Draw N points uniformly at random from a disk and bin them into a grid of
voxels. Each voxel's count is a Poisson-ish random variable with mean
μ = points per voxel, so its relative noise is
1/√μ — independent of how big the disk is. Sweep N and the
resolution below to see when the sampled disk stops looking speckled and starts
looking like a disk.
Exact density: each voxel shaded by the fraction of it inside the disk.
Counts from random points, on the same color scale.
Error in units of the expected standard deviation, (count − ideal) / √ideal.
Why this matters for MRI simulation. Discretizing a continuous object into
isochromats is exactly this experiment: the signal in a voxel is a sum over the
isochromats that landed in it, so random placement injects noise of order
1/√μ on top of the physics. Statistics are computed only over
voxels lying entirely inside the disk — boundary voxels have a smaller
expected count and would otherwise inflate the measured spread. Note that the noise
panel looks like white noise at every setting: what improves with more points is not
its character but its size relative to the signal.