MEASUREMENT PRECISION AND ACCURACY OF THE CENTRE LOCATION OF AN ELLIPSE BY WEIGHTED CENTROID METHOD

Abstract. Circular targets are often utilized in photogrammetry, and a circle on a plane is projected as an ellipse onto an oblique image. This paper reports an analysis conducted in order to investigate the measurement precision and accuracy of the centre location of an ellipse on a digital image by an intensity-weighted centroid method. An ellipse with a semi-major axis a, a semi-minor axis b, and a rotation angle θ of the major axis is investigated. In the study an equivalent radius r = (a²cos²θ + b²sin²θ)^1/2 is adopted as a measure of the dimension of an ellipse. First an analytical expression representing a measurement error (ϵ_x, ϵ_y,) is obtained. Then variances V_x of ϵ_x are obtained at 1/256 pixel intervals from 0.5 to 100 pixels in r by numerical integration, because a formula representing V_x is unable to be obtained analytically when r > 0.5. The results of the numerical integration indicate that V_xwould oscillate in a 0.5 pixel cycle in r and V_x excluding the oscillation component would be inversely proportional to the cube of r. Finally an effective approximate formula of V_x from 0.5 to 100 pixels in r is obtained by least squares adjustment. The obtained formula is a fractional expression of which numerator is a fifth-degree polynomial of {r−0.5×int(2r)} expressing the oscillation component and denominator is the cube of r. Here int(x) is the function to return the integer part of the value x. Coefficients of the fifth-degree polynomial of the numerator can be expressed by a quadratic polynomial of {0.5×int(2r)+0.25}.