The tutorial describes essential features of moire patterns, as well as the circumstances, when the moire patterns appear and how to estimate their characteristics (parameters) such as the orientation and period. The moire effect is described in two domains, the image space (spatial domain) and in the spectral domain using the complex numbers. The tutorial covers the indicial equation method, the coplanar and noncoplanar sinusoidal gratings, the moire effect in a spatial object (a cylinder), as well as explains the moire wave vector, the moire spectra, the spectral trajectories, and summarizes behavior of the visible patterns in moved/rotated gratings.

https://www.spiedigitallibrary.org/journals/Optical-Engineering/volume-57/issue-3/030803/Moir%c3%a9-effect-in-displays-a-tutorial/10.1117/1.OE.57.3.030803.pdf

Moiré effect in displays: a tutorial

Equations for spectral peaks and trajectories are found for N superposed one-dimensional gratings. The equations of trajectories are represented using the complex numbers. The number of geometric elements in the spectrum is found under various conditions and in the matrix form. The derivatives of trajectories are obtained. The orthogonal case is investigated in details, in particular, the regular structures (the square and the octagon) are found in the spectrum. The numerical simulation is in a good agreement with the theory. The proposed technique seems to be helpful in estimation of occurrence of moire patterns in visual displays.

Theoretical estimation of moiré effect using spectral trajectories

We describe an optical phenomenon of unmovable moire patterns in sliding (moving) grids and gratings. The phenomenon was observed visually in the planar straight movement of the black-and-white gratings with a period of several mm. This is a velocity-independent effect confirmed analytically and in a computer simulation based on the spatial averaging. We found the static directions of the moire patterns in the regular grids, but our technique can be also applied to other objects. The orientation and period of the static moire patterns are not obvious, especially in the presence of the distance effect. The phenomenon can be practically used in security applications.

/pdf/static-moire-patterns-in-moving-grids-2j92xzhqrd.pdf

Static moiré patterns in moving grids.

The moire effect in multi-layered cylindrical objects was studied theoretically and experimentally. In theory, the central projection of double-layered convex and concave half-cylinders was considered. Based on the projected period expressed analytically in terms of the incidence angles, the period of the moire patterns on the axis was obtained as the closed-form expression; the off-axis effect was analyzed qualitatively. The distance was discovered, where the period as the function of the angle is nearly constant. The experiments performed with two cylindrical devices confirm the theory. The results can be applied to nanoparticles and curved/flexible displays.

Moiré effect in double-layered coaxial cylinders

If two patterns are combined with each other they create a pattern of their own. Interference between two sets of fine pattern grids, the scanner samples and the geometric structure in the original image cause moire patterns. These artifacts are generally accepted as an inevitable quality degradation of the screening process. In this paper two patterns were combined to provide a moire- the first one with RGB sensor, the second one with the RG-BG sensors, both with the Siemens' star.

Moire in a scanned image

The fourth-order differential operator has been designed including its coefficients B i and C i in such a way, that its first
three eigenfunctions Φ i (λ i E) are almost identical those obtained by orthogonalising the colour matching functions x (λ) ,
y (λ), and z (λ), CIE 1964. This operator gives an exact description of colour perception. Any changes to the spectral sensitivity
of the cone system and B i , C i coefficients in our mathematical model result in a description of colour blindness in
the observer. The principal aim of this paper is to mathematically describe the above-mentioned operator.

Miroslav Dohnal

Papers

Moire in a scanned image

Colour-matching functions as an eigenproblem solution