Eliminating Nonuniform Smearing and Suppressing the Gibbs Effect on Reconstructed Images

Abstract

In this work, the problem of eliminating a nonuniform rectilinear smearing of an image is considered, using a mathematical- and computer-based approach. An example of such a problem is a picture of several cars, moving with different speeds, taken with a fixed camera. The problem is described by a set of one-dimensional Fredholm integral equations (IEs) of the first kind of convolution type, with a one-dimensional point spread function (PSF) when uniform smearing, and by a set of new one-dimensional IEs of a general type (i.e., not the convolution type), with a two-dimensional PSF when nonuniform smearing. The problem is also described by a two-dimensional IE of the convolution type with a two-dimensional PSF when uniform smearing, and by a new two-dimensional IE of a general type with a four-dimensional PSF when nonuniform smearing. The problem of solving the Fredholm IE of the first kind is ill-posed (i.e., unstable). Therefore, IEs of the convolution type are solved by the Fourier transform (FT) method and Tikhonov’s regularization (TR), and IEs of the 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 one-dimensional IEs is preferable to one two-dimensional IE in the case of nonuniform smearing. In the inverse problem (i.e., image restoration), the Gibbs effect (the effect of false waves) in the image may occur. This may be an edge or an inner effect. The edge effect is well suppressed by the proposed technique, namely, “diffusing the edges”. The inner effect is difficult to eliminate, but the image smearing itself plays the role of diffusion 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.