenault, "Rotation and Scale Invariance With Polar and Log-Polar Coordinate Transformations," Optics Comm., vol. 104, pp. 391-404, Jan. 1994.
[4] J.K. Brousil and D.R. Smith, "A Threshold-Logic Network for Shape Invariance," IEEE Trans. Computers, vol. 16, pp. 818-828, 1967.
[5] D. Casasent and D. Psaltis, "Position, Rotation and Scale-Invariant Optical Correlation," Applied Optics, vol. 15, pp. 1,793-1,799, 1976.
[6] B.R. Frieden and C. Oh, "Integral Logarithmic Transform: Theory and Applications," Applied Optics, vol. 31, no. 8, pp. 1,138-1,145, Mar. 1992.
[7] A.S. Rojer and E.L. Schwartz, "Design Considerations for a Space-Variant Visual Sensor With Complex-Logarithmic Geometry," Proc. Int'l Conf. Pattern Recognition, ICPR-10, vol. 2, pp. 278-285, 1990.
[8] G. Sandini, F. Bosero, F. Bottino, and A. Ceccherini, "The Use of an Antropomorphic Visual Sensor for Motion Estimation and Object Tracking," Proc. OSA Topical Meeting Image Understanding and Machine Vision, 1989.
[9] J. van der Spiegel, F. Kreider, C. Claiys, I. Debusschere, G. Sandini, P. Dario, F. Fantini, P. Belluti, and G. Soncini, "A Foveated Retina-Like Sensor Using CCD Technology," Analog VLSI Implementations of Neural Networks, C. Mead and M. Ismail, eds. Boston: Kluwer, 1989.
[10] R.D. Juday, "Log-Polar Dappled Target," Optics Letters, vol. 20, no. 21, pp. 2,234-2,236, Nov. 1995.
[11] G. Engel, D. Greve, J. Lubin, and E. Schwartz, "Space-Variant Active Vision and Visually Guided Robotics: Design and Construction of a High-Performance Miniature Vehicle," Proc. Int'l Conf. Pattern Recognition, ICPR-12, 1994.
[12] G. Bonmassar and E. Schwartz, "Geometric Invariance in Space-Variant Vision," Proc. Int'l Conf. Pattern Recognition, ICPR-12, 1994.
[13] D.F. Elliot and K.R. Rao, Fast Transforms: Algorithms, Analyses, Applications.New York: Academic Press, 1982.
[14] P. Cavanagh, "Size and Position Invariance in the Visual System," Perception, vol. 7, pp. 167-177, 1978.
[15] E.L. Schwartz, "Spatial Mapping in Primate Sensory Projection: Analytic Structure and Relevance to Perception," Biological Cybernetics, vol. 25, pp. 181-194, 1977.
[16] P. Kellman and J.W. Goodman, "Coherent Optical Implementation of 1-D Mellin Transforms," Applied Optics, vol. 16, pp. 2,609-2,610, 1977.
[17] Y. Sheng and H.H. Arsenault, "Experiments on Pattern Recognition Using Invariant Fourier-Mellin Descriptors," J. Optical Soc. Am., A, vol. 3, pp. 771-776, 1986.
[18] A. Papoulis, "Error Analysis in Sampling Theory," Proc. IEEE, vol. 54, pp. 947-955, July 1966.
[19] D.C. Stickler, "An Upper Bound on Aliasing Error," Proc. IEEE, vol. 55, pp. 418-419, 1967.
[20] J.L. Brown Jr., "A Least Upper Bound for Aliasing Error," IEEE Trans. Automatic Control, vol. 13, pp. 754-755, 1968.
[21] A.W. Splettstosser, "Error Estimates for Sampling Approximation of Non-Bandlimited Functions," Math. Methods in the Applied Sciences, vol. 1, pp. 127-137, 1979.
[22] H.S. Shapiro and R.A. Silverman, "Alias-Free Sampling of Random Noise," J. Soc. Industrial and Applied Math., vol. 8, no. 2, pp. 225-249, June 1960.
[23] F.J. Beutler, "Alias-Free Randomly Timed Sampling of Stochastic Processes," IEEE Trans. Information Theory, vol. 16, no. 2, pp. 147-152, Mar. 1970.
[24] A.J. Jerry, "The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review," Proc. IEEE, vol. 65, no. 11, pp. 1,565-1,596, Nov. 1977.
[25] B. Van Der Pol, "The Fundamental Principles of Frequency Modulation," J. IEE (London), vol. 93, pt. 3, no. 23, pp. 153-158, May 1946.
[26] A. Papoulis, Signal Analysis. McGraw-Hill, 1977.
