Methods for Improving Image Quality for Contour and Textures Analysis Using New Wavelet Methods

Abstract

The fidelity of an image subjected to digital processing, such as a contour/texture highlighting process or a noise reduction algorithm, can be evaluated based on two types of criteria: objective and subjective, sometimes the two types of criteria being considered together. Subjective criteria are the best tool for evaluating an image when the image obtained at the end of the processing is interpreted by man. The objective criteria are based on the difference, pixel by pixel, between the original and the reconstructed image and ensure a good approximation of the image quality perceived by a human observer. There is also the possibility that in evaluating the fidelity of a remade (reconstructed) image, the pixel-by-pixel differences will be weighted according to the sensitivity of the human visual system. The problem of improving medical images is particularly important in assisted diagnosis, with the aim of providing physicians with information as useful as possible in diagnosing diseases. Given that this information must be available in real time, we proposed a solution for reconstructing the contours in the images that uses a modified Wiener filter in the wavelet domain and a nonlinear cellular network and that is useful both to improve the contrast of its contours and to eliminate noise. In addition to the need to improve imaging, medical applications also need these applications to run in real time, and this need has been the basis for the design of the method described below, based on the modified Wiener filter and nonlinear cellular networks.

1. Introduction

Norbert Wiener’s theory of optimal filtering of continuous signals is the basis of linear least squares linear error filters dependent on input data. Wiener studied in his 1949 paper “Extrapolation, Interpolation and Smoothing of Stationary Time Series”, considering stationary signals, the problem of estimation in terms of the least squares error in the continuous case of time series. The extension of Wiener’s theory to the discrete case is simple and has a great practical utility because it led to the implementation of this type of filter using digital signal processors or specialized circuits in this class. Wiener filters play an important role in a wide range of applications [1] such as linear prediction, signal encoding, echo cancellation, signal recovery, channel equalization, signal identification or additive noise suppression, a number of studies focusing on their use in practical applications. The coefficients of the Wiener filter are calculated so as to minimize the average square error between the filter output and the useful signal. Randy Gomez, Tatsuya Kawahara and Kazuhrio Nakadai [2] are researching with the purpose of improving the late reflection, noise power and the speech from a contaminated signal and improve it with a novel scheme. The solutions include an acoustic model (AM) and proves its robustness and effectiveness through experimental evaluation.

The system proposed by Saeed Ayat, M. T. Manzuri-Shalmani and Roohollah Dianat [3] involves a WaveShrink module and proposed an improved system of wavelet-based enhancement with a thresholding algorithm and a novel method for the selection of the threshold. The results are presenting, comparatively with similar approaches, an improvement in the system performances.

Israel Cohen [4] presents a new approach with an improved minima controlled recursive averaging (IMCRA) for noise estimation, low input signal to noise ratio (SNR) weak speech components. The IMCRA method proved to be effective and it has a lower estimation error compared with other methods, provides a lower residual noise and better speech quality.

Boll, S. [5] is using a stand-alone noise suspension algorithm to reduce spectral effects of added noise. This system can be used for speech recognition systems, speaker authentication systems or narrow band voice as a pre-processor.

Babul Islam, Hamidul Islam and Monsor Rahman [6] are using Wiener filter to identify and estimate the mel frequency axis Mel-LPC spectra in a medium with additive noises. After the filter is applied, the transform of inverse wavelet is applied and it obtains the signal of enhanced time domain speech. The result has presented an improvement on the word accuracy by 17.22%.

S. G. Mallat [7] uses a pyramidal algorithm with quadrature mirror filters. The representation presents several spatial orientations. This representation of the application may be used in fractal analysis, texture discrimination and image coding.

M. Kivanc Mihcak, I. Kozintsev and K. Ramchandran [8] want to apply first the maximum likelihood (ML) rule and then the estimation procedure MMSE—minimum mean squared error. Comparative with other results from literature, this method presents favorable result.

P. Moulin and Juan Liu [9] research on thresholding and minimax/global wavelet shrinkage methods has proven to be very good in different scenarios. They are introducing a new group of priors with the base on Rissanen universal prior on integers. The theoretical and experimental results are presenting the robustness of some wrong specifications of others on image processing.

Eero P. Simoncelli and Edward H. Adelson [10] are analyzing the noise removing problem starting from its solution, the Wiener solution and develop its extension, a Bayesian estimator. This estimator is nonlinear and performs a “coring” operation. Based on steerable wavelet pyramid, they create for subband statistics a simple model semiblind noise-removal algorithm.

Martin J. Wainwright and Eero P. Simoncelli [11] proposed for the modeling of natural images the Gaussian scale mixture. Using multiscale bases are two surprising types of non-Gaussian behavior. A procedure of nonlinear “normalization” can be used as Gaussian coefficients.

M. Kazubek [12] presents an error occurring on the approximate analysis in the empirical Wiener. They observed that the performance of the filter Wiener can be increased fixing the problems produced by the thresholding operation.

Shi Zhong and Vladimir Cherkassky [13] are analyzing under the framework of the theory of statistical learning the image denoising to present the wavelet thresholding. They propose a tree structure based on spatial location, scale and magnitude.

Dohono, D. L. [14] is using an empirical wavelet transform and proves two results: smooth and adapt. The first one has high probability and the second has precision. This method highlights new information about abstract statistical inference and the possibility of connecting it with an optimal recovery model.

H. Zhang, A. Nosratinia and R.O. Wells [15] are developing an extension of the Minim Mental State Examination (MMSE) pixel wise wavelet denoising. For autocorrelations they use an exponential decay model and for Fir Wiener filtering a parametric solution in the wavelet domain. This filter proves a notable denoising performance.

H.G. Senel, R.A. Peters and B. Dawant [16] are using a novel median filter. They are combining in fuzzy connectedness already existing ideas and new ones in order to improve the extraction of edges. The edge detection proved, through qualitative and statistical analyses, to have better accuracy.

Suresh Kumar et al. [17] want to remove the noise captured in images among the capturing or those injected during image transmission. Their attention is centered on the salt and pepper noise, Gaussian noise and speckle noise through the Wiener and performance comparison of median filters.

Mei-Sen Pan, Jing-Tian Tang and Xiao-Li Yang [18] are applying a filter method in order to improve the quality of the image and to remove the noise. The medical images are analyzed and when noises are identified the noise granularity function (NGF) is applied, the size of the filtering window is adaptively adjusted and then with the median filter are eliminated from the MI—median the current noise marked pixel, amid with the noise mark cancellation. This method is efficient in the imagine preservation and to remove the noises [19].

San-li Yi, Zhen-Cheng Chen and Hong-li Ling [20] are proposing a modified Wiener filter that uses diffusion weighted images (DWIs) with multiboundary. This modified Wiener and the classical Wiener filter are compared and analyzed in detail. The modified method has lower mean square error, more accurate DTIs and a lower nonpositive percentage.

Hosseini, H.; Hessar, F. and Marvasti, F. [21] are presenting a method to reduce noise suppression with a high-density impulse in real-time. This method has better performance on visual quality and peak signal to noise ratio (PSNR) compared with the best performing methods.

Wink, A. M. and Roerdink, J. B. [22] are presenting a denoising scheme based on general wavelet. The method was verified through the spot temporal SNR and by its shape. The method introduces more false positives to produce smooth images, less errors than the Gaussian method or other wavelet methods.

