I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because of the greater flexibility, I want to solve it by interpreting it as a least squares problem, which I define as follows:

the variables correspond to the cells of a rectangular grid, that covers the relevant range of $f$'s arguments

each equation corresponds to a different line(-segment), where each variable is weighted with the length of the intersection of that line(-segment) with the rectangular cell that corresponds to the variable and the right hand side corresponds to the integral of $f(x,y)$ along that line(-segment).

If the line misses a cell, then the length of the intersection is defined to be $0$.

Now, the problem and freedom I have, is to determine a set of line(-segments), from which the cell-weights that approximate the function values can be determined.

It is clear, that the number of equations must at least equal the number of cells but, as there are no other restrictions, there are various options for choosing the line(-segments) that yield the equations, with different implications on the numerical properties of the resulting least squares problem.

Question:

what can be recommended for generating potentially infinite sequences of lines for setting up the least squares problems, whose solutions will yield increasingly better approximations of $f(x,y)$,

i.e. how to select the angles and anchor points, that define the line(segment)s.

For the directions of the lines, the following options came to my mind:

- distribute them evenly over the range $[-\pi,\pi)$
generate them via a low-discrepancy sequence over $[-\pi,\pi)$ e.g., van der Corput sequence.

generate them via a sequence of Pythagorean triples $(a,b,c)\in\mathbb{N}^3: a^2+b^2=c^2$, setting $\phi := arctan(\frac{a}{b})$

For the anchor points the following strategies seem reasonable:

use the center of cell

use a corner of a cell

if the line integrals happen to be integers, then combining the "Pythagorean directions" with cell-corners as anchor points guarantees, that the tile weights are also integers, which may increase numeric precision.

isprimarily a numeric one and, evenly spaced sampling is the simplest strategy, but not the best as can be learned from approximating functions by polynomials. $\endgroup$