Inpainting missing data

Missing data in an image can be an issue, especially when one wants to perform Fourier analysis. This tutorial explains how to fill-up missing pixels with values which looks “realistic” and introduce as little perturbation as possible for subsequent analysis. The user should keep the mask nearby and only consider the values of actual pixels and never the one inpainted.

This tutorial will use fully synthetic data to allow comparison between actual (syntetic) data with inpainted values.

The first part of the tutorial is about the generation of a challenging 2D diffraction image with realistic noise and to describe the metric used, then comes the actual tutorial on how to use the inpainting. Finally a benchmark is used based on the metric determined.

Creation of the image

A realistic challenging image should contain:

• Bragg peak rings. We chose LaB6 as guinea-pig, with very sharp peaks, at the limit of the resolution of the detector

• Some amorphous content

• strong polarization effect

• Poissonian noise

One image will be generated but then multiple ones with different noise to discriminate the effect of the noise from other effects.

%matplotlib inline
# Used for documentation to inline plots into notebook
# %matplotlib widget
# uncomment the later for better UI
from matplotlib.pyplot import subplots
import numpy

import pyFAI
print("Using pyFAI version: ", pyFAI.version)
from pyFAI.gui import jupyter
import pyFAI.test.utilstest
from pyFAI.calibrant import get_calibrant
import time
start_time = time.perf_counter()

Using pyFAI version:  2025.1.0-dev0

detector = pyFAI.detector_factory("Pilatus2MCdTe")
mask = detector.mask.copy()
nomask = numpy.zeros_like(mask)
detector.mask=nomask
ai = pyFAI.load({"detector":detector})
ai.setFit2D(200, 200, 200)
ai.wavelength = 3e-11
print(ai)

Detector Pilatus CdTe 2M         PixelSize= 172µm, 172µm         BottomRight (3)
Wavelength= 3.000000e-11 m
SampleDetDist= 2.000000e-01 m   PONI= 3.440000e-02, 3.440000e-02 m      rot1=0.000000  rot2=0.000000  rot3=0.000000 rad
DirectBeamDist= 200.000 mm      Center: x=200.000, y=200.000 pix        Tilt= 0.000° tiltPlanRotation= 0.000° 𝛌= 0.300Å

LaB6 = get_calibrant("LaB6")
LaB6.wavelength = ai.wavelength
print(LaB6)
r = ai.array_from_unit(unit="q_nm^-1")
decay_b = numpy.exp(-(r-50)**2/2000)
bragg = LaB6.fake_calibration_image(ai, Imax=1e4, W=1e-6) * ai.polarization(factor=1.0) * decay_b
decay_a = numpy.exp(-r/100)
amorphous = 1000*ai.polarization(factor=1.0)*ai.solidAngleArray() * decay_a
img_nomask = bragg + amorphous
#Not the same noise function for all images two images
img_nomask = numpy.random.poisson(img_nomask)
img_nomask2 = numpy.random.poisson(img_nomask)
img = numpy.random.poisson(img_nomask)
img[numpy.where(mask)] = -1
fig,ax = subplots(1,2, figsize=(10,5))
jupyter.display(img=img, label="With mask", ax=ax[0])
jupyter.display(img=img_nomask, label="Without mask", ax=ax[1])

LaB6 Calibrant with 640 reflections at wavelength 3e-11

<Axes: title={'center': 'Without mask'}>

Note the aliassing effect on the displayed images.

We will measure now the effect after 1D intergeration. We do not correct for polarization on purpose to highlight the defect one wishes to whipe out. We use a R-factor to describe the quality of the 1D-integrated signal.

kwargs = {"npt":2000, "unit":"q_nm^-1", "method":("full", "histogram", "cython"), "radial_range":(0,210)}
wo = ai.integrate1d(img_nomask, **kwargs)
wo2 = ai.integrate1d(img_nomask2, **kwargs)
wm = ai.integrate1d(img, mask=mask, **kwargs)
ax = jupyter.plot1d(wm , label="with_mask")
ax.plot(*wo, label="without_mask")
ax.plot(*wo2, label="without_mask2")
ax.plot(wo.radial, wo.intensity-wm.intensity, label="delta")
ax.plot(wo.radial, wo.intensity-wo2.intensity, label="relative-error")
ax.legend()
print("Between masked and non masked image R= %s"%pyFAI.utils.mathutil.rwp(wm,wo))
print("Between two different non-masked images R'= %s"%pyFAI.utils.mathutil.rwp(wo2,wo))