Mitsuru Ikeda, Reiko Makino and Kuniharu Imai [23] use an evaluation method for image reduction to preserve the noise free components. In this study it is proved that for the computed tomography (CT), chest phantom images may be applied to the algorithm BayesShrink, which will work well. This solution is expected to have enough robustness for the chest phantom images.

Indulekha N.R and Sasikumar M [24] are decomposing the image in eight subbands using thresholding methods with bilateral filter. They are evaluating the performance of the denoising through root mean square error (RMSE), peak signal to noise ratio and structural similarity index (SSIM).

Zuo Feng Zhou and Peng Lang Shui [25], in order to eliminate the noise from the images, use directional windows and the contourlet based image algorithm. This algorithm has better performance compared with other image denoising algorithms.

R. Pavithra, R. Ramya and G. Alaiyarasi [26] are analyzing how increasing the contrast of a picture makes it easier to analyze. The noise that corrupts the image of magnetic resonance imaging (MRI) is coming from the acquisition’s moment or the transmission process. Seven algorithms have been compared on their effectiveness on MRI images.

Satheesh S and Prasad K. [27] are using the contourlet transform algorithm to analyze the magnetic resonance imaging denoising. The algorithm has shown that this algorithm obtains higher PSNR (peak signal to noise ratio) compared with the wavelet algorithms in the presence of additive white Gaussian noise (AWGN) on MRI.

Abbas Hanon AlAsadi [28] are using the contourlet transform using directional and multiscale filter banks. This algorithm contains time—frequency—localization and multiscale and high degree directionality.

Taujuddin N S A M and Ibrahim R. [29] are highlighting the importance of the correct reading of medical images. This research’s purpose is to use the threshold values in a compression algorithm, which is expected to decrease the image size.

Amjad Ali, S., Vathsal, S. and Lal Kishore, K. [30] are analyzing the denoising using computed tomography image that is dependable on image quality. The used method has three stages of processing. In conclusion, the multiwavelet based denoising technique can be used to denoise and obtain quality CT images.

Guang Yang et al. [31] are proposing a super resolution (SR) method to recover high resolution (HR) medical images. The method is coupled with dual-tree complex wavelet transform (DTCWT) and modeled and applied on multiple medical images. The solution proved to be more qualitative and quantitative compared with other methods.

Yang, A.F. et al. [32] are proposing the denoising of astronomical images named local sparse representation (LSR) using iterative shrinkage-thresholding algorithm (ISTA). Using the optimization method, they were able to update the dictionary and the sparse coding vectors and to prove the superiority of the method.

Chandrika Saxena and Deepak Kourav [33] are working to provide high quality images eliminating multiple types of noises, like Brownian noise, speckle noise, pepper noise and the Gaussian noise. Applying the appropriate algorithm to the specific type of noise is a very important element. They are coupling the type of noise with the best solution, for example for the corruption with salt and paper noise the best method to apply is the filtering.

Zhou Wang et al. [34] are approaching the images from the degradation of the structural information view. To extend the limitation of the error sensitivity is used as an alternative to the structural similarity as the structural similarity index measure (SSIM) index is developed and proves the validity of the method. They are investigating multiple issues and try to identify the appropriate solutions for it.

Shah, A. J.; Makwana, R. and Gupta, S. B. [35] are considering the quality of the images from medical science, satellite or high-definition television (HDTV). They propose a method using non-subsampled contourlet transform (NSCT) based learning and an interior-point barrier (IBP) to optimize the cost function. They have undertaken multiple simulations and proved the validity of the method.

Alex Apotsos et al. [36] tested multiple parametric models and calibrated them. Using data records and default values, the tuned models they built model evaluation and parametrization. The results are presenting that the errors are reduced.

Chen, G. Y., Bui, T. D., and Krzyżak, A. [37] are approaching the noised images and identify the customized wavelet filters, specific for the type of noise, using simulated annealing (SA). Compared with the VisuShrink image the method they developed shows better experimental results, the VisuShrink is outperformed in all the experiments.

Ding Liu et al. [38] are proposing two solutions for image denoising, a convolutional neural network and a deep neural network. There are two tasks presented, to denoise images and use deep learning for high level vision tasks. They are demonstrating that the methods have the capability to beat the performance degradation and produce more visually appealing results.

Jing Liu et al. [39] use 3D block-matching and 3D filtering to identify in the entire image the similar blocks. They want to identify among the edge direction all the matching blocks and to perform the denoising shrinking of the 3D transform coefficients applied on the groups. This method improves the visual quality, numerical results and preserves the textures and edges. This research paper has as main objective the presentation of the results obtained in the case study with reference to the improvement of image quality by Wiener filtering in the field of wavelet and cellular neural networks for the preservation of contours and the improvement of image quality.

In the present article, the authors present the results of research work on improving image quality through two powerful procedures for improving images obtained by soft truncation of wavelet coefficients correlated with nonlinear cellular networks. New filters are proposed: one based on the use of the translation invariant wavelet transform, which mediates the results obtained by repeating this procedure for several sets of wavelet coefficients and method based on Wiener filtering in the wavelet domain. The Wiener filters in the wavelet domain were called multiple Wiener filter, mediated Wiener filter and iterated Wiener filter. The practical applicability of the proposed methodology is addressed generically to the medical act, respectively, the medical imaging essential especially for neurosurgery. It should be noted that CT is performed at the macroscopic level, while the algorithm proposed in this research work contributes to the analysis of medical imaging at the microscopic level so that tumors and areas of transition from diseased tissue to healthy tissue are highlighted much better.

The subsequent sections have the structure organized as follows: Section 2 presents the steps followed as a research methodology, namely Section 2.1. Improving image quality by Wiener empirical filtering in the wavelet domain; Section 2.2. Wiener filter in the field transformed; Section 2.3. Empirical Wiener Wavelet Filter; Section 2.4. Multiple Wiener filter in wavelet domain; Section 2.5. Conjugated use of the invariant wavelet transform in translation and Wiener filtering in the wavelet domain to improve the image quality obtained by soft truncation of the wavelet coefficients; Section 2.6. Reconstruction of contours from images using nonlinear cellular network (NCN) algorithm. In Section 3, the results are highlighted and critically analyzed based on the proposed research methodology, namely algorithm testing and the combined use of the two methods covered in this article, the results being superior to those obtained by individual use of each method. Finally, the main conclusions of this research work are presented in Section 4.

