Dot trajectories in the superposition of random screens: analysis and synthesis

Abstract

Moiré effects that occur in the superposition of aperiodic layers such as correlated random dot screens are known as Glass patterns. One of the most interesting properties of such moiré effects, which clearly distinguish them from their periodic counterparts, is undoubtedly the appearence in the superposition of intriguing microstructure dot alignments, also known as dot trajectories. These dot trajectories may have different geometric shapes, depending on the transformations undergone by the superposed layers. In the case of simple linear transformations such as layer rotations or layer scalings, the resulting dot trajectories are rather simple (circular, radial, spiral, elliptic, hyperbolic, linear, etc.); but in more complex layer transformations the dot trajectories can have much more interesting and surprising shapes. A full mathematical analysis of the dot trajectories, their morphology, and their various properties is provided. Furthermore, it is shown how the approach also allows us to synthesize correlated random screens that give in their superposition dot trajectories having any desired geometric shapes. Finally, it is also explained why such dot trajectories are visible only in superpositions of aperiodic screens but not in superpositions of periodic screens.

Continuous Glass Patterns for Painterly Rendering

“…, a Taylor expansion yields (21) As well known, a singular point is a center of a closed curve iff the eigenvalues of have zero real part. Basic algebra shows that, for the vector field defined in (19), the eigenvalues of are equal to (22) where is the trace of .…”

“…3). In general, the geometrical structure exhibited by the GP defined in (3) is related to the trajectories which solve (2) [21].…”

“…On the other hand, microstructure concerns the orientation field that is induced by the strong correlation between the two superposed point sets that compose a GP. Microstructural properties of GP can be studied naturally in the framework of the dynamical systems theory [20], [21] and algorithms able to synthesize any microstructure have been provided [21].…”

High spatial sensitivity method based on moiré interferences derived from a single layer. Case study: The Moiré Pendulum

“…When working and superimposing two layers with periodic patterns, a moiré interference layout with a periodic pattern is also generated. On the contrary, if the superimposed layers have aperiodic patterns, the pattern of the resultant layer will also be aperiodic [2,3]. Not all the superpositions of periodic layers result in an interference layout with the moiré effect.…”

Glass Patterns and Artistic Imaging

“…On the other hand, microstructure concerns the orientation field that is induced by the strong correlation between the two superposed point sets that compose a GP. Microstructural properties of GP can be studied naturally in the framework of the dynamical systems theory [30,31] and algorithms able to synthesize any microstructure have been provided [31]. In this paper, we propose a simple mathematical operator which transfers the microstructure of a GP to a natural image, thus resulting in a nice artistic effect.…”