Fourier analysis - Lin Yangchen

©Lin Yangchen

Fast Fourier transformation is widely used in time series analysis. It converts temporal oscillations or repetitive spatial patterns into a power spectrum showing the amplitudes of oscillations at different frequencies.

Whole mint stamps were photographed with a Canon MP-E 65 mm lens mounted on a Canon EOS-1D X single-lens reflex camera. The optical system has a flat field of focus, no distortion and a resolution much higher than the finest engraved line on the stamp. The camera sensor was leveled with a two-dimensional spirit level. The analogue signal was converted to png format through raw and tiff, with no compression at any stage.

Stamp perforations were cropped out, leaving only the printed design for analysis. The images were converted to greyscale and contrast was boosted using Photoshop curves to increase the signal-to-noise ratio without blowing out the highlights.

Signal processing and visualization were programmed using the R Language for Statistical Computing. Each image was converted to matrices of grey pixel values and put through FFT. This produces complex numbers that represent the magnitude and phase shift of the signal. The magnitude component was used to plot the power spectrum.

In the plots, the lowest frequencies are in the centre, increasing outwards. The crossed lines passing through the centre are derived from the edges of the original image. Shown in the following plots are the lowest frequencies from 1 to 250 cycles across each dimension, which exceeds the Nyquist criterion as confirmed by a visual examination of the engraved lines. The power spectra show that the essay (first row) has more clutter across different spatial scales compared with the issued stamps in which large-scale features stand out from the fine detail.


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