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 (**PC**

_{z}*) becomes*

**and**the further the pattern center is shifted from the center of the image, i.e.**the bigger****Δ**= (PC

_{x}-½)² + (PC

_{y}-½)²

*.*

*This is one reason for the later implemented, special treatment of TKD pattern indexing, especially if PC*

_{y}<0.To **reduce** these **curvature**, first the **electron energy** can be** increased** which reduces, however, the spatial resolution of the technique, and **PC _{z}** 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 PC

_{z}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**.