to the diffraction signal (“Background aware") in addition to a support constraint on the electron density consisting of a fixed number of voxels rather than a static mask (“Voxel number support").

The iterate $\Psi$ is comprised of both the real space density ${\rho(\mathbf{x})}$ and background ${B(\mathbf{q})}$ $$\Psi = \left\{\rho(\mathbf{x}), B(\mathbf{q})\right\}$$ In practice this consists of two 3D volumes, one for the real-space electron density and the other for the square root of the background intensity. The calculated intensity is the sum of the intensity from the particle plus the background, $$I_\text{calc}[\Psi](\mathbf{q}) = \left|\mathcal{F}[\rho](\mathbf{q})\right|^2 + B^2(\mathbf{q})
\label{eq:icalc}$$ where $\mathcal{F}[\rho]$ is the discrete Fourier transform of the electron density $\rho$. The modulus projection rescales both terms by the ratio to the measured Fourier magnitude, $$P_M[\Psi] = \left\{\mathcal{F}^{-1}\left[\sqrt{\frac{I_\text{meas}(\mathbf{q})}{I_\text{calc}(\mathbf{q})}} \mathcal{F}[\rho](\mathbf{q})\right], \sqrt{\frac{I_\text{meas}(\mathbf{q})}{I_\text{calc}(\mathbf{q})}}B(\mathbf{q})\right\}
\label{eq:pmod}$$ where $I_\text{meas}(\mathbf{q})$ is the measured intensity.

The support projection imposes two different constraints on the two halves of the iterate, $\rho$ and $B$. A constant $N$ is chosen at the beginning representing the number of voxels inside the particle for which the density is allowed to be non-zero. In this case we chose $N=2000$. The modulus-squared electron density values are sorted and the highest $N$ are left unchanged while the rest are 