Space-variant filtering through the Wigner distribution function

Abstract

Space-invariant and space-variant filtering of discrete images with unidimensional variation is performed in this paper through the Wigner distribution function (WDF). Low-pass, bandpass, and high-pass filtering is used in the Fourier domain and in the Wigner distribution, to compare their different behavior in both cases. A space-variant filter is generated to modify the WDF of an object to obtain, after inversion, a spatially variant filtered image. Finally, spatially variant defocused images generated through the Wigner distribution are restored by Wiener based filters applied in the Wigner domain. The method applied to recover the filtered images can easily be generalized to bidimensional images.

Cloud covering denoising through image fusion

“…where H 1 (n, m) can be interpreted as a pseudo-filtering function, since g might not exists [13,14]. For a thorough discussion see Ref.…”

A two incoherent sources resolution criterion in the phase space

“…Considering the right side of Eq (9) it is straightforward that the dependence of the cosine term is with the d parameter and the u variable; consequently, since we are looking for a resolution value along the x coordinate we can make 0 = = v y , without loss of significant evidence and make ) , (18) where the inequality of Eq. (11) has been used and the variable n being an integer number.…”

“…In optics, the WDF has been introduced by Bastiaans 16 and the most common optical set up used to display it has been implemented by Bartelt, et al 17 for the 1-D and 2-D spatial functions. A hybrid processor for image processing has been proposed by Gonzalo, et al 18 Other applications in optics can be found in the analysis of complex amplitude pupil functions 19 and in the explanation of paraxial polychromatic optics 20 . In most of the WDF applications implicitly are involved the linear, paraxial and isoplanatic conditions, and the mathematical properties of this function can be found in the classical paper of Claasen and Mecklenbräuker 21 .…”

The convolution theorem for the continuous wavelet tranform

“…This process has been carried out with the complex spectrogram [34] and the Wigner-Ville distribution [2,11,34]. The continuous wavelet transform (CWT) [12,15] stands for a scale-space representation and is an e ective way to analyze non-stationary signals and images.…”

“…which is also known as a space-variant ÿltering in optics and engineering literature [2,11], as opposed to the space invariant ÿltering given by the convolution deÿned in (12). Given a ÿxed function ∈ L 2 (R), called wavelet, consider its translations and dilations deÿned as…”