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Publication# Synthèse d'images moiré

Abstract

We explore a new approach for synthesizing moiré images that can be used for authentication of documents. For synthesizing moiré images, we need two layers: a base layer made of replicated bands or parallelograms and a revealing layer made of transparent lines (or dots). When superposing a base layer and a revealing layer, moiré images appear. Our method enables us to create dynamically moving messages incorporating text or color elements. In a simple layout of the base layer, the same pattern is replicated in each base band or parallelogram. Since the base layer's replication periods and the revealing layer period are similar, the revealed image is formed by the replicated tiny patterns, enlarged and possibly transformed. By considering the formation of the moiré image as a line sampling process, we derive the linear transformation between the base layer and the moiré image. We obtain the geometric layout of the resulting moiré image, i.e. its orientation, size and displacement direction when moving the revealing layer on top of the base layer. Interesting moiré images can be synthesized by applying geometric transformations to both the base and revealing layers. We propose a mathematical model describing the geometric transformation that a moiré image undergoes, when its base layer and its revealing layer are subject to different freely chosen non linear geometric transformations. We derive the geometric transformation to apply to the base and revealing layers, in order to obtain a desired moiré image transformation. We also derive layer transformations which yield periodic moiré images despite the fact that both the base and the revealing layers are curved. The approach for deriving the relationship between the respective geometric transformations of base layer, revealing layer and moiré images, relies on the comparison between the relative shifts of the revealing layer, the base layer and the moiré image. We generalize the 1D moiré synthesis approach to the 2D moirés. The base layer is made of replicated tiny parallelograms. The revealing layer is made of transparent dots. New interesting effects are obtained by applying geometric transformations such as independent moiré movements in two different orientations. A further method is presented in chapter 5, which relies on elevation profile level lines. A set of lines is modified according to an elevation profile. When superposing the modified base layer and the revealing layer, the level lines are revealed by a succession of various intensity or color lines. This elevation profile can either be a mathematical function, or can be created from an image containing text or symbols. Most moiré effects are created thanks to an interactive Photoshop plug-in called MoiréImaging.

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Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

Geometric transformation

In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations).

Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments.

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