Truncated projection computer tomography closed-form reconstruction algorithm stability

William J. Dallas

doi:10.1117/12.387720

6 June 2000 Truncated projection computer tomography closed-form reconstruction algorithm stability

William J. Dallas

Proceedings Volume 3979, Medical Imaging 2000: Image Processing; (2000) https://doi.org/10.1117/12.387720
Event: Medical Imaging 2000, 2000, San Diego, CA, United States

Abstract

Truncated Projection Computer Tomography (TPCT) is similar to conventional Computer Tomography (CT) in that it acquires projection information. The difference is that it weights and/or truncates the projections along their integration paths. The problem is succinctly stated as an integral equation termed the incomplete Radon transform. TPCT image reconstruction is equivalent to inversion of the incomplete Radon transform. We presented an inversion formula for this transform at SPIE Medical Imaging 1999. The solution involves the application of an integral transform that is equivalent to a combination of a square-root geometrical distortion and a Fourier transform. Subsequent to the 1999 presentation, we recast part of the inversion derivation in terms of Lagrange multipliers. Doing so led us to the realization that the incomplete Radon transform is a member of a much larger class of integral equations that lend themselves to the TPCT inversion techniques. In the present work we touch briefly on several topics. We will exhibit the larger class of integral equations in the form of Hilbert-space scalar products. They define generalized spatial filtering operations anchored simultaneously in direct space and any one of a multitude of transform spaces. In the examples relating to the TPCT reconstruction algorithm's stability, we perform computer simulations using off-axis Gaussian distributions as objects.

Citation Download Citation

William J. Dallas "Truncated projection computer tomography closed-form reconstruction algorithm stability", Proc. SPIE 3979, Medical Imaging 2000: Image Processing, (6 June 2000); https://doi.org/10.1117/12.387720

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KEYWORDS

Computed tomography

Tomography

Radon transform

Reconstruction algorithms

Condition numbers

Space operations

Spatial filters

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