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

applied sciences

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

Catalin Dumitrescu 1 , Maria Simona Raboaca 2,3,4 and Raluca Andreea Felseghi 3,4,*

1 Department Telematics and Electronics for Transports, University “Politehnica” of Bucharest, 060042 Bucharest, Romania; [email protected] 2 ICSI Energy, National Research and Development Institute for Cryogenic and Isotopic Technologies, 240050 Ramnicu Valcea, Romania; [email protected] 3 Faculty of Electrical Engineering and Computer Science, “¸Stefancel Mare” University of Suceava, 720229 Suceava, Romania 4 Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania * Correspondence: [email protected]

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 Citation: Dumitrescu, C.; Raboaca, in assisted diagnosis, with the aim of providing physicians with information as useful as possible M.S.; Felseghi, R.A. Methods for in diagnosing diseases. Given that this information must be available in real time, we proposed Improving Image Quality for Contour a solution for reconstructing the contours in the images that uses a modified Wiener filter in the and Textures Analysis Using New Wavelet Methods. Appl. Sci. 2021, 11, wavelet domain and a nonlinear cellular network and that is useful both to improve the contrast of 3895. https://doi.org/10.3390/ its contours and to eliminate noise. In addition to the need to improve imaging, medical applications app11093895 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. Academic Editor: Athanasios Nikolaidis Keywords: wavelet adaptive methods; contours preservation; wavelet multiresolution analysis

Received: 30 March 2021 Accepted: 22 April 2021 Published: 25 April 2021 1. Introduction Norbert Wiener’s theory of optimal filtering of continuous signals is the basis of linear Publisher’s Note: MDPI stays neutral least squares linear error filters dependent on input data. Wiener studied in his 1949 with regard to jurisdictional claims in paper “Extrapolation, Interpolation and Smoothing of Stationary Time Series”, considering published maps and institutional affil- stationary signals, the problem of estimation in terms of the least squares error in the iations. 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, Copyright: © 2021 by the authors. echo cancellation, signal recovery, channel equalization, signal identification or additive Licensee MDPI, Basel, Switzerland. noise suppression, a number of studies focusing on their use in practical applications. The This article is an open access article coefficients of the Wiener filter are calculated so as to minimize the average square error distributed under the terms and between the filter output and the useful signal. Randy Gomez, Tatsuya Kawahara and conditions of the Creative Commons Kazuhrio Nakadai [2] are researching with the purpose of improving the late reflection, Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ noise power and the speech from a contaminated signal and improve it with a novel 4.0/).

Appl. Sci. 2021, 11, 3895. https://doi.org/10.3390/app11093895 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3895 2 of 23

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 Di- anat [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 Appl. Sci. 2021, 11, 3895 3 of 23

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. Appl. Sci. 2021, 11, 3895 4 of 23

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. Us- ing 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 Krzyzak,˙ 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 Appl. Sci. 2021, 11, 3895 5 of 23

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: Section2 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 Section3, 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 Section4.

2. Materials and Methods 2.1. Improving Image Quality by Wiener Empirical Filtering in the Wavelet Domain Consider a sequence of input data y = x + n, consisting of the sum of the samples N N of the useful signal x = {xi}i = 1 and the additive white Gaussian noise n = {ni}i = 1, both the useful signal x and the noise n having zero averages was considered (E(x) = 0, E(n) = 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 (xˆ = Gy),

2 Gwiener = arg min E kx − xˆk (1) G

The autocorrelation matrix Rxx of the useful signal, with Rnn that of the noise and with Ryy that of the input signal was noted:

T T T Rxx = E xx ,Rnn = E nn ,Ryy = E yy (2)

The useful signal x, and the noise n, are uncorrelated processes, therefore:

T T E xn = E nx = 0N (3)

In this case, the mean square error made in estimating the signal x will be: E kx − xˆk2 = E (xˆ − x)T(xˆ − x) (4)

which through further development will be able to be written: E kx − xˆk2 = E yTGTGy − 2E yTGTx + E kxk2 (5) Appl. Sci. 2021, 11, 3895 6 of 23

where G must be chosen so that the mean square error of estimation is minimal, the condition must be satisﬁed: 2 ∇GE kx − xˆk = 0N (6) that is, the gradient of the mean square estimation error relative to the matrix G is zero. This is how the relationship is reached:

