Pattern processing

The raw EBSD signal can be empirically evaluated as a superposition of a Kikuchi diffraction pattern and a smooth background intensity. For pattern indexing, the latter intensity is usually undesirable, while for virtual backscatter electron (VBSE) imaging, this intensity can reveal topographical, compositional or diffraction contrast. This section details methods to enhance the Kikuchi diffraction pattern and manipulate detector intensities in patterns in an EBSD object.

Most of the methods operating on EBSD objects use functions that operate on the individual patterns (numpy.ndarray). These single pattern functions are available in the kikuchipy.pattern module.

Almost all methods operate inplace (indicated in their docstrings), meaning it overwrites the patterns in the EBSD object. If a new object is desired, create a deepcopy() of the original object and perform the operation on this:

>>> s2 = s.deepcopy()
>>> s2.remove_static_background()

Rescale intensity

Vendors usually write patterns to file with 8 (uint8) or 16 (uint16) bit integer depth, holding [0, 2^8] or [0, 2^16] gray levels, respectively. To avoid loosing intensity information when processing, we often change data types to e.g. 32 bit floating point (float32). However, only changing the data type with change_dtype() does not rescale pattern intensities, leading to patterns not using the full available data type range:

>>> print(s.data.dtype, s.data.max())
uint8 255
>>> s.change_dtype(np.uint16)
>>> print(s.data.dtype, s.data.max())
uint16 255
>>> s.plot(vmax=1000)

In these cases it is convenient to rescale intensities to a desired data type range, either keeping relative intensities between patterns in a scan or not. We can do this for all patterns in a scan (EBSD object) with kikuchipy.signals.EBSD.rescale_intensity():

>>> s.rescale_intensity(relative=True)
>>> print(s.data.dtype, s.data.max())
uint16 65535
>>> s.plot(vmax=65535)

Or, we can do it for a single pattern (numpy.ndarray) with kikuchipy.pattern.rescale_intensity():

>>> p = s.inav[0, 0].data
>>> p2 = kp.pattern.rescale_intensity(p)
_images/rescale_intensities.jpg

A pattern, initially with uint8 data type, cast to uint16 (left), with intensities not filling the available gray levels (left). The same pattern after rescaling (right).

We can also stretch the pattern contrast by removing intensities outside a range passed to in_range or at certain percentiles by passing percents to percentiles:

>>> s.rescale_intensity(in_range=(5, 250))
>>> print(s.data.min(), s.data.max())
5 250
>>> s.rescale_intensity(percentiles=(0.5, 99.5))
>>> print(s.data.min(), s.data.max())
0 255
_images/contrast_stretching.jpg

A pattern before (left) and after (right) stretching its contrast to a range given by the lowest 0.5% and highest 0.5% intensities.

This can reduce the influence of outliers with exceptionally high or low intensities, like hot or dead pixels.

Normalize intensity

It can be useful to normalize pattern intensities to a mean value of \(\mu = 0.0\) and a standard deviation of e.g. \(\sigma = 1.0\) when e.g. comparing patterns or calculating the image quality. Patterns can be normalized with normalize_intensity():

>>> np.mean(s.data)
146.0670987654321
>>> s.change_dtype(np.float32)  # Or passing dtype_out=np.float32 to s.no...
>>> s.normalize_intensity(num_std=1)  # Default
>>> np.mean(s.data)
2.6373216e-08
_images/normalize_intensity.jpg

Histogram of pattern intensities in a scan before normalization (left) and after normalization (right).

Background correction

Remove the static background

Effects which are constant, like hot pixels or dirt on the detector, can be removed by either subtracting or dividing by a static background via remove_static_background():

>>> s.remove_static_background(operation='subtract', relative=True)
_images/static_correction.jpg

The same pattern as acquired (left) and after removing the static background (right).

Here, the static background pattern is assumed to be stored as part of the signal metadata, which can be loaded via set_experimental_parameters(). The static background pattern can also be passed to the static_bg parameter. Passing relative=True (default) ensures that relative intensities between patterns are kept when they are rescaled after correction to fill the available data range. In this case, for a scan of data type uint8 with data range [0, 255], the highest pixel intensity in a scan is stretched to 255 (and the lowest to 0), while the rest is rescaled keeping relative intensities between patterns. With relative=False, all patterns are stretched to [0, 255].

The static background pattern intensities can be rescaled to each individual pattern’s intensity range before removal by passing scale_bg=True, which will result in the relative intensity between patterns to be lost (passing relative=True along with scale_bg=True is not allowed).

