Linjesums

Linjesums, often translated as line sums in English, is a term used in mathematical and data-analytic contexts to describe the sum of values along a straight line that intersects a grid or matrix. In two dimensions, a linjesum is obtained by adding all entries that lie on a line of a fixed slope, commonly including the horizontal and vertical lines (rows and columns) as well as diagonals or other directions chosen for analysis.

In discrete tomography, linjesums serve as projection data that summarize an object arranged on a finite grid. By knowing the line sums for several directions, one aims to reconstruct the original grid of values, which is typically binary or nonnegative. This reconstruction problem mirrors the idea of imaging by projections, but in a discrete and finite setting. The choice of directions and the number of line sums strongly influence whether a unique reconstruction is possible.

Reconstruction from linjesums often leads to solving a system of linear equations with integer or binary constraints. In general, the problem can be ill-posed and computationally challenging, especially when limited directions are available. Algorithms range from algebraic and iterative methods to optimization-based approaches that incorporate prior information about the expected structure of the grid.

Applications of linjesums include discrete tomography, materials science (e.g., reconstructing microstructures from projection data), non-destructive testing, and certain combinatorial problems in which arrays are characterized by prescribed line sums. The concept is closely related to line projections and the discrete Radon transform in the study of finite grids.

See also: discrete tomography, line projection, Radon transform, binary matrix reconstruction.