This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It should be emphasized that since the Fourier transform is reversible, no information is gained or lost by its application. It only reveals certain image features which were present but not explicitly apparent before the image was transformed.
- 2.
Note that throughout this book we adopt the Fourier transform conventions that are commonly used in optics (see [Bracewell86 p. 241] or [Gaskill78 p. 128]); thus, the Fourier transform of a function f(x,y) and its inverse are given by:
$${\bf{f}}( {u,v} ) = \in{\bf{T}}_{ - \infty }^\infty {\in{\bf{T}}_{ - \infty }^\infty f ( {{\bf{x}},y} )e^{ - i2\pi ( {ux + vy} )} dx\,dy,\,\,f( {{\bf{x}},y} ) = \in{\bf{T}}_{ - \infty }^\infty {\in{\bf{T}}_{ - \infty }^\infty F ( {u,v} )e^{ - i2\pi ( {ux + vy} )} dx\,dy.} } $$For alternative definitions used in literature and the relationships between them see [Bracewell86 pp. 7 and 17] or [Gaskill78 pp. 181-183].
- 3.
A short survey of the spectral Fourier representation of periodic functions is also provided in Appendix A
- 4.
It should be stressed that the direction θ of the impulse, i.e., the direction of the corresponding periodic component in the image, is perpendicular to the corrugations of the periodic component (see, for example, gratings (a) and (b) and their spectra (d) and (e) in Fig. 2.2).
- 5.
In fact, as shown in Appendix C.2, the amplitude of the DC impulse represents the average intensity level of the image (which is, in our case, a number between 0 and 1, since our images can only take values between 0 and 1).
- 6.
For the sake of completeness we mention here that this conjugate symmetry property in the spectrum only breaks up in the case of complex-valued images. For example, a single impulse at the point (u,v) in the spectrum corresponds to the complex-valued function p(x,y) =; e −2πi(ux+vy)in the image domain. We will rarely be concerned with such cases, since all physically realizable images are purely real.
- 7.
For a more detailed account on the human visual system and its properties the reader is referred to specialized references on this subject such as [Cornsweet70], [Wandell95] or Chapter 34 in [Boff86].
- 8.
Note that the situation is inversed when the angle between f 2 and f 1 is obtuse, since in this case the length of f 1+f 2 is smaller than the length of f 1−f 2.
- 9.
The moiré which corresponds to the frequency difference is often called in the literature a difference moiré or a subtractive moiré, whilst the moiré which corresponds to the frequency sum is called an additive moiré. We will usually prefer not to use these terms in order to avoid any possible confusion with the superposition rules (additive superposition, etc.; see Remark 2.1).
- 10.
A superposition operation S[f,g] which takes a pair of functions f(x) and g(x) and returns their superposition according to a given mathematical rule is called linear if for any functions f(x) and g i (x) and constants a and b we have S[af 1+bf 2, g] =; aS[f 1,g] + bS[f 2,g] and S[f, ag 1 +bg 2] =; aS[f,g 1] + bS[f,g 2]. In our case the additive superposition operation S 1[f,g] =; f + g is linear, but the multiplicative superposition operation S 2[f,g]=; fg is not. Note that if we apply to S 1[f,g] a non-linearity, as in S 3[f,g] =; (S 1[f,g])2 =; (f + g)2, the resulting superposition operation S 3 is non-linear.
- 11.
Note that the moiré angle formulas found in literature may vary according to the angle conventions being used.
- 12.
Note that in the following expressions we tacitly use the impulse indexing notation that will be formally introduced in Sec. 2.7.
- 13.
Note that by convention, a moiré continues to exist even when its fundamental impulse exceeds the visibility circle or has a low amplitude, and the moiré is no longer visible.
- 14.
Note, however, that in some cases angle or period changes may cause impulses in the spectrum convolution to fall on top of each other, in which case their individual amplitudes are summed. Such combined impulses will be called in Chapter 6 compound impulses, and it will be shown there that they only occur in singular states (see Sec. 2.9 below).
- 15.
The stress is on individual impulse amplitudes, since in cases where compound impulses are generated in the spectrum convolution (see the previous footnote) it is clear that the summed amplitude of a compound impulse only exists at the precise angle and frequency combination in which the compound impulse is generated.
- 16.
These expressions will be presented in their final form in Proposition 2.3, after introducing the impulse indexing notation in Sec. 2.7.
- 17.
Note, however, that such attempts to interpret the different moirés by counting line intersections in the image domain cannot be extended to superpositions involving more than two gratings (see, for example, Fig. 2.8(h)).
- 18.
