Both transformations have their advantages and disadvantages.
Hough (or Radon) transformation
Purely projection-technically the Hough transformation fits better to the gnomonic projection of the crystal lattice on the planar detector screen. The drawback, however, is that we do not see the lattice projection, but diffraction phenomena of the lattice produce two hyperbola-shaped intensity drops, which are taken as evidence of geometrically predicted band edges.
The Hough peaks become more blurred as the deviations between straight lines and hyperbola-shaped curvature becomes stronger.
The curvature of the hyperbolas resulting from the gnomonic projection of small circles (band edges) grows, the larger the angular distance to the pattern center as well as the Bragg angles become.
According to Bragg’s law, the Bragg angle changes: (1) inversely proportional with the squareroot of the acceleration voltage and/or (2) proportional with the lattice parameters of the phase, (3) with the indexing of the lattice plane (hkl), and (4) with the diffraction order n.
The angular distance to the projection center PC, on the other hand, grows the smaller the distance to the detection screen (PCz) becomes and the further the pattern center is shifted from the center of the image, i.e. the bigger Δ = (PCx-½)² + (PCy-½)².
This is one reason for the later implemented, special treatment of TKD pattern indexing, especially if PCy<0.
To reduce these curvature, first the electron energy can be increased which reduces, however, the spatial resolution of the technique, and PCz can be increased which increases the necessary dwell time and so a possible charging. Since the applied technique works preferably for large angle Kikuchi patterns, an increase of PCz is from this point of view limited.
Please note: For high-accurate techniques is valid: The Hough peak does not define the lattice plane trace, except it crosses the pattern center. With increasing angular distance to the pattern center the deviation increases.
The biggest advantage of the currently used Hough transform is that PC is not needed. This explains why the pattern quality of pattern is always available, even if PC or the diffracting phase are unknown.
One could also easily improve the Hough transformation considering PC. Then gnomonic distortions could be immediately corrected so that equivalent bands would have equivalent peak widths. However, lattice planes of a zone would then be aligned along sine-shaped curves which is certainly not easy to interprete as great circles in the Funk transformation.
Funk transformation
The main disadvantage of the Funk transformation is that the recorded Kikuchi pattern cannot be used immediately, but must first be correctly projected onto a spherical surface what requires PC.
By this spherical projection alone, lattice planes become great circles and diffraction cones become small circles, i.e., their shape is unambiguously determined, which considerably limits possible errors in analyses.
The Funk transformation converts great circles into points and points into great circles. More generally one could say that it transforms small circles with the opening angle χ into small circles with the opening angle 90°-χ, because for a great circle χ=90° is valid.
All critical distortions mentioned above in Hough transformation are inherently compensated. It represents some kind of projection of the reciprocal lattice, where
- the cone axes (angular center of the circle) indicate reciprocal lattice directions [hkl]*,
- the great circles between them display reciprocal lattice planes (uvw)*, and
- the circle diameter represent the distance to one reciprocal lattice point hkl.