Finite Element Method for Rectification of Architectural Heritage Images of National Importance
The parametric approach uses a reverse ray tracing to determine the grey values for all pixel positions in the rectified image. Re-sampling of the grey values requires the position of corresponding position in the distorted image. This can be calculated by the collinearity equations using interior and exterior orientation parameters and the position of the corresponding point in the object coordinate system. The interior orientation may be known from a calibration protocol of the image and a measurement of fiducial marks or a reseau. The exterior orientation parameters can be calculated by a resection in space using at least three control points or a bundle adjustment. Interior and exterior orientation parameters can also be estimated together within a spatial resection. In this case, the spatial distribution of the control points plays an important role in order to obtain the accurate results.
In case of non-parametric approaches, a planar coordinate transformation between the matrices of the distorted and the rectified image is used. The object coordinates of the control points have to be first reduced to 2D-coordinates on the projection surface.
In both the methods, least square adjustment is used, if number of control points are more than the required points. Further, in these methods control points should be distributed all over the image in a systematic way, so that the major portion of the image lies inside the control points, as these methods propagate errors in extrapolation. Generally, control points are available with varying accuracy, and therefore with the help of least square adjustment, error is distributed in all the control points even with better accuracy.
The finite element method was originally developed for structural analysis, but the general nature of the theory on which it is based, has made its application possible in other fields of engineering also. In finite element method, it is possible to express the displacements of any point within the element in terms of its nodal displacements using the shape functions. In other words, if a rectangular image is transformed into any arbitrary quadrilateral shape, the new pixel coordinates for each pixel in the parent image can be computed, if its nodal displacements are known.
Common Analytical Rectification Methods
1. Affine Transformation: assumes that there exist two different scales along both the directions X and Y.
This has six unknown parameter, a’s and the b’s, although the real unknowns are only five, viz., S
x , S
y , K, and X0 Y0
