=Paper=
{{Paper
|id=Vol-2984/paper7
|storemode=property
|title=Enhancing the spatial resolution of digital images using circular scanning (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2984/paper7.pdf
|volume=Vol-2984
|authors=Aleksandr Reznik,Aleksandr Soloviev,Andrey Torgov
|dblpUrl=https://dblp.org/rec/conf/itams/ReznikST21
}}
==Enhancing the spatial resolution of digital images using circular scanning (short paper)==
https://ceur-ws.org/Vol-2984/paper7.pdf
Enhancing the spatial resolution of digital images using circular
scanning
Aleksandr Reznik 1, Aleksandr Soloviev 1, Andrey Torgov 1
1
Institute of Automation and Electrometry of the Siberian Branch of the Russian Academy of Sciences (IA&E SB
RAS), Academician Koptyug ave. 1, Novosibirsk, 630090, Russia
Abstract
The article describes original algorithms developed to improve the spatial resolution of
digital images using a circular scan procedure. Examples of image reconstruction in the
presence and absence of noise are given, and the error in reconstructing the original signal is
calculated.
Keywords 1
Image processing, circular scanning, enhancing of spatial resolution.
1. Introduction
In many scientific and technical fields and practical applications of computer science, it becomes
necessary to obtain an ideal initial signal (image) from the results of measurements carried out using
low-resolution photo matrices. The use of photo matrixes with high spatial resolution is often
hampered by high cost, and sometimes (for example, for infrared matrices) difficulties are associated
with technological problems in the manufacture of such receivers.
The complexity of increasing the resolution of the observed original image is determined by the
required reconstruction accuracy and depends on the applied registration method, which in the general
case is described by two characteristics - the trajectory of movement of the recording photo matrix
and the shooting frequency. To date, many authors have proposed [1-7] a large number of high-
performance schemes and algorithms for increasing the resolution of digital images using various
computational methods and optimization procedures. In a number of publications on this topic [8-10],
it is assumed that the set of observed data, which should be used to reconstruct the original high-
resolution image, was obtained by sequentially moving a low-resolution photo matrix in two mutually
perpendicular directions by an amount less than the linear dimensions of the integrating element of
the photo matrix.
But this approach is not effective enough in the operation of technological control systems, where
the recording photo matrix cannot move along the axes of a rectangular coordinate system, but a
circular scanning trajectory is required. This article is devoted to this variant of reconstruction
(estimation) of the initial noisy field. Below is a technique for solving the entire chain of problems
associated with circular scanning of images, starting from the development of the necessary
mathematical model and ending with the construction of final programs and algorithms for optimal
subpixel reconstruction of original digital images with improved spatial resolution.
2. Enhancing spatial resolution using circular scanning
Within the framework of the model under consideration, it is assumed that the original digital
image to be reconstructed (see Figure 1) is a two-dimensional numeric array Fij corresponding to the
zero-order approximation f(0)(x, y) of a function of two variables f(x, y). In other words, there is a set
ITAMS 2021 – Information Technologies: Algorithms, Models, Systems, September 14, 2021, Irkutsk, Russia
EMAIL: reznik@iae.nsk.su (A. 1); soloviev@iae.nsk.su (A. 2); torgov@iae.nsk.su (A. 3)
ORCID: 0000-0003-3466-5433 (A. 1); 0000-0002-5352-222X (A. 2); 0000-0002-8037-1746 (A. 3)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
of samples regularly specified on a square lattice, which corresponds to a piecewise discontinuous
function f(0)(x, y), which is a constant inside each elementary pixel (i, j) with area S∆=∆2:
f(0)(x,y)=Fij, (i-1)∆∆, i.e. an aperture, the linear
dimensions of which are (Ω/∆) times greater than the linear dimensions of the pixel of the
reconstructed image. In this case, the trajectory of movement of the photo matrix is a circle of radius
R (Fig. 2).
Figure 2: The trajectory of element's movement of the low-resolution photo matrix.
Below, for simplicity, we will assume that both the photo matrix as a whole and each of its
elements sequentially “run through” K positions uniformly distributed around a circle of radius R
(each element of the photo matrix has its own circle and its own discrete set of K positions on these
circles). For simplicity, it is assumed that immediately during registration, the circular motion of the
photo matrix is suspended, so that at the moment of “exposure” it remains motionless, and the
transition from one point of the circle to the next occurs instantly. Thus, the possible “blur” of the
image associated with the circular motion of the photo matrix is not taken into account at the first
stage. The result of such registration is K digital images of low resolution (the resolution for each
coordinate is (Ω/∆) times lower than the original image), which we need to reconstructed the original
digital image Fij.
The developed approach and the software system implemented on its basis allows using simulation
tools to test the algorithms for the synthesis and reconstruction of digital images from a set of images
of lower resolution obtained in the process of circular scanning of the original two-dimensional field
on a computer. The input data for programs that reconstruct images with the original (i.e. increased)
spatial resolution are:
1. A set of K observed images (parameter K is controlled by software).
2. L = Ω/∆ - the ratio of the linear size of the element of the low-resolution photo matrix to the
linear size of the pixel of the reconstructed image (adjustable in the program).
3. R/∆ - radius of circular scanning, specified in relative values. In this case, the optimal value of
the parameter R/∆ should be determined based on the results of calculations.
So, schematically, the process of software simulation of the circular scanning process with
subsequent reconstruction of the original digital image can be described as follows. For each fixed k, a
separate sample is, as already noted, the result of “integrating” the original digital image Fij over the
aperture (Ω×Ω) centered at the point
2π ( k − 1) 2π ( k − 1)
( xst( k ) , yst( k ) ) = ( xst( 0) + R cos ; yst( 0) + R sin ).
