Characterization

The value ofn_s sets the position of the streaks and depends on the choice of coordinate system. As only the absolute of the Fourier componenents is considered, we may assume without loss of generality, that n_s= 0, which results in

I˜_t(k) =A

The resulting value of ˇS is

Sˇ= 1

By taking into account, that intensity values are always real and non-negative, the definition of the Fourier transform implies, that ˜I_f(0) ≥ I˜_f(n). This implies for a renormalized Streak Finder, that dividing ˇS by the number N_SF of considered Fourier modes (here NSF = 6) results in a renormalized anisotropy paramter S which fulfills 0≤S≤1.

Until here,NSF = 6 was chosen, but this is not necessarily the only possible choice. NSF

allows to adapt for different signal shapes:

S_NSF := 1

SmallerNSF can account for broader peaks and higher numbers of NSF allow a more accurate sampling of peaks with a small full width at half maximum.

4.3. Characterization of the Streak Finder

In this section, regions of insensitive pixels in the Streak Finder ROI will play a major role, therefore the following nomenclature is introduced: If q_r,max exceeds the detector

CHAPTER 4. STREAK FINDER 4.3. CHARACTERIZATION

size, empty regions will occur in the ROI, this case shall be calledrectangular detector.

Otherwise, round detector will be used, see also Fig. 4.2. Test data were generated to test and characterize the Streak Finder. The constructed intensity distribution has the form of a two dimensional Gaussian distribution

I_test(x, y) = A

where σ denotes the distribution width, µ the

ROI ROI

q_r,max q_r,max

rectangular

detector round detector

Figure 4.2.: On the definition of nomenclature for ROIs larger or smaller than the detector.

centre of the distributions andA an amplitude to the distribution function. The constant B allows to add a constant background. The two directions are indexed for the components in parallel and perpendicular direction with respect to the main orientation of the streak. The data are first gen-erated and then rotated with the matlab function imrotate to match the desired orientation angle.

A mask can be applied to simulate dead regions in the data as found in diffraction images from the Pilatus detector. The empty circle around the

beam centre corresponds to the inner radius of the ROI that is considered.

Optimal conditions A first test was made without the intermodule gaps of the Pilatus detector or dead regions from the sides. The transformed and reweighted intensity distribution is shown in Fig. 4.3 a). The streak orientation was varied and the resulting anisotropy parameter and streak angle plotted vs. the simulated streak orientation.

The angle was determined accurately (within the discretisation of the sampling rate of χ, given by the number N of angular segments, here N = 360) and the anisotropy parameter shows variations on the order of 10⁻³ of its value. This proves the correct functioning of the algorithm. The slight variation could be due to few empty pixels, that result from the coordinate transformation at lowqr or numerical residues. Secondly, the contributions of the areas without intensity information c) resulting from the sides of the detector are simulated, the output of the Streak Finder algorithm is presented in d). The resulting shape of the anisotropy parameter curve is found to be characteristic for strongly anisotropic intensity distributions with a decay alongq (see following paragraph).

CHAPTER 4. STREAK FINDER 4.3. CHARACTERIZATION

0 streak orientation ¹³⁵χ

1

0 streak orientation ¹³⁵χ_S 1

Figure 4.3.: Test data without intermodule gaps from a) a round and c) a reactangular detector.

The resulting anisotropy parameterS and angular deviation ∆χ_S are shown in b) for the round detector and d) for the rectangular one with respect to the streak orientation. The test data had a background ofB= 2, streak widths ofσ_k= 100 andσ_⊥= 15 and an intensity ofA= 10⁶.

CHAPTER 4. STREAK FINDER 4.3. CHARACTERIZATION

Figure 4.4.: Characterization of Streak Finder algorithm with respect to the aspect ratio AR = ^σ_σ^⊥

k of the intensity distribution. Test data (a,b) with σ_k = 100px and varyingσ_⊥∈[1. . .99px] are processed in the Streak Finder algorithm, the resulting anisotropy parameter and deviation of the actual orientation are plotted as a function of the aspect ratio (c). The small images below a) and b) are the reweighted intensity distributions obtained from the data. The angle of the main axis was fixed atχ = 30^◦, with a background B = 2 and fixed streak intensity A= 10⁶.

Distribution width A distribution withσ_k= 100pxand varyingσ⊥∈[1px,99px] was processed with the Streak Finder algorithm. The background is set toB = 2 and the amplitude to A= 10⁶. Fig. 4.4 (a+b) shows two of the resulting test datasets with the exemplary aspect ratios AR = ^σ_σ^⊥

k of (a) AR = 0.01 and (b) AR= 0.6. All test data were generated with a fixed streak orientation of χs = 30^◦. The resulting anisotropy parameter and angular deviation ∆χ_s (i. e., the difference between the calculated angle and the actual angle defined in the test data) are shown in Fig. 4.4 (c). The anisotropy parameter decays in a slope from S= 0.42 to≈0. The angle deviates from the chosen angle by no more than ∆χ_s≤6^◦.

Orientation and background Fig. 4.5 (c-f) illustrate the behaviour ofS and ∆χ_s with varyingχs and for different background levels. An example of the intensity distribution put to test is shown in a) with the corresponding transformed and reweighted intensity distribution in b).

Compared with Fig. 4.3, the overal trend is retained, only the insensitive detector regions add some high frequency features. The anisotropy parameter graphs of different

CHAPTER 4. STREAK FINDER 4.3. CHARACTERIZATION

0 orientation angle χ¹³⁵ -1

0 orientation angle χ¹³⁵ 1

0 orientation angle χ¹³⁵ -1

Figure 4.5.: Characterization of the Streak Finder algorithm for different background levels.

Test distributions (A= 10⁶, σ_k= 100, σ_⊥= 5, a) and b) show the case forB = 2) are fed into the Streak Finder algorithm with different background levelsB. The resultingS and ∆χS are shown in c)-f).

0 orientation angle χ¹³⁵ -1

Figure 4.6.: A box shaped distribution that is 1000 inside and 0.01 outside the 5px wide streak is tested and the result plotted in c).