Effects of Window Size and Shape on Accuracy of Subpixel Centroid Estimation of Target Images

A new algorithm is presented for increasing the accuracy of subpixel centroid estimation of (nearly) point target images in cases where the signal-to-noise ratio is low and the signal amplitude and shape vary from frame to frame. In the algorithm, the centroid is calculated over a data window that is matched in width to the image distribution. Fourier analysis is used to explain the dependency of the centroid estimate on the size of the data window, and simulation and experimental results are presented which demonstrate the e ects of window size for two di erent noise models. The e ects of window shape have also been investigated for uniform and Gaussian-shaped windows. The new algorithm has been developed to improve the dynamic range of a close-range photogrammetric tracking system that provides feedback for control of a large gap magnetic suspension system (LGMSS). Introduction Centroid-estimation algorithms have long been used in digital imaging to locate target images to subpixel accuracies. Applications of centroid estimation include star tracking (ref. 1), point and edge detection for machine vision (ref. 2), close-range photogrammetry (ref. 3), and motion analysis. The e ects of sampling and noise on the accuracy of the centroid estimate for point source images, images of extended sources, and edge detection have been analyzed previously and documented by several authors (refs. 4 to 8). The systematic errors due to undersampling that have been described for centroid estimation are common to all interpolation algorithms and have been analyzed from the point of view of performing image reconstruction (refs. 9 and 10). In these previous analyses, experimental approaches as well as analytical approaches based on Fourier techniques have been used to quantify the errors due to noise, quantization, and sample spacing. To date, the e ect of window size on the accuracy of subpixel centroid estimation has been limited to a qualitative analysis derived from experiments that measured the error in centroid estimation as a function of di erent N -point algorithms (ref. 4). In this paper, Fourier techniques are used to analyze the dependency of the systematic error on window size. In addition, the e ects of window size and shape on subpixel centroid-estimation accuracy in the presence of noise are studied. It is shown that there can be an advantage to using a shaped window for centroid estimation of point target images for signals that vary in amplitude and width, provided the pixel-to-pixel noise is independent of signal amplitude. A brief review of the e ects of the optical point spread function (PSF), target size, and sample spacing on systematic error is provided in order to compare and contrast these e ects with those attributable to the data window. Quantization e ects, however, are not addressed in this paper. In many applications involving centroid estimation, the signal shape and amplitude are either controllable or xed. In these cases, the optimum sample spacing and window size relative to the target image distribution are known a priori, and a correction can be applied for systematic errors (ref. 7). In other applications where the noise is small or the image is averaged over several frames, a larger window (relative to the distribution) can be used in the centroid calculation. This eliminates systematic errors arising because of truncation of the signal. In the application discussed in this paper, centroid estimation is used to locate images of point targets along a linear charge-coupled device (CCD) detector, where the signal-to-noise ratio is small and, because the targets are moving, the (one-dimensional) images vary in size and amplitude. For these reasons, it is not possible to apply a correction for the systematic errors or to calculate the centroid using a large xed data window. The application discussed herein is the optical measurement system (OMS) for the large gap magnetic suspension system (LGMSS) ( g. 1). In the OMS, small infrared light-emitting diode (LED) targets have been embedded in the top surface of a rigid cylindrically shaped element that contains a permanent magnet core. The element is magnetically levitated above a planar array of electromagnets. Sixteen linear CCD cameras arranged in pairs are located symmetrically about and above the levitated cylinder. A total of eight targets are located along the top surface of the cylinder ( g. 2), and the targets are multiplexed in time for target identi cation. Triangulation techniques are used to determine the position and attitude of the levitated cylinder from the locations of the projected target images in the 16 cameras. The position and attitude information is supplied to the electromagnet controller to stabilize levitation of the cylinder and to control motion in six degrees of freedom. The position and attitude of the cylinder are determined using weighted least squares. An estimate of the error in each computed centroid value is passed along with the centroid value and is used to establish a weighting factor for the particular camera measurement. In order to achieve the required accuracy in the estimate of position and attitude, it is necessary to locate the centroids of at least 6 of the target images to 1/15 of a pixel in a minimum of 12 of the 16 cameras. As the cylinder, and hence each target, moves over the eld of view, both the amplitude and the width of the target images vary ( g. 3). If the centroid location of a target image is determined with a xed window size (which is the same for all 16 cameras), and the window size is optimum for those image distributions falling in the midrange of possible values, then as the target images vary in amplitude and width, the error in the centroid estimate grows for those images for which the distribution falls outside the midrange. Thus, with a xed window size, the accuracy of the centroid estimate falls o because of noise or systematic error, and the accuracy of the position and attitude estimate correspondingly decreases. In order to increase the dynamic range of the system, an algorithm has been developed to adjust the width of the centroid window as the light intensity distribution of the image varies. This algorithm provides the minimum error in centroid estimation over the maximum range of signal amplitude and shape. This paper analyzes the dependency of systematic and noise-induced errors on the size and shape of the data window. Experimental results are presented and compared with the results of numerical simulations. The following analysis is limited to one spatial dimension. It is assumed that the PSF of the imaging optics can be approximated by a Gaussian function. Because a two-dimensional Gaussian function is separable in x and y, the results of the one-dimensional analysis can readily be extended to two dimensions. Symbols and Abbreviations b half-width of Gaussian distribution used to simulate the optical point spread of the imaging optics comb(x=xs) sampling function f(x) light intensity distribution of onedimensional target image F ( ) Fourier transform of f(x) e F ( ) = F ( )R( ) FRW = FR * W FSW( ) Fourier transform of fSW(x) F 0 SW( ) rst derivative of FSW( ) with respect to fSW(x) sampled and windowed version of f(x) g(x xo) one-dimensional light intensity distribution of target located at xo H( ) general function of N number of pixels corresponding to half the window width PSF(x) point spread function of imaging optics R( ) Fourier transform of r(x) r(x) pixel response function T ( ) general function of W ( ) Fourier transform of w(x) w(x) window function x spatial variable x centroid of continuous onedimensional light intensity distribution xc amount by which f(x) is shifted with respect to the sampling grid xp pixel location of the peak signal xs sample spacing xSW centroid of fSW(x) c systematic error, x xSW ; ; variables of integration Abbreviations: CCD charge-coupled device LED light-emitting diode rms root-mean-square SNR = Peak signal Standard deviation of background signal A prime on a symbol denotes the rst derivative. An asterisk used as an operational sign denotes convolution of the quantities.