Multiplexing and Demultiplexing Signals for Radiography Application Using the Discrete Fourier Transform
Our goal is to develop an X-ray phase-contrast imaging system that can provide excellent soft tissue contrast of phase, attenuation, and small-angle scatter. We propose to replace the common system of G0, G1, and G2 gradings with a biprism array to replace the G1 grading and introduce a novel X-ray tube designed to replace the motion of the phase stepping grading G2. The proposed X-ray tube uses temporal multiplexing to provide simultaneous virtual “electronic phase stepping.” In this work the discrete Fourier transform is used to separate from the composite measurement individual X-ray phase contrast measurements sampled at different frequencies. The method performs a discrete Fourier transform of a composite refence sequence to obtain using the frequency amplitudes calibration factors needed to extract the X-ray phase contrast measurement amplitudes from the composite image. The composite reference sequence is the sum of the individual sequences, at different frequencies, with amplitudes of one. The method takes the discrete Fourier transform of this composite reference sequence; whereby, the amplitude of each frequency component is compared with the total sum of its stand-alone sequence amplitude. A calibration factor is determined so that the amplitude of this composite reference frequency times the calibration factor must equal the total sum of the sequence amplitude—the zero-frequency amplitude of the discrete Fourier transform of its stand-alone sequence. To demultiplex the composite measured signal these calibration factors are multiplied by the amplitudes of the frequency components of the discrete Fourier transform of the composite X-phase-contrast measurement to obtain the amplitude of each frequency encoded measurement. Using these calibration factors, we demonstrate with the discrete Fourier transform in Mathematica the extraction of individual images from a composite image that one would expect obtaining from our proposed new X-ray phase contrast imaging system. We then demonstrate as an example how using images from X-ray phase contrast data one can calculate phase, attenuation and the dark field images using grading phase step data supplied to use from Microworks, GmbH in Karlsruhe, Germany.