Kinetic theory

In physics, mathematics, and art, a moiré pattern (Template:IPAc-en; French: Template:IPA-fr) is a secondary and visually evident superimposed pattern created, for example, when two identical (usually transparent) patterns on a flat or curved surface (such as closely spaced straight lines drawn radiating from a point or taking the form of a grid) are overlaid while displaced or rotated a small amount from one another.

Etymology

The term originates from moire (moiré in its French adjectival form), a type of textile, traditionally of silk but now also of cotton or synthetic fiber, with a rippled or 'watered' appearance.

The history of the word moiré is complicated. The earliest agreedTemplate:By whom origin is the Arabic mukhayyar (مُخَيَّر in Arabic, which means chosen), a cloth made from the wool of the Angora goat, from khayyara (خيّر in Arabic), 'he chose' (hence 'a choice, or excellent, cloth'). It has also been suggested that the Arabic word was formed from the Latin marmoreus, meaning 'like marble'. By 1570 the word had found its way into English as mohair. This was then adopted into French as mouaire, and by 1660 (in the writings of Samuel Pepys) it had been adopted back into English as moire or moyre. Meanwhile the French mouaire had mutated into a verb, moirer, meaning 'to produce a watered textile by weaving or pressing', which by 1823 had spawned the adjective moiré. Moire (pronounced "mwar") and moiré (pronounced "mwar-ay") are now used somewhat interchangeably in English, though moire is more often used for the cloth and moiré for the pattern.

"Watered textile" refers to laying part of the textile on top of another part, and pressing the two layers when wet. The similarity of the spacing of individual threads (warp and woof), which is, however, not perfect spacing, creates characteristic patterns when the layers are pressed together; when dry, the patterns remain.

Pattern formation

Moiré patterns are often an undesired artifact of images produced by various digital imaging and computer graphics techniques, for example when scanning a halftone picture or ray tracing a checkered plane (the latter being a special case of aliasing, due to undersampling a fine regular pattern). This can be overcome in texture mapping through the use of mipmapping and anisotropic filtering.

The drawing on the upper right shows a moiré pattern. The lines could represent fibers in moiré silk, or lines drawn on paper or on a computer screen. The nonlinear interaction of the optical patterns of lines creates a real and visible pattern of roughly parallel dark and light bands, the moiré pattern, superimposed on the lines.

More complex line moiré patterns are created if the lines are curved or not exactly parallel. Moiré patterns revealing complex shapes, or sequences of symbols embedded in one of the layers (in form of periodically repeated compressed shapes) are created with shape moiré, otherwise called band moiré patterns. One of the most important properties of shape moiré is its ability to magnify tiny shapes along either one or both axes, that is, stretching. A common 2D example of moiré magnification occurs when viewing a chain-link fence through a second chain-link fence of identical design. The fine structure of the design is visible even at great distances.

Calculations

Moiré of parallel patterns

Geometrical approach

Let us consider two patterns made of parallel and equidistant lines, e.g., vertical lines. The step of the first pattern is $p$, the step of the second is $p+\delta p$, with $0<\delta <1$.

If the lines of the patterns are superimposed at the left of the figure, the shift between the lines increase when going to the right. After a given number of lines, the patterns are opposed: the lines of the second pattern are between the lines of the first pattern. If we look from a far distance, we have the feeling of pale zones when the lines are superimposed, (there is white between the lines), and of dark zones when the lines are "opposed".

The middle of the first dark zone is when the shift is equal to ${\frac {p}{2}}$. The $n$^th line of the second pattern is shifted by $n\cdot \delta p$ compared to the $n$^th line of the first network. The middle of the first dark zone thus corresponds to

$n\cdot \delta p={\frac {p}{2}}$

that is

$n={\frac {p}{2\delta p}}$.

The distance d between the middle of a pale zone and a dark zone is

$d=n\cdot p={\frac {p^{2}}{2\delta p}}$

the distance between the middle of two dark zones, which is also the distance between two pale zones, is

$2d={\frac {p^{2}}{\delta p}}$

From this formula, we can see that :

• the bigger the step, the bigger the distance between the pale and dark zones;
• the bigger the discrepancy $\delta p$, the closer the dark and pale zones; a great spacing between dark and pale zones mean that the patterns have very close steps.

Of course, when $\delta p={\frac {p}{2}}$, we have a uniformly grey figure, with no contrast.

The principle of the moiré is similar to the Vernier scale.

Mathematical function approach

Moiré pattern (bottom) created by superimposing two grids (top and middle)

The essence of the moiré effect is the (mainly visual) perception of a distinctly different third pattern which is caused by inexact superimposition of two similar patterns. The mathematical representation of these patterns is not trivially obtained and can be somewhat arbitrary. In this section we shall give a mathematical example of two parallel patterns whose superimposition forms a moiré pattern, and show one way (of many possible ways) these patterns and the moiré effect can be rendered mathematically.