2. Materials and Methods

2.1. Improving Image Quality by Wiener Empirical Filtering in the Wavelet Domain

Consider a sequence of input data $\mathrm{y}=\mathrm{x}+\mathrm{n}$, consisting of the sum of the samples of the useful signal $\mathrm{x}={\left\{{\mathrm{x}}_{\mathrm{i}}\right\}}_{\mathrm{i}=1}^{\mathrm{N}}$ and the additive white Gaussian noise $\mathrm{n}={\left\{{\mathrm{n}}_{\mathrm{i}}\right\}}_{\mathrm{i}=1}^{\mathrm{N}}$, both the useful signal x and the noise $\mathrm{n}$ having zero averages was considered ($\mathrm{E}\left(\mathrm{x}\right)=0$, $\mathrm{E}\left(\mathrm{n}\right)=0$, where E represents the statistical average). There is the problem of determining the Wiener filter, a filter that minimizes the average square error between the useful signal and its estimate ($\widehat{\mathrm{x}}=\mathrm{Gy}$),

$${\mathrm{G}}_{wiener}=\underset{\mathrm{G}}{\arg \min } \mathrm{E}\left({\Vert \mathrm{x}-\widehat{\mathrm{x}}\Vert }^2\right)$$

(1)

The autocorrelation matrix ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$ of the useful signal, with ${\mathrm{R}}_{\mathrm{n}\mathrm{n}}$ that of the noise and with ${\mathrm{R}}_{\mathrm{y}\mathrm{y}}$ that of the input signal was noted:

$${\mathrm{R}}_{\mathrm{xx}}=\mathrm{E}\left({\mathrm{xx}}^{\mathrm{T}}\right), {\mathrm{R}}_{\mathrm{nn}}=\mathrm{E}\left({\mathrm{nn}}^{\mathrm{T}}\right), {\mathrm{R}}_{\mathrm{yy}}=\mathrm{E}\left({\mathrm{yy}}^{\mathrm{T}}\right)$$

(2)

The useful signal $\mathrm{x}$, and the noise $\mathrm{n}$, are uncorrelated processes, therefore:

$$\mathrm{E}\left(\mathrm{x}{\mathrm{n}}^{\mathrm{T}}\right)=\mathrm{E}\left(\mathrm{n}{\mathrm{x}}^{\mathrm{T}}\right)=0_{\mathrm{N}}$$

(3)

In this case, the mean square error made in estimating the signal $\mathrm{x}$ will be:

$$\mathrm{E}\left({\Vert \mathrm{x}-\widehat{\mathrm{x}}\Vert }^2\right)=\mathrm{E}\left({\left(\widehat{\mathrm{x}}-\mathrm{x}\right)}^{\mathrm{T}}\left(\widehat{\mathrm{x}}-\mathrm{x}\right)\right)$$

(4)

which through further development will be able to be written:

$$\mathrm{E}\left({\Vert \mathrm{x}-\widehat{\mathrm{x}}\Vert }^2\right)=\mathrm{E}\left({\mathrm{y}}^{\mathrm{T}}{\mathrm{G}}^{\mathrm{T}}\mathrm{G}\mathrm{y}\right)-2\mathrm{E}\left({\mathrm{y}}^{\mathrm{T}}{\mathrm{G}}^{\mathrm{T}}\mathrm{x}\right)+\mathrm{E}\left({\Vert \mathrm{x}\Vert }^2\right)$$

(5)

where $\mathrm{G}$ must be chosen so that the mean square error of estimation is minimal, the condition must be satisfied:

$${\nabla }_{\mathrm{G}}\mathrm{E}\left({\Vert \mathrm{x}-\widehat{\mathrm{x}}\Vert }^2\right)=0_{\mathrm{N}}$$

(6)

that is, the gradient of the mean square estimation error relative to the matrix $\mathrm{G}$ is zero.

This is how the relationship is reached:

$$2{\mathrm{G}}_{wiener}\left(\mathrm{E}\left(\mathrm{y}{\mathrm{y}}^{\mathrm{T}}\right)\right)-2\mathrm{x}{\mathrm{y}}^{\mathrm{T}}=0_{\mathrm{N}}$$

(7)

which leads further to:

$${\mathrm{G}}_{wiener}\left(\mathrm{E}\left(\mathrm{x}{\mathrm{x}}^{\mathrm{T}}+\mathrm{n}{\mathrm{n}}^{\mathrm{T}}\right)\right)-\mathrm{x}{\mathrm{x}}^{\mathrm{T}}=0_{\mathrm{N}}$$

(8)

By introducing the autocorrelation matrices, we will have:

$${\mathrm{G}}_{wiener}\left({\mathrm{R}}_{\mathrm{x}\mathrm{x}}+{\mathrm{R}}_{\mathrm{n}\mathrm{n}}\right)={\mathrm{R}}_{\mathrm{x}\mathrm{x}}$$

(9)

or

$${\mathrm{G}}_{wiener}={\mathrm{R}}_{\mathrm{x}\mathrm{x}}{\left({\mathrm{R}}_{\mathrm{x}\mathrm{x}}+{\mathrm{R}}_{\mathrm{n}\mathrm{n}}\right)}^{-1}$$

(10)

This relation establishes the optimal filter matrix according to the second order statistical properties of the useful signal and of the noise, respectively, according to the autocorrelation matrices of the useful signal, ${\mathrm{R}}_{\mathrm{xx}}$ and of the noise, ${\mathrm{R}}_{\mathrm{nn}}$.

If the filter thus obtained is characterized by a diagonal matrix:

$${\mathrm{G}}_{wiener}={\mathrm{diag}(\mathrm{g}}_{0,0},{\mathrm{g}}_{1,1},\dots ,{\mathrm{g}}_{\mathrm{N}-1,\mathrm{N}-1})$$

(11)

we have a scalar filter.

The importance of scalar filters lies in the fact that they simplify the calculation operations for estimating the useful signal. In general, however, the matrix ${\mathrm{G}}_{\mathrm{wiener}}$ is not a diagonal matrix, in which case we are dealing with a vector filter.

2.2. Wiener Filter in the Field Transformed

If the Wiener filter is not scalar, the calculations required to estimate the useful signal are very laborious. The approach of Wiener filtration in the transformed domain, when working with an orthogonal transformation, has the advantage that in this case we can work with a scalar filter, or which can be approximated by a scalar filter.

At the same time, Gaussian additive white noise retains its second-order statistical properties when an orthogonal transformation is used, which means that the noise autocorrelation matrix can be expressed by ${\mathrm{R}}_{\mathrm{nn}}={\mathsf{\sigma }}^2{\mathrm{I}}_{\mathrm{N}}$, $\mathsf{\sigma }$ are noise dispersion, and ${\mathrm{I}}_{\mathrm{N}}$ the N-order unit matrix.

The matrix corresponding to the Wiener filter ${\mathrm{G}}_{\mathrm{wiener}}={\mathrm{G}}_{\mathrm{w}}$ is given by:

$${\mathrm{G}}_w={\mathrm{R}}_{\mathrm{x}\mathrm{x}}{\left({\mathrm{R}}_{\mathrm{x}\mathrm{x}}+{\sigma }^2{\mathrm{I}}_{\mathrm{N}}\right)}^{-1}$$

(12)

where ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$ is the autocorrelation matrix of the signal. The development of the matrix ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$ according to the eigenvalues and eigenvectors is given by:

$${\mathrm{R}}_{\mathrm{x}\mathrm{x}}=\mathrm{U}\mathrm{\Lambda }{\mathrm{U}}^{\mathrm{T}}={\displaystyle \sum _{k=1}^{{\mathrm{N}}_s}{\mathrm{\lambda }}_k{\mathrm{u}}_k{\mathrm{u}}_k^{\mathrm{T}}}$$

