ORTHOGONAL 3-D RECONSTRUCTION USING VIDEO IMAGES
Ilkka Niini
Researcher
Helsinki University of Technology
Institute of Photogrammetry and Remote Sensing
Otakaari 1
SF-02150 Espoo
Finland
IWG V/III - Image Sequence Analysis
KEY WORDS: 3-D Reconstruction, Camera Calibration, Radial Distortion, Close-Range, Block Adjustment, Orientation
ABSTRACT
This paper deals with the projective block adjustment method developed at Helsinki University of Technology. Especially, the
question how the radial distortion is corrected in the method is discussed. The method is based on the projective singular correlations
between the images in the block, and it can be used to obtain a 3-D orthogonal coordinate system for an arbitrary block of images.
Linear distortions do not cause any problems in the computation of the singular correlations, and they can be determined from the
singular correlation parameters afterwards. Nonlinear image errors, such as radial distortion, instead, cause significant error in the
determination of the correlation parameters. This effect makes it also possible to determine the radial distortion simultaneously with
the singular correlation parameters.
The solution of the radial distortion requires approximate interior orientation to be known, and an improved interior orientation is
computed afterwards from the singular correlation parameters. The cycle can then be repeated, and a rigid solution can be obtained,
bringing the whole block into an orthogonal 3-D coordinate system which is free from nonlinear distortions caused by the radial
distortion of the images. The resulting coordinate frame can then be utilized in further 3-D reconstruction of the scene, for example,
using digitized video images.
1. INTRODUCTION
This article describes a method to obtain full orthogonal 3-D
model coordinate system using pure image information only.
The method is based on the projective singular correlation
between the images taken from the same object. Singular
correlation is also known in the computer vision society as the
epipolar transformation /Maybank et al., 1992/.
The principle of the method was presented in /Niini, 1994/, and
in /Niini, 1995/. Now, the method has been extended so that
possible radial distortion of the images can also be computed.
This is a significant improvement since radial distortion is
usually quite large when video cameras are used.
The original projective block adjustment method has four
sequential parts: solution of the singular correlations, solution of
the interior orientations, solution of the rotation matrices, and
the solution of the relative positions of the images along with
the model coordinates. The original method assumed the images
to be free from nonlinear distortions, or possible nonlinear
distortions had to be corrected in advance /Niini, 1994,1995/.
2. NEW BLOCK ADJUSTMENT
The solution of the radial distortion requires approximate interior
orientation to be known. This means that the solution of radial
distortion and singular correlations, and the solution of interior
orientations should actually proceed simultaneously.
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
In the new block adjustment method, however, to keep the
method similar to the older version, they are solved in a cyclic
iteration. First, solve the singular correlations with the radial
distortion coefficients using approximate interior orientation.
Second, solve the interior orientation only. Usually with small
radial distortion, only the approximate image center is needed.
Repeat the procedure until the radial distortion and interior
orientation do not any more change significantly.
Starting from reasonable approximate values, the total iteration
converges satisfactorily in a few loops. However, the total
number of iterations may still be much larger due to the nested
iterations since both steps are also iterative.
The radially undistorted image coordinates are also obtained,
and the resulting interior orientation corresponds to the one
which transforms the undistorted image coordinates to the ideal
image coordinates which strictly fulfil the collinearity
conditions. The least squares solution of the system is based on
the minimization of the noise in the original image observation
space.
The third and fourth parts of the block adjustment method, the
solution of the image rotation matrices, base vectors and model
coordinates remain the same as before, except that the image
coordinates used in these steps are now free from the radial
distortion making it possible to obtain pure orthogonal 3-D
model coordinate system without control points or a priori
information about the block geometry.
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