T T 2Gwiener E yy − 2xy = 0N (7)

which leads further to:

T T T Gwiener E xx + nn − xx = 0N (8)

By introducing the autocorrelation matrices, we will have:

Gwiener(Rxx + Rnn) = Rxx (9)

or −1 Gwiener = Rxx(Rxx + Rnn) (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, Rxx and of the noise, Rnn. If the ﬁlter thus obtained is characterized by a diagonal matrix:

Gwiener = diag(g 0,0, g1,1, . . . , gN−1,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 Gwiener 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 auto- 2 correlation matrix can be expressed by Rnn = σ IN, σ are noise dispersion, and IN the N-order unit matrix. The matrix corresponding to the Wiener filter Gwiener = Gw is given by:

−1 2 Gw = Rxx Rxx + σ IN (12)

where Rxx is the autocorrelation matrix of the signal. The development of the matrix Rxx according to the eigenvalues and eigenvectors is given by:

Ns T T Rxx = UΛU = ∑ λkukuk (13) k = 1

where uk and λk are the eigenvectors and the eigenvalues of the matrix Rxx, NS < N, respectively, being the number of eigenvalues. The matrix U consists of the eigenvectors of

the matrix Rxx, being a matrix of dimensions N × Ns, U = [u1, u2, . . . , uNs ]. The matrix

Λ is a diagonal matrix containing eigenvalues, Λ = diag[λ1, λ2,..., λNs ]. The matrix U deﬁnes the Karhunen–Loewe transformation for the signal x and, as a result, decorates Appl. Sci. 2021, 11, 3895 7 of 23

the input data, respectively, the matrix Rxx, 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:

Ns T λk T Gw = UΛeU = u u (14) ∑ λ + 2 k k k = 1 k σ

where λ λ λ = 1 2 Ns Λe diag 2 , 2 ,..., 2 (15) λ1 + σ λ2 + σ λNs + σ This formulation indicates that the Wiener filter in the transformed domain first performs the Karhunen–Loewe transformation of the data by, UT then processes, using the optimal weighting matrix Λe, individually each coefficient. Finally, the data thus processed are returned to the primary space by 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 λ a first threshold Nλ < 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 Nλ weighting coefficients of the filter Hhard representing this operation are: 1, i = 1, 2, . . . , Nλ hhard(i) = (16) 0, i = Nλ + 1, Nλ + 2, . . . , N

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

θ2(i) hw = (17) θ2(i) + σ2

θ(i) being the wavelet coefficients corresponding to the useful signal. Assuming a perfect estimate of the estimated subspace (in particular Nλ = Ns), 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: MSE N h = λ N (18) MSEw τ ∑i = 1 hw

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 Figure1. The authors called this method WienerShrink. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 23

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 con- Appl. Sci. 2021, 11, 3895 8 of 23 tours, both in peak signal-to-noise ratio (PSNR) and visual terms. This can also be seen from the data presented in Table 1.