(13)

where ${\mathrm{u}}_k$ and ${\mathrm{\lambda }}_k$ are the eigenvectors and the eigenvalues of the matrix ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$, ${\mathrm{N}}_S<\mathrm{N}$, respectively, being the number of eigenvalues. The matrix $\mathrm{U}$ consists of the eigenvectors of the matrix ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$, being a matrix of dimensions $\mathrm{N}\times {\mathrm{N}}_{\mathrm{s}}$, $\mathrm{U}=\left[{\mathrm{u}}_1,{\mathrm{u}}_2,\dots ,{\mathrm{u}}_{{\mathrm{N}}_{\mathrm{s}}}\right]$. The matrix $\mathrm{\Lambda }$ is a diagonal matrix containing eigenvalues, $\mathrm{\Lambda }=\mathrm{diag}\left[{\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2,\dots ,{\mathrm{\lambda }}_{{\mathrm{N}}_{\mathrm{s}}}\right]$. The matrix $\mathrm{U}$ defines the Karhunen–Loewe transformation for the signal x and, as a result, decorates the input data, respectively, the matrix ${\mathrm{R}}_{\mathrm{x}\mathrm{x}}$, concentrating the signal energy in the smallest possible subspace. In terms of development according to eigenvalues and eigenvectors, the Wiener filter can be expressed as:

$${\mathrm{G}}_w=\mathrm{U}\tilde{\mathrm{\Lambda }}{\mathrm{U}}^{\mathrm{T}}={\displaystyle \sum _{k=1}^{{\mathrm{N}}_s}\frac{{\mathrm{\lambda }}_k}{{\mathrm{\lambda }}_k+{\sigma }^2}{\mathrm{u}}_k{\mathrm{u}}_k^{\mathrm{T}}}$$

(14)

where

$$\tilde{\mathrm{\Lambda }}=\mathrm{diag}\left[\frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_1+{\sigma }^2},\frac{{\mathrm{\lambda }}_2}{{\mathrm{\lambda }}_2+{\sigma }^2},\dots ,\frac{{\mathrm{\lambda }}_{{\mathrm{N}}_s}}{{\mathrm{\lambda }}_{{\mathrm{N}}_s}+{\sigma }^2}\right]$$

(15)

This formulation indicates that the Wiener filter in the transformed domain first performs the Karhunen–Loewe transformation of the data by, ${\mathrm{U}}^{\mathrm{T}}$ then processes, using the optimal weighting matrix $\tilde{\mathrm{\Lambda }}$, individually each coefficient. Finally, the data thus processed are returned to the primary space by $\mathrm{U}$.

From the wavelet transformation that approximates the deceleration of the input data and the concentration of the signal energy in a small number of coefficients, it can be used as an approximation of the Karhunen–Loewe transformation, i.e., the wavelet development can approximate the development of a wide class of signals. Additionally, the procedure of hard truncation of the wavelet coefficients, by retaining only a fraction of all the development coefficients, implicitly presupposes this approximation.

It is considered that the procedure of hard truncation of the wavelet coefficients is supposed to be retained following the operation of comparison with $\mathrm{\lambda }$ a first threshold ${\mathrm{N}}_{\mathrm{\lambda }}<\mathrm{N}$ (as a result of an operation of rearrangement of the coefficients, since the wavelet transformation unlike other transformations such as Fourier, discrete cosine or even Karhunen–Loewe does not generate decreasing spectra).

In the procedure of hard truncation of wavelet coefficients, the ${\mathrm{N}}_{\mathrm{\lambda }}$ weighting coefficients of the filter ${\mathrm{H}}_{hard}$ representing this operation are:

$$h_{hard}\left(i\right)=\{\begin{array}{c}1, i=1, 2, \dots , {\mathrm{N}}_{\mathrm{\lambda }}\\ 0,i={\mathrm{N}}_{\mathrm{\lambda }}+1, {\mathrm{N}}_{\mathrm{\lambda }}+2,\dots , \mathrm{N}\end{array}$$

(16)

as long as the optimal Wiener weights to minimize the least squares error are given by:

$$h_w=\frac{{\theta }^2\left(i\right)}{{\theta }^2\left(i\right)+{\sigma }^2}$$

(17)

$\theta \left(i\right)$ being the wavelet coefficients corresponding to the useful signal.

Assuming a perfect estimate of the estimated subspace (in particular ${\mathrm{N}}_{\mathrm{\lambda }}={\mathrm{N}}_s$), the increase in the mean square error due to the approximation made by using the hard truncation of the wavelet coefficients instead of using the Wiener filter in the wavelet domain will be:

$$\frac{MS{E}_h}{MS{E}_w}=\frac{{\mathrm{N}}_{\mathrm{\lambda }}}{{{\displaystyle \sum }}_{i=1}^{{\mathrm{N}}_{\tau }}{h}_w}$$

(18)

2.3. Empirical Wiener Wavelet Filter

However, for any signal, optimal processing in the sense of the mean square error is performed by the Wiener filter, which leads to a substantial improvement in performance. SP Ghael, AM Sayeed and RG Baraniuk [40] proposed an algorithm in the wavelet domain for removing white Gaussian additive noise from images, an algorithm that uses an estimate obtained by hard or soft truncation of wavelet coefficients to design a Wiener filter in the wavelet domain. The scheme of this filter is presented in Figure 1. The authors called this method WienerShrink.

A particularity of this algorithm is the use of two different wavelet bases, one for obtaining the estimate and another for designing the Wiener filter. We implemented this type of filter using both hard and soft truncation of wavelet coefficients and semisoft truncation to obtain the estimated solution [41]. Both the results reported by S. P. Ghael, A. M. Sayeed and R. G. Baraniuk, and their own simulations, show that the use of this algorithm leads to an improvement of the filtered image quality but also to the preservation of contours, both in peak signal-to-noise ratio (PSNR) and visual terms. This can also be seen from the data presented in

Table 1. Dependence on performance obtained by processing with empirical Wiener filter.

Lena				Wavelet Coefficients	PSNR (dB)	C (%)
Initial		Estimated Solution		Daubechies 6	30.03	63.67
PSNR (dB)	C (%)	PSNR (dB)	C (%)	Daubechies 12	30.56	64.65
				Coifman 6	30.49	64.75
26.00	63.94	29.74	61.87	Coifman 12	30.56	64.65
Camera				Wavelet Coefficients	PSNR (dB)	C (%)
Initial		Estimated Solution		Daubechies 6	30.39	63.11
PSNR (dB)	C (%)	PSNR (dB)	C (%)	Daubechies 12	30.97	63.66
				Coifman 6	30.91	63.22
26.24	65.72	29.98	62.20	Coifman 12	30.97	63.66

.

In the case of noise reduction algorithms, the problem is to evaluate the amount of noise removed, the distortions introduced and the loss of information. One such measure can be the mean square error (MSE) between the pixel values in the original image and those in the remade image:

$$MSE=\frac{1}{\mathrm{M}\mathrm{N}}{\displaystyle \sum _{i=1}^{\mathrm{M}}{\displaystyle \sum _{j=1}^{\mathrm{N}}{\left(\mathrm{x}\left(i,j\right)-\widehat{\mathrm{x}}\left(i,j\right)\right)}^2}}$$

(19)

where M and N are the image sizes; $\mathrm{x}\left(i,j\right)$—pixel intensities in the original image; $\widehat{\mathrm{x}}\left(i,j\right)$—pixel intensities in the restored image.

