A box spline calculus for the discretization of computed tomography reconstruction problems

Area of Science:

• Image processing and analysis
• Applied mathematics
• Medical imaging

Background:

• B-splines are effective for continuous-domain biomedical image representation.
• The Radon transform is fundamental in tomographic imaging.

Purpose of the Study:

• To prove box splines are closed under the Radon transform.
• To derive explicit formulae for box spline Radon transforms.
• To extend this framework to non-Cartesian lattices and tomographic reconstruction.

Main Methods:

• Mathematical proof of closure under the Radon transform for extended box splines.
• Derivation of explicit formulae for these transforms.
• Application to tomographic reconstruction problems in a box spline basis.

Main Results:

• The extended family of box splines is closed under the Radon transform.
• Explicit formulae for box spline Radon transforms are derived.
• The 2-D Radon transform of an N-direction box spline is a polynomial spline of degree N-1.
• The framework supports non-Cartesian lattices.

Conclusions:

• The proposed box spline framework enables proper discretization of tomographic reconstruction.
• This approach offers practical advantages for improving quality and efficiency in medical imaging modalities like X-ray computed tomography and cryo-electron microscopy.