New Carré Equation

Abstract

The present work offers new equations for phase evaluation in measurements. Several phase-shifting equations with an arbitrary but constant phase-shift between captured intensity signs are proposed. The equations are similarly derived as the so called Carré equation. The idea is to develop a generalization of the Carré equation that is not restricted to four images. Errors and random noise in the images cannot be eliminated, but the uncertainty due to their effects can be reduced by increasing the number of observations. An experimental analysis of the errors of the technique was made, as well as a detailed analysis of errors of the measurement. The advantages of the proposed equation are its precision in the measures taken, speed of processing and the immunity to noise in signs and images.

Separation of Isochromatics and Isoclinics Phasemaps for the Photoelastic Technique with use Phase Shifting and a Large Number of High Precision Images

“…The motivation for choosing this mathematical model is the success achieved in [11] with a similar model. The idea of the mathematical model is to maximize the coefficients (b r,s , c r,s , e r,s , f r,s ) so that their values are large enough (not close to zero) to make them significant in the equation obtained.…”

“…By analogy with the equations of phase calculation used by other authors and the mathematical model proposed in [11], we had the idea to try a new general model for the equations of phase in photoelasticity. After many different attempts, a general equation for calculating the phase for any number, N, of images is proposed: (10) where N is the number of images, b r,s and e r,s are coefficients of the numerator, c r,s and f r,s are coefficients of the denominator, and r and s are the indices of the sum [11].…”

“…where M is the number of pixels of the image and  i e and  i e are the exact value calculated by (6)(7)(8)(9)(10)(11)(12) for the disk. The values of  i and  i are calculated by the new equation.…”

“…After many different attempts, a general equation for calculating the phase for any number, N, of images is proposed: (10) where N is the number of images, b r,s and e r,s are coefficients of the numerator, c r,s and f r,s are coefficients of the denominator, and r and s are the indices of the sum [11].…”

“…3 shows an example of this procedure. To compare the new equations for calculating the phase, nine sets of photos with Step set to 3,4,6,7,10,11,16,19, and 31 were generated. In each set, the angle  of the analyzer is varied (): 45º, 30º, 18º, 15º, 10º, 9º, 6º, 5º, and 3º, respectively.…”