The visibility of these patterns is dependent on the medium or substrate in which they appear, and these may be opaque (as, e.g. on paper) or transparent (as, e.g., in plastic film). For purposes of discussion we shall assume the two primary patterns are each printed in grey scale ink on a white sheet, where the opacity (e.g., shade of grey) of the "printed" part is given by a value between 0 (white) and 1 (black) inclusive, with 1/2 representing neutral grey. Any value less than 0 or greater than 1 using this grey scale is essentially "unprintable".

We shall also choose to represent the opacity of the pattern resulting from printing one pattern atop the other at a given point on the paper as the average (i.e. the arithmetic mean) of each pattern's opacity at that position, which is half their sum, and, as calculated, does not exceed 1. (This choice is not unique. Any other method to combine the functions that satisfies keeping the resultant function value within the bounds [0,1] will also serve; arithmetic averaging has the virtue of simplicity -- with hopefully minimal damage to one's concepts of the printmaking process.)

We now consider the "printing" superimposition of two almost similar, sinusoidally varying, grey-scale patterns to show how they produce a moiré effect in first printing one pattern on the paper, and then printing the other pattern over the first, keeping their coordinate axes in register. We represent the grey intensity in each pattern by a positive opacity function of distance along a fixed direction (say, the x-coordinate) in the paper plane, in the form

$f={\frac {1+\sin(kx)}{2}}$

where the presence of 1 keeps the function positive definite, and the divide by 2 prevents function values greater than 1.

The quantity $k$ represents the periodic variation (i.e., spatial frequency) of the pattern's grey intensity, measured as the number of intensity cycles per unit distance. Since the sin function is cyclic over argument changes of $2\pi$, the distance increment $\Delta x$ per intensity cycle (the wavelength) obtains when $k\Delta x=2\pi$, or $\Delta x={\frac {2\pi }{k}}$.

Consider now two such patterns where one has a slightly different periodic variation from the other:

$f_{1}={\frac {1+\sin(k_{1}x)}{2}}$

$f_{2}={\frac {1+\sin(k_{2}x)}{2}}$

such that $k_{1}\approx k_{2}$.

The average of these two functions, representing the superimposed printed image, evaluates as follows:

$f_{3}={\frac {f_{1}+f_{2}}{2}}$

$={\frac {1}{2}}+{\frac {\sin(k_{1}x)+\sin(k_{2}x)}{4}}$

$={\frac {1+\sin(Ax)\cos(Bx)}{2}}$

where it is easily shown that

$A={\frac {k_{1}+k_{2}}{2}}$

and

$B={\frac {k_{1}-k_{2}}{2}}$.

This function average, $f_{3}$, clearly lies in the range [0,1]. Since the periodic variation $A$ is the average of and therefore close to $k_{1}$ and $k_{2}$, the moiré effect is distinctively demonstrated by the sinusoidal envelope "beat" function $\cos(Bx)$, whose periodic variation is half the difference of the periodic variations $k_{1}$ and $k_{2}$ (and evidently much "slower").

Other one-dimensional moire effects include the classic beat frequency tone which is heard when two pure notes of almost identical pitch are sounded simultaneously. This is an acoustic version of the moiré effect in the one dimension of time: the original two notes are still present -- but the listener's perception is of two pitches that are the average of and half the difference of the frequencies of the two notes. Aliasing in sampling of time-varying signals also belongs to this moiré paradigm.

A concluding note for this section: it is popularly thought that the moiré effect results from "interference" between two superimposed patterns. However, interference (as understood in the physics sense) requires the summing and squaring of dynamically moving wave amplitudes, and these superposed amplitudes and their products can also have negative values. Those dynamic attributes are never present in the static patterns printed on opaque or transparent sheets which are superimposed to produce a moiré effect. That is, superimposition is not the same as superposition. The moiré effect is therefore not a true interference phenomenon, even though its appearance can sometimes seem similar to pictures of actual visible fringe patterns produced by combining (and interfering) physical waves taken at any one instant of time.

Rotated patterns

Let us consider two patterns with the same step $p$, but the second pattern is turned by an angle $\alpha$. Seen from far, we can also see dark and pale lines: the pale lines correspond to the lines of nodes, that is, lines passing through the intersections of the two patterns.

If we consider a cell of the "net", we can see that the cell is a rhombus: it is a parallelogram with the four sides equal to $d={\frac {p}{\sin \alpha }}$; (we have a right triangle which hypothenuse is $d$ and the side opposed to the $\alpha$ angle is $p$).

The pale lines correspond to the small diagonal of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the pale line makes an angle equal to ${\frac {\alpha }{2}}$ with the perpendicular of the lines of each pattern.

