1080 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 10, OCTOBER 1997 Space-Variant Fourier Analysis: The Exponential Chirp Transform Giorgio Bonmassar, Student Member, IEEE, and Eric L. Schwartz 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. Index Terms���Logpolar mapping, rotation scale and shift invariance, attention, space-variant image processing, Fourier analysis, nonuniform sampling, real-time imaging, warped template matching. ������������������������������ ��� ������������������������������ 1 INTRODUCTION N this paper, we describe an algorithm for estimating the Fourier transform of an image which has been remapped by an arbitrary invertible C1 map function, in the range co- ordinates of the map function. After stating this general result, we specialize the discussion by applying the log- polar map function. This is because the log-polar image format has been shown to approximate that used by higher vertebrate [1] visual systems, and has been the basis of many investigations in machine vision during the past sev- eral decades [2], [3], [4], [5], [6], [7], [8], [9], [10]. Our main focus in this paper is computer vision, and the motivation for interest in the log-polar form of image architecture is that it provides a continuous multiresolution representation of an image in a form which provides a large (but lossy) reduction in image size, (up to four orders of magnitude in the human visual system [7] and up to two or three orders of magnitude in several recently constructed computer vi- sion systems [11]). This image representation is space- variant, due to a progressive decrease in the spatial sam- pling rate with distance from the center of vision, or ���fovea.��� Nevertheless, it provides a large effective work- space with a simultaneous ability to perform high resolu- tion vision, while maintaining a small amount of pixels in the image. There are many contexts in machine vision in which this is a desirable trade-off. In biology, some form of foveal strategy is the only image architecture that is in use in higher vertebrate systems. There has been little systematic development of image processing tools that are explicitly designed for this space- variant vision. Conventional frequency domain methods are of no utility. The shift invariant property of the FFT does not hold when applied directly to a log-polar image format, since translation symmetry in the image domain is broken by the space-variant properties of the map. In this paper, we provide a transform which solves the seemingly paradoxical problem of achieving a form of shift-invariance on a strongly space-variant architecture. We call this trans- form the exponential chirp transform (ECT). Our principle practical result is that we have found an algorithm which allows a resampled form of the ECT to be computed with the conventional FFT, which we call the FECT. This allows us to combine the favorable space complexity of the log- polar mapping (i.e., small images) with the speed of the FFT, to achieve a general frequency domain analysis tool which is applicable in space-variant applications. Using this approach, we estimate that current generation DSP archi- tectures are sufficient for frame-rate full-field image corre- lation and filtering. The structure of this paper is first to present the general form of transformed Fourier Integral. Then we introduce the exponential chirp transform, and give one- and two- dimensional examples of it. Next, we demonstrate a fre- quency domain coordinate transform (log map in fre- quency) which allows us to express the exponential chirp integral as a complex correlation, which can then be effi- ciently evaluated with the FFT. We describe a variational method for computing the antialiasing and normalization filters needed for the FECT, and demonstrate their efficacy by showing the inversion of the transform. Finally, we pro- vide a series of examples from an image database of tem- plate matching and filtering using the FECT implementation, and briefly discuss the practical significance of this work. 0162-8828/97/$10.00 �� 1997 IEEE �������������������������������� ��� G. Bonmassar is with the Department of Biomedical Engineering, Boston University, Boston, MA 02215. E-mail: giorgio@engc.bu.edu. ��� E.L. Schwartz is with the Department of Cognitive and Neural Systems, Boston University, Boston, MA 02146. E-mail: eric@thing4.bu.edu. Manuscript received 15 Jan. 1996 revised 26 Aug. 1997. Recommended for acceptance by J. Daugman. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number 105666. I