Figure 1. Scheme of the empirical Wiener filter in the wavelet domain proposed by S. P. Ghael, A. M. Sayeed and R. G. Figure 1. Scheme of the empirical Wiener filter in the wavelet domain proposed by S. P. Ghael, A. Baraniuk [40]. M. Sayeed and R. G. Baraniuk [40]. A particularity of this algorithm is the use of two different wavelet bases, one for In the caseobtaining of noise the reduction estimate andalgorithms, another the for problem designing is theto evaluate Wiener filter. the amount We implemented of this noise removed,type the of distortions filter using introduced both hard and and the soft loss truncation of information. of wavelet One such coefficients measure and semisoft can be the meantruncation square to error obtain (MSE) the estimatedbetween the solution pixel values [41]. Both in the the original results reportedimage and by S. P. Ghael, those in the remadeA. M. Sayeed image: and R. G. Baraniuk, and their own simulations, show that the use of this algorithm leads to an improvementM N of the filtered image quality but also to the preservation of contours, both in peak= 1 signal-to-noise()() ratio− () (PSNR)2 and visual terms. This can also be MSE  x i, j xˆ i, j (19) seen from the data presentedMN in== Table1. i 11j x()i, j whereTable 1.MDependence and N are on the performance image sizes; obtained by processing—pixel intensities with empirical in the Wiener original filter. image; () xˆ i, j Wavelet —pixelLena intensities in the restored image. PSNR (dB) C (%) Another measure of the similarity between the twoCoefficients images is the ratio between the Initialpeak value of the signal and Estimated the mean Solution square error (MSE), Daubechies defined 6 by: 30.03 63.67 ()Daubechiesk − 2 12 30.56 64.65 PSNR (dB) C (%) PSNR (dB) C= (%) 2 1 PSNR 10 log10 Coifman 6 30.49(20) 64.75 MSE 26.00 63.94 29.74 61.87 Coifman 12 30.56 64.65 where k is the bits number with which the pixel intensity is quantized. Wavelet PSNR isCamera a metric widely used both in the evaluation of image compressionPSNR (dB) algo- C (%) Coefficients rithms and in the evaluation of image noise reduction algorithms. Quantitative evaluation Initialof the quality of images proc Estimatedessed Solution based on MSE or Daubechiesother derived 6 metrics 30.39 does not pro- 63.11 vide sufficient information on the preservation of imageDaubechies contours. 12 Thus, images 30.97 with the 63.66 PSNR (dB) C (%) PSNR (dB) C (%) same PSNR may have different visual qualities. If theCoifman PSNR can 6 be used 30.91to evaluate the 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:

1 M N MSE = ∑ ∑ (x(i, j) − xˆ(i, j))2 (19) MN i = 1 j = 1 Appl. Sci. 2021, 11, 3895 9 of 23

where M and N are the image sizes; x(i, j)—pixel intensities in the original image; xˆ(i, j)— 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:

2 2k − 1 PSNR = 10 log (20) 10 MSE 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:

N C = c (21) N0

where Nc 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:

Fmerit = PSNR(dB) + 100 · 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 ¸siR. G. Baraniuk is shown in Figure1. Table1 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 σ = 0.05. The estimated solution was obtained using Daubechies coefficient 4. From the data presented in Table1 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 Table2 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. Appl. Sci. 2021, 11, 3895 10 of 23

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.

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

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 Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 23

Appl. Sci. 2021, 11, 3895 11 of 23 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 usetexture of such analysis. a filter Likewise,achieves a different stronger signalsimprovement are poorly in image represented quality while by different improving wavelet con- toursbases. and From textures, the analysis compared of to the the data case presented of the Wiener in Table empirical1, it iswavelet possible filter. to choose for the implementationIn the implementation of the that Wiener was filter performed such a according wavelet base to this so scheme, that the only signal seven is poorly types ofrepresented wavelet coefficients with its help, were in which used case(Haar, the Daubechies4, performance ofDaubec filteringhies6, and Daubechies8, highlighting Daubechies12,contours and textures Coifman6, will beand poorer. Coifman12) and dyadic wavelet transform were used. However,To avoid the suchsimulations a situation, show we that propose by usin a newg several Wiener types filter of in thewavelet wavelet coefficients domain, the resultsstructure obtained of which areis better. shown in Figure2. The same estimated solution, obtained by soft modificationThe fact ofthat the the wavelet Wiener coefficients, filter implemen is processedted using in parallel the wavelet by several transform Wiener in filterspackets in leadsthe wavelet to better domain. results Eachin terms filter of beingpreserving implemented contours using and highlighting other wavelet textures coefficients, compared and tothe the outputs Wiener of filter these implemented filters are mediated. using the Results dyadic obtained wavelet using transform, this type and of filterthe latter show leads that tothe better use of results such a in filter PSNR achieves terms, a strongersuggests improvement the idea of mediation in image qualityof the images while improving obtained throughcontours the and two textures, filters. compared to the case of the Wiener empirical wavelet filter.

Figure 2. TheThe proposed architecture for the multiple Wiener ﬁlterfilter in thethe waveletwavelet domain.domain.

