ALBULA  4.0.1
Statistics

Region of Interest

Line Plot

  • Select line using the right mouse button

  • For every pixel $i$ that is crossed by the line $I_i$ is defined to be the pixel-intensity and $\ell_i$ the length of the line segment within the $i$th pixel. The following quantities are computed:

    Integral $\sum_i I_i \ell_i$
    Length $\sum_i \ell_i$
    Mean $(\sum_i I_i \ell_i)/(\sum_i \ell_i)$
    Sigma $\sqrt{(\sum_i I_i^2 \ell_i)/(\sum_i \ell_i)-\mbox{Mean}^2}$
    Min $\mbox{min}_i I_i$
    Max $\mbox{max}_i I_i$

    Note: Except for the length we only take pixels into account that have a valid pixel value.

  • The values shown between two reflections in the plot provide an estimation of one possible lattice distance. The computation is based on the assumption that either the $h$, $k$ or $l$ values of the reflections differ by one and the other two are identical. This assumption is commonly fulfilled if the reflections are arranged along a straight line. In particular for two reflections that are separated by the angle $\theta$ the computed value is $\lambda/\sqrt{2(1-\cos\theta)}$, which takes the form $|\mathbf{a}| \cos\gamma $ if (without loss of generality) the $h$ values of the reflections differ by one and the $k$ and $l$ values are identical. $\mathbf{a}$ is the lattice vector which is related to $h$ and $\gamma$ is the angle between $\mathbf{a}$ and the plane spanned by the other lattice vectors $\mathbf{b}$ and $\mathbf{c}$.

Box Plot

  • Select box using the right mouse button

  • For every pixel $i$ that is covered by the box $I_i$ is defined to be the pixel-intensity and $A_i \le 1$ the part of the pixel, that is covered by the box. The following quantities are computed:

    Integral $\sum_i I_i A_i$
    Area $\sum_i A_i$
    Mean $(\sum_i I_i A_i)/(\sum_i A_i)$
    Sigma $\sqrt{(\sum_i I_i^2 A_i)/(\sum_i A_i)-\mbox{Mean}^2}$
    Min $\mbox{min}_i I_i$
    Max $\mbox{max}_i I_i$

    Note: Except for the area we only take pixels into account that have a valid pixel value.

  • Plots: The $x$-projection plot shows for every $x$-value inside the box the sum of all pixel-values $\sum_{y=y_\text{bottom}}^{y_\text{top}} I_{xy}$, where the sum ranges from the bottom to the top of the selected area. The $y$-projection plot is defined analogously.

Circle Plot

  • Select circle using the right mouse button

  • The selected region consists out of an inner and an outer circle, where pixels in the inner circle ( $A_{\text{refl.}}$) contribute to the reflection intensity and the outer circle ( $A_{\text{bg}}$) is used to compute the background. For every pixel $i$ that is covered by the selected region $I_i$ is defined to be the pixel-intensity and $A_i \le 1$ the part of the pixel, that is covered by the circle. The following quantities are computed:

    Integral $\sum_{i\in A_{\text{refl.}}} I_i A_i$
    Background $\sum_{i\in A_{\text{bg}}} I_i A_i$
    Intensity $\text{Integral}-\text{Background}$
    Intensity error (Squared sum of Poisson error and uncertainty of background estimation) $\sqrt{\text{Integral}+ A_{\text{refl.}}\text{Var}_{\text{bg}}}$
    Area $\sum_{i\in A_{\text{refl.}}} A_i$
    Mean $(\sum_{i\in A_{\text{refl.}}} I_i A_i)/(\sum_{i\in A_{\text{refl.}}} A_i)$
    Sigma $\sqrt{(\sum_{i\in A_{\text{refl.}}} I_i^2 A_i)/ (\sum_{i\in A_{\text{refl.}}} A_i)-\mbox{Mean}^2}$
    Min $\mbox{min}_{i\in A_{\text{refl.}}} I_i$
    Max $\mbox{max}_{i\in A_{\text{refl.}}} I_i$

    Note: Except for the area we only take pixels into account that have a valid pixel value.\

  • The plot shows the profile of the pixel-values along the line in the inner circle.

Powder Plot

  • Select circle around beam centre using the right mouse button

  • For every pixel $i$ that is covered by the selected region $I_i$ is defined to be the pixel-intensity and $A_i \le 1$ the part of the pixel, that is covered by the selected region. The following quantities are computed:

    Integral $\sum_i I_i A_i$
    Area $\sum_i A_i$
    Mean $(\sum_i I_i A_i)/(\sum_i A_i)$
    Sigma $\sqrt{(\sum_i I_i^2 A_i)/(\sum_i A_i)-\mbox{Mean}^2}$
    Min $\mbox{min}_i I_i$
    Max $\mbox{max}_i I_i$

    Note: Except for the area we only take pixels into account that have a valid pixel value.

  • Plot: The graph plots the pixel intensity averaged over concentric rings around the beam centre as function of $\theta = \arctan(r/d)$. Here $r$ is the radius and $d$ the distance of the detector from the crystal. If $d$ cannot been found in the image header the x-axis shows $r$ rather than $\theta$.

Max Finder

Find intensity maxima in the current image. Entering a number $n$ in the spinbox will mark the $n$ greatest maxima in the image. The pixel intensities will be listed together with their image coordinates in descending order. Clicking on a line in the list will highlight the corresponding peak in the image.

The maxima finder algorithm looks for local maxima within the image. In this context a pixel at $(x,y)$ is a local maximum if its intensity is greater or equal than its neighbor pixels, explicitly $I_{x,y} \ge I_{x\pm 1,y\pm 1}$.