The application of another transformation than Hough or Radon commonly used for band detection goes back to [1]. There, this function is called SCRUFF transformation but seem to be similar to the Funk transformation used in CALM.
The edge-filtered Funk transformation as shown below is exclusively used to indicate [hkl]*, i.e., to discover as many as possible Kikuchi bands. It is not and cannot be used for band profile extraction.
Phenomenological description assuming a complete signal
If the diffraction signal is projected onto a spherical surface, the intensity of any direction [x,y,z] in the Funk transformation is equal to the summed intensity along the great circle perpendicular to [x,y,z].
As well-known, each Kikuchi band can be interpreted by means of a very flat double cone with half opening angle of (90-θ). In the stereographic projection (left image below) the band edges represent so-called small circles very close to the great circle describing the band-forming lattice plane trace.
After Funk transformation (right image) of the complete diffraction signal the band becomes again a double cone. The visible circle has now, however, a diameter of 2θ. The cone axis indicates the lattice plane normal [hkl]*.
sterographic projection of the master pattern 
edge corrected Funk transformation of the master pattern
This perfect condition of complete and well-shaped circles displaying the band widths undistorted is only available for a complete simulation as used in a digital twin (“master pattern”).
The typical, incomplete signal (Kikuchi pattern)
If only a part of the signal is available, e.g. a single cube plane (left image below), all circles in the edge-filtered Funk transformation (right) degenerate to circle-like features which are still well detectable.
If we reduce the captured segment further using the typical size and shape of a Kikuchi pattern (only between 12-15% of the entire signal) with a resolution of 320×230 pixels only and shift the projection center to PC = [0.5,0.25,0.667], the Funk transformation becomes asymmetric. Reducing the sector size the empty region in the Funk transformation (right) grows and the degeneration of circles increases since the respective bands become shorter an shorter. The projection center is responsible for the shift of the empty area.
The medium image resolution has no considerable effect on the Funk transformation.
Experimental pattern
In comparison to simulated patterns an experimental Kikuchi pattern is slightly more blurred. In experiments, the wavelength is represented by a distribution and not a distinct value, but also the crystal is not as perfect as described in simulations.
Nevertheless, the Funk transformation is a great tool because of the strong constraints between [hkl]* and (uvw)*.
experimental pattern from Al 
Experimental influences like excess-deficiency will certainly affect the alignment of [hkl]* along great circles displaying (uvw)* and also the Bragg angles but the impact is small. In case of bigger deviations, either PC is incorrect, or considerable image distortions occur.
TKD
Also TKD patterns can be processed in CALM. However, it should be clear that the captured sector is commonly smaller than for EBSD geometry. The following pattern has been aquired for a comparatively unusual PC=[0.521,-0.334,0.6].
Experimental TKD pattern from Al 
[1] Day, A. P.: Spherical EBSD, J. Microscopy, 2008, 230, 472-486