TheIn the results implementation obtained by implementing that was performed such a filter, according which to we this call scheme, the mediated only Wiener seven filtertypes, much of wavelet more efficient coefficients in terms were of used computational (Haar, Daubechies4, effort than Daubechies6, the multiple Daubechies8,Wiener filter, areDaubechies12, presented in Coifman6, Table 3. andIt is Coifman12)observed that and superior dyadic wavelet results transformare obtained were compared used. How- to thoseever, theobtained simulations by using show each that of the by usingtwo filters several that types form of it. wavelet coefficients the results obtainedAs can are be better. seen from the analysis of the data in Tables 2 and 3, the mediated Wiener filter Theleads fact to results that the close Wiener to those filter obtained implemented by using using the multiple the wavelet Wiener transform filter but in requires packets aleads much to smaller better results volume in of terms calculations. of preserving This ca contoursn be considered and highlighting an example textures of the compared trade-off betweento the Wiener the performance filter implemented of the filtering using theproc dyadicess and wavelet the computational transform, and effort the required. latter leads to betterAt the results output in of PSNR the filter terms, in suggestsFigure 2, thefor a idea better of mediationreconstruction of the of the images contours obtained and texturesthrough in the the two images, filters. the result obtained after the Wiener mediated filter can be intro- ducedThe in resultsa nonlinear obtained cellular by network implementing (NCN) algorithm. such a filter, which we call the mediated Wiener filter,NCN much algorithms more efficient will incontribu terms ofte to computational the reconstruction effort thanof damaged the multiple contours Wiener by using filter, theare presentedinterpolation in Table function3. It is if observed the values that are superior known results only to are a obtainedpart of the compared image elements to those which,obtained in bymost using cases, each are of arranged the two filters irregularly, that form and it. it is necessary to obtain the missing elementsAs can by becalculation. seen from the analysis of the data in Tables2 and3, the mediated Wiener filterBasically, leads to results an observer close to who those is obtainednot familiar by using with the multipleoriginal image Wiener does filter not but even requires no- ticea much that smallerthe image volume has been of calculations. restored. The This outp canut be image considered will have an examplethe same of size the and trade-off reso- between the performance of the filtering process and the computational effort required. lution as the input image. At the output of the filter in Figure2, for a better reconstruction of the contours At the same time, by reconstructing the contours in the images it is possible to obtain and textures in the images, the result obtained after the Wiener mediated filter can be special effects, removing parts unwanted, highlighting textures, or analyzing multireso- introduced in a nonlinear cellular network (NCN) algorithm. lution objects in images. Appl. Sci. 2021, 11, 3895 12 of 23

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

Wiener Filter in Soft Truncation of Wiener Filter in the Wavelet Wavelet the Wavelet Domain Wiener Mediated Initial Coefﬁcients with Domain Using Using the Dyadic Filter Optimal Transformed Images σ Wavelet Threshold Values Wavelet Packets Transform PSNR C PSNR C PSNR C PSNR C PSNR C (dB) (%) (dB) (%) (dB) (%) (dB) (%) (dB) (%) 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

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 Table4. Appl. Sci. 2021, 11, 3895 13 of 23

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.

Multiple Wiener Soft Truncation Filter with Soft Truncation with Optimal Estimate by Soft Initial with Optimal Threshold Values, Scheme A Scheme B Truncation with Threshold Values Invariant Version Images Optimal at Translation Threshold Values PSNR C PSNR C PSNR C PSNR C PSNR C PSNR C (dB) (%) (dB) (%) (dB) (%) (dB) (%) (dB) (%) (dB) (%) 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 Table4, 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 I0(x, y), 2 I0 :R → R. Processing this image, based on an algorithm based on a certain operator F, can be described by the following differential equation:

∂I = F[I(x, y, t)] (23) ∂t where t is an artificial parameter, and F is the characterizing operator/function desired processing algorithm, F :R2 → 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{MinIE(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). Appl. Sci. 2021, 11, 3895 14 of 23

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

∂I ∂I = F [I(x, y, t)] and = F [I(x, y, t)] (25) ∂t 1 ∂t 2 A combined processing of the distinct processes can be described by combining the two equations, namely: ∂I = F + λF (26) ∂t 1 2 + where λ ∈ R . If I1 and I2 result from minimizing the energies E1 and E1, the differential Equation (27) is solved by minimizing the energy E1 + λE2. 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: 2 E(I, G) = x k∇Ik dxdy + λ|G| (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 use- 2 ful signal. Consider an original grayscale image by Φ0(x, y), Φ0(x, y) : R → R, and Ω = (x, y) : x ∈ [1, M], y ∈ [1, N], M and N ∈ R+ . The processing of this image according to an operator-based algorithm can be described by the partial differential equation:

