Moiré pattern
Encyclopedia
In physics, a moiré pattern (icon; French: mwaʁe) is an interference
pattern created, for example, when two grids are overlaid at an angle, or when they have slightly different mesh sizes.
(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.
The history of the word moiré is complicated. The earliest agreed 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 and moiré 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.
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.
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 nth line of the second pattern is shifted by n·δp compared to the nth line of the first network. The middle of the first dark zone thus corresponds to
that is
The distance d between the middle of a pale zone and a dark zone is
the distance between the middle of two dark zones, which is also the distance between two pale zones, is
From this formula, we can see that :
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
.
(the steps are respectively p1 = 1/k1 and p2 = 1/k2), when the patterns are superimposed, the resulting intensity (interference
) is
with the Euler's formula
:
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
if we consider that's p1 = p and p2 = p+δp:
.
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 as 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.
, 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 = p/sin α; (we have a right triangle
which hypothenuse is d and the side opposed to the α 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 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:
id est
thus
When α is very small (α < π/6), the following approximations can be done:
thus
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
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.
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.
jacket, due to sampling problems in the television camera. 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, more commonly known as 'strobing'.
Photographs of a TV
screen taken with a digital camera
often exhibit moiré patterns.
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
.
driver
program
s 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 image
s.
Interference
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have...
pattern created, for example, when two grids are overlaid at an angle, or when they have slightly different mesh sizes.
Etymology
The term originates from moireMoire (fabric)
In textiles, a moire is a fabric with a wavy appearance produced mainly from silk, but also wool, cotton and rayon. The watered appearance is usually created by the finishing technique called calendering...
(or moiré in its French form), a type of textile
Textile
A textile or cloth is a flexible woven material consisting of a network of natural or artificial fibres often referred to as thread or yarn. Yarn is produced by spinning raw fibres of wool, flax, cotton, or other material to produce long strands...
, traditionally of silk
Silk
Silk is a natural protein fiber, some forms of which can be woven into textiles. The best-known type of silk is obtained from the cocoons of the larvae of the mulberry silkworm Bombyx mori reared in captivity...
but now also of cotton
Cotton
Cotton is a soft, fluffy staple fiber that grows in a boll, or protective capsule, around the seeds of cotton plants of the genus Gossypium. The fiber is almost pure cellulose. The botanical purpose of cotton fiber is to aid in seed dispersal....
or synthetic fiber
Synthetic fiber
Synthetic fibers are the result of extensive research by scientists to improve on naturally occurring animal and plant fibers. In general, synthetic fibers are created by forcing, usually through extrusion, fiber forming materials through holes into the air, forming a thread...
, with a rippled or 'watered' appearance.
The history of the word moiré is complicated. The earliest agreed origin is the Arabic mukhayyar (مُخَيَّر in Arabic, which means chosen), a cloth made from the wool of the Angora goat
Angora goat
The Angora goat is a breed of domestic goat that originated in Ankara , Turkey and its surrounding region in central Anatolia...
, 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
Mohair
Mohair usually refers to a silk-like fabric or yarn made from the hair of the Angora goat. The word "mohair" was adopted into English before 1570 from the Arabic: mukhayyar, a type of haircloth, literally 'choice', from khayyara, 'he chose'. Mohair fiber is approximately 25-45 microns in...
. This was then adopted into French as mouaire, and by 1660 (in the writings of Samuel Pepys
Samuel Pepys
Samuel Pepys FRS, MP, JP, was an English naval administrator and Member of Parliament who is now most famous for the diary he kept for a decade while still a relatively young man...
) 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.
"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 imagesDigital image
A digital image is a numeric representation of a two-dimensional image. Depending on whether or not the image resolution is fixed, it may be of vector or raster type...
produced by various digital imaging
Digital imaging
Digital imaging or digital image acquisition is the creation of digital images, typically from a physical scene. The term is often assumed to imply or include the processing, compression, storage, printing, and display of such images...
and computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
techniques, for example when scanning
Image scanner
In computing, an image scanner—often abbreviated to just scanner—is a device that optically scans images, printed text, handwriting, or an object, and converts it to a digital image. Common examples found in offices are variations of the desktop scanner where the document is placed on a glass...
a halftone
Halftone
Halftone is the reprographic technique that simulates continuous tone imagery through the use of dots, varying either in size, in shape or in spacing...
picture or ray tracing a checkered plane (the latter being a special case of aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
, due to undersampling
Undersampling
In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass filtered signal at a sample rate below the usual Nyquist rate In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass filtered signal at a sample rate...
a fine regular pattern). This can be overcome in texture mapping through the use of mipmapping and anisotropic filtering
Anisotropic filtering
In 3D computer graphics, anisotropic filtering is a method of enhancing the image quality of textures on surfaces that are at oblique viewing angles with respect to the camera where the projection of the texture appears to be non-orthogonal In 3D computer graphics, 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
Line moiré
Line moiré is one type of moiré pattern; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move...
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
Shape moiré
Shape moiré is one type of moiré patterns demonstrating the phenomenon of moiré magnification. 1D shape moiré is the particular simplified case of 2D shape moiré. 1D patterns may appear when superimposing an opaque layer containing tiny horizontal transparent lines on top of a layer containing a...
