Correction of obliquity

Even in expert hands a perfect transversal cut is almost impossible, and a certain degree of obliquity always remains. In that case, most fibers appear as ovals instead of disks. Fiber orientation provides an estimation of the obliquity. It can be evaluated by inspecting the principal axes of the fibers (see figure ). If the long axes are globally aligned, this global alignment corresponds to the orientation of the cut. The ratio between the average length of the long and short axis denotes the angle of the cut.

  

Figure 4.7: Obliquity parameters of a fiber

Practically, we define the obliquity vector $\vec{v_{i}}$ for the ith fibre as follows:

$$\vec{v_{i}} = \vert v_{i}\vert e^{j\theta_{i}}\\$$ (4.2)

where $\theta_{i}$ is the angle of the longest axis with the horizontal and |v_i| is

$$\vert v_{i}\vert = \frac{\sigma_{ai}-\sigma_{bi}}{\sigma_{bi}}$$ (4.3)

with $\sigma_{ai}$ and $\sigma_{bi}$ the lengths of the long and short principal axis of fiber i, respectively. The average obliquity over the N fibers of the image is

$$\vec{v_{mean}} \vert v_{mean}\vert e^{j\theta_{mean}}$$ = (4.4)

$$\mathrm{where} \; \; \; \; \vert v_{mean}\vert e^{j2\theta_{mean}} \sum_{i=1}^N \omega_{i}\vert v_{i}\vert e^{j2\theta_{i}}$$ = (4.5)

with weighting factors $\omega_{i}$. In practice, we use $\omega_{i} = \sigma_{bi}$, because larger fibers provide a more reliable estimate of the obliquity. In order to correct the obliquity of the cut, all fibers are contracted along the direction $\theta_{mean}$ by a factor $\frac{1}{(1+\vert v_{mean}\vert)}$, as illustrated at figure .

  

Figure 4.8: Correction of obliquity. From left to right: a) Fibers found on an oblique section; b) Principal axes of fibers c) Oblique-corrected fibers