In this paper, an effective realization of the α-rooting method of image enhancement by splitting-signals is
proposed. The splitting-signals completely determine the image and split its spectrum by disjoint subsets of
frequencies. Image enhancement is reduced to processing separate splitting-signals. We focus on processing
only one specified splitting-signal, to achieve effective image enhancement that in many cases exceeds the
enhancement by known a-rooting and wavelet methods. An effective realization of enhancement of image
(N × N) is achieved by using one coefficient, instead of N/2 such coefficients for splitting-signals in the split
α-rooting and N × N in traditional α-rooting. The proposed method does not require Fourier transforms, its
realization is performed with N multiplications. The processing of the splitting-signal leads to the change of
the image along the parallel lines by N different values, which leads to the concept of directional images and
their application in enhancing the image along directions. A novel method of combining paired transform (pre-step
of SMEME (spectral spatial maximum exclusive mean) filter) by wavelet transforms is proposed. While
denoising directional clutters, the most corrupted splitting-signal is estimated and found, depending on the
angle of long-waves.
In this paper, applications of the tensor and paired representations of an image are presented for image enhancement. The proposed methods are based on the fact that the 2-D image can be represented by a set of 1-D "independent" signals that split the 2-D discrete Fourier transform (DFT) of the image into different groups of frequencies. Each splitting-signal carries information of the spectrum in a specific group. Rather than enhance the image by traditional methods of the Fourier transform (or other transforms), splitting-signals can be processed separately and the 2-D DFT of the processed image can be defined by 1-D DFTs of new splitting-signals. The process of splitting-signals related to the paired representation is very effective, because of no redundancy of spectral information carrying by ifferent splitting-signals. The effectiveness of such approach is illustrated through processing the image by the a-rooting method of enhancement. Images can be enhanced by processing only a few splitting-signals, to achieve enhancement that in many cases exceeds the enhancement by the α-rooting method and other known methods. The selection of such splitting-signals is described.
A new concept of weighted thresholding is considered and a new set-theoretical representation of signals and images is described, that can be used for design of new nonlinear and morphological filters. Such representation maps many operations of nonbinary signal and image processing to the union of simple operations over the binary signals. The weighted thresholding is invariant under the morphological transformation, including such basic operations as the erosion and dilation. The main idea of using the weighted thresholding is in the choice of a special few levels of thresholding on which we can process the signals and images. We focus on the arithmetical weighted thresholding, but other thresholding, including the geometrical, probability-based, and the so-called Fibonacci series based thresholding, are also considered. Properties of these kinds of thresholding are described. Experimental results show that the weighted thresholding is very promising and can be used for many applications, such as image enhancement and edge detection.