∂Ω = F[Φ(x, y, t)] (28) ∂t where an artificial parameter t was introduced, F being the operator that characterizes the desired processing algorithm (F:R2 → 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: Z E(Φ) = |∇Φ|2(x)dx (29)

which is associated with the equation:

∂Φ (x, t) = ∇Φ(x) (30) ∂t Equation (27) becomes:

M,N∈R+" # ∂I 2 ∂I 2 E(I, G) = k∇Ik2dxdy + λ|G| = + dxdy + λ|G| (31) x x ∂x ∂y x,y∈Ω

The energy function E has two terms: a term corresponding to the uniformity con- straint 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 Appl. Sci. 2021, 11, 3895 15 of 23

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 λ 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:

 0 0.25 0  A =  0.25 0 0.25  (32) 0 0.25 0

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):

 0 −1 0  B =  −1 4 −1  (33) 0 −1 0

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:

 0 0.25 0   0 −λ 0  A =  0.25 0 0.25  B =  −λ 4λ −λ  z = 0 (34) 0 0.25 0 0 −λ 0

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 Table2. 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 Figure3. Due to its structure, we will call this structure iterated Wiener filter. Appl.Appl. Sci. Sci. 20212021,, 1111,, x 3895 FOR PEER REVIEW 1616 of of 23 23

FigureFigure 3. 3. ModifiedModified Wiener Wiener filter filter in in the the wavelet wavelet domain domain ( (schematicschematic ofof the the iterated iterated Wiener Wiener filter). filter).

WienerTable5 filtersshows have the resultsthe property obtained of reduci by implementingng noise in images the iterated and enhancing Wiener filter. contours, The beinggain ofused this in filter deconvolution in terms of applications improving to the remove image or quality reduce evaluated blurring. in terms of PSNR is insignificantUsing together starting the proposed with the secondWiener stagefilter in of the processing, wavelet domain but by performingand a cellular several neu- ralprocessing network stepswe obtain (iterations) an adaptive is gained improvement in terms of of contour the images quality. and the restoration of the contours. Table 5. Results obtained by using the iterated Wiener filter. The test images were 256 × 256 pixels. The method of improving the contours and textures of images resulting from processing with the Wiener filter proposed in theLena article (Figure 4), followed Bridge by the application Processingof the NCN algorithm,Wavelet using the AB template, improves the contours of images by Stage Coefficients PSNR C PSNR C equalizing and adapting the histogram.(dB) Additionally,(%) the evaluation(dB) of the performance(%) of the processing process presented above can be done if we evaluate the image resulting after Initial - 26.04 62.97 26.12 66.41 edge detection, for an initial image without noise but with low contrast, the image obtained by globalEstimated scaling and the image obtained after preprocessing with AB template (Figures 5 solution by soft Daubechies 4 29.76 60.49 27.52 62.45 and 6). Thus, it is found that edge detection is correct and robust only in the image obtained truncation by global scaling and in the image obtained after preprocessing with the AB template. InStage addition 1 to the Coifman need 6to improve 30.84 imaging, medical 63.92 applications 27.80 also need these 63.77 ap- plicationsStage to 2 run in Coifmanreal time 12 and this 31.10necessity was the 64.52 basis for the 27.84 design of the 63.49method describedStage in 3 the article, Daubechies based 6on the modified 31.13 Wiener 65.15 filter and nonlinear 27.88 cellular network 63.86 . TheStage proposed 4 Daubechiesmethod for 8 improving 31.09 images with 65.08 modified Wiener 27.84 filter and 63.03 nonlinear cellular network can be applied with success also with low contrast CT (computer tomo- Daubechies graph)Stage images. 5 The improvement of 31.17tomographic image 65.76 quality aims 27.89 at obtaining 63.02 supe- 12 rior visibility of the image components, the adaptive increase of the contrast, in order to be interpreted as easily as possible by the specialist doctor. Figures 7–9 show the results obtainedThe in simulations the case of performed such images. 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. Appl. Sci. 2021, 11, 3895 17 of 23

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 (Figure4), 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 (Figures5 and6). 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 Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 23 AB template.

