Sizikov, V.; Dovgan, A.; Lavrov, A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers2020, 9, 30.
Sizikov, V.; Dovgan, A.; Lavrov, A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers 2020, 9, 30.
Sizikov, V.; Dovgan, A.; Lavrov, A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers2020, 9, 30.
Sizikov, V.; Dovgan, A.; Lavrov, A. Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images. Computers 2020, 9, 30.
Abstract
In this work, the problem is considered for eliminating a non-uniform rectilinear smearing of an image by mathematical and computer-based way, for example, a picture of several cars taken with a fixed camera and moving with different speeds. The problem is described by a set of 1-dimensional Fredholm integral equations (IEs) of the first kind of convolution type with a 1-dimensional point spread function (PSF) at uniform smearing and by a set of new 1-dimensional IEs of a general type (not convolution type) with a 2-dimensional PSF at non-uniform smearing. The problem is also described by one 2-dimensional IE of convolution type with a 2-dimensional PSF at uniform smearing and by a new 2-dimensional IE of a general type with a 4-dimensional PSF at non-uniform smearing. The problem for solving Fredholm IE of the first kind is ill-posed (unstable). Therefore, IEs of convolution type are solved by the Fourier transform (FT) method and Tikhonov's regularization (TR), and IEs of general type are solved by the quadrature/cubature and TR methods. Moreover, the magnitude of the image smear Δ is determined by the original “spectral method”, which increases the accuracy of image restoration. It is shown that the use of a set of 1-dimensional IEs is preferable to one 2-dimensional IE in the case of non-uniform smearing. In the inverse problem (image restoration), the Gibbs effect (the effect of false waves) in the image may occur. It can be edge or inner. The edge effect is well suppressed by the proposed technique “diffusing the edges". In the case of an inner effect, it is eliminated with difficulty, but the image smearing itself plays the role of diffusing and suppresses the inner Gibbs effect to a large extent. It is shown (in the presence of impulse noise in an image) that the well-known Tukey median filter can distort the image itself, and the Gonzalez adaptive filter also distorts the image (but to a lesser extent). We propose a modified adaptive filter. A software package was developed in MatLab and illustrative calculations are performed.
Keywords
smeared image; non-uniform rectilinear smear; integral equations; spectral method; edge and inner Gibbs effects; MatLab
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
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