Another measure of the similarity between the two images is the ratio between the peak value of the signal and the mean square error (MSE), defined by:

$$\mathrm{PSNR}=10{\log }_{10}\frac{{\left(2^k-1\right)}^2}{MSE}$$

(20)

where k is the bits number with which the pixel intensity is quantized.

PSNR is a metric widely used both in the evaluation of image compression algorithms and in the evaluation of image noise reduction algorithms. Quantitative evaluation of the quality of images processed based on MSE or other derived metrics does not provide sufficient information on the preservation of image contours. Thus, images with the same PSNR may have different visual qualities. If the PSNR can be used to evaluate the amount of noise removed from the image, instead for the quantitative evaluation of images processed in terms of preserving contours this metric is not sufficiently suggestive [42]. Therefore, for the purpose of quantitative evaluation of the images under this aspect, the coefficient C is introduced, defined by:

$$\mathrm{C}=\frac{{\mathrm{N}}_c}{{\mathrm{N}}_0}$$

(21)

where N_c is the number of common pixels, determined as belonging to the contours in both the original and the processed image, and N₀ is the total number of pixels determined as belonging to the contours in the original image. The coefficient C will be expressed as a percentage, the images with a higher coefficient C have a better visual quality. This provides a way to quantitatively evaluate the conservation of contours in an image.

Depending on the PSNR and the coefficient C, a merit factor of the processed image can be defined, expressed by:

$$F_{merit}=\mathrm{PSNR}\left(dB\right)+100\cdot \mathrm{C}$$

(22)

If it will proceed to optimize the parameters of a filter considering as a criterion the maximization of the merit factor, different results are obtained than those obtained by maximizing the PSNR.

The scheme of this filter proposed by S. P. Ghael, A. M. Sayeed şi R. G. Baraniuk is shown in Figure 1.

Table 1 shows the dependence of the performances obtained by processing with the empirical Wiener filter proposed by S. P. Ghael, A. M. Sayeed and R. G. Baraniuk depending on the wavelet coefficients used for its implementation. The test images were 256 × 256 pixels and were degraded with white noise Gaussian dispersion additive $\sigma =0.05$. The estimated solution was obtained using Daubechies coefficient 4.

From the data presented in Table 1 it is observed that in the case of each image, the results obtained differ depending on the wavelet coefficients used to implement the Wiener filter. Additionally, the wavelet coefficients that lead to the best results differ from image to image.

The results presented in

Table 2. Results obtained by Wiener filtration in the wavelet domain. The test images were 256 × 256 pixels and were degraded with white Gaussian additive noise.

Images	$\mathit{\sigma }$	Initial		Soft Truncation of Wavelet Coefficients with Optimal Threshold Values Wiener Filter in the Wavelet Domain with Estimation by Soft Truncation with Optimal Threshold Values		Wiener Filter in Wavelet Domain with Soft Truncation Estimation with Optimal Threshold Values Using Dyadic Wavelet Transformation		Wiener Filter in Wavelet Domain with Soft Truncation Estimation with Optimal Threshold Values Using Transformed Wavelet in Packets		Multiple Wiener Filter in Wavelet Domain with Soft Truncation Estimation with Optimal Threshold Values
		PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)
Lena	0.05	26.00	63.94	29.74	61.92	30.47	64.79	30.32	65.06	30.89	66.05
	0.1	20.07	49.02	26.14	45.06	26.98	47.99	26.89	47.34	27.52	49.40
Camera	0.05	26.24	65.72	29.98	62.46	30.89	63.28	30.64	63.46	31.30	64.82
	0.1	20.38	52.48	25.91	48.58	27.07	49.21	26.82	50.39	27.47	51.44
Goldhill	0.05	26.00	58.53	29.10	58.01	29.54	59.07	29.46	59.85	29.87	60.18
	0.1	20.11	41.02	25.79	38.27	26.35	38.27	26.36	39.78	26.73	42.46
Peppers	0.05	26.09	63.62	30.17	61.04	30.95	62.06	30.66	62.93	31.43	63.84
	0.1	20.15	49.85	26.45	46.96	27.41	48.00	27.14	48.53	27.98	50.50
Einstein	0.05	26.09	50.99	30.25	51.46	31.18	51.84	31.01	52.71	31.53	53.35
	0.1	20.05	38.87	26.51	37.14	27.82	37.33	27.72	37.87	28.14	39.78
USA Force	0.05	26.04	70.57	29.52	66.35	30.37	68.86	30.11	68.48	30.79	69.87
	0.1	20.23	55.64	25.78	52.45	26.74	54.84	26.60	54.79	27.21	56.78
Wheel	0.05	26.25	74.34	28.40	72.99	28.70	74.14	28.63	73.72	29.13	73.89
	0.1	20.37	58.15	24.55	57.59	25.11	59.45	25.04	59.72	25.59	60.53
House	0.05	26.05	63.83	31.18	64.24	32.19	66.20	32.17	66.30	32.67	68.05
	0.1	20.15	50.78	27.59	51.25	28.78	53.10	28.74	56.09	29.37	55.60
Boat	0.05	26.03	64.03	30.43	62.65	31.22	65.56	31.38	66.73	31.78	65.92
	0.1	20.06	46.57	26.75	47.01	27.63	44.97	27.79	47.41	28.18	50.65
Bridge	0.05	26.04	63.69	27.48	62.05	27.52	62.16	27.42	62.80	27.86	62.57
	0.1	20.19	47.49	23.39	45.48	23.69	44.22	23.64	45.82	23.97	46.62

highlight that the implementation of the Wiener filter in the wavelet domain by using the wavelet transformation in packets leads to poorer results in PSNR terms than the implementation by using the dyadic wavelet transformation but ensures better contour preservation.

The simulations confirm that the Wiener empirical wavelet filter leads to images with superior performance to those obtained by soft truncation of wavelet coefficients, thus being an effective tool for improving the quality of images obtained by truncating wavelet coefficients.

2.4. Multiple Wiener Filter in Wavelet Domain

The simulations performed showed that the performances obtained by using the Wiener filter in the wavelet domain, described in Section 2.3, depend on the pair of wavelet coefficients (Wiener filter estimate) used. At the same time, the use of the empirical Wiener filter in the wavelet domain leads to the improvement of the image quality obtained by soft truncation of the wavelet coefficients when it is used as an estimate both in terms of PSNR and in terms of contour conservation. However, there are cases when there is no improvement in contours. The quality of contours is especially important in the perception of images by the human visual system [43], so obtaining with the reduction of noise and an improvement in terms of contours and textures is an important goal in image processing.

The fact that not the same wavelet coefficients provide the best results for all images is explainable by the fact that different signals can best be represented by different wavelet bases, an idea that underlies both compression and reduction algorithms of noise and texture analysis. Likewise, different signals are poorly represented by different wavelet bases. From the analysis of the data presented in Table 1, it is possible to choose for the implementation of the Wiener filter such a wavelet base so that the signal is poorly represented with its help, in which case the performance of filtering and highlighting contours and textures will be poorer.

To avoid such a situation, we propose a new Wiener filter in the wavelet domain, the structure of which is shown in Figure 2. The same estimated solution, obtained by soft modification of the wavelet coefficients, is processed in parallel by several Wiener filters in the wavelet domain. Each filter being implemented using other wavelet coefficients, and the outputs of these filters are mediated. Results obtained using this type of filter show that the use of such a filter achieves a stronger improvement in image quality while improving contours and textures, compared to the case of the Wiener empirical wavelet filter.

