# 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 signal.

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.

Let’s import the necessary libraries and read the Nickel EBSD test data set

```
[1]:
```

```
# Exchange inline for notebook or qt5 (from pyqt) for interactive plotting
%matplotlib inline
import hyperspy.api as hs
import matplotlib.pyplot as plt
import numpy as np
import kikuchipy as kp
s = kp.data.nickel_ebsd_small() # Use kp.load("data.h5") to load your own data
```

```
WARNING:hyperspy.api:The ipywidgets GUI elements are not available, probably because the hyperspy_gui_ipywidgets package is not installed.
WARNING:hyperspy.api:The traitsui GUI elements are not available, probably because the hyperspy_gui_traitsui package is not installed.
```

Most methods operate inplace (indicated in their docstrings), meaning they overwrite the patterns in the EBSD signal. If we instead want to keep the original signal and operate on a new signal, we can create a deepcopy() of the original signal. As an example here, we create a new EBSD signal from a small part of the original signal:

```
[2]:
```

```
s2 = s.deepcopy()
np.may_share_memory(s.data, s2.data)
```

```
[2]:
```

```
False
```

## 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():

```
[3]:
```

```
s2.remove_static_background(operation="subtract", relative=True)
```

```
Removing the static background:
[########################################] | 100% Completed | 0.1s
```

```
[4]:
```

```
fig, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s.inav[0, 0].data, cmap="gray")
ax[0].set_title("As acquired")
ax[0].axis("off")
ax[1].imshow(s2.inav[0, 0].data, cmap="gray")
ax[1].set_title("Static background removed")
ax[1].axis("off")
fig.tight_layout(w_pad=0-8)
```

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].

Warning

The `Acquisition_instrument.SEM.Detector.EBSD`

and `Sample.Phases`

metadata nodes are deprecated and will be removed in v0.6.

There are three main reasons for this change: the first is that only the static background array stored in the `Acquisition_instrument.SEM.Detector.EBSD.static_background`

node is used internally, and so the remaining metadata is unnecessary. The background pattern will be stored in its own `EBSD.static_background`

property instead. The second is that keeping track of the unnecessary metadata makes writing and maintaining input/ouput plugins challenging. The third is that the
EBSD.xmap and EBSD.detector properties, which keeps track of the CrystalMap and EBSDDetector for a signal, respectively, should be used instead of the more “static” metadata.

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`

:

```
[5]:
```

```
s3 = s2.deepcopy()
s3.remove_dynamic_background(
operation="subtract", # Default
filter_domain="frequency", # Default
std=8, # Default is 1/8 of the pattern width
truncate=4, # Default
)
# _ means we don't want the output
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s2.inav[0, 0].data, cmap="gray")
ax[0].set_title("Static background removed")
ax[1].imshow(s3.inav[0, 0].data, cmap="gray")
_ = ax[1].set_title("Static + dynamic background removed")
```

```
Removing the dynamic background:
[########################################] | 100% Completed | 0.1s
```

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 an EBSD signal by calling get_dynamic_background():

```
[6]:
```

```
bg = s.get_dynamic_background(filter_domain="frequency", std=8, truncate=4)
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s.inav[0, 0].data, cmap="gray")
ax[0].set_title("As acquired")
ax[1].imshow(bg.inav[0, 0].data, cmap="gray")
_ = ax[1].set_title("Dynamic background")
```

```
Getting the dynamic background:
[########################################] | 100% Completed | 0.1s
```

## Average neighbour patterns¶

The signal-to-noise ratio in patterns in an EBSD signal `s`

can be improved by averaging patterns with their closest neighbours within a window/kernel/mask with average_neighbour_patterns():

```
[7]:
```

```
s4 = s3.deepcopy()
s4.average_neighbour_patterns(window="gaussian", std=1)
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s3.inav[0, 0].data, cmap="gray")
ax[0].set_title("Static + dynamic background removed")
ax[1].imshow(s4.inav[0, 0].data, cmap="gray")
_ = ax[1].set_title("After neighbour pattern averaging")
```

```
Averaging with the neighbour patterns:
[########################################] | 100% Completed | 0.1s
```

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 [GW17]

