IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 6, JUNE 2000 1769 A Noise-Robust Frequency Domain Technique for Estimating Planar Roto-Translations Luca Lucchese and Guido Maria Cortelazzo Abstract—This work presents a new method for estimating planar roto-translations that operates in the frequency domain and, as such, is not based on features. Since the proposed technique uses all the image information, it is very robust against noise, and it can be very accurate; estimation errors on the rotational angle range from a few hundredths to a few tenths of a degree, depending on the noise level. In the presence of not-too-large translational displacements, it may work, though with less accuracy, in the case of cropped images as well. Experimental evidence of this performance is presented, and the mathematical reasons behind these characteristics are explained in depth. Another remarkable feature of the algorithm consists in that it works in Cartesian coordinates, bypassing the need to transform data from the Cartesian to the polar domain, which, typically, is a numerically delicate and computationally onerous task. The proposed technique can become an effective tool for unsupervised estimation of roto-translations by means of implementations based on FFT algorithms. Index Terms—Fast Fourier transform, Fourier transform, Her- mitian symmetry, image registration, phase correlation, signal-to- noise ratio, two-dimensional roto-translations. I. INTRODUCTION T HE ESTIMATION of relative translations and rotations between images finds applications in several image processing tasks, such as image registration [1]–[11], pattern recognition [12]–[18], motion compensation [19]–[21], and video coding [22]–[26]. In these fields, general planar rigid transformations (both translational and rotational), commonly referred to as roto-translations may well represent, in a wide range of cases, a valid model for relating images taken from the same scene before more sophisticated models, such as affine or projective transformations, are called for. The frequency domain consideration of rigid motion has sev- eral advantages, which have been recognized for a long time [1], [2], [6] and could be useful for digital implementations as well. There are two main reasons that may favor the frequency tech- niques over standard feature-based methods: robustness to noise and separability of the rotational and translational components. Indeed, the separability of rotation and translation is intrinsic in the structure of the Fourier representation of signals [27]. Rota- tion affects only the relationship between the magnitudes of two Manuscript received July 24, 1998; revised November 26, 1999. The associate editor coordinating the review of this paper and approving it for publication was Prof. Arnab K. Shaw. L. Lucchese is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 USA, on leave from the De- partment of Electronics and Informatics, University of Padova, Padova, Italy. G. M. Cortelazzo is with the Department of Electronics and Informatics, Uni- versity of Padova, Padova, Italy. Publisher Item Identifier S 1053-587X(00)04063-0. relatively translated and rotated versions of the same object; in particular, the magnitudes are rotated with respect to each other at the origin of the spatial frequencies by the same angle as their spatial domain counterparts. Accordingly, we may first estimate the rotational component from the magnitudes of the Fourier transforms, and then, after compensating for rotation, we may easily estimate the translational component, e.g., by using phase correlation techniques [2]. The computational burden of trans- forming the data in the frequency domain (a problem that does not exist in the case of optical implementations) can be effec- tively alleviated in the case of digital implementations by mul- tidimensional FFT algorithms, which nowadays are also hard- ware supported. Frequency domain methods use all the information available and not just the information associated with sets of features. Consequently, the delicate issues associated with the correspon- dence problem can be bypassed, and the methods can be rather robust against noise or image impairment as the analysis of [10] shows in the case of translational displacement, and the analysis and results of this work are also highlighted for the case of rota- tional displacement. Furthermore, if the algorithmic steps nec- essary to extract the motion parameters are able to preserve the original information, these techniques may be potentially highly accurate as well. The algorithmic idea of this work, which was originally pre- sented in [28] and [29], rests on the following property: Given two relatively roto-translated images, the difference between the Fourier transform (FT) magnitude of one image and the mir- rored version, with respect to either frequency axis, of the FT magnitude of the other can be proved to have a pair of orthog- onal zero-crossing lines. These lines are rotated with respect to the frequency axes by an angle that is half the rotational angle. Therefore, the estimate of a planar rotation turns out to be equiv- alent to the detection of these two zero-crossing lines, which are a very distinctive feature of the difference function we have de- fined. We present an original method for solving this specific problem; the estimation of the rotational angle is accomplished by a three-stage coarse-to-fine procedure that gives estimates with a precision of up to hundredths of a degree. A technique of a type of phase correlation is subsequently adopted in order to disambiguate between pairs of possible solutions for the rotational angle and to estimate the translational displacement as well. We show, both analytically and experimentally, that the two zero-crossing lines are very robust to noise at low frequencies. This enables our algorithm to deliver estimates of the rotational angle, with errors within a few tenths of a degree, even with 1053–587X/00$10.00 © 2000 IEEE