Remove the dynamic background

Uneven intensity in a static background subtracted pattern can be corrected by subtracting or dividing by a dynamic background obtained by Gaussian blurring. This so-called flat fielding is done with remove_dynamic_background(). A Gaussian window with a standard deviation set by std is used to blur each pattern individually (dynamic) either in the spatial or frequency domain, set by filter_domain. Blurring in the frequency domain is effectively accomplished by a low-pass Fast Fourier Transform (FFT) filter. The individual Gaussian blurred dynamic backgrounds are then subtracted or divided from the respective patterns, set by operation:

>>> s.remove_dynamic_background(
...     operation='subtract',  # Default
...     filter_domain="frequency",  # Default
...     std=8,  # Default is 1/8 of pattern width
...     truncate=4.0  # Default
... )
_images/dynamic_correction.jpg

The same pattern after removal of the static background (left), followed by removing the dynamic background pattern produced by Gaussian blurring in the frequency domain (right).

The width of the Gaussian window is truncated at the truncated number of standard deviations. Output patterns are rescaled to fill the input patterns’ data type range.

Get the dynamic background

The Gaussian blurred pattern removed during dynamic background correction can be obtained as it’s own EBSD object:

>>> s
<EBSD, title: patterns Scan 1, dimensions: (3, 3|60, 60)>
>>> bg = s.get_dynamic_background(
...     filter_domain="frequency",
...     std=8,
...     truncate=4,
... )
>>> bg
<EBSD, title: , dimensions: (3, 3|60, 60)>
_images/get_dynamic_background.jpg

The pattern as acquired (left) and the same pattern after Gaussian blurring in the frequency domain, showing only the large scale variations and no Kikuchi pattern features.

Average neighbour patterns

The signal-to-noise ratio in patterns in an EBSD scan s can be improved by averaging patterns with their closest neighbours within a window/kernel/mask with average_neighbour_patterns():

>>> s.average_neighbour_patterns(window="gaussian", shape=(3, 3), std=1)
_images/average_neighbour_pattern.jpg

An example pattern before (left) and after (right) averaging with the nearest neighbour patterns in a (3 x 3) Gaussian window with \(\sigma\) = 1.

The array of averaged patterns \(g(n_{\mathrm{x}}, n_{\mathrm{y}})\) is obtained by spatially correlating a window \(w(s, t)\) with the array of patterns \(f(n_{\mathrm{x}}, n_{\mathrm{y}})\), here 4D, which is padded with zeros at the edges. As coordinates \(n_{\mathrm{x}}\) and \(n_{\mathrm{y}}\) are varied, the window origin moves from pattern to pattern, computing the sum of products of the window coefficients with the neighbour pattern intensities, defined by the window shape, followed by normalizing by the sum of the window coefficients. For a symmetrical window of shape \(m \times n\), this becomes [Gonzalez2017]

\[g(n_{\mathrm{x}}, n_{\mathrm{y}}) = \frac{\sum_{s=-a}^a\sum_{t=-b}^b{w(s, t) f(n_{\mathrm{x}} + s, n_{\mathrm{y}} + t)}} {\sum_{s=-a}^a\sum_{t=-b}^b{w(s, t)}},\]

where \(a = (m - 1)/2\) and \(b = (n - 1)/2\). The window \(w\), a Window object, can be plotted:

>>> w = kp.filters.Window(window="gaussian", shape=(3, 3), std=1)
>>> w.plot(cmap="inferno")
_images/window_gaussian_std1.png

A Gaussian averaging window with \(\sigma\) = 1 and the origin in the window centre.

Any 1D or 2D window with desired coefficients can be used. This custom window can be passed to the window parameter in average_neighbour_patterns() or Window as a numpy.ndarray or dask.array.Array. Additionally, any window in scipy.signal.windows.get_window() passed as a string via window with the necessary parameters as keyword arguments (like std=1 for window="gaussian") can be used. To demonstrate the creation and use of an asymmetrical circular window (and the use of make_circular(), although we could create a circular window directly by calling window="circular" upon window initialization):

>>> w = kp.filters.Window(window="rectangular", shape=(5, 4))
>>> w
Window (5, 4) rectangular
[[1. 1. 1. 1.]
 [1. 1. 1. 1.]
 [1. 1. 1. 1.]
 [1. 1. 1. 1.]
 [1. 1. 1. 1.]]
>>> w.make_circular()
>>> w
Window (5, 4) circular
[[0. 0. 1. 0.]
 [0. 1. 1. 1.]
 [1. 1. 1. 1.]
 [0. 1. 1. 1.]
 [0. 0. 1. 0.]]
>>> s.average_neighbour_patterns(w)
>>> figure, image, colorbar = w.plot()
_images/window_circular_54.png

A circular averaging window. Note the location of the origin (0, 0).

Note

Neighbour pattern averaging increases the virtual interaction volume of the electron beam with the sample, leading to a potential loss in spatial resolution. Averaging may in some cases, like on grain boundaries, mix two or more different diffraction patterns, which might be unwanted. See [Wright2015] for a discussion of this concern.