In the implementation that was performed according to this scheme, only seven types of wavelet coefficients were used (Haar, Daubechies4, Daubechies6, Daubechies8, Daubechies12, Coifman6, and Coifman12) and dyadic wavelet transform were used. However, the simulations show that by using several types of wavelet coefficients the results obtained are better.

The fact that the Wiener filter implemented using the wavelet transform in packets leads to better results in terms of preserving contours and highlighting textures compared to the Wiener filter implemented using the dyadic wavelet transform, and the latter leads to better results in PSNR terms, suggests the idea of mediation of the images obtained through the two filters.

The results obtained by implementing such a filter, which we call the mediated Wiener filter, much more efficient in terms of computational effort than the multiple Wiener filter, are presented in

Table 3. Results obtained by using the Wiener mediated filter.

Images	$\mathit{\sigma }$	Initial		Soft Truncation of Wavelet Coefficients with Optimal Threshold Values		Wiener Filter in the Wavelet Domain Using the Dyadic Wavelet Transform		Wiener Filter in the Wavelet Domain Using Transformed Wavelet Packets		Wiener Mediated Filter
		PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)
Einstein	0.05	26.10	50.99	30.25	51.42	31.18	51.82	31.01	52.59	31.44	52.74
Goldhill	0.05	26.04	58.25	29.09	57.94	29.54	59.63	29.44	59.69	29.84	60.04
Aerial	0.05	20.07	70.83	28.15	68.82	28.41	68.90	28.41	62.89	28.75	69.85
Peppers	0.05	26.09	63.25	30.18	60.55	30.94	62.64	30.65	61.55	31.18	62.36

. It is observed that superior results are obtained compared to those obtained by using each of the two filters that form it.

As can be seen from the analysis of the data in Table 2 and Table 3, the mediated Wiener filter leads to results close to those obtained by using the multiple Wiener filter but requires a much smaller volume of calculations. This can be considered an example of the trade-off between the performance of the filtering process and the computational effort required.

At the output of the filter in Figure 2, for a better reconstruction of the contours and textures in the images, the result obtained after the Wiener mediated filter can be introduced in a nonlinear cellular network (NCN) algorithm.

NCN algorithms will contribute to the reconstruction of damaged contours by using the interpolation function if the values are known only to a part of the image elements which, in most cases, are arranged irregularly, and it is necessary to obtain the missing elements by calculation.

Basically, an observer who is not familiar with the original image does not even notice that the image has been restored. The output image will have the same size and resolution as the input image.

At the same time, by reconstructing the contours in the images it is possible to obtain special effects, removing parts unwanted, highlighting textures, or analyzing multiresolution objects in images.

In principle, there may be similarities in the way an image is processed in the case of NCN contour reconstruction and filtering. However, these two processing approaches are differentiated by the fact that in the case of restoring the contours of images with RNC the unknown portions may be larger in size, which generally do not contain any information, while in the case of noise reduction and preservation of the contours, the image pixels contain information that also includes additive noise.

2.5. Conjugated Use of the Invariant Wavelet Transform in Translation and Wiener Filtering in the Wavelet Domain to Improve the Image Quality Obtained by Soft Truncation of the Wavelet Coefficients

The obtained results show that both the use of the translation invariant wavelet transform and the Wiener filtering in the wavelet domain led to the improvement of the image quality obtained by soft truncation of the wavelet coefficients. The combined use of the two efficient filtering techniques in the wavelet domain is expected to lead to superior results of each. In this sense, we proposed and tested two schemes for combining the two methods. The first scheme (which we call scheme A) of the combined use of the two methods consists in using the image obtained by applying the soft truncation in the invariant version to translation as an estimate for the multiple Wiener filter. The second schematic scheme (which we call scheme B) consists in the implementation of the multiple Wiener filter in invariant version at translation. The results obtained are presented in

Table 4. Results obtained by the combined use of undecided wavelet transformation and Wiener filtration in the wavelet domain. Test images are 256 × 256 pixels. A total of 16 circular displacements were used to calculate the translation invariant wavelet transform.

Images	Initial		Soft Truncation with Optimal Threshold Values		Soft Truncation with Optimal Threshold Values, Invariant Version at Translation		Multiple Wiener Filter with Estimate by Soft Truncation with Optimal Threshold Values		Scheme A		Scheme B
	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)	PSNR (dB)	C (%)
Lena	26.00	63.94	29.74	61.96	30.70	64.12	30.91	66.05	31.13	66.32	31.22	66.81
Boat	26.00	63.28	30.38	61.83	31.66	64.56	31.74	64.79	32.05	66.24	32.09	66.19
House	26.05	63.91	31.21	61.59	32.45	64.95	32.69	65.83	32.98	67.19	33.03	66.95
Aerial	26.01	64.93	28.12	67.66	29.00	70.00	28.80	69.09	28.96	69.92	29.09	70.52
Camera	26.24	65.86	29.98	62.00	30.96	64.17	31.24	64.36	31.48	65.19	31.58	65.23

.

As can be seen from the data presented in Table 4, Scheme B ensures in all cases the best results, but this scheme also requires the largest number of calculations. At the same time, it is observed that in the case of rich detail images (Aerial), the use of the invariant wavelet transformation in translation leads, for an approximately equal number of calculations, to better results than the multiple Wiener filter. Instead, it ensures better results for smoother images.

2.6. Reconstruction of Contours from Images Using Nonlinear Cellular Network (NCN) Algorithm

In general, image processing methods can be described and implemented by differential equations with partial derivatives. Image processing applications that can be described as equations require a high computational power and a long period of calculation. One of the solutions to reduce computation time is the use of nonlinear cellular networks.

With the development of available hardware structures, the available computing power has increased and with it the interest in using differential equations in image processing and analysis is increasing. To illustrate the principle of applying differential equations in image processing, we will consider an image with gray levels ${\mathrm{I}}_0\left(\mathrm{x},\mathrm{y}\right)$, ${\mathrm{I}}_0:{\mathrm{R}}^2\to \mathrm{R}$. Processing this image, based on an algorithm based on a certain operator F, can be described by the following differential equation:

$$\frac{\partial \mathrm{I}}{\partial t}=F\left[\mathrm{I}\left(\mathrm{x},\mathrm{y},t\right)\right]$$

(23)

where t is an artificial parameter, and F is the characterizing operator/function desired processing algorithm, $F:{\mathrm{R}}^2\to \mathrm{R}$. In general, the operator/function F, depends on the image as well as its 1st and 2nd order spatial derivatives. The final image, processed, is obtained as a solution of this differential equation, with partial derivatives.

The solution of the differential equation that characterizes the desired processing can be obtained regarding this processing as a problem of variational calculation [44]. Thus, the final image follows minimizing a cost function of the form:

$$arg\{Mi{n}_{\mathrm{I}}\mathrm{E}(\mathrm{I})\}$$

(24)

where E is a function of energy that characterizes the image. Argument I for which at least the energy E is obtained, can also be obtained from the condition F(I) = 0, from Equation (24).