K K
Figure 3 shows an example of the original image Fij (a fragment of the chessboard).
Figure 3: The original image (a fragment of a chessboard).
Figure 4 shows two of 16 images with low spatial resolution, obtained by emulating the circular
scanning process of the original image 3 with the following parameters: K = 16; L = 89/29 (≈3.069);
R = 1.1.
Figure 4: Two out of 16 low-resolution simulated images; the original high-resolution image was
reconstructed from them.
Physically, the process of circular scanning of images, i.e. the process of analog photoreading of a
real three-dimensional spatially distributed signal using a low-resolution photo matrix is inevitably
accompanied by interference (they are associated, on the one hand, with dynamic spatially distributed
random noise, and, on the other hand, with the noise of the recording path itself). Therefore, in our
case, computer simulation of the process of synthesis of recorded images included, as a component
stage, a software simulation of this kind of pseudo-noise. The presented program model provides for
the generation of random noise of two types: multiplicative, the level of which is correlated with the
level of the integrated signal, and additive random noise, the level of which does not depend on the
recorded signal, but only characterizes the recording path itself.
As a result of modeling the circular scanning process, a system of linear equations arises that
connects each of the expressions U(k)st with the samples Fij of the original image to be estimated, that
fall into the integration aperture (Ω×Ω) centered at a point (x(k)st ,y(k)st).
In the general case, the number of linear equations connecting the multidimensional array of
registered observations U(k)st with the two-dimensional reconstructed array of unknowns Fij exceeds
the number of unknowns N2, i.e. the resulting system of linear equations is overdetermined. It is
solved by the standard least squares method, so that the final expression for each of the components
Fij of the solution vector F is a linearly weighted combination of the observed values with a matrix of
weight coefficients W(1:N2; 1:(K×[entier(N/(Ω/∆)]2)).
This means that the matrix W for the given parameters of circular scanning can be calculated once,
and then used to reconstruct the unknown original high-resolution image by multiplying by the
observation vector U.
3. Results
As an illustration, Fig. 5 shows one of the examples of reconstruction of a noisy image by the
circular scanning method.
Circular scan options:
radius R = 1.2;
relative integration aperture Ω⁄∆ = 3.069;
number of low-resolution images = 16;
additive noise level = 0.2;
multiplicative noise level = 0.01.
reconstruction quality obtained = 0.07851.
Figure 5: An example of reconstructing the original image in the presence of noise: a) the original
image; b), c) two out of 16 low-resolution images, which were used for reconstruction; d) the
reconstructed image
The developed software system allows us not only to carry out the image reconstruction process
with increased resolution, but also to calculate the optimal parameters of the circular scanning process
itself. The data from Table 1 can be an example of such optimization.
Table 1. Calculation of the error in reconstructing the original signal (noise/signal ratio) depending
on the scanning radius.
Scanning radius 0,9 1 1,1 1,2 1,25 1,3 1,4 1,5
Realization №
Realization 1 0,105 0,074 0,077 0,073 0,073 0,071 0,238 3,792
Realization 2 0,276 0,076 0,112 0,137 0,087 0,076 0,170 3,149
Realization 3 0,142 0,136 0,114 0,160 0,073 0,112 0,185 2,794
Realization 4 0,190 0,260 0,115 0,081 0,073 0,115 0,126 5,773
Realization 5 0,447 0,133 0,118 0,106 0,086 0,132 0,103 4,608
Realization 6 0,317 0,086 0,188 0,103 0,071 0,103 0,204 2,669
Realization 7 0,493 0,235 0,200 0,110 0,064 0,066 0,218 4,436
Realization 8 0,270 0,274 0,101 0,116 0,068 0,128 0,118 7,441
Realization 9 0,294 0,146 0,106 0,119 0,077 0,131 0,172 3,742
Realization 10 0,620 0,082 0,254 0,102 0,074 0,056 0,100 5,738
Average error 0,315 0,150 0,139 0,111 0,075 0,099 0,163 5,414
(min)
The purpose of the calculations was to determine the optimal scanning radius at a fixed value of
the relative integration aperture Ω/∆. The criterion for the optimality of the reconstruction algorithm
was the ratio σ1/σ0, where σ12 is the variance of the difference image (difference between the
reconstructed and the original images), and σ02 is the variance of the original image.
From the data given in Table 1, it can be seen, for example, that to improve the spatial resolution
in both coordinates by a factor of 89/29 ≈ 3.069, the optimal radius of the circular path should be R =
1.25∆. This is clearly manifested in the form of a minimum on the last line of Table 1. The numerical
value of this minimum, equal to 0.075, corresponds to the noise/signal value and indicates a high
quality of the original signal reconstruction achieved at Ropt = 1.25∆ (control calculations were carried
out in the presence of multiplicative and additive noise. To obtain statistically reliable estimates, all
results were averaged over 10 realizations of random noise).
4. Conclusion
New efficient algorithms for increasing the spatial resolution of photo matrix images based on
optimal digital processing of circular scanning results have been constructed. Testing has
demonstrated the stable operation of the algorithms in the presence of additive and multiplicative
noise.
5. Acknowledgements
This work was partially supported by the Russian Foundation for Basic Research (project No. 19-
01-00128) and the Ministry of Science and Higher Education of the Russian Federation (project No.
AAA-A17-117052410034-6).
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