Wright2015

S. I. Wright, M. M. Nowell, S. P. Lindeman, P. P. Camus, M. De Graef, M. A. Jackson, “Introduction and comparison of new EBSD post-processing methodologies,” Ultramicroscopy 159 (2015), doi: https://doi.org/10.1016/j.ultramic.2015.08.001.

Adaptive histogram equalization

Enhancing the pattern contrast with adaptive histogram equalization has been found useful when comparing patterns for dictionary indexing [Marquardt2017]. With adaptive_histogram_equalization(), the intensities in the pattern histogram are spread to cover the available range, e.g. [0, 255] for patterns of uint8 data type:

>>> s.adaptive_histogram_equalization(kernel_size=(15, 15))
_images/adapthist.jpg

The same pattern after dynamic correction (left) followed by adaptive histogram equalization (right).

The kernel_size parameter determines the size of the contextual regions. See e.g. Fig. 5 in [Jackson2019], also available via EMsoft’s GitHub repository wiki, for the effect of varying kernel_size.

Marquardt2017

K. Marquardt, M. De Graef, S. Singh, H. Marquardt, A. Rosenthal, S. Koizuimi, “Quantitative electron backscatter diffraction (EBSD) data analyses using the dictionary indexing (DI) approach: Overcoming indexing difficulties on geological materials,” American Mineralogist 102 (2017), doi: https://doi.org/10.2138/am-2017-6062.

Jackson2019

M. A. Jackson, E. Pascal, M. De Graef, “Dictionary Indexing of Electron Back-Scatter Diffraction Patterns: a Hands-On Tutorial,” Integrating Materials and Manufacturing Innovation 8 (2019), doi: https://doi.org/10.1007/s40192-019-00137-4.

Filtering in the frequency domain

Filtering of patterns in the frequency domain can be done with fft_filter(). This method takes a spatial kernel defined in the spatial domain, or a transfer function defined in the frequency domain, in the transfer_function argument as a numpy.ndarray or a Window. Which domain the transfer function is defined in must be passed to the function_domain argument. Whether to shift zero-frequency components to the centre of the FFT can also be controlled via shift, but note that this is only used when function_domain="frequency".

Popular uses of filtering of EBSD patterns in the frequency domain include removing large scale variations across the detector with a Gaussian high pass filter, or removing high frequency noise with a Gaussian low pass filter. These particular functions are readily available via Window:

>>> pattern_shape = s.axes_manager.signal_shape[::-1]
>>> w_low = kp.filters.Window(
...     "lowpass",
...     cutoff=22,
...     cutoff_width=10,
...     shape=pattern_shape
... )
>>> w_high = kp.filters.Window(
...     "highpass",
...     cutoff=3,
...     cutoff_width=2,
...     shape=pattern_shape
... )
>>> w = w_low * w_high
>>> import matplotlib.pyplot as plt
>>> plt.imshow(w)
>>> plt.colorbar()
>>> plt.figure()
>>> plt.plot(w[pattern_shape[0] // 2:, :])
_images/fft_filter_highlowpass.jpg

The product of the combined high and low pass transfer functions defined in the frequency domain (left), and the intensity profile across its centre (right).

Then, to multiply the FFT of each pattern with this transfer function, and subsequently computing the inverse FFT (IFFT), we use fft_filter(), and remember to shift the zero-frequency components to the centre of the FFT:

>>> s.fft_filter(
...     transfer_function=w, function_domain="frequency", shift=True)
_images/fft_filter_highlowpass_result.jpg

The same pattern before (left) and after (right) filtering with a combined high and low pass Gaussian transfer function.

Note that filtering with a spatial kernel in the frequency domain, after creating the kernel’s transfer function via FFT, and computing the IFFT, is, in this case, the same as spatially correlating the kernel with the pattern. Let’s demonstrate this by attempting to sharpen a pattern with a Laplacian kernel in both the spatial and frequency domains and comparing the results (this is a purely illustrative example, and perhaps not that practically useful):

>>> w_laplacian = np.array([[-1, -1, -1], [-1, 8, -1], [-1, -1, -1]])
>>> p = s.inav[0, 0].deepcopy().data.astype(np.float32)
>>> s.fft_filter(transfer_function=w_laplacian, function_domain="spatial")

>>> from scipy.ndimage import correlate
>>> p_filt = correlate(w, weights=w_laplacian)
>>> p_filt_resc = kp.util.pattern.rescale_intensity(
...     p_filt, dtype_out=np.uint8)
_images/fft_filter_laplacian.jpg

The result of correlating a pattern with a Laplacian sharpening kernel (left). The exact same result is obtained by filtering in the frequency domain with the kernel’s transfer function and subsequently computing the IFFT (right).

Note also that fft_filter() performs the filtering on the patterns with data type np.float32, and therefore have to rescale back to the pattern’s original data type if necessary.