The formulation of processing an image using differential equations allows the description under a similar form of complex processing of the same image. For example, if two distinct processes are described by the equations:

$$\frac{\partial \mathrm{I}}{\partial t}=F_1\left[\mathrm{I}\left(\mathrm{x},\mathrm{y},t\right)\right] and \frac{\partial \mathrm{I}}{\partial t}=F_2\left[\mathrm{I}\left(\mathrm{x},\mathrm{y},t\right)\right]$$

(25)

A combined processing of the distinct processes can be described by combining the two equations, namely:

$$\frac{\partial \mathrm{I}}{\partial t}=F_1+\mathrm{\lambda }{F}_2$$

(26)

where $\mathrm{\lambda }\in {\mathrm{R}}^{+}$. If ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$ result from minimizing the energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_1$, the differential Equation (27) is solved by minimizing the energy ${\mathrm{E}}_1+\mathrm{\lambda }{\mathrm{E}}_2$. To minimize such energies in as short a time as possible, nonlinear cellular network can be used. For the use of cellular neural networks in minimizing energies in applications of processing images with grayscale, appropriate operators (templates) must be designed.

The proposed method of improving the contours in the images assumes that the corresponding transfer functions depend on the local, or sometimes regional, level of intensity or contrast.

When implementing the proposed method, we start from an energy function E, of general form:

$$\mathrm{E}\left(\mathrm{I},\mathrm{G}\right)=\iint \Vert \nabla {\mathrm{I}\Vert }^{2}d\mathrm{x}d\mathrm{y}+\mathrm{\lambda }\left|\mathrm{G}\right|$$

(27)

In image analysis and processing an important problem is the reconstruction of an original image Φ₀, from the acquired degraded image Φ (observed and degraded image). Two phenomena can contribute to image damage: one is related to the acquisition method (for example, the acquisition of a CT image by projection or blurring caused by motion), and the other is the inherent random noise that is associated with any useful signal. Consider an original grayscale image by ${\mathsf{\Phi }}_0\left(\mathrm{x},\mathrm{y}\right)$, ${\mathsf{\Phi }}_0\left(\mathrm{x},\mathrm{y}\right):{\mathrm{R}}^2$→ R, and $\mathsf{\Omega } =\left\{\left(\mathrm{x},\mathrm{y}\right):\mathrm{x}\in \left[1, \mathrm{M}\right],\mathrm{y}\in \left[1,\mathrm{N}\right], \mathrm{M} \mathrm{and} \mathrm{N}\in {\mathrm{R}}^{+}\right\}$. The processing of this image according to an operator-based algorithm can be described by the partial differential equation:

$$\frac{\partial \mathsf{\Omega }}{\partial t}=F\left[\mathsf{\Phi }\left(\mathrm{x},\mathrm{y},t\right)\right]$$

(28)

where an artificial parameter t was introduced, F being the operator that characterizes the desired processing algorithm ($\mathrm{F}:{\mathrm{R}}^2\to \mathrm{R}$). In general, the function F depends on the initial image, its first and second spatial derivatives. The final image, obtained after processing, results as a solution of this partial differential equation.

Applying the Dirichlet integral:

$$\mathrm{E}\left(\mathsf{\Phi }\right)=\int {\left|\nabla \mathsf{\Phi }\right|}^2\left(\mathrm{x}\right)d\mathrm{x}$$

(29)

which is associated with the equation:

$$\frac{\partial \mathsf{\Phi }}{\partial t}\left(\mathrm{x},t\right)=\nabla \mathsf{\Phi }\left(\mathrm{x}\right)$$

(30)

Equation (27) becomes:

$$\mathrm{E}\left(\mathrm{I},\mathrm{G}\right)=\iint \Vert \nabla {\mathrm{I}\Vert }^{2}d\mathrm{x}d\mathrm{y}+\mathrm{\lambda }\left|\mathrm{G}\right|=\underset{\mathrm{x},\mathrm{y}\in \mathsf{\Omega }}{\overset{\mathrm{M},\mathrm{N}\in {\mathrm{R}}^{+}}{{\displaystyle \iint }}}\left[{\left(\frac{\partial \mathrm{I}}{\partial \mathrm{x}}\right)}^2+{\left(\frac{\partial \mathrm{I}}{\partial \mathrm{y}}\right)}^2\right]d\mathrm{x}d\mathrm{y}+\mathrm{\lambda }\left|\mathrm{G}\right|$$

(31)

The energy function E has two terms: a term corresponding to the uniformity constraint and a term to penalize the discontinuities that, in general, correspond to the contours. Thus, the minimization of the energy E, implies the minimization of the two terms: the term that imposes a continuity in the plane of the image I and the term that favors or penalizes the discontinuities. The latter term is implemented, in most cases, by means of a binary field of lines G, and $\mathrm{\lambda }$ is a scalar parameter. Thus, depending on the chosen parameters, the energy function in Equation (32) formalizes a compromise between the uniformity of the image (favoring the elimination of noise from image) and an improvement and restoration of the contours (accentuate edges).

To be able to implement the procedure on a single-layer cellular neural network structure, in a first approximation, in relation (32) it is considered that the function G ensures the preservation of the similarity (fidelity) of the processed image with the initial image. In the initial design conditions, by minimizing the mediation term from energy E, it results in the template A, which includes only the operator A:

$$\mathrm{A}=\left(\begin{array}{ccc}0& 0.25& 0\\ 0.25& 0& 0.25\\ 0& 0.25& 0\end{array}\right)$$

(32)

In principle, the cost function corresponding to the penalty (accentuation) term of the edges can have a similar shape as the mediation term, obviously with a changed sign. Under the same design conditions, minimizing this component results in template B which includes only non-zero operator B (A = 0 and z = 0):

$$\mathrm{B}=\left(\begin{array}{ccc}0& -1& 0\\ -1& 4& -1\\ 0& -1& 0\end{array}\right)$$

(33)

Consequently, for the improvement of the algorithm with nonlinear cellular network of images based on variational calculation (relations (26), (27), (32)), the final AB template of the form results:

$$\mathrm{A}=\left(\begin{array}{ccc}0& 0.25& 0\\ 0.25& 0& 0.25\\ 0& 0.25& 0\end{array}\right) \mathrm{B}=\left(\begin{array}{ccc}0& -\mathrm{\lambda }& 0\\ -\mathrm{\lambda }& 4\mathrm{\lambda }& -\mathrm{\lambda }\\ 0& -\mathrm{\lambda }& 0\end{array}\right) \mathrm{z}=0$$

(34)

where λ is a scalar parameter that determines the ratio between uniformity and improvement contrast and contours, respectively. Uniformity is dominant for its subunit values λ, while for supraunitary values the improvement of the contrast predominates.

3. Results

The results obtained by simulations show that the use of this algorithm leads to an improvement of the filtered image quality both from the PSNR point of view and visually. This can also be seen from the data presented in Table 2. The results show that the implementation of the Wiener filter in the wavelet domain by using the wavelet transformation in packets leads to weaker results in PSNR terms than the implementation using the wavelet dyadic transformation but ensures better conservation of contours.

Another possible Wiener filter structure in the wavelet domain, based on the mediated Wiener filter, is shown in Figure 3. Due to its structure, we will call this structure iterated Wiener filter.