[27] H.P. Kramer, "A Generalized Sampling Theorem," J. Math. Physics, vol. 38, pp. 68-72, 1959.
[28] J.J. Clark, M.R. Palmer, and P.D. Lawrence, "A Transformation Method for the Reconstruction of Functions From Nonuniformly Spaced Samples," IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 33, no. 4, p. 1,151, Oct. 1985.
[29] H. Stark, "Sampling Theorems in Polar Coordinates," J. Optical Soc. Am., vol. 69, no. 11, pp. 1,519-1,525, Nov. 1979.
[30] C.F.R. Weiman, "Video Compression via Log-Polar Mapping," Proc. SPIE Symp. OE-Aereospace Sensing, Apr. 1990.
[31] B. Fischl, M. Cohen, and E.L. Schwartz, "The Local Structure of Space-Variant Images," Neural Networks, vol. 10, no. 5, pp. 815-831, 1997.
[32] G. Bonmassar and E. Schwartz, "Lie Groups, Space-Variant Fourier Analysis and the Exponential Chirp Transform," Proc. Computer Vision and Pattern Recognition 96, pp. 229-237, 1996.
[33] P.J. Burt and E.H. Adelson, “The Laplacian Pyramid as a Compact Image Code,” IEEE Trans. Comm., vol. 31, no. 4, pp. 532-540, 1983.
[34] S.L. Tanimoto, "Template Matching in Pyramids," Computer Vision, Graphics, and Image Processing, vol. 16, pp. 356-369, 1981.
[35] S.G. Mallat,“A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 674-693, 1989.
[36] J.G. Daugman, "Two-Dimensional Spectral Analysis of Cortical Receptive Field Profile," Vision Research, vol. 20, pp. 847-856, 1980.
[37] J.G. Daugman, “Complete Discrete 2D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, no. 7, 1988.
[38] R. Wallace, P.-W. Ong, B. Bederson, and E. Schwartz, "Space Variant Image Processing," Int'l J. Computer Vision, vol. 13, no. 1, pp. 71-90, 1994.
[39] E.L. Schwartz, "Image Processing Simulations of the Functional Architecture of Primate Striate Cortex," Investigative Ophthalmic and Vision Research (Supplement), vol. 26, no. 3, pp. 164, 1985.
[40] G. Bonmassar, "The Exponential Chirp Transform," PhD Thesis, Biomedical Eng. Dept., Boston Univ., 1997.
[41] R.L. DeValois and K.K. DeValois, Spatial Vision. Oxford Univ. Press, 1988.
[42] J.A. Nelder and R. Mead, "A Simplex Method for Function Minimization," Computer J., vol. 7, pp. 308-313.
[43] J.-C. Liu and H.-C. Chiang, "Fast High-Resolution Approximation of the Hartley Transform at Arbitrary Frequency Estimator," Signal Processing, vol. 44, no. 2, pp. 211, June 1995.
[44] Jose A. Ferrari, "Fast Hankel Transform of Order Zero," J. Optical Soc. Am., A, vol. 12, no. 8, pp. 1,812, Aug. 1995.
[45] J. Strain, "A Fast Laplace Transform Based on Laguerre Functions," Math. Computation, vol. 58, no. 197, pp. 275-283, Jan. 1992.
[46] L. Greengard and J. Strain, "The Fast Gauss Transform," SIAM J. Scientific Statistical Computing, vol. 12, no. 1, pp. 79-94, Jan. 1991.
[47] J.F. Yang, S.C. Shaih, and B.L. Bai, "Fast Two-Dimensional Inverse Discrete Cosine Transform for HDTV or Videophone Systems," IEEE Trans. Consumer Electronics, vol. 39, no. 4, pp. 934-940, Nov. 1993.
[48] B.T. Kelly and V.K. Madisetti, "The Fast Discrete Radon Transform—I: Theory," IEEE Trans. Image Processing, vol. 2, no. 3, pp. 382-400, July 1993.
[49] C.M. Rader, "Discrete Fourier Transform When the Number of Data Samples is Prime," Proc. IEEE, vol. 56, no. 6, pp. 1,107-1,108, June 1968.
[50] J.H. McClellan and C.M. Rader, Number Theory in Digital Signal Processing.Englewood Cliffs, N.J.: Prentice-Hall, 1979.

Index Terms:
Logpolar mapping, rotation scale and shift invariance, attention, space-variant image processing, Fourier analysis, nonuniform sampling, real-time imaging, warped template matching.
Giorgio Bonmassar, Eric L. Schwartz, "Space-Variant Fourier Analysis: The Exponential Chirp Transform," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 10, pp. 1080-1089, Oct. 1997, doi:10.1109/34.625108
Usage of this product signifies your acceptance of the Terms of Use.