Linjesum

Linjesum is a concept in discrete mathematics and image processing referring to the sum of elements of a finite grid or matrix along lines of fixed direction. For a two-dimensional array A with indices i and j, a line L is a set of cells that lie on a line of chosen slope (for example horizontal, vertical, or diagonal). The linjesum of L is the sum of A[i,j] over all (i,j) in L. The collection of such sums for all lines in the chosen directions is called the linjesum transform of A.

In practice, linjesums generalize row sums and column sums. They can be taken for several directions to form a line-sum projection of the image. This concept is central to discrete tomography, where an unknown grid is reconstructed from its line sums. Linear properties apply: the linjesum operator is linear, so the linjesum of a sum of grids equals the sum of their linjesums, and scaling a grid scales its linjesums accordingly.

Inverse problems and applications: Reconstructing a grid from its line sums is often under-determined, depending on grid size and the set of directions used. There are uniqueness results for specific cases, such as certain binary grids and direction sets. Algorithms for reconstruction include algebraic methods, linear programming, and iterative techniques inspired by backprojection, commonly used in imaging and materials analysis.

Example: consider a 2x2 grid A with entries [ [1, 2], [3, 4] ]. Horizontal line sums are 1+2 = 3 and 3+4 = 7. Vertical line sums are 1+3 = 4 and 2+4 = 6. The main diagonal sum is 1+4 = 5, and the anti-diagonal sum is 2+3 = 5. These line sums illustrate the basic idea of linjesum as a line-based projection of a grid.