Between masked and non masked image R= 5.674909248404255
Between two different non-masked images R'= 0.2819170356887315

# Effect of the noise on the delta image
fig, ax = subplots()
jupyter.display(img=img_nomask-img_nomask2, label="Delta due to noise", ax=ax)
ax.figure.colorbar(ax.images[0])

<matplotlib.colorbar.Colorbar at 0x7fe24ca50850>

Inpainting

This part describes how to paint the missing pixels for having a “natural-looking image”. The delta image contains the difference with the original image

#Inpainting:
inpainted = ai.inpainting(img, mask=mask, poissonian=True)
fig, ax = subplots(1, 2, figsize=(12,5))
jupyter.display(img=inpainted, label="Inpainted", ax=ax[0])
jupyter.display(img=img_nomask-inpainted, label="Delta", ax=ax[1])
ax[1].figure.colorbar(ax[1].images[0])

<matplotlib.colorbar.Colorbar at 0x7fe24c7d33d0>

# Comparison of the inpained image with the original one:
wm = ai.integrate1d(inpainted, **kwargs)
wo = ai.integrate1d(img_nomask, **kwargs)
ax = jupyter.plot1d(wm , label="inpainted")
ax.plot(*wo, label="without_mask")
ax.plot(wo.radial, wo.intensity-wm.intensity, label="delta")
ax.legend()
print("R= %s"%pyFAI.utils.mathutil.rwp(wm,wo))

R= 1.2587407796205665

One can see by zooming in that the main effect on inpainting is a broadening of the signal in the inpainted region. This could (partially) be adressed by increasing the number of radial bins used in the inpainting.

Benchmarking and optimization of the parameters

The parameter set depends on the detector, the experiment geometry and the type of signal on the detector. Finer detail require finer slicing.

#Basic benchmarking of execution time for default options:
%timeit inpainted = ai.inpainting(img, mask=mask)

1.12 s ± 3.01 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

wo = ai.integrate1d(img_nomask, **kwargs)
for m in (("no", "histogram", "cython"), ("bbox", "histogram","cython")):
    for k in (512, 1024, 2048, 4096):
        ai.reset()
        for i in (0, 1, 2, 4, 8):
            inpainted = ai.inpainting(img, mask=mask, poissonian=True, method=m, npt_rad=k, grow_mask=i)
            wm = ai.integrate1d(inpainted, **kwargs)
            print(f"method: {m} npt_rad={k} grow={i}; R= {pyFAI.utils.mathutil.rwp(wm,wo):.3f}")

