In two dimensions, we consider the problem of reconstructing a vector field from partial knowledge of its zeroth and first moment ray transforms.Different from existing works the data is known on a subset of lines, namely the ones intersecting a given arc.The problem is non-local and, for partial data, severely ill-posed.We present a reconstruction method which recovers the vector field in the convex hull of the arc.An algorithm based on this method is implemented on some numerical experiments.While still ill-posed the discretization stabilizes the numerical reconstruction.
This paper discusses a quantitative partial imaging problem in vector tomography, specifically focusing on reconstructing a vector field from limited data provided by zeroth and first moment ray transforms defined over a subset of lines. The authors present a reconstruction method that, while still considered ill-posed, stabilizes numerical reconstruction by judiciously choosing discretization techniques. The introduction highlights the general problem framework, while later sections detail the theoretical foundation and implementation of a numerical algorithm alongside two numerical experiments to validate the proposed methodology. Results indicate challenges in accuracy, particularly near boundaries, yet demonstrate the method's ability to reasonably recover the overall structural properties of the vector field, even with noisy data.
This paper employs the following methods:
- Bukhgeim-Beltrami equations
- Cauchy type singular integral equations
The following datasets were used in this research:
- The reconstruction method stabilizes despite the problem being ill-posed.
- Numerical experiments indicate reconstruction errors of 48.1% and 80.9% relative errors in L2 for different test cases.
The authors identified the following limitations:
- The reconstruction is less accurate near the boundary due to ill-posedness of the singular integral equation.
- Increased order of ill-posedness makes reconstruction for higher order tensors more difficult.
- Number of GPUs: None specified
- GPU Type: None specified
- Compute Requirements: None specified