θ kð Þ¼arctg ky ffiffiffiffiffiffiffiffiffiffiffiffiffifffii k2 x þk 2 y q ð1:20Þ At every given point of a cosinusoidal fringe field defined by the position vector r ¼xi þyj there is a phasor represented by Eq. (1.1) and the local orientation of the signal defined by the angle θ. The addition of one more parameter, the angle θ requires to extend the definition of the gray levels as a potential scalar functionVr • in a 2D space, wherer ¼ xi þyjis an upper script that indicates that the potential corresponds to a point inℛ2, the 2D continuum. Following the same steps applied in two dimensions and recalling that levels of gray are scalar quantities, gradVr • ¼G3 rð Þ¼ ∂Vr • ∂xc i) þ ∂Vr • ∂yc j) þ ∂Vr • ∂zc k) ð1:21Þ where the versors i), j)and k)indicate a Cartesian coordinates system in a 3D complex space, the subscript “3” indicates that one is dealing with a 3D vector in the complex space. The divergence of the field is determined as: ∇•G3 rð Þ¼ ∂2 Vr • ∂xc 2 þ ∂2 Vr • ∂yc 2 þ ∂2 Vr • ∂zc 2 ð 1:22Þ Calling Vr xc • ¼∂V r • ∂xc , Vr yc • ¼∂V r • ∂yc , Vr zc • ¼∂V r • ∂zc , and computing the vectorial product of the ∇operator with the vector G3(r), it follows: ∇ G3 rð Þ¼ ∂Vr zc • ∂yc ∂Vr yc • ∂zc 0 @ 1 A i) þ ∂Vr xc • ∂zc ∂Vr zc • ∂xc 0 @ 1 A j) þ ∂Vr yc • ∂xc ∂Vr xc • yc 0 @ 1 A k) ð1:23Þ Since we are dealing with a scalar potential, the divergence is zero. Hence, it can be written: ∂2 Vr • ∂xc 2 þ ∂2 Vr • ∂yc 2 þ ∂2 Vr • ∂zc 2 ¼ 0 ð1:24Þ Equation (1.24) indicates that the potential Vr • satisfies the Laplace’s equation in the complex 3D space. The meaning of this equation is the same as for two dimensions. The fact that the rotor is zero implies, ∂Vr xc • ∂yc ¼ ∂Vr yc • ∂zc ¼ ∂2 Vr • ∂yc∂zc ð 1:25Þ ∂Vr xc • ∂zc ¼ ∂Vr zc • ∂xc ¼ ∂2 V • ∂xc∂zc ð 1:26Þ ∂Vr yc • ∂xc ¼ ∂Vr xc • ∂yc ¼ ∂2 Vr • ∂xc∂yc ð 1:27Þ Equations (1.26)–(1.28) are the conditions for the existence of a scalar potential in the 3D complex space and are equivalent to the Cauchy-Riemann conditions in the two dimensional case. The above derivations indicate that the information contained in a 2D cosinusoidal fringe pattern is described mathematically by a conservative 3D vectorial field in the complex space. Similarly with the one dimensional case the derivatives that appear in the preceding developments can be computed in the 2D real space as recorded in the sensor. The difference with the one dimensional case is now that the information is in the form of a Monge’s type surface where the gray level is of the form, 6 C.A. Sciammarella and L. Lamberti
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