The Paired Representation Method of Signal De-noising
In this work, a method of restoration of noisy one-dimensional (1-D) discrete-time signals and two-dimensional (2-D) signals such as the case of grayscale images with sharp feature content is proposed, which provides a computationally efficient means to deal with the shortcomings of the linear space-invariant (LSI) filtration -- the blurring of sharp content -- without sacrificing the computational simplicity of the fast Fourier transform (FFT) approach. The method relies on the advanced representation of the signal as a set of 1-D splitting-signals by the paired transformation, whose spectral composition is non-overlapping and completely found in them; the 1-D paired transformation is used for the splitting of the 1-D signals and the 2-D paired transformation is used for splitting of the 2-D signals into 1-D splitting-signals. The filtration of the signal is posed as the filtration of each splitting-signal independently of the other members of the set, by separate LSI finite-impulse response (FIR) filters; the set of relations of circular convolutions have been derived to show that the direct filtration is not equal to the paired method.
Although the proposed method is applicable for any LSI-FIR filters, the performance of the method is proved by the optimal linear filtration, the Wiener smoothing filter tested on a range of signals corrupted by additive white Gaussian noise. The evaluation of the method on a wide class of 1-D signals using two measures: the minimum mean-squared error (M.M.S.E) and the relative cumulative spectral distribution revealed better edge preservation quality than the case of direct smoothing of the signal itself; the M.M.S.E is close to the error in the direct case, whereas the high-frequency edge content is preserved. The evaluation of the 2-D case on high-resolution grayscale images using two measures: the M.M.S.E and by the visual inspection of focal-blurring, revealed that the filter trades its error performance with the preservation of sharp feature content: sharp-texture and sharp-edges.