International Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 5. Hakodate 1998 
ALGORITHM OF SPACE RESECTION AND ITS EVALUATION 
Nobuo MAKIMOTO f 
Jun-ichi TAGUCHI ' 
! Systems Development Laboratory, Hitachi Ltd., Kawasaki, Japan 
Commission V, Working Group 1 
KEYWORDS: SPACE RESECTION, ORIENTATION, SATELLITE IMAGE PROCESSING, 
COMPUTER VISION, NONLINEAR OPTIMIZATION, NUMERICAL ANALYSIS 
ABSTRACT 
Space resection is a technique for calibrating camera parameters, mainly the angle and the position, by looking 
at the images of GCPs (points on the ground with known position) on a picture taken by the camera and is an 
indispensable technique in the field of satellite image processing. In this paper we describe a new algorithm for 
space resection. First, and most important, we derive space resection equations in a new form. Specifically, we 
split the least-square equation into two parts by square completion: one is an equation only of the angle; the 
other is an explicit formula to calculate the position directly from the angle. We also derive formulae for the 
variances of the errors of the least-square estimators. We also describe a method for numerical solution of the 
angle equation. The difficulty is that the equation has a few tens of fake solutions. To find the true solution 
(global minimum) among these fake solutions, we propose some techniques of nonlinear optimization. Evaluation 
of the proposed method by computer simulation shows that the global minimum is obtained efficiently when 
four or more GCPs are available. The simulation results also show that the variance formulae are valid, and 
this means we can guarantee the precision of the estimated camera parameters. The proposed optimization 
techniques are applicable to a wide range of nonlinear optimization problems other than space resection. 
1 INTRODUCTION 
e Error analysis: 
We derive practical variance formulae and prove 
1.1 Background and Summary the approximate efficiency of the estimator. 
Space resection, a technique for finding the angle À numerical simulation shows that the proposed nu- 
and the position of a camera from a picture taken merical solution is very stable and that we can guar- 
by it, is indispensable in the field of satellite image antee the precision of the estimator by the variance 
processing. formulae. 
It is usually formulated as a nonlinear least-squares 
estimation, but is not easy to solve because of its : 
) y 1.2 The Problem of Space Resection 
search space is large and there is a large number of 
local minima. In this paper we focus on the follow- First let us introduce some notations. The camera 
ing aspects of space resection (though some are in the model is illustrated in Fig.l. Xy is the position, or 
appendix). the viewpoint, of the camera. The orthonormal frame 
: R = (eq, ez, e3) represents the orientation of the cam- 
e New equations: : : : 
era. The image plane is spanned by ej, es and is sep- 
arated from X, by the focal length c. A GCP (ground 
control point) is a point on the ground whose posi- 
We split the conventional equation into two small 
parts by some elementary linear algebra. 
e Numerical solution: tion (latitude, longitude, and height) is known in ad- 
We propose "absolute Newton method” and some vance. Each GCP X; is imaged onto the point x; on 
other techniques of nonlinear optimization. the plane by the central projection with its center Xj. 
110 
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