APPENDIX D:

INDUCED MOIRE PATTERN BACKGROUND

Large-scale moiré patterns occur when two regularly repeating patterns with slightly different periods are superimposed. The spacing of the resulting moiré pattern is most striking when the frequencies are in the ratio of small integers.

A simple illustration of induced moiré can be seen using the concentric-circle pattern shown in Figure D-1. To observe an induced moiré pattern, place a pocket comb one to two inches above the concentric-circle pattern. In this example, the comb serves a sampling function which analogous to the sampling that occurs in digital imaging systems.

FIGURE D-1 Concentric-circle pattern.



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OCR for page 115
COUNTERFEIT DETERRENT FEATURES FOR THE NEXT-GENERATION CURRENCY DESIGN APPENDIX D: INDUCED MOIRE PATTERN BACKGROUND Large-scale moiré patterns occur when two regularly repeating patterns with slightly different periods are superimposed. The spacing of the resulting moiré pattern is most striking when the frequencies are in the ratio of small integers. A simple illustration of induced moiré can be seen using the concentric-circle pattern shown in Figure D-1. To observe an induced moiré pattern, place a pocket comb one to two inches above the concentric-circle pattern. In this example, the comb serves a sampling function which analogous to the sampling that occurs in digital imaging systems. FIGURE D-1 Concentric-circle pattern.

OCR for page 115
COUNTERFEIT DETERRENT FEATURES FOR THE NEXT-GENERATION CURRENCY DESIGN The mathematical formulation of the moiré phenomenon can be done in several ways. A Fourier series representation can be made of the object and the sampling function in frequency space. The presence of a sampling function leads to the generation of aliased frequencies, which, when reconstructed, represent the moiré pattern. The same calculation can be performed in the spatial domain using discrete sampling. If there exists a certain ratio relationship between the period of an object and the sampling function, then false data (moiré pattern) will be generated that are fixed in space and related to the sampling ratio.