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Notes
- 1.
Note that in Chapters 10 and 11 we sometimes find it convenient to use the classical terms subtractive moiré and additive moiré (not to be confused with additive superposition!). These terms designate moirés which correspond, respectively, to frequency differences or frequency sums in the spectrum. For example, the (1,−1)-moiré is subtractive, while the (1,1)-moiré is an additive moiré.
- 2.
It can be shown that this equation leads, indeed, to the classical formulas (2.9) that give the period and the angle of the (1,−1)-moiré between two superposed line gratings (see [Oster64 p. 170]).
- 3.
Note that a moiré of order > 1, too, is the locus of points of intersection (see, for example, the dotted lines of the (2,−1)-moiré in Fig. 11.1(a)). But since usually the density of points of intersection along these loci is lower than in a first-order moiré, a higher order moiré is usually less clearly visible.
- 4.
Their indicial equations even include the phases of the different gratings and of the resulting moirés. Note, however, that unlike in the spectral approach, analysis of the phase in the indicial equations method is rather limited; for example, it fails to discriminate between black and white zones of a zone grating (see [Leifer73 p. 40] and [Walls75 p. 596]).
- 5.
Note that we adopt here the convention that the frequencies f 1 and f 2 (or their reciprocals T 1 and T 2) are already incorporated into the functions g 1(x,y) and g 2(x,y), respectively (see Sec. 10.2).
- 6.
In these points the cosine term equals 1 and the imaginary sine term vanishes. Note that when the periodic-profiles of the gratings are symmetric the sine term is identically zero anyway.
- 7.
The function g(x,y) may represent any physical quantity which can be encoded in the departure of the grating lines from straightness: In strain analysis g(x,y) is the in-plane deformation of the object under load, in moiré topography it is the out-of-plane deformation, etc. [Patorski93 p. 4]. The distortion g(x,y) is considered as the information which is contained in the deformed grating.
- 8.
Further information about the local frequency, the local magnitude and the local phase of a signal can be found in [Hahn96 pp. 44–46].
- 9.
Note that the direction of the local frequency vector is perpendicular to the local (tangential) direction of the corrugations.
- 10.
This example has been readapted from [Rogers77 pp. 4–5] to our present mathematical formulation; it may clearly illustrate the difference between our formulation of the local frequency method and that of the original paper, which uses purely geometric considerations.
- 11.
In particular, Eq. (11.17) is simply the gradient to the level-lines of the curved surface (11.9) which represents the (k 1,…,k m )-moiré.
- 12.
The local frequency vector f(x 0,y 0) indicates the location (u 0,v 0) of the fundamental impulse in the “local Fourier spectrum” which corresponds to the infinitesimal portion of the curvilinear grating at (x 0,y 0).
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Amidror, I. (2009). Other possible approaches for moiré analysis. In: Amidror, I. (eds) The Theory of the Moiré Phenomenon. Computational Imaging and Vision, vol 38. Springer, London. https://doi.org/10.1007/978-1-84882-181-1_11
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