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- 1.
In terms of the spectral domain, the first level considerations only take into account impulse locations (or frequency vectors) within the u,v plane, while the second level considerations also take into account the impulse amplitudes.
- 2.
The most general form of the Fourier series development of the ( k 1, k 2 )-moiré, which incorporates the explicit values of θ M and T M and also covers the case of non-symmetric or shifted gratings, will be given in a concise and elegant way in Chapter 6 (see Eq. (6.3)). This will become possible after having introduced the algebraic notations of Chapter 5 and their Fourier interpretation in Chapter 6.
- 3.
T-convolution (also called cyclic convolution) is the periodic analog of the normal convolution with integration limits of (−∞,∞). Note that normal convolution cannot be used in the case of periodic functions [see Gaskill78 pp. 157–158]. In general, the normal convolution of a single period of p 1 with a single period of p 2 is not equal to a single period of the T-convolution p 1*p 2. Such an equality only occurs in cases in which the normal convolution of the two single periods is not longer than the period T; otherwise the outer ends which exceed the boundaries of each convolution period T inevitably penetrate (additively) into the neighbouring periods in the T-convolution, thus generating a cyclic wrap-around effect which does not exist in the case of normal convolution. The discrete counterpart of the cyclic convolution is widely used in the discrete Fourier transform theory [Bracewell86 p. 362].
- 4.
In fact, in the case of (1,-1)-moiré it may be more appropriate to use the term T-cross-correlation of p 1(x) and p 2(x), which is defined, following [Gaskill78 p. 172], as p 1(x)☆p 2(x) = p 1(x)*p 2(−x). The reason is that in the case of (1,-1)-moiré we have \(d_n = a^{\left( 1 \right)} _n a^{\left( 2 \right)} _{ - n,} \) which means that the second comb in the term-by-term product is reflected about the origin, and therefore represents in the image domain the reflected image p 2(−x); the resulting moiré-profile is therefore the T-cross-correlation of p 1(x) and p 2(x). However, for the sake of consistency in the general case of the (k 1,…,k m )-moiré, where some of the indices are positive and others are negative, we prefer to stick to the terminology of T-convolution, understanding that for any negative index in the list the image it represents must be reflected. In the common case where the original images are symmetric about the origin, the two terms coincide.
- 5.
Obviously, some (or even most) of the nailbed impulses may have a zero amplitude, as in the case of f(x,y) = cos(x) + cos(y), for instance.
- 6.
Note that this impulse is generated in the convolution by the (k 1,k 2)-impulse in the spectrum R 1(u,v) of the first image and the (k 3,k 4)-impulse in the spectrum R 2(u,v) of the second image.
- 7.
The explicit mathematical expression of the function \(m_{k_1,k_2,k_3,k_4,} \left( {{\bf{x}},y} \right)\) can be given in the form of a 2D Fourier series, which is a 2D extension of Eq. (4.3). However, we prefer to wait for this expression until Chapter 6, where we will be able to give it in a much more concise and elegant way (see Eq. (6.8)), thanks to the algebraic notations of Chapter 5 and their Fourier interpretation in Chapter 6.
- 8.
This has already been illustrated, in the case of 2-gratings superposition, by the difference between Fig. 2.5(c) (the image superposition) and Fig. 4.2(a) (the extracted intensity profile of the (1,−1)-moiré).
- 9.
The explicit Fourier series development of the (k 1,…,k m )-moiré in the most general case, which includes non-symmetric or shifted dot-screens, will be given in a concise and elegant way in Secs. 6.7 and 6.8.
- 10.
Remember that the 2D (1,0,−1,0)-moiré between two screens is geometrically equivalent to the moiré between two pairs of gratings; referring to Fig. 2.10(a), the gratings A and C generate a (1,−1) moiré, and the gratings B and D generate a second, perpendicular (1,−1)-moiré.
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© 2009 Springer-Verlag London Limited
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Amidror, I. (2009). The moiré profile form and intensity levels. 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_4
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DOI: https://doi.org/10.1007/978-1-84882-181-1_4
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