Application of Image Reconstruction by Means of Chirp z-Transform

The optical C T (Computerized Tomography), which utilises light transmitted through a semi-opaque object extends vision to the inside of objects. The projection profiles of the optical C T are smeared due to the scattering of light in a translucent medium. In the image reconstruction for MRI (Magnetic Resonance Imaging), FID (Free Induction Decay) signals consisting of decaying sinusoids inevitably produce smeared spectra. The chirpz transform provides an effective means which reduces such smearing, when the DFT (Discrete Fourier Transform) used in different stages of image reconstruction processes, is replaced by the CZT (Chirp z-Transform). This paper discusses the peak value and the bandwidth of the improved spectrum by CZT in relation to the ordinary DFT (Discrete Fourier Transform) method and proper selection of a contour for a given set of data. The validity of replacing DFT by CZT in different image reconstruction schemes is also discussed. The improvement in image quality is demonstrated for the experimental data obtained from an optical C T system and that obtained from MRI specifically for the purpose of chemical-shift imaging. The Chirp z-Transform DFT evaluates z-transforms on the unit circle in the z-plane, whereas the chirp z-transform evaluates them on a spiral contour inside or outside the unit circle. Since one can adjust the contour such that the contour passes through or comes close to the poles of a signal under study, the spectrum is sharpened, and its frequency resolution is appreciably improved. When a sequence of N data x,, n = O...N 1 is sampled from an analog signal, the z-transform of this finite sequence is given by