Instrumentation & Measurement Magazine 23-2 - 67

Fig. 8. (a) Simulated out-of-focus image and (b) its reconstruction, assuming the distortion is known.

A successful application of this principle was to improve an ultra high speed sampling oscilloscope as a
reference instrument at the National Institute of Standards
and Technology (NIST, USA). The sampling system measures
repetitive pulses (reference waveform) in equivalent-time
sampling mode. The waveform of the reference system was
numerically post-processed, and different errors were numerically compensated. Among them, the lowpass filtering
effect of averaging together with aperture jitter (fluctuation
of the sampling instance with respect to the nominal time instance) of equivalent-time sampling was corrected by means
of inverse filtering [9]. This effect can be modeled as a convolution of the measured waveform with the time-domain
probability density function of the jitter (i.e., the probability
that sampling instance differs from the nominal time instance
by a certain time).

Compensation of Faulty, Non-Repeatable
Measurements
The third application area addresses faulty measurements.
The measurement instrument could provide the required accuracy, but-as a mistake-the settings of the setup are wrong,
resulting in a distorted observation. Once it is realized, certainly, the best is to repeat the measurement with correct
settings. However, there are cases when it cannot be repeated.
Not because of the instrument, but because the event we are
going to capture cannot be replayed. In this case-if the faulty
setting is known, or the distortion can be identified-the numerical post processing (inverse algorithm) can at least partly
recover the information.
We face such mistakes often with photography. The picture is blurred because it is out of focus. Even if the camera
has an autofocus mechanism, it is many times fooled by the
environment (e.g., there is an object in the foreground). The
shoot can be repeated with correct focus, but the event might
be gone. We can miss the funny face expression of our child or
be late for repeating the shot at a sport event (miss the score).
April 2020	

In these cases, if the distortion can be identified, it can also be
corrected. Out of focus distortion can be described by a twodimensional convolution with the so-called point spread
function (two-dimensional version of the impulse response,
that is, how the intensity of a point like source is distributed
on the image). Similar aberrations that can be described by
convolution are camera shake or spherical aberration. Their
inversion is two-dimensional deconvolution (in most cases
ill-posed one). Such a simulated out-of-focus image is shown
in Fig. 8, with 8-bit resolution (producing a non-negligible
quantization noise), together with its reconstruction. The inverse filter in the two-dimensional frequency space takes the
following form:

	

(

)

K fx , fy , γ =

(

PSF fx , f y

(

PSF fx , f y

)

2

)

∗

(

+ γ D fx , fy

)

2

	(15)

where PSF(fx,fy) stands for the 2D FFT of point spread function,
and D(fx,fy) represents the FFT of the 2D second order difference operator.
In this example, the point spread function describing
the blur was assumed to be known. This is a good assumption if the (bad) settings of the camera and object distance
are known. In that case, identification measurements can
be accomplished with the bad and correct camera settings,
that allow the determination of the point spread function.
The deconvolution problem is called "blind," if the distortion is not known, that is, we do not have a measurement
of the point spread function. The inverse problem without any additional constraint becomes underdetermined.
One of the most widespread solutions for this is to assume
some model for the point spread function (e.g., Gaussian
blur), and adjust model parameters of both the inverse filter and that of the distortion. The enhancement depends
strongly on how close the assumption of the distortion was
to the reality.

IEEE Instrumentation & Measurement Magazine	67



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