IIIT Hyderabad Publications
Tomographic Image Reconstruction in Noisy and Limited Data Settings.
Author: Syed Abbas Tabish
Report no: IIIT/TH/2016/39
Reconstruction of images from projections lays the foundations for computed tomography (CT). Tomographic image reconstruction, due to its numerous real world applications, from medical scanners in radiology and nuclear medicine to industrial scanning and seismic equipment, is an extensively studied problem. The study of reconstructing function from its projections/line integrals, is around a century old. The classical tomographic reconstruction problem was originally solved 1917 by J. Radon, proposing and inversion method now known as filtered backprojection (FBP). It was later shown that infinitely many projections are required to reconstruct an image perfectly. It is understood that incomplete data would leads to artifacts in the reconstructed images. In addition to the artifact problem, arising due to limited data availability, the reconstructed images are known to be corrupted by noise. We study these two problems of noisy and incomplete data in the follwoing two setups. Nuclear imaging modalities like Positron emission tomography (PET) are characterized by a low SNR value due to the underlying signal generation mechanism. Given the significant role images play in current-day diagnostics, obtaining noise-free PET images is of great interest. With its higher packing density and larger and symmetrical neighbourhood, the hexagonal lattice offers a natural robustness to degradation in signal. Based on this observation, we propose an alternate solution to denoising, namely by changing the sampling lattice. We use filtered back projection for reconstruction, followed by a sparse dictionary based denoising and compare noise-free reconstruction on the Square and Hexagonal lattices. Experiments with PET phantoms (NEMA, Hoffman) and the Shepp-Logan phantom show that the improvement in denoising, post reconstruction, is not only at the qualitative but also quantitative level. The improvement in PSNR in the hexagonal lattice is on an average between 2 to 10 dB. These results establish the potential of the hexagonal lattice for reconstruction from noisy data, in general. In the limited data scenario we consider the Circular arc Radon Transform (CAR). Circular arc Radon transforms associate to a function, its integrals along arcs of circles. The transforms involve the integrals of a function $f$ on the plane along a family of circular arcs. These transforms arise naturally in the study of several medical imaging modalities including thermoacoustic and photoacoustic tomography, ultrasound, intravascular, radar and sonar imaging. The inversion of such transforms is of natural interest. Unlike the full circle counterpart -- the circular Radon transform -- which has attracted significant attention in recent years, the circular arc Radon transforms are scarcely studied objects. We present an efficient algorithm that gives a numerical inversion of such transforms for the cases in which the support of the function lies entirely inside or outside the acquisition circle. The numerical algorithm is non-iterative and is very efficient as the entire scheme, once processed, can be stored and used repeatedly for reconstruction of images.
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