, otherwise called band moiré patterns. One of the most important properties of shape moiré
Shape moiré
Shape moiré is one type of moiré patterns demonstrating the phenomenon of moiré magnification. 1D shape moiré is the particular simplified case of 2D shape moiré. 1D patterns may appear when superimposing an opaque layer containing tiny horizontal transparent lines on top of a layer containing a...
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.
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+δp, with 0<δ<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 p/2. The nth line of the second pattern is shifted by n·δp compared to the nth line of the first network. The middle of the first dark zone thus corresponds to
- n·δp = p/2
that is
-
.
The distance d between the middle of a pale zone and a dark zone is
the distance between the middle of two dark zones, which is also the distance between two pale zones, is
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
Vernier scale
A vernier scale is an additional scale which allows a distance or angle measurement to be read more precisely than directly reading a uniformly-divided straight or circular measurement scale...
.
Interferometric approach
Let us consider now two transparent patterns with a contrast I that varies with a sinusoidal law:(the steps are respectively p1 = 1/k1 and p2 = 1/k2), when the patterns are superimposed, the resulting intensity (interference
Interference
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have...
) is
with the Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
:
We can see that the resulting intensity is made of a sinus law with a high "spatial frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the 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
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
λ of this component is the inverse of the spatial frequency
if we consider that's p1 = p and p2 = p+δp:
.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 as 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.
Rotated patterns
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 nodesNode (physics)
A node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the...
, 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
Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
: it is a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
with the four sides equal to d = p/sin α; (we have a right triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
which hypothenuse is d and the side opposed to the α angle is p).
The pale lines correspond to the small diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
of the rhombus. As the diagonals are the bisectors
Bisection
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...
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
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
gives:
- (2D)2 : d2(1+cos α)2 + p2
id est
thus
When α is very small (α < π/6), the following approximations can be done:
- sin α ≈ α
- cos α ≈ 1
thus
- D ≈ p / α.
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.
Printing full-color images
In graphic artsGraphic arts
A type of fine art, graphic art covers a broad range of art forms. Graphic art is typically two-dimensional and includes calligraphy, photography, drawing, painting, printmaking, lithography, typography, serigraphy , and bindery. Graphic art also consists of drawn plans and layouts for interior...
and prepress
Prepress
Prepress is the term used in the printing and publishing industries for the processes and procedures that occur between the creation of a print layout and the final printing...
, the usual technology for printing full-color images involves the superimposition of halftone
Halftone
Halftone is the reprographic technique that simulates continuous tone imagery through the use of dots, varying either in size, in shape or in spacing...
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
Manufacturing
Manufacturing is the use of machines, tools and labor to produce goods for use or sale. The term may refer to a range of human activity, from handicraft to high tech, but is most commonly applied to industrial production, in which raw materials are transformed into finished goods on a large scale...
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
Moiré patterns are commonly seen on television screens directly when a person is wearing a shirt or jacket of a particular weave or pattern, such as a houndstoothHoundstooth
Houndstooth, houndstooth check or hound's tooth , also known as dogstooth, dogtooth or dog's tooth, is a duotone textile pattern characterized by broken checks or abstract four-pointed shapes, often in black and white, although other colours are used...
jacket, due to sampling problems in the television camera. 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, more commonly known as 'strobing'.
Photographs of a TV
Television
Television is a telecommunication medium for transmitting and receiving moving images that can be monochrome or colored, with accompanying sound...
screen taken with a digital camera
Digital camera
A digital camera is a camera that takes video or still photographs, or both, digitally by recording images via an electronic image sensor. It is the main device used in the field of digital photography...
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 you 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 (50°51′21.63"N 1°19′44.77"W). Similar Moiré effect beacons can be used to guide mariners to the centre point of an oncoming bridge (here you would position the vessel so that you could see vertical lines - denoting the 'centreline').Strain measurement
The moiré effect can be used in strainStrain (materials science)
In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...
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
Holographic interferometry
Holographic interferometry is a technique which enables static and dynamic displacements of objects with optically rough surfaces to be measured to optical interferometric precision . These measurements can be applied to stress, strain and vibration analysis, as well as to non-destructive testing...
.
Image processing
Some image scannerImage scanner
In computing, an image scanner—often abbreviated to just scanner—is a device that optically scans images, printed text, handwriting, or an object, and converts it to a digital image. Common examples found in offices are variations of the desktop scanner where the document is placed on a glass...
driver
Device driver
In computing, a device driver or software driver is a computer program allowing higher-level computer programs to interact with a hardware device....
program
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...
s provide an optional filter
Filter (software)
A filter is a computer program to process a data stream. Some operating systems such as Unix are rich with filter programs. Even Windows has some simple filters built into its command shell, most of which have significant enhancements relative to the similar filter commands that were available in...
, called a "descreen" filter, to remove Moiré-pattern artifacts which would otherwise be produced when scanning printed halftone
Halftone
Halftone is the reprographic technique that simulates continuous tone imagery through the use of dots, varying either in size, in shape or in spacing...
images to produce digital image
Digital image
A digital image is a numeric representation of a two-dimensional image. Depending on whether or not the image resolution is fixed, it may be of vector or raster type...
s.