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A coherent optical Fourier transform system is studied for which masks of curves only are used as input. A general theory for this problem is discussed, resulting in the functional connection between input and output data and limitations due to thickness, curvature, and extension of the curve. The general result can be simplified for a curve which represents a signal modulated by a carrier. In this case the Fourier transform of the signal can be found optically in first-order diffraction. The method is applied to interferogram curves arising in Fourier spectroscopy.