Elsevier

Pattern Recognition

Volume 35, Issue 12, December 2002, Pages 2993-2999
Pattern Recognition

Using Fourier/Mellin-based correlators and their fractional versions in navigational tasks

https://doi.org/10.1016/S0031-3203(02)00018-3Get rights and content

Abstract

The navigational tasks of computing time to impact and controlling movements in specific range are addressed here. We show how time to impact can be obtained via the Mellin-based correlator, and we introduce the concept of fractional based correlators. The fractional Fourier/Mellin based correlators can be used in detecting or controlling specific range of movements. Also, both the conventional and the fractional-based correlators can be easily implemented optically in lenses, thus providing correlation images directly at image acquisition time. Lenses are considered cost-effective and therefore, the optical correlators are optimal in both senses, speed and cost.

Introduction

Correlators are well known for their role in object recognition. Here, their possible applications to navigational tasks were investigated. We will show that when the motion is assumed to consist of rotation and scaling, the Mellin-based correlator can be used to compute time to impact. In addition, we will introduce the concept of fractional correlators. The fractional Fourier-based correlator can be used to control the range of translation, while the Fractional Mellin-based correlator, can be used to control the range of rotation and scaling.

Algorithms concerning the above mentioned navigational tasks are usually known to be computationally heavy and considered as time consuming [1], [2], [3], [4], [5], [6], [7]. Therefore, physical sensors that provide as much information as possible directly at acquisition time have an utmost importance. The major advantage of the correlators is the fact that they can be easily implemented optically. Here, we suggest the common use of cameras, but with optically implemented functions. This means that the correlators can be implemented within the camera, by using special purpose lenses, and can be obtained at image acquisition time. The special lenses can be formed by different combinations of regular Fourier lenses and special purpose designed filters [8], [9], [10]. In addition to the high speed in which relevant data can be supplied, lenses are usually cost effective. Thus, this solution is optimal in both senses, speed and cost.

The usage of optical correlators in real time vehicle navigation was previously demonstrated by Psaltis [11]. There, an opto-electronic information processing system based on holographic memory database (realized using the DuPont HRF-150 photopolymer) is applied to perform the navigational tasks. In contrast to the database technique, this paper uses transformations that are optically implemented, and thus, shows how translation, rotation, and scaling can be computed simply, without the requirement for a digital processor of high capacity.

This paper is organized as follows: Section 2 discusses the Mellin correlator and its role in time to impact estimation. Section 3 presents the theoretical concept of the fractional based correlators and their role in range control. Section 4 shows and discusses experiments that demonstrate these concepts, and Section 5 is a summary.

Section snippets

Time to impact estimation using the Mellin-based correlator

The correlation image provides information concerning the translation of a tested pattern in relation to a centered reference pattern. The correlation of two functions, h and g, is given by:h⊗g=FT−1{HG}=−∞h(x′)g(x′−x)dx′where H and G are the Fourier transform of h and g, respectively, H is the complex conjugate of H, and FT denotes the Fourier transform. The correlation of the pattern with itself will produce a maximum (called ‘correlation-peak’) in the center of the correlation image. If

Fractional Fourier-based correlators and their application to range control

The common correlators provide information regarding the translation of a certain object or about its scaling and rotation. In some visual systems there is no need to know every type of motion, and the only interesting movements are limited to a specific range. For example, if a robot is moving around in a room consisting of known obstacles, it should not come near an obstacle if it is at a certain distance away. We would like to be able to detect only the nearby obstacles and ignore the

Time-to-impact computation

Fig. 1a depicts the schematic implementation of the Fourier-based correlators in optics, used in detecting translations, and Fig. 1b depicts the schematic implementation of the Mellin-based correlator in optics, used in detecting scaling and rotation. Here, we concentrate on the Mellin based correlator and its usage in estimating time to impact. Although we use it in a computational fashion, it can be incorporated into a single lens commercially, and therefore, can be obtained at image

Summary

This paper introduced different types of correlators that can find applications in common tasks of visual navigation. The major advantage of the correlators is the fact that they can be implemented optically, and therefore can be obtained at image acquisition time and can be obtained at real time. The correlators discussed here are: (a) The Mellin based correlator that can be used in computation of time to impact. (b) The Fractional Mellin and fractional Fourier-based correlators (first defined

About the Author—DIDI SAZBON received her B.Sc. degree in Computer Science and her M.Sc. degree in Biomedical Engineering from the Technion, Israel Institute of Technology. Currently she is a Ph.D. student in the Computer Science Department at the Technion, Israel Institute of Technology. Her current research interests are in computer vision and robot navigation.

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About the Author—DIDI SAZBON received her B.Sc. degree in Computer Science and her M.Sc. degree in Biomedical Engineering from the Technion, Israel Institute of Technology. Currently she is a Ph.D. student in the Computer Science Department at the Technion, Israel Institute of Technology. Her current research interests are in computer vision and robot navigation.

About the Author—ZEEV ZALEVSKY received his B.Sc. degree in Electrical Engineering from the Tel-Aviv University, Israel, in 1993. He completed his Ph.D. in the Department of Physical Electronics at the Tel-Aviv University in 1996. His major research interests are optical signal processing, optical pattern recognition, super resolution, temporal optics, computer vision, and optic communication.

About the Author—EHUD RIVLIN received his B.Sc. and his M.Sc. degrees in Computer Science and his M.B.A. degree from the Hebrew University in Jerusalem, Israel, and his Ph.D. degree from the University of Maryland. Currently he is an Associate Professor in the Computer Science Department at the Technion, Israel Institute of Technology. His current research interests are in machine vision and robot navigation.

About the Author—DAVID MENDLOVIC received his B.Sc. and his Ph.D. degrees in Electrical Engineering from the Tel-Aviv University, Israel, in 1987 and 1990, respectively. At present, he is a Professor in the Department of Physical Electronics. His research interests include optics for computers, optical signal processing, diffractive optics, three-dimensional surface sensing and temporal optics.

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