BioIngenium Group
Centro de Telemedicina
Universidad Nacional de Colombia
 
     
  Distributed Genetic Algorithm for Subtraction Radiography  
     
  2. Methodology  
 
Image Acquisition
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.

Preprocessing
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.

Parametric Transformations
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.

Interpolation Approach
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.

Similarity Measure
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 as
 
 
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.
 
     
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