Moiré pattern

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A moiré pattern, formed by two sets of parallel lines, one set inclined at an angle of 5° to the other

Original image

A moiré pattern formed by incorrectly downsampling the image above. (Details)

A moiré pattern is an interference pattern created, for example, when two grids are overlaid at an angle, or when they have slightly different mesh sizes.

The drawing on the 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 horizontal dark and light bands, the moiré pattern, superimposed on the lines. More complex moiré patterns are created if the lines are curved or not exactly parallel.

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

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. This cause of moiré is a special case of aliasing, due to undersampling a fine regular pattern.

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;" i.e. 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.

1 Etymology
2 Moirés in digital images of TV screens
3 Calculations
- 3.1 Moiré of parallel patterns
  - 3.1.1 Geometrical approach
  - 3.1.2 Interferometric approach
- 3.2 Rotated patterns
4 Application to strain measurement
- 4.1 Uniaxial traction
- 4.2 Shear strain
5 External links

[edit] Etymology

The history of the word moiré is complicated. The earliest agreed origin is the Arabic-Persian mukhayyar, a cloth made from the wool of the Angora goat, from khayyana, '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 and moiré are now used somewhat interchangeably in English, though moire is more often used for the cloth and moiré for the pattern.

[edit] Moirés in digital images of TV screens

Moiré patterns from a TV

Taking a picture of a TV screen with a digital camera often produces severe moiré patterns. In the example shown, two distinct patterns are visible.

The first consists of broad, dark, horizontal, slightly curved bands. They are caused by the superimposition of (the image of) the TV scan lines on the array of CCD pixels in the digital camera used to shoot the screen. Since the camera was not absolutely perpendicular to the screen, the moiré lines converge to the right. They are crooked because the scan lines on a TV (tube) screen are seldom perfectly straight. Note the arrow at left—below that, the image consists of two interlaced scans, and the moiré disappears due to tighter spacing of the TV scan lines.

The second set of moiré lines can clearly be seen only in the full-size image. They are much narrower than the horizontal lines, and strongly curved, especially at lower left. They are caused by the regular pattern of the colored phosphor dots on the TV screen and the pixels in the camera. As before, the curvature is due to imperfect alignment of camera and screen.

Both these patterns are the result of not following the sampling theorem. In this case, the resolution of a single pixel is not less than half the highest spatial frequency on the TV screen. The highest resolution corresponds to the black borders of each pixel and between the red, green, and blue sub-pixels. Since the sampling theorem has not been satisfied, aliasing can be expected and is evident as the moiré patterns are present.

[edit] Calculations

[edit] Moiré of parallel patterns

[edit] Geometrical approach

the patterns are superimposed in the mid-height of the figure

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+δp, with δp>0.

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 p/2. The n^th line of the second pattern is shifted by n·δp compared to the n^th line of the first network. The middle of the first dark zone thus corresponds to

n·δp = 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 δp, 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 δp = p/2, we have a uniformly grey figure, with no contrast.

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

[edit] Interferometric approach

Let us consider now two transparent patterns with a contrast I that varies with a sinus law:

$I_1(x) = I_0 \cdot \sin (2\pi \cdot k_1 \cdot x)$

$I_2(x) = I_0 \cdot \sin (2\pi \cdot k_2 \cdot x)$

(the steps are respectively p₁ = 1/k₁ and p₂ = 1/k₂), when the patterns are superimposed, the resulting intensity (interference) is

$I(x) = I_0 \cdot ( \sin (2\pi \cdot k_1 \cdot x) + \sin (2\pi \cdot k_2 \cdot x) )$

with the Euler's formula:

$I(x) = I_0 \cdot 2 \cos \left ( 2\pi \frac{(k_1-k_2)}{2} \cdot x \right ) \cdot \sin \left ( 2\pi \frac{(k_1+k_2)}{2} \cdot x \right )$

We can see that the resulting intensity is made of a sinus law with a high "spatial frequency" (wave number) which is the average of the spatial frequencies of the two patterns, and of a sinus law with a low spatial frequency which is the half of the difference between the spatial frequencies of the two patterns. This second component is an "envelope" for the first sinus law. The wavelength λ of this component is the inverse of the spatial frequency

$\frac{1}{\lambda} = \frac{k_1 - k_2}{2} = \frac{1}{2} \cdot \left ( \frac{1}{p_1} - \frac{1}{p_2} \right )$

if we consider thats p₁ = p and p₂ = p+δp:

$\lambda = 2\frac{p_1 p_2}{p_1 - p_2} \approx 2\frac{p^2}{\delta p}$ .

The distance between the zeros of this envelope is λ/2, and the maxima of amplitude are also spaced by λ/2; we thus obtain the same results ad the geometrical approach, with a discrepancy of p/2 which is the uncertainty linked to the reference that is considered: pattern 1 or pattern 2. This discrepancy is negligible when δp << p.

This phenomenon is similar to the stroboscopy.

[edit] Rotated patterns

Moiré obtained by the superimposition of two similar patterns rotated by an angle α

Let us consider two patterns with the same step p, but the second pattern is turned by an angle α. Seen from far, we can also see dark and pale lines: the pale lines correspond to the lines of nodes, i.e. 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 = p/sin α; (we have a right triangle which hypothenuse is d and the side opposed to the α angle is p).

Unit cell of the "net"; "ligne claire" means "pale line"

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 α/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 α) and p. The Pythagorean theorem gives:

(2D)² : d²(1+cos α)² + p²

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

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

When α is very small (α << 2π), the following approximations can be done:

sin α ≈ α

cos α ≈ 1

thus

D ≈ p / α

α is of course in radian.

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

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

α ≈ 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.

[edit] Application to 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 and strain tensor.

[edit] Uniaxial traction

Let us consider an object with a length of l, and we draw a pattern with a step p; the lines of the pattern are perpendicular to the axis of traction.

We the object is under tension, its length becomes l·(1+ε), where ε is the strain (relative stretch). The step of the pattern becomes p·(1+ε), and thus δp = p·ε.

The spacing between the center of two dark zones is:

$2d = \frac{p}{\varepsilon}$

this spacing allows the determination of the strain. However, the determination of the center of a dark zone is not accurate, because the zone is wide. We can instead count the number N of dark zones: on a length of l, there are

$N = \frac{l \cdot \varepsilon}{p}$

darke zones, i.e.

$\varepsilon = N \cdot \frac{p}{l}$

The accuracy of the determination is the difference of strain between the apparition of two dark zones, i.e.

$\Delta \varepsilon = \frac{p}{l}$

[edit] Shear strain

In the case of a pure shear strain, we draw a pattern which lines are perpendicular to the shear forces. The pattern on the deformed object is turned by the shear angle γ compared to the reference object (undeformed object).

For the same reason as for uniaxial traction, we can count the pale zones, as long as

γ is very small,
the object is rectangular, and
the forces are parallel to the sides (the pale lines are then almost parallel to the sides of the object).

When the width of the object (dimension perpendicular to the forces) is l, then the number N of pale lines is:

$N = l / D = l \cdot \gamma / p$