Fig 1.
Schematic of a typical experimental setup used for SBI.
The X-ray beam is patterned by the diffuser mounted on a scanning stage upstream of the sample stage. A detector system consisting of a pixel array detector scintillator-coupled to an optical microscope, giving sufficient resolving power to resolve the speckle pattern, is used to collect the images.
Fig 2.
Graphical representation of the directional dark-field UMPA model: The speckle pattern is locally modulated by the sample transmittance (T), a convolution with a blurring kernel (Kα) and a lateral shift (u).
The transmittance corresponds to an image similar to the one that would have been acquired without the sandpaper in the beam. The lateral shift is used to generate differential phase-contrast images, while the parameters of Kα quantify the dark-field signal. (The blurring kernel and vector shifts shown have been exaggerated).
Fig 3.
Results of the UMPA-DDF analysis for 25 diffuser positions and a 7 × 7 pixel window.
a) Transmittance T. c) and e) Horizontal and vertical refraction angles u/d = (ux/d, uy/d) in μrad. The right column shows the magnitude, directionality, and orientation of the extracted directional dark-field signal: b) Average kernel width σ, in pixels. d) Directionality, given by the eccentricity ε of the kernel’s exponential argument. f) Orientation θ of the larger scattering axis relative to the horizontal, given in degrees.
Fig 4.
Composite image using the HSV colour space (mapping illustrated with the inset colour-cylinder).
The hue H shows the orientation θ of the scattering, saturation S represents the directionality ε, and the value V shows the mean kernel width σ.
Fig 5.
Precision of the retrieved scattering orientation.
a) The orientation map highlighting the 150 × 150 pixel regions of interest used for the analysis. b) One example of a polar histogram obtained with 25 diffuser positions and a 7 x 7 pixel analysis window. The central region acts as a colourbar for a). c) Standard deviation of the scattering angles as a function of the number of independent pixel measurements P, given by the number of pixels in the window multiplied by the number of diffuser positions used. The error bars are given by the standard error across the four regions. Sample histograms are shown as insets for values of P = 125, 225, 961 and 24025. Power-law regression to the data with P > 50 yields an exponent of −0.538 ± 0.003 (slope of the black line).
Table 1.
Measured tube angles ϕ and associated main scattering directions θ (both relative to the horizontal axis of the detector).
Results were obtained with 25 diffuser positions and analysis window size of 31 × 31 pixels.