Pattern Recognition 38 (2005) 2314 – 2322 www.elsevier.com/locate/patcog Rotation invariant pattern recognition using ridgelets, wavelet cycle-spinning and Fourier features G.Y. Chen, T.D. Bui ∗ , A. Krzy˙ zak Department of Computer Science, Concordia University, 1455 De MaisonneuveWest, Montreal, Quebec, Canada H3G 1M8 Received 21 April 2004; received in revised form 18 February 2005; accepted 18 February 2005 Abstract In this paper, we propose a rotation-invariant descriptor for pattern recognition by using ridgelets, wavelet cycle-spinning, and the Fourier transform. Ridgelets have been developed recently and have many advantages over wavelets in applications to image processing. However, the current implementation of ridgelets cannot be applied to pattern recognition directly. In order to overcome this problem, we have successfully extracted ridgelet features within the circle surrounding the pattern we are trying to recognize.Wavelet cycle-spinning and Fourier spectrum magnitudes are used to achieve rotation invariance. The main motivation of using ridgelets is that we have a much better tool for the extraction of features based on line singularities as compared to point singularities as in the case of wavelets. Based on this observation, important features can be extracted. Our experiments show that our proposed descriptor is very robust to Gaussian noise and it achieves high recognition rates.  2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. Keywords: Ridgelets; Wavelets; Cycle-spinning; Fourier transform; Feature extraction; Pattern recognition 1. Introduction Recently the ridgelet transform has been successfully pro- posed to analyze digital images [1–8]. Unlike wavelet trans- forms, the ridgelet transform processes data by first com- puting integrals over different orientations and locations. A ridgelet is constant along the lines x 1 cos  + x 2 sin  = C. Transversing to these ridges it is a wavelet. Ridgelet transform could be successfully applied in invariant pat- tern recognition. However, no known paper has considered this new transform for pattern recognition. This is because the ridgelet transform defined on a square is not suitable for extracting rotation-invariant features. In order to extract ∗ Corresponding author. Tel.: +1 514 848 3014; fax: +1 514 848 2830. E-mail addresses: bui@cs.concordia.ca (G.Y. Chen), guang_c@cs.concordia.ca (T.D. Bui), krzyzak@cs.concordia.ca (A. Krzy˙ zak). 0031-3203/$30.00  2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2005.02.008 rotation-invariant features, we implement our ridgelet trans- form on the pattern region that falls inside the circle sur- rounding the pattern we want to recognize. This means our ridgelets work on a disk instead of a square. In this paper, we present a novel descriptor for pattern recognition by using a combination of ridgelets, wavelet cycle-spinning and the Fourier transform. For the patterns in the pattern database, our descriptor can be described as fol- lows. First, we normalize the pattern so that it is translation- and scale-invariant. Second, we discard all those pixels of the pattern that fall outside the circle surrounding the pat- tern to be recognized. Third, we project the pattern on each slice segment that passes through the center of the circle and ends at the boundary of the circle. These slice segments are equally spaced in angle. This is the Radon transform, but it works on a disk rather than a square. Fourth, we perform 1D wavelet transform along each Radon slice so that we get the ridgelet coefficients. Fifth, we conduct wavelet cycle- spinning along the angle direction, perform 1D Fourier transform on each wavelet subband and get the magnitude