Note that formulas (2.9) can be generalized to the moiré caused by any given k 1 f 1+k 2 f 2 impulse, by deriving them from Eq. (2.23) using the given values of k 1,k 2 instead of k 1 =; 1, k 2 =; −1. Such a formula has been obtained in [Tollenaar64 p. 626] using the geometric approach; however, without the help of the spectral approach, i.e., without having a panoramic view of the spectrum, we cannot know which integers k 1,k 2 correspond to the dominant moiré and should be therefore used in the formula.
- 19.
Note, however, that this notation is only unique up to a sign, since the twin impulse (−k 1,…,−k m ) also spans the same comb.
- 20.
In fact, the distinction between “valid” and “degenerate” moirés is just a matter of convention; in some circumstances it can be more convenient to consider all of them as periodic components (cosines) in the superposition and to treat them all on an equal basis.
- 21.
Note that although to each 2D moiré there corresponds in the spectrum a unique impulse-cluster, the notation of a 2D moiré is not necessarily unique, since several different pairs of impulse-twins within the visibility circle may span the same 2D cluster. The reason is that a dot-lattice (the support of the impulse-cluster on the u,v plane) does not have a unique basis (see Sec. 5.2.1).
- 22.
Note also that a (k 1,…,k m )-impulse whose index-vector is an integer linear combination of two index-vectors of a lower order does not correspond to a “valid” but rather to a “degenerate” 2-fold periodic moiré. For example, the (2,0,−1,1)-impulse is simply a higher harmonic impulse in the impulse-cluster spanned by the (1,1,−1,0)-impulse and the (1,−1,0,1)-impulse: (2,0,−1,1) =; (1,1,−1,0) + (1,−1,0,1), and therefore it does not correspond to a “valid”, independent moiré effect. This remark is, in fact, a 2D generalization of point (2) above.
- 23.
Note that in Figs. 2.9(g)–(i) we have approximated the unbounded logarithmic function D =; −log10 r by D =; −log10 (0.9r + 0.1), whose values for the reflectance r, 0 ≤ r ≤ 1, vary between 0 (for r =; 1) and 1 (for r =; 0).
- 24.
Note, however, that singular states are mainly of interest for (k 1,…,k m )-impulses which represent “valid” moirés, i.e., impulses which satisfy conditions (1) and (2) in Sec. 2.8.
- 25.
As we will see in Remark 5.1 (Sec. 5.6.2), Remark 2.7 can be expressed in terms of the linear dependence over Z of the frequency vectors f i of the superposed layers: if f 1,…,f m are all linearly independent over Z in the u,v plane, then the superposition is regular; if f 1,…,f m are linearly dependent over Z, then the superposition is singular; and if rankZ(f 1,…,f m ) =; r < m, then the singularity is of order m−r.
- 26.
It should be mentioned that the range [0,1] of reflectance values is only respected by the precise profile reconstruction which takes into account all the Fourier terms up to infinity. An approximation using only a finite number of terms, such as the DC plus the first harmonic cosine (×2), may somewhat exceed the range of [0,1].
- 27.
If (k 1,k 2,k 3,k 4) and −(k 1,k 2,k 3,k 4) are an impulse pair (in the sense of Fig. 2.1) in the spectrum of two superposed square grids, then their perpendicular impulse pair (or orthogonal twin) is the impulse pair consisting of the impulses (−k 2,k 1,−k 4,k 3) and (k 2,−k 1,k 4,−k 3). This is further explained in Problem 2-15.
- 28.
We prefer to avoid the term “square screen” which could be coined from “square grid”, in order to avoid confusion with the dot shape of the screen, which may be square, circular, or anything else.
- 29.
If the scanner's aperture cannot be considered as a theoretical impulse-like pinhole, the nailbed R 2(u,v) simply takes the envelope shape of the Fourier transform of the aperture. This can be pictorially illustrated by Fig. 2.12, where (a) is interpreted as a single aperture, (b) is its continuous spectrum, and (e) is an infinite array of such apertures that represents r 2(x,y). As we can see in (f), the spectrum R 2(u,v) of this aperture-array is a bell-shaped nailbed having the envelope form of (b).
- 30.
Or in other words, if the spectrum R 1(u,v)R 1(u,v) of the original function r 1(x,y) contains frequencies that exceed the region \( - \frac{1}{2}f < u < \frac{1}{2}f,\, - \frac{1}{2}f < v < \frac{1}{2}f\)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag London Limited
About this chapter
Cite this chapter
Amidror, I. (2009). Background and basic notions. 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_2
Download citation
DOI: https://doi.org/10.1007/978-1-84882-181-1_2
Publisher Name: Springer, London
Print ISBN: 978-1-84882-180-4
Online ISBN: 978-1-84882-181-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)