Phase-only Correlation Function by Means of Hartley Transform

In image processing or pattern recognition, Fourier Transform is widely used for frequency-domain analysis. In particular, the Phase Only Correlation (POC) method demonstrates high robustness and accuracy in the pattern matching and the image registration. However, there is a disadvantage in required memory machine because of the calculation of 2D-FFT. In this case, Hartley transform can be a very good substitute for more commonly used Fourier transform when the real input data are concerned. The Hartley transform is similar to the Fourier transform, but it is free from the need to process complex numbers. It also has some distinctive features that make it an interesting choice when a greater efficiency in memory requirements is needed. In this paper we show the correspondence between the Phase-Only Correlation (POC) function obtained by means of FFT and by FHT. ABBREVIATIONS FFT: Fast Fourier Transform; FHT: Fast Harley Transform; POC: Phase-Only Correlation; PIV: Particle Image Velocimetry INTRODUCTION The automatic determination of similarity between two structured data sets is fundamental to the disciplines of pattern recognition and image processing [1,2].The cross-correlation between two data sets is a good method to measure their similarity. The cross-correlation function is used extensively in pattern recognition and signal detection. We know that projecting one signal onto another is a means of measuring how much of the second signal is present in the first. Since a digital image can be considered as a dataset, cross-correlation can be used to detect the similarity and the lagging between the images. In general, the standard cross correlation [3,4] yields several broad peaks and a main peak whose maximum is not always; therefore, it is difficult to locate the maximum; see Figure 1 (c). One of the alternatives to the cross-correlation function is the Fourier Phase-Only Correlation (POC) function [5,6]. The POC function yields an even sharp maximum at the best match point, as shown in Figure 1(d); therefore, it is easy to locate the maximum. The Phase-Only Correlation (POC) method demonstrates high robustness and accuracy in the pattern matching and the image registration. However, there is a disadvantage in required memory machine because of the calculation of 2D-FFT. The Fast Hartley Transform (FHT) can be a valid alternative to Fast Fourier Transform (FFT) [7].The Hartley transform [8,9] resembles a Fourier transform but it is free from the need to process complex numbers. The Hartley transform also has some better properties and faster algorithms than the Fourier one, therefore it can represent a valid alternative, particularly interesting when a greater efficiency in memory requirements is needed. In this work we present a Hartley transform algorithm for accurate and fast elaboration of POC. This paper is organized as follows: Section 2 give the definition of POC function and its basic properties, the properties of Harley transform and the definition of POC in Hartley space. Section 3 presents a set of experiments for evaluating performance of the proposed methods. In section 4, we end with some conclusions. MATERIALS AND METHODS Phase-Only Correlation (POC) The Phase-Only Correlation (POC) function (or simply “phasecorrelation”) has been successfully applied to high-accuracy image registration tasks for computer vision applications [10,11], for Fingerprint Matching [12], for Iris Recognition [13], Palmar Recognition [14],in PIV analysis [15,16] and in security application [17-19]. Phase correlation algorithm uses the cross-power spectrum to get the translation factor between two images. Assume that there are two images ( ) 1 , I x y and ( ) 2 , I x y and the translation between them is as following: ( ) ( ) 2 1 0 0 , , I x y I x x y y = − − . (1) The Fourier transformation: ( ) ( ) ( ) 2 1 0 0 , , exp F u v F u v j u x v y = ⋅ − +     . (2) In equation (2), ( ) 1 , F u v and ( ) 2 , F u v are the Fourier transform of ( ) 1 , I x y and ( ) 2 , I x y . The cross-power Central Cozzella et al. (2014) Email: JSM Math Stat 1(1): 1004 (2014) 2/8 spectrum will be: ( ) ( ) ( ) ( ) ( ) 1 2 0 0 1 2 * * * * , , exp , , F u v F u v j u x v y F u v F u v ⋅ = − +     ⋅ (3) with ( ) 1 , F u v the conjugate function of ( ) 1 , F u v . We can get an impulse function ( ) 0 0 , x x y y δ − − about the value of translation invariant 0 x and 0 y by using Fourier inverse transforming to equation (3). In other words, the phase correlation surface is defined as: ( ) ( ) ( ) ( ) ( ) 1 1 2