Additionally, the spacing between two pale lines is $D$, the half of the big diagonal. The big diagonal $2D$ is the hypothenuse of a right triangle and the sides of the right angle are $d(1+\cos \alpha )$ and $p$. The Pythagorean theorem gives:

$(2D)^{2}=d^{2}(1+\cos \alpha )^{2}+p^{2}$

id est

$(2D)^{2}={\frac {p^{2}}{\sin ^{2}\alpha }}(1+\cos \alpha )^{2}+p^{2}=p^{2}\cdot \left({\frac {(1+\cos \alpha )^{2}}{\sin ^{2}\alpha }}+1\right)$

thus

$(2D)^{2}=2p^{2}\cdot {\frac {1+\cos \alpha }{\sin ^{2}\alpha }}$ or $D={\frac {p}{2}}/\sin {\frac {\alpha }{2}}$.

When $\alpha$ is very small ($\alpha <{\frac {\pi }{6}}$), the following approximations can be done:

$\sin \alpha \approx \alpha$

$\cos \alpha \approx 1$

thus

$D\approx {\frac {p}{\alpha }}$.

We can see that the smaller the $\alpha$, the farthest the pale lines; when the both patterns are parallel ($\alpha =0$), the spacing between the pale lines is "infinite" (there is no pale line).

There are thus two ways to determine $\alpha$: by the orientation of the pale lines and by their spacing

$\alpha \approx {\frac {p}{D}}$

If we choose to measure the angle, the final error is proportional to the measurement error. If we choose to measure the spacing, the final error is proportional to the inverse of the spacing. Thus, for the small angles, it is best to measure the spacing.

Implications and applications

Printing full-color images

File:BeatTrackMoirePattern.ogg In graphic arts and prepress, the usual technology for printing full-color images involves the superimposition of halftone screens. These are regular rectangular dot patterns—often four of them, printed in cyan, yellow, magenta, and black. Some kind of moiré pattern is inevitable, but in favorable circumstances the pattern is "tight;" that is, the spatial frequency of the moiré is so high that it is not noticeable. In the graphic arts, the term moiré means an excessively visible moiré pattern. Part of the prepress art consists of selecting screen angles and halftone frequencies which minimize moiré. The visibility of moiré is not entirely predictable. The same set of screens may produce good results with some images, but visible moiré with others.

In manufacturing industries, these patterns are used for studying microscopic strain in materials: by deforming a grid with respect to a reference grid and measuring the moiré pattern, the stress levels and patterns can be deduced. This technique is attractive because the scale of the moiré pattern is much larger than the deflection that causes it, making measurement easier.

Television screens and photographs

Template:Double image Moiré patterns are commonly seen on television screens when a person is wearing a shirt or jacket of a particular weave or pattern, such as a houndstooth jacket. This is due to interlaced scanning in televisions and non-film cameras, referred to as Interline Twitter. As the person moves about, the Moiré pattern is quite noticeable. Because of this, newscasters and other professionals who appear on TV regularly are instructed to avoid clothing which could cause the effect.

Photographs of a TV screen taken with a digital camera often exhibit moiré patterns.

Marine navigation

The Moiré effect is used in shoreside beacons to mark underwater hazards (usually pipelines or cables). The Moiré effect creates arrows that 'point' towards an imaginary line marking the hazard; as navigators pass over the hazard, the arrows on the beacon appear to become vertical bands before 'changing' back to arrows pointing in the reverse direction. An example can be found in the UK on the East shore of Southampton water, opposite Fawley oil refinery (Template:Coord). Similar Moiré effect beacons can be used to guide mariners to the centre point of an oncoming bridge; when the vessel is aligned with the centreline, vertical lines are visible.

Strain measurement

Use of the moiré effect in strain measurement: case of uniaxial traction (top) and of pure shear (bottom); the lines of the patterns are initially horizontal in both cases

The moiré effect can be used in strain measurement: the operator just has to draw a pattern on the object, and superimpose the reference pattern to the deformed pattern on the deformed object.

A similar effect can be obtained by the superposition of an holographic image of the object to the object itself: the hologram is the reference step, and the difference with the object are the deformations, which appear as pale and dark lines.

See also: theory of elasticity, strain tensor and holographic interferometry.

Image processing

Some image scanner driver programs provide an optional filter, called a "descreen" filter, to remove Moiré-pattern artifacts which would otherwise be produced when scanning printed halftone images to produce digital images.

Currency

Many currencies exploit the tendency of digital scanners to produce moiré patterns by including fine circular or wavy designs that are likely to exhibit a moiré pattern when scanned and printed.

Animation

The Moiré pattern can be used to animate images. An acetate overlay containing vertical stripes is placed over the image and slowly moved from right to left. These are sold in sets under the brand name Scanimation. It was invented by Rufus Butler Seder, and patented in 2006 under US Patent #7151541.

Super-resolution microscopy

The Moiré pattern can be used to obtain images with a resolution higher than the diffraction limit using a technique known as Structured Illumination Microscopy.

See also

• Aliasing
• Angle-sensitive pixel
• Beat frequency
• Kan't Kopy paper
• Multidimensional sampling

References

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• A series of oil paintings based on Moiré principles by British artist, Pip Dickens
• A live demonstration of the Moiré effect that stems from interferences between circles
• An interactive example of various Moiré patterns Use arrow keys and mouse to manipulate layers.