Ten intra-oral radiographs pairs, taken at different occasions, were randomly selected from an unrelated study of
periodontal therapy. Patients were placed within a support that maintained the head in a fixed and comparable position.
Geometrical registration of radiographs was obtained using occlusal registration film holder devices. The film holders
were coupled mechanically to the cone of an x-ray machine. Radiographs were then digitized in a HP 3570 scanner using
a transparent material adapter at 600 x 600 DPI resolution, producing 797 x 637 pixel images.
Even though acquisition conditions are standardized as much as possible, illumination differences are inevitable. Thus,
the histogram of the floating image is equalized by using the reference image luminances. This transformation first computes
the histogram of each image and then luminances are homogeneously distributed in the floating image according to the levels
found in the reference one.
Small tissue deformations are conveniently modeled using affine or projective transformations. The affine transformation
implemented in this analysis is defined as
where S is the scale matrix, R the rotation
matrix and t the displacement vector. The parameters involved are therefore the scale factor, the rotation angle
and the horizontal and vertical translations.
In terms of linear interpolation, the reconstructed signal is obtained by convolution of the discrete signal (defined
as a sum of Dirac functions) with a convenient selected kernel. We used spline interpolation due to its accuracy and acceptable
computing speed. Spline interpolation of order n is uniquely characterized in terms of a B-spline expansion
which involves integer shifts of the central B-spline. The parameters of the spline are the coefficients c(k).
In the case of images with regular grids, they are calculated at the beginning of the procedure by recursive filtering.
A three order approximation was used in the present work.
The concept of functional dependence, fundamental in statistics, provides the framework for computation of similarity
between two images. This means that we consider the image as a random variable and its histogram as the probability density
function. Furthermore, the 2-D histogram of one pair of images is considered as the joint probability density function.
Thus when a pixel is randomly selected from an image X having N pixels, the probability of getting an intensity
i is proportional to the number of pixels, Ni, in X having intensity i, i.e.
The measure of the functional dependence between two random variables used was the correlation ratio (A. Roche, G. Malandain,
X. Pennec and N. Ayache, "Multimodal Image Registration by Maximization of the Correlation Ratio"), defined
Unlike the correlation coefficient which measures the linear dependence between two variables, the correlation ratio measures
the functional dependence. In practice the correlation ratio is calculated from
This last equation expresses the fact that the variance can be decomposed as a sum of two energy terms (A. Roche, idem):
a first term Var[ E( Y/X ) ] that is the variance of the conditional expectation and measures the part of Y
which is predicted by X, and a second term Ex[ Var( Y/X ) ] which is the conditional variance
and stands for the part of Y which is functionally independent of X.