method: ('no', 'histogram', 'cython') npt_rad=512 grow=0; R= 7.837
method: ('no', 'histogram', 'cython') npt_rad=512 grow=1; R= 7.837
method: ('no', 'histogram', 'cython') npt_rad=512 grow=2; R= 7.836
method: ('no', 'histogram', 'cython') npt_rad=512 grow=4; R= 7.837
method: ('no', 'histogram', 'cython') npt_rad=512 grow=8; R= 7.838
method: ('no', 'histogram', 'cython') npt_rad=1024 grow=0; R= 8.085
method: ('no', 'histogram', 'cython') npt_rad=1024 grow=1; R= 8.085
method: ('no', 'histogram', 'cython') npt_rad=1024 grow=2; R= 8.085
method: ('no', 'histogram', 'cython') npt_rad=1024 grow=4; R= 8.085
method: ('no', 'histogram', 'cython') npt_rad=1024 grow=8; R= 8.085
method: ('no', 'histogram', 'cython') npt_rad=2048 grow=0; R= 8.205
method: ('no', 'histogram', 'cython') npt_rad=2048 grow=1; R= 8.205
method: ('no', 'histogram', 'cython') npt_rad=2048 grow=2; R= 8.205
method: ('no', 'histogram', 'cython') npt_rad=2048 grow=4; R= 8.205
method: ('no', 'histogram', 'cython') npt_rad=2048 grow=8; R= 8.205
method: ('no', 'histogram', 'cython') npt_rad=4096 grow=0; R= 8.280
method: ('no', 'histogram', 'cython') npt_rad=4096 grow=1; R= 8.280
method: ('no', 'histogram', 'cython') npt_rad=4096 grow=2; R= 8.280
method: ('no', 'histogram', 'cython') npt_rad=4096 grow=4; R= 8.280
method: ('no', 'histogram', 'cython') npt_rad=4096 grow=8; R= 8.280
method: ('bbox', 'histogram', 'cython') npt_rad=512 grow=0; R= 3.166
method: ('bbox', 'histogram', 'cython') npt_rad=512 grow=1; R= 2.904
method: ('bbox', 'histogram', 'cython') npt_rad=512 grow=2; R= 2.707
method: ('bbox', 'histogram', 'cython') npt_rad=512 grow=4; R= 2.618
method: ('bbox', 'histogram', 'cython') npt_rad=512 grow=8; R= 2.539
method: ('bbox', 'histogram', 'cython') npt_rad=1024 grow=0; R= 1.686
method: ('bbox', 'histogram', 'cython') npt_rad=1024 grow=1; R= 1.324
method: ('bbox', 'histogram', 'cython') npt_rad=1024 grow=2; R= 1.316
method: ('bbox', 'histogram', 'cython') npt_rad=1024 grow=4; R= 1.276
method: ('bbox', 'histogram', 'cython') npt_rad=1024 grow=8; R= 1.271
method: ('bbox', 'histogram', 'cython') npt_rad=2048 grow=0; R= 0.900
method: ('bbox', 'histogram', 'cython') npt_rad=2048 grow=1; R= 0.660
method: ('bbox', 'histogram', 'cython') npt_rad=2048 grow=2; R= 0.643
method: ('bbox', 'histogram', 'cython') npt_rad=2048 grow=4; R= 0.640
method: ('bbox', 'histogram', 'cython') npt_rad=2048 grow=8; R= 0.641
method: ('bbox', 'histogram', 'cython') npt_rad=4096 grow=0; R= 0.575
method: ('bbox', 'histogram', 'cython') npt_rad=4096 grow=1; R= 0.425
method: ('bbox', 'histogram', 'cython') npt_rad=4096 grow=2; R= 0.431
method: ('bbox', 'histogram', 'cython') npt_rad=4096 grow=4; R= 0.429
method: ('bbox', 'histogram', 'cython') npt_rad=4096 grow=8; R= 0.430

#Inpainting, best solution found:
ai.reset()
%time inpainted = ai.inpainting(img, mask=mask, poissonian=True, method=("full", "csr", "cython"), npt_rad=4096, grow_mask=1)
fig, ax = subplots(1, 2, figsize=(12, 5))
jupyter.display(img=inpainted, label="Inpainted", ax=ax[0])
jupyter.display(img=img_nomask-inpainted, label="Delta", ax=ax[1])
ax[1].figure.colorbar(ax[1].images[0])
pass

CPU times: user 4.53 s, sys: 191 ms, total: 4.72 s
Wall time: 1.6 s

# Comparison of the inpained image with the original one:
wm = ai.integrate1d(inpainted, **kwargs)
wo = ai.integrate1d(img_nomask, **kwargs)
ax = jupyter.plot1d(wm , label="inpainted")
ax.plot(*wo, label="without_mask")
ax.plot(wo.radial, wo.intensity-wm.intensity, label="delta")
ax.legend()
print("R= %s"%pyFAI.utils.mathutil.rwp(wm,wo))

R= 0.9561842586052813

Conclusion

Inpainting is one of the only solution to fill up the gaps in detector when Fourier analysis is needed. This tutorial explains basically how this is possible using the pyFAI library and how to optimize the parameter set for inpainting. The result may greatly vary with detector position and tilt and the kind of signal (amorphous or more spotty).

print(f"Execution time: {time.perf_counter()-start_time:.3f} s")

Execution time: 45.533 s