u displacement field is applied to the surface and a final distribution of intensities If(x,y) is obtained. It is assumed that the light intensity changes are only a function of the displacement field and as it is the case in all experimental methods noise is present. Noise is indicated as all the changes of intensity that are not caused by the displacement field. The displacement field is defined by the function [39], [41], ∧ ∧ + = D(x, y) u(x, y) i v(x, y) j (29) From the preceding assumption, n i i i i i i i fI (x u,y v) I(x,y) I(x u,y v) I +Δ +Δ +Δ + + + = (30) In (22) ΔI is the change of intensity caused by the rigid body motion plus the local deformation of the analyzed surface. In (30) the assumption that the light intensity is modified only by the displacements is implicit. The term In refers to all other causes of change of intensity. The validity of (30) boils down to the signal to noise ratio. To develop the model one has to postulate that the signal content of In is small and hence can be neglected. The problem to be solved is to find u (x,y) and v (x,y) knowing Ii(x,y) and If(x+u,y+v). The solution of the above problem requires the regularity of the functions u(x,y) and v(x,y) implicit in the Theory of the Continuum. One can formulate the problem as an optimization problem, that is find the best values of these two functions that minimize or maximize a real function, the objective function of the optimization process. There are many criteria that can be utilized for this purpose. One criterion is the minimum squares; the difference of the intensities of the two images must be minimized as a function of the experienced displacements. Calling Φ(u,v) the optimization function. [ ] I (x u,y v) I (x ,y ) dxdy (u,v) 2 i i i i i f + + − Φ =∫∫ (31) For small u(x,y) and v(x,y) the above expression can be expanded in a Taylor series and limiting the expansion to the first order and using vectorial notation as, [ ] I (r) I (r) D(r) I (r) dxdy (D) 2 f i f − + •∇ Φ =∫∫ (32) In the above equation r is the spatial coordinate; D(r) is the displacement vector and∇is the gradient operator. Equation (32) tells us that the gradient of If provides the following information; the displacement information is associated with the gradient of the intensity distribution. If is a scalar function (light intensity), the gradient is a vector and going back to Figure 28 the vectors displacements are plotted following the vectors joining the centers of correlation peaks of the sub-images. Hence the displacement information can be retrieved following the gradient function of the light intensity.The minimization of the objective functions is then a central problem of the image digital correlation technique. In the technical literature there is a large variety of approaches to this problem. One can utilize criteria other than the minimum squares [42]-[44]. (a) Figure 29. Field for the correlation process. (a) Dotted rectangle, NsxNs sub-element, δ mesh of the region of interest. (b) Displacement experienced by the sub-image with components u and v. (b) 173
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