Background Fitting (Parametric, Iterative 2D)

Parametric fitting models the diffuse background as an explicit 2D function and subtracts it, in contrast to the non-parametric methods (White-top-hats, 2D Convex Hull, etc.) documented on Background Subtraction, which estimate the background numerically. It targets the two structures that dominate a folded muscle pattern: the anisotropic equatorial streak near the beam and the broad, near-isotropic general background. Use it when you need to subtract the background on the whole pattern or when a smooth analytic background that does not follow the peaks is preferable.

Fitting runs in the Iterative 2D Background Fitting window, opened from the Parametric Background Fitting panel in the Results tab. It operates on the quadrant-folded image. When the result is applied, the residual (image minus fitted background) replaces the current result.

Model

The fitted background is the sum of an equator component and a general component.

Equator component — a single elliptical-coordinate peak (Murthy et al., Polymer 1997), fit over an equatorial band with the beam and the equatorial Bragg peaks masked out. The pattern is remapped into elliptical coordinates (u, v) sharing a focal distance A; the peak is a modified Lorentzian (Pearson VII) along u — centered at u0 with radial width uw and shape exponent m — multiplied by the sum of four modified-Lorentzian lobes along v, replicated to the four quadrant-symmetric angular positions (v0, 180°−v0, 180°+v0, 360°−v0) with a common angular width vw. The fitted values are an amplitude, a radial (u) position and width, an angular (v) position and width, and a shape exponent, for example:

EQUATOR (elliptical streak)
  Ellipticity A :      0.000 px
  Baseline      :     -0.019 counts
  Equatorial streak:
    Amplitude   :      1.875 counts
    u-position  :     13.317 px
    u-width     :    106.933 px (FWHM)
    v-position  :      0.838 deg
    v-width     :     10.824 deg (FWHM)
    m           :      2.622

General component — an isotropic, slightly elongated background of the form exponential + component 2 + constant baseline:

Component 1 is a fixed exponential.
Component 2 is selectable: lorentzian, powerlaw, stretched, or auto. With auto, the first round tries all three kernels and keeps the one with the best anti-oversubtraction score, then pins it for the remaining rounds. The default is lorentzian as it works best in most cases; however, for some patterns with changed structure a different kernel may be better.
Component 3 is a constant baseline.

GENERAL (exp + lorentzian + baseline)
  Baseline      :      0.019 counts
  Radial rmin   :     35.000 px
  Comp1 (exponential):
    amp         :      1.122 counts
    scale (eff) :      1.171 px
  Comp2 (lorentzian):
    amp         :     16.909 counts
    scale (eff) :      0.820 px

How the fit runs

Fitting is alternating block-coordinate descent. A hidden step 0 seeds the first equator fit with a rough projection background (a per-sector cone-model). Each round then alternates:

Fit the equator to (image − current general background).
Fit the general background to (image − current equator).

The general fit penalizes oversubtraction (when the fit goes above the data, so the residual is negative). After the configured number of rounds, the fit selects the best iteration: the one with the fewest oversubtracted pixels among the rounds whose equator formed the expected two lobes. If no round forms two lobes, the fit falls back to the least-oversubtracted round and warns to inspect the fitted equator before applying.

Fitting is done on a center-crop of size 2 x fit size and optionally downsampled for speed; the fitted background is reconstructed at full resolution. For faster processing is it recommended to use lower Fit size, as long as the crop still covers the data.

Masks

The fit uses a general mask and an equator mask, built from the same evaluation-mask parameters as the non-parametric optimizer and mirrored into the fitting window under Mask parameters. Editing a parameter here updates the matching Background Subtraction control and rebuilds the masks; select General mask or Equator mask in the View dropdown to inspect them before running.

Equator band height — height of the equatorial band kept for the equator fit.
Beam Center Radius — radius of the central beam removed from both masks.
Layer line spacing (M1) and Layer line width — mask the Bragg layer lines so they do not bias the general fit.
R-max — outer radius of the rmin-rmax annulus used for the masks and for every oversubtraction measurement.
Equator peaks and Equator peak width — number and width of equatorial Bragg peaks detected and removed from the equator fit (local to this window).

Advanced Settings

Under Advanced Settings (collapsed by default):

Number of rounds — equator/general alternating rounds.
Equator max fit iters / General max fit iters — least-squares evaluation limits per stage.
Fit size (r-max*) — fit radius; the image is center cropped to twice this value. Lower is faster, as long as it still covers the data. Defaults to 0.8 × r-max.
Downsample — downsample factor during fitting. Lower is faster, the default value of 2 is usually sufficient. The fitted background is reconstructed at full resolution.
General reduction / Equator reduction — scale each fitted background down by this fraction before subtracting, to guard against oversubtraction. The general reduction is always applied (default 5%). Editing a reduction after a fit rebuilds the background and residual with no re-fit. It is recommended to increase these value if there is oversubtraction and apply a smooth non-parametric method on the residual, rather than trying to fit the background more aggressively.
Auto-reduce — increase both reductions on top of the fixed values until the oversubtracted-pixel fraction stops improving. After a fit the reduction spinboxes show the values actually used.

Views and fitted parameters

The View dropdown shows Original, Fitted background (equator+general), Equator component, General component, Residual (background removed), Equator / Meridian profiles, and the two mask overlays. Before a fit, only the mask overlays are available.

The Fitted parameters panel reports the selected iteration, the component-2 kernel, the oversubtracted-pixel fraction over r-min/r-max, the reductions used, and the full equator and general parameter values.

Applying and running for a folder

Apply & Close replaces the current result with the residual and enables the checkbox Subtract fitted before non-parametric, so a non-parametric method can then run on top of the fitted residual.

Run Fitting with Default Settings and Apply (Parametric Background Fitting panel) runs the fit headlessly with the current settings and applies it in one click, without opening the window.
Fit background for each image in folder fits and subtracts the parametric background for every image during folder processing, using the current fitting parameters, instead of reusing a single applied fit.

Every applied fit is saved to qf_results/bg_fit_params/: the equator, general, and residual images as TIFFs, and the fit parameters as <image>_bgfit_params.npz. Fit parameters are also cached per image, so reopening the window on a processed image reloads the applied fit without re-running it.

Headless mode

Parametric (iterative 2D) fitting is controlled by bgfit_* flags mirroring the fitting window: bgfit_comp2, bgfit_iters, bgfit_eq_max_nfev, bgfit_gen_max_nfev, bgfit_fit_size (the fit radius; doubled internally for the crop), bgfit_downsample, bgfit_use_step0, bgfit_general_reduction, bgfit_equator_reduction, and bgfit_auto_reduce. These are written when settings are saved from the GUI and are used for per-image fitting during batch runs when Fit background for each image in folder is enabled.

See How to use — Headless Mode for the general headless workflow.