Table 5. Results obtained by using the iterated Wiener filter. The test images were 256 × 256 pixels.

Processing Stage	Wavelet Coefficients	Lena		Bridge
		PSNR (dB)	C (%)	PSNR (dB)	C (%)
Initial	-	26.04	62.97	26.12	66.41
Estimated solution by soft truncation	Daubechies 4	29.76	60.49	27.52	62.45
Stage 1	Coifman 6	30.84	63.92	27.80	63.77
Stage 2	Coifman 12	31.10	64.52	27.84	63.49
Stage 3	Daubechies 6	31.13	65.15	27.88	63.86
Stage 4	Daubechies 8	31.09	65.08	27.84	63.03
Stage 5	Daubechies 12	31.17	65.76	27.89	63.02

shows the results obtained by implementing the iterated Wiener filter. The gain of this filter in terms of improving the image quality evaluated in terms of PSNR is insignificant starting with the second stage of processing, but by performing several processing steps (iterations) is gained in terms of contour quality.

The simulations performed on many images highlighted the efficiency of implementing the Wiener filter in the transformed domain when this domain is the wavelet domain. If better performance is achieved by using the empirical Wiener filter in the wavelet domain due to the superior properties of the wavelet transformation in terms of energy compaction in as few coefficients as possible, the success of multiple, mediated, and iterated Wiener filters is due to the possibility that the domain wavelet provides it in terms of choosing the basis for signal representation.

It should be noted that in almost all images, the reduction of noise is also accompanied by an improvement of the contours of the processed image, given that usually the reduction of noise is accompanied by a degradation of the contours.

Wiener filters have the property of reducing noise in images and enhancing contours, being used in deconvolution applications to remove or reduce blurring.

Using together the proposed Wiener filter in the wavelet domain and a cellular neural network we obtain an adaptive improvement of the images and the restoration of the contours.

The method of improving the contours and textures of images resulting from processing with the Wiener filter proposed in the article (Figure 4), followed by the application of the NCN algorithm, using the AB template, improves the contours of images by equalizing and adapting the histogram. Additionally, the evaluation of the performance of the processing process presented above can be done if we evaluate the image resulting after edge detection, for an initial image without noise but with low contrast, the image obtained by global scaling and the image obtained after preprocessing with AB template (Figure 5 and Figure 6). Thus, it is found that edge detection is correct and robust only in the image obtained by global scaling and in the image obtained after preprocessing with the AB template.

In addition to the need to improve imaging, medical applications also need these applications to run in real time and this necessity was the basis for the design of the method described in the article, based on the modified Wiener filter and nonlinear cellular network.

The proposed method for improving images with modified Wiener filter and nonlinear cellular network can be applied with success also with low contrast CT (computer tomograph) images. The improvement of tomographic image quality aims at obtaining superior visibility of the image components, the adaptive increase of the contrast, in order to be interpreted as easily as possible by the specialist doctor. Figure 7, Figure 8 and Figure 9 show the results obtained in the case of such images.

As can be seen, the proposed method provides robust results in eliminating noise from images, as well as improving image contrast and contour detection. The method implemented has the advantage of a small computation time because the nonlinear cellular network used has a single layer, and the operators used are of size 3 × 3, which allows the calculations to be performed in a single step. The method can be extended to more complex processing that can be described by differential equations, considering the relations (26, 27) ÷ (32).

Following the simulations performed, it is found that both methods lead to significant improvements both in terms of the amount of noise removed from the image, but also in terms of the quality of the contours in the processed images. At the same time, the combined use of the two methods that are the subject of this article is tested, the results obtained being superior to those obtained by using each of them.

The interest in such filters is determined by the fact that much of the existing information in the images is provided by contours, and most of the noise removal methods cause them to fade. In the present case, at the cost of removing a smaller amount of noise in the contour-containing regions, it can be better preserved, as shown by the results shown in Figure 7, Figure 8 and Figure 9. There is thus a trade-off between the amount of noise removed and the quality of preserving the contours in the image.

It should be noted that in the results presented in Figure 8 and Figure 9, the noise reduction is accompanied by an improvement of the contours of the processed image, leading to a microscopic analysis, highlighting very well the contours of tumor formations from diseased tissues to healthy tissues. Such an estimate is useful for the medical act, especially for surgery. Additionally, the proposed method, in addition to the contour enhancement property, highlights the deconvolution applications for removing or reducing the blurring of energy areas in images by highlighting their vascularization elements (Figure 8 and Figure 9).

Imaging techniques are especially useful for interpreting biomedical CT/MRI images. The improvement of the quality of the tomographic images aims at obtaining a superior visibility of the image components, the adaptive increase of the contrast, to be interpreted as easily as possible by the specialist doctor. In the case of CT images, when the contrast is satisfactory ($\mathrm{\lambda }=0.05)$, it is necessary to preprocess the image using the Wiener filter proposed in the article and nonlinear cellular network. Thus, image analysis after edge detection in preprocessed images can be facilitated. Based on the presented results, the validity of the proposed method for improvement of medical images can be found by the simultaneous use of the modified Wiener filter described in the article and cellular neural networks, for concrete applications in CT medical imaging.

4. Conclusions

The article presents two powerful procedures for improving images obtained by soft truncation of wavelet coefficients correlated with nonlinear cellular network. The first method consists in using the invariant wavelet transform in translation, and the second in designing Wiener filters in the wavelet domain that use as an estimate the image obtained by soft truncation of the wavelet coefficients. For both methods, the theoretical substantiation, implementation, and testing are presented. Thus, both methods are characterized based on simulations performed on many images. New filters are proposed: one based on the use of the translation invariant wavelet transform, which mediates the results obtained by repeating this procedure for several sets of wavelet coefficients and method based on Wiener filtering in the wavelet domain. The Wiener filters in the wavelet domain were called multiple Wiener filter, mediated Wiener filter and iterated Wiener filter. Although close, the results of the methods from Wiener wavelet filters differ depending on the computational effort involved. The largest computational effort requires the multiple Wiener filter, and the lowest, the mediated Wiener filter.

The experimental results presented for testing nonlinear cellular network methods for image processing were obtained using the Matlab development environment. The proposed algorithm was tested, in a semiparallel variant, on Field Programmable Gate Array (FPGA) platform for digital emulation of the nonlinear cellular network, using FPGA Xilinx Series-7. The proposed variant, of semiparallel implementation, leads to obtaining an optimal ratio between the processing speed and the necessary hardware resources. Thus, the computation time depends only on the time constant of the hardware structure and the maximum number of steps of the algorithm (equal to the maximum displacement), being independent of the dimensions of the processed images (Figure 10).

The obtained results showed that the unit tact frequency is only to a small extent dependent on the size of the template image, practically the tact frequency of 400 MHz can be reached. The calculation time thus estimated may be a few orders of magnitude smaller than in the case of serial implementation.

Following the simulations performed, it was found that the proposed method leads to significant improvements both in terms of the amount of noise removed from the image, but also in terms of the quality of the contours in the filtered images.

Considering several circular displacements so that we have approximately the same computational complexity, the Wiener filter in the wavelet domain together with the AB template in the nonlinear cellular network leads to multiresolution results, superior to those obtained by using the translation invariant wavelet transform and of classical neural networks in the case of medical images and satellite imagery.