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

```
[8]:
```

```
w = kp.filters.Window(window="gaussian", shape=(3, 3), std=1)
w.plot()
```

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 a `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):

```
[9]:
```

```
w = kp.filters.Window(window="rectangular", shape=(5, 4))
w
```

```
[9]:
```

```
Window (5, 4) rectangular
[[1. 1. 1. 1.]
[1. 1. 1. 1.]
[1. 1. 1. 1.]
[1. 1. 1. 1.]
[1. 1. 1. 1.]]
```

```
[10]:
```

```
w.make_circular()
w
```

```
[10]:
```

```
Window (5, 4) circular
[[0. 0. 1. 0.]
[0. 1. 1. 1.]
[1. 1. 1. 1.]
[0. 1. 1. 1.]
[0. 0. 1. 0.]]
```

```
[11]:
```

```
s5 = s3.deepcopy()
s5.average_neighbour_patterns(w)
```

```
Averaging with the neighbour patterns:
[########################################] | 100% Completed | 0.1s
```

```
[12]:
```

```
w.plot()
```

But this `(5, 4)`

averaging window cannot be used with our `(3, 3)`

navigation shape signal.

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 [WNL+15] for a discussion of this concern.

## Adaptive histogram equalization¶

Enhancing the pattern contrast with adaptive histogram equalization has been found useful when comparing patterns for dictionary indexing [MDeGraefS+17]. 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:

```
[13]:
```

```
s6 = s3.deepcopy()
s6.adaptive_histogram_equalization(kernel_size=(15, 15))
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s3.inav[0, 0].data, cmap="gray")
ax[0].set_title("Static + dynamic background removed")
ax[1].imshow(s6.inav[0, 0].data, cmap="gray")
_ = ax[1].set_title("After adaptive histogram equalization")
```

```
Adaptive histogram equalization:
[########################################] | 100% Completed | 0.1s
```

The `kernel_size`

parameter determines the size of the contextual regions. See e.g. Fig. 5 in [JPDeGraef19], also available via EMsoft’s GitHub repository wiki, for the effect of varying `kernel_size`

.

## 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`

:

```
[14]:
```

```
pattern_shape = s.axes_manager.signal_shape[::-1]
w_low = kp.filters.Window(
window="lowpass",
cutoff=23,
cutoff_width=10,
shape=pattern_shape
)
w_high = kp.filters.Window(
window="highpass",
cutoff=3,
cutoff_width=2,
shape=pattern_shape
)
fig, ax = plt.subplots(figsize=(22, 6), ncols=3)
fig.subplots_adjust(wspace=0.05)
im0 = ax[0].imshow(w_low, cmap="gray")
ax[0].set_title("Lowpass filter", fontsize=22)
fig.colorbar(im0, ax=ax[0])
im1 = ax[1].imshow(w_high, cmap="gray")
ax[1].set_title("Highpass filter", fontsize=22)
fig.colorbar(im1, ax=ax[1])
im2 = ax[2].imshow(w_low * w_high, cmap="gray")
ax[2].set_title("Lowpass * highpass filter", fontsize=22)
_ = fig.colorbar(im2, ax=ax[2])
```

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:

```
[15]:
```

```
s7 = s3.deepcopy()
s7.fft_filter(
transfer_function=w_low * w_high,
function_domain="frequency",
shift=True
)
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s3.inav[0, 0].data, cmap="gray")
ax[0].set_title("Static + dynamic background removed")
ax[1].imshow(s7.inav[0, 0].data, cmap="gray")
_ = ax[1].set_title("After FFT filtering")
```

```
FFT filtering:
[########################################] | 100% Completed | 0.3s
```

