Correction Implementations

The mathematical implementations and the academic reference of each of the implemented corrections are tabulated below for the convenience of those utilizing the package.

In all of the below operations, \(I_j\) denotes the uncorrected photon count of detector pixel \(j\) whilst \(I_{j,c}\) denotes the corrected photon count.

Correction Implementations

Correction	Operation	Parameters	Reference
correct_deadtime	\[\begin{split}I_{j,c} &= \frac{W_0(-I_jT)}{T} \\[8pt] T &= \frac{\tau_1+\tau_2}{\delta t}\end{split}\]	\(\delta t\): Count time \(\tau_1\): Minimum pulse separation \(\tau_2\): Minimum arrival separation	Everything SAXS: small-angle scattering pattern collection and correction
correct_dark_current	\[I_{j,c} = I_j - D_a - D_b \delta t - D_cI_j\]	\(\delta t\): Count time \(D_a\): Base dark current \(D_b\): Temporal dark current \(D_c\): Flux dependant dark current	Everything SAXS: small-angle scattering pattern collection and correction
normalize_frame_time	\[I_{j,c} = \frac{I_j}{\delta t}\]	\(\delta t\): Count time	Everything SAXS: small-angle scattering pattern collection and correction
normalize_transmitted_flux	\[I_{j,c} = \frac{I_j}{\unicode{x222F}{I_j}}\]		The modular small-angle X-ray scattering data correction sequence
correct_self_absorption	\[\begin{split}I_{j,c} &= I_j\frac{1-T^{\sec{2\theta}-1}}{\ln{T}(1-\sec{2\theta})} \\[8pt] T &= e^{-\mu t} \\[8pt] {2\theta}_j &= \arctan{\frac{\sqrt{{\delta x}_j^2 + {\delta y}_j^2}}{\Delta z}}\end{split}\]	\(\mu\): Absorption coefficient \(t\): Sample thickness \({\delta x}_j\): Horizontal pixel beam separation \({\delta y}_j\): Vertical pixel beam separation \(\Delta z\): Detector sample separation	Everything SAXS: small-angle scattering pattern collection and correction
average_all_frames	\[I_{j,c} = \sum{I_{n,j}}\]	\(I_{n,j}\): Intensity of pixel \(j\) at frame \(n\)	The modular small-angle X-ray scattering data correction sequence
subtract_background	\[I_{j,c} = I_j - I_{j,b}\]	\(I_{j,b}\): Background frame intensity of pixel \(j\)	Everything SAXS: small-angle scattering pattern collection and correction
correct_flatfield	\[I_{j,c} = I_j F_j\]	\(F_j\): Flatfield correction factor of pixel \(j\)	The modular small-angle X-ray scattering data correction sequence
correct_angular_efficiency	\[\begin{split}I_{j,c} &= \frac{I_j}{1-e^{\frac{-\mu t}{\cos{2\theta}}}} \\[8pt] {2\theta}_j &= \arctan{\frac{\sqrt{{\delta x}_j^2 + {\delta y}_j^2}}{\Delta z}}\end{split}\]	\(\mu\): Absorption coefficient \(t\): Sample thickness \({\delta x}_j\): Horizontal pixel beam separation \({\delta y}_j\): Vertical pixel beam separation \(\Delta z\): Detector sample separation	The modular small-angle X-ray scattering data correction sequence
correct_solid_angle	\[\begin{split}I_{j,c} &= \frac{I_j}{\cos{2\theta}^3} \\[8pt] {2\theta}_j &= \arctan{\frac{\sqrt{{\delta x}_j^2 + {\delta y}_j^2}}{\Delta z}}\end{split}\]	\({\delta x}_j\): Horizontal pixel beam separation \({\delta y}_j\): Vertical pixel beam separation \(\Delta z\): Detector sample separation	Everything SAXS: small-angle scattering pattern collection and correction
correct_polarization	\[\begin{split}I_{j,c} &= I_j[PK_v+(1-P)K_h] \\[8pt] K_v &= 1-(\sin{\psi}\sin{2\theta})^2 \\[8pt] K_h &= 1-(\cos{\psi}\sin{2\theta})^2 \\[8pt] {2\theta}_j &= \arctan{\frac{\sqrt{{\delta x}_j^2 + {\delta y}_j^2}}{\Delta z}} \\[8pt] \psi &= \arctan{\frac{\delta x}{\delta y}}\end{split}\]	\(P\): Horizontal polarization \({\delta x}_j\): Horizontal pixel beam separation \({\delta y}_j\): Vertical pixel beam separation \(\Delta z\): Detector sample separation	Everything SAXS: small-angle scattering pattern collection and correction
normalize_thickness	\[I_{j,c} = \frac{I_j}{t}\]	\(t\): Sample thickness	Everything SAXS: small-angle scattering pattern collection and correction
correct_displaced_volume	\[I_{j,c} = I_j (1 - v_d)\]	\(v_d\): Displaced volume	The modular small-angle X-ray scattering data correction sequence