Date of Award

2025

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Evan Randles

Second Advisor

Stephanie Dodson

Abstract

This thesis develops the mathematical foundations of computed tomography (CT) reconstruction through the lens of harmonic analysis. Beginning with the Schwartz class, we introduce the Fourier transform and its role in expressing the Radon transform and its inversion via a fractional Laplacian. After constructing the Radon transform in general dimension R^d, we specialize to the cases d = 2 and d = 3, demonstrating explicit inversion formulas and the associated instability in lower dimensions. For its computational advantages, we study filtered back-projection using classical low-pass filters (Ram-Lak, Shepp–Logan, Cosine, Gaussian) and formulate a discrete reconstruction algorithm grounded in convolution and interpolation. Quantitative evaluation on synthetic phantoms with spatially varying Gaussian noise is performed using metrics such as Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and a Laplacian-of-Gaussian (LoG) edge preservation index. We then introduce a spatially adaptive post-processing filter guided by local LoG responses, which improves structural accuracy and noise suppression beyond frequency-domain methods. Our results establish a rigorous and computational framework for CT reconstruction, connecting classical analysis with applied imaging.

Keywords

Computed Tomography, Radon Transform, Fourier Analysis

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