Note that filtering with a spatial kernel in the frequency domain, after creating the kernel’s transfer function via FFT, and computing the inverse FFT (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):

```
[16]:
```

```
s8 = s3.deepcopy()
w_laplacian = np.array([[-1, -1, -1], [-1, 8, -1], [-1, -1, -1]])
s8.fft_filter(transfer_function=w_laplacian, function_domain="spatial")
```

```
FFT filtering:
[########################################] | 100% Completed | 0.1s
```

```
[17]:
```

```
from scipy.ndimage import correlate
p_filt = correlate(s3.inav[0, 0].data.astype(np.float32), weights=w_laplacian)
p_filt_resc = kp.pattern.rescale_intensity(p_filt, dtype_out=np.uint8)
_, ax = plt.subplots(figsize=(13, 6), ncols=2)
ax[0].imshow(s8.inav[0, 0].data, cmap="gray")
ax[0].set_title("Filter in frequency domain")
ax[1].imshow(p_filt_resc, cmap="gray")
ax[1].set_title("Filter in spatial domain")
np.sum(s8.inav[0, 0].data - p_filt_resc) # Which is zero
```

```
[17]:
```

```
0
```

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.

## 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 losing 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:

```
[18]:
```

```
s9 = s3.deepcopy()
print(s9.data.dtype, s9.data.max())
s9.change_dtype(np.uint16)
print(s9.data.dtype, s9.data.max())
plt.figure(figsize=(6, 5))
plt.imshow(s9.inav[0, 0].data, vmax=1000, cmap="gray")
plt.title("16-bit pixels w/ 255 as max intensity", pad=15)
_ = plt.colorbar()
```

```
uint8 255
uint16 255
```

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 an EBSD signal with kikuchipy.signals.EBSD.rescale_intensity():

```
[19]:
```

```
s9.rescale_intensity(relative=True)
print(s9.data.dtype, s9.data.max())
plt.figure(figsize=(6, 5))
plt.imshow(s9.inav[0, 0].data, cmap="gray")
plt.title("16-bit pixels w/ 65535 as max. intensity", pad=15)
_ = plt.colorbar()
```

```
Rescaling the image intensities:
[########################################] | 100% Completed | 0.5s
uint16 65535
```

Or, we can do it for a single pattern (`numpy.ndarray`

) with kikuchipy.pattern.rescale_intensity():

```
[20]:
```

```
p = s3.inav[0, 0].data
print(p.min(), p.max())
p2 = kp.pattern.rescale_intensity(p, dtype_out=np.uint16)
print(p2.min(), p2.max())
```

```
0 255
0 65535
```

We can also stretch the pattern contrast by removing intensities outside a range passed to `in_range`

or at certain percentiles by passing percentages to `percentiles`

:

```
[21]:
```

```
s10 = s3.deepcopy()
print(s10.data.min(), s10.data.max())
s10.rescale_intensity(out_range=(10, 245))
print(s10.data.min(), s10.data.max())
```

```
0 255
Rescaling the image intensities:
[########################################] | 100% Completed | 0.1s
10 245
```

```
[22]:
```

```
s10.rescale_intensity(percentiles=(0.5, 99.5))
print(s10.data.min(), s3.data.max())
```

```
Rescaling the image intensities:
[########################################] | 100% Completed | 0.1s
0 255
```

```
[23]:
```

```
fig, ax = plt.subplots(figsize=(13, 5), ncols=2)
im0 = ax[0].imshow(s3.inav[0, 0].data, cmap="gray")
ax[0].set_title("Static + dynamic background removed", pad=15)
fig.colorbar(im0, ax=ax[0])
im1 = ax[1].imshow(s10.inav[0, 0].data, cmap="gray")
ax[1].set_title("After clipping lowest/highest intensities", pad=15)
_ = fig.colorbar(im1, ax=ax[1])
```

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():

```
[24]:
```

```
s11 = s3.deepcopy()
np.mean(s11.data)
```

```
[24]:
```

```
113.94771604938272
```

```
[25]:
```

```
# s11.change_dtype(np.float32) # Or pass `dtype_out` as below
s11.normalize_intensity(num_std=1, dtype_out=np.float32)
np.mean(s11.data)
```

```
Normalizing the image intensities:
[########################################] | 100% Completed | 0.6s
```

```
[25]:
```

```
-1.9191225e-08
```

```
[26]:
```

```
_, ax = plt.subplots(figsize=(13, 4), ncols=2)
ax[0].hist(s3.data.ravel(), bins=255)
ax[0].set_title("Static + dynamic background removed")
ax[1].hist(s11.data.ravel(), bins=255)
_ = ax[1].set_title("After intensity normalization")
```