Finite Element Method for Rectification of Architectural Heritage Images of National Importance
2. Polynomial Transformation: In view of the propagated error or orderly and systematic deformation often a polynomial transformation is found to be convenient.
3. Projective Transformation: It uses following equations:
This indicates that the projectivity between two planes is uniquely determined if a total of eight coefficients are known. This requires at least four points whose both X and Y coordinates in both the spaces are known. (Theodoropoulou)
The Finite Element Concept
In finite element methods, the variation of displacements etc., in the element is expressed as a function of its nodal values, which is known as displacement function, or interpolation function or shape function. The representation of geometry in terms of (non linear) shape functions can be considered as a mapping procedure, which transforms a regular shape, like a straight-sided rectangle in local coordinate system into a distorted shape, like a curved sided rectangle in global cartesian coordinate system. A displacement function is commonly assumed in a polynomial form, and practical considerations limit the number of terms that can be retained in the polynomial. An exact solution for displacement u (x), is approximated by various degree polynomials of the general form, as given below (Reddy, 1985):
n(x)=a
1+a
2x+a
3x
2+a
4x
3.................a
n+1x
n (4)
In Eq. 4, x is one-dimensional variable and ai’s are (4) coefficients. The greater the number of terms included in the approximation, the more closely the exact solution is represented (obviously, this last statement does not apply to the case wherein the exact solution is a polynomial of some finite order ‘m’. Here terms in excess of ‘m’ do not improve the representation).
