. September 1-3, 2010
In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds), IAPRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3, 2010
w \ . ■ /V Vs.;'* * -
EFFECT OF SAMPLING IN CREATING A DIGITAL IMAGE
ON MEASUREMENT ACCURACY OF CENTER LOCATION OF A CIRCLE
R. Matsuoka a,b ' *, M. Sone a , N. Sudo a . H. Yokotsuka a , N. Shirai b
a Tokai University Research & Information Center, 2-28-4 Tomigaya, Shibuya-ku, Tokyo 151-0063. JAPAN
(ryuji. sone3)@.yoyogi.ycc.u-tokai.ac.jp, (sdo, yoko)@keyaki.cc.u-tokai.ac.jp
b Kokusai Kogyo Co., Ltd.. 2-24-1 Harumi-cho, Fuchu. Tokyo 183-0057, JAPAN
(ryujimatsuoka, naoki shirai) @kkc.co.jp
Commission III, WG I1I/1
KEY WORDS: Analysis, Sampling, Measurement, Accuracy, Digital
ion (second row)
ABSTRACT:
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Our previous paper reported an experiment conducted in order to evaluate measurement methods of the center location of a circle by
using simulated images of various sizes of circles. The variances of the measurement errors of the centroid methods in the
experiment appeared to oscillate on a one-pixel cycle in diameter. This paper reports an analysis of the dependence of the
measurement accuracy of the center location of a circle by the centroid methods on its diameter. Two centroid methods: intensity-
weighted centroid method (WCM) and unweighted centroid method using a binary image created by thresholding (BCM) are
investigated. Since general expressions representing the measurement accuracy by both WCM and BCM are unable to be obtained
analytically, the variances of the measurement errors by both methods are obtained by numerical integration. From the results by
the numerical integration, we conclude that sampling in digitization would cause the measurement accuracy of the center location of
a circle by both WCM and BCM to oscillate on a one-pixel cycle in diameter. The results show' that the variance of measurement
errors by WCM can be expressed by the combination of the inverse proportion to the cube of the diameter and the oscillation on a
one-pixel cycle in diameter. On the other hand, the variance of the measurement errors by BCM should approximate to the
combination of the inverse proportion to the diameter and the oscillation on a one-pixel cycle in diameter.
1. INTRODUCTION
Circular targets are often utilized in photogrammetry,
particularly in close range photogrammetry. Since a circle is
radially symmetrical, circular targets are w'ell suited for
photogrammetric use such as camera calibration and 3D
measurement. It is said that determination of the center of a
circular target by digital image processing techniques is
rotation-invariant and scale-invariant over a wide range of
image resolutions. The center of a circular target can be
estimated by centroid methods, by matching with a reference
pattern, or by analytical determination of the circle center
(Luhmann et ah, 2006).
Our previous paper (Matsuoka et a!., 2009) reported an
experiment conducted in order to evaluate measurement
methods of the center location of a circle by using simulated
images of various sizes of circles. We investigated tw'o centroid
methods: intensity-weighted centroid method and unweighted
centroid method using a binary image created by thresholding,
and least squares matching in the experiment. We made the
experiment by the Monte Carlo simulation using 1024
simulated images of w'hich the centers were randomly
distributed in one pixel for each circle. The radius of a circle
was examined at 0.1 pixel intervals from 2 to 40 pixels. The
variances of measurement errors by both centroid methods in
the experiment appeared to oscillate on a 0.5 pixel cycle in
radius, even though the formula to estimate the center of a
circle by each centroid method does not seem to produce such
cyclic measurement errors. We wondered whether the
oscillation of the measurement accuracy by the centroid
methods might be caused by sampling in creating a digital
image.
Bose and Amir (1990) reported the investigation of the effect of
the shape and size of a square, a diamond, and a circle on the
measurement accuracy of its center location by the unweighted
centroid method using a binarized image. They conducted the
analysis of the measurement accuracy of the center location of a
square and showed the standard deviations of the measurement
errors of the center location of a square derived from the
variances of the measurement errors of the center location of a
line segment. How'ever, w'e confirmed that their study would be
incomplete and the measurement accuracy of the center location
of a square from 2 to 22 pixels in side shown in their paper is
that when the side of a square is infinite. Moreover, they
executed the simulation on the measurement accuracy of the
center location of a circle. In their simulation, 400 binarized
circles were placed at 0.05 pixel intervals covering a range of
one pixel in x and y direction, and the radius of a circle was
examined at merely 0.25 pixel intervals. Consequently, there
was no mention finding cyclic measurement errors of the center
location of a circle in their paper.
This paper reports an analysis of the dependence of the
measurement accuracy of the center location of a circle by
centroid methods on its diameter. Two centroid methods:
intensity-weighted centroid method and unweighted centroid
method using a binary image created by thresholding were
investigated.
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