ELSEVIER Pattern Recognition Letters 17 (1996) 1001-1015 Pattem Recognition Letters Representations that uniquely characterize images modulo translation, rotation, and scaling Robert D. Brandt a,l, Feng Lin h,, a Intelligent Devices, Inc., 465 Whittier Avenue, Glen Ellyn, IL 60137, USA b Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, USA Received 15 June 1995; revised 25 March 1996 Abstract Representations that uniquely characterize images modulo the planar geometric similarity transformations are presented. Such representations are invariant with respect to translation, rotation, scaling, and their combination. We will first discuss basic invariants, that is, the invariants for translation, rotation, and scaling only. We will then discuss hybrid invariants, that is, invariants with respect to combinations of translation, rotation, and scaling. Keywords: Invariant; Translation; Rotation; Scaling 1. Introduction In this paper, we consider the problem of recognizing isolated objects from their images, regardless of the position, orientation, and scale at which they are represented. Such a recognition process is said to be invariant with respect to the (planar) geometric similarity transformations: translation, rotation, and scaling. The most natural representation of an image is as a non-negative function of the Cartesian coordinates of the image plane. The value of the function usually represents the intensity of light reflected to the eye. We refer to images represented in this way as raw images. In order to directly verify that two raw images are of the same object, comparison of the two images must be made for every possible similarity transformation until a match is determined. Such comparisons could be performed either by using the cross-correlation of the images or by using a metric, such as the L2 metric (Ballard and Brown, 1982; Gonzalez and Woods, 1992; Rogers et al., 1990). The number of computations required to perform a digital approximation of this idealized matching process depends on the resolution (sample density) of the sampled images and the accuracy required to justify the confirmation of a match. Match of the four (quantized) transformation parameters (two for translation, one for rotation, and one for scaling) contributes to the number of calculations that must be performed. In applications with a large number of candidate objects, such brute force comparisons are unfeasible. * Corresponding author. E-mail: flin@ece.eng.wayne.edu, ] Currently visiting the Beckman Institute, University of Illinois, Urbana, IL 61801, USA. 0167-8655/96/$12.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved. PII S0167-8655 (96)00062-1