Figure 4. ProcessingFigure 4. Processing a degraded a degraded image image with with Gaussian Gaussian additive additive white white noise noise with with the mu theltiple multiple Wiener filter Wiener in the filter wavelet in the wavelet domain. (a) Lena image,× 512 × 512 pixels, degraded,σ σ = 0.075, PSNR = 22.50 dB, C = 44.92%. (b) Image obtained by soft domain. (a)truncation Lena image, with 512optimal 512thresh pixels,old values, degraded, PSNR = 29.83= 0.075,dB, C = PSNR41.01%; =this 22.50 image dB, is the C =estimate 44.92%. for (Wienerb) Image filters. obtained (c) by soft truncation withImage optimal obtained thresholdby processing values, with the PSNR empirical = 29.83Wiener dB, filter C in = the 41.01%; wavelet thisdomain image having is as the estimate estimate the image for Wiener from filters. (c) Image obtainedpoint byb, PSNR processing = 30.08 dB, with C = the 41.67%. empirical (d) Image Wiener obtained filter by processing in the wavelet with the domain multiple havingWiener filter as estimatein the wavelet the image from domain having as estimate the image from point b, PSNR = 31.50 dB, C = 45.15%. point b, PSNR = 30.08 dB, C = 41.67%. (d) Image obtained by processing with the multiple Wiener filter in the wavelet domain having as estimate the image fromAs can point be seen, b, PSNR the proposed = 31.50 dB,method C = 45.15%.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.

Appl. Sci. 2021, 11, 3895 18 of 23 Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 23 Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 23

(a) Local histograms (b) Local histograms (a) Local histograms (b) Local histograms

(c) Local histograms (d) Local histograms (c) Local histograms (d) Local histograms Figure 5. Original image with noise: (a–d) local histograms. FigureFigure 5. 5.Original Original image image with with noise: noise: ( a(–ad–d)) local local histograms. histograms.

Figure 6. Image obtained using Wiener ﬁlter and cellular nonlinear network: (a,b) local histograms for λ = 1; (c) global Figure 6. Image obtained using Wiener filter and cellular nonlinear network: (a,b) local histograms for λ = 1; (c) global histogram.Figure 6. Image The effect obtained of preprocessing using Wiener an imagefilter and by applying cellular nonlinear the method network: described (a, aboveb) local can histograms be highlighted for λ by = 1; evaluating (c) global histogram. The effect of preprocessing an image by applying the method described above can be highlighted by evaluating histogram. The effect of preprocessing an image by applying the method described above can be highlighted by evaluating locallocal histogramshistograms from from the the samesame interestinterest areasareas ofof thethe initialinitial imageimage (Figure(Figure5 5),), respectively, respectively, from from the the image image obtained obtained by by using using local histograms from the same interest areas of the initial image (Figure 5), respectively, from the image obtained by using thethe ABAB templatetemplate forforλ λ= = 11 (Figure(Figure6 6).). Nonlinear Nonlinear cellular cellular network network processing processing based based on on the the described described method method achieves achieves a a local local the AB template for λ = 1 (Figure 6). Nonlinear cellular network processing based on the described method achieves a local balancingbalancing ofof thethe intensityintensity level.level. balancing of the intensity level. 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 Appl. Sci. 2021, 11, x FOR PEER REVIEW 19 of 23 Appl. Sci. 2021, 11, x FOR PEER REVIEW 19 of 23

Appl. Sci. 2021, 11, 3895 19 of 23 The interest in such filters is determined by the fact that much of the existing infor- mationThe in interest the images in such is provided filters is determinedby contours, by and the mostfact that of the much noise of removalthe existing methods information in the images is provided by contours, and most of the noise removal methods cause them to fade.tomograph) In the present images. case, The at improvementthe cost of removing of tomographic a smaller imageamount quality of noise aims at obtaining cause them to fade. In the present case, at the cost of removing a smaller amount of noise in the contour-containingsuperior visibility regions, of it the can image be better components, preserved, the adaptiveas shown increase by theof results the contrast, in order in the contour-containing regions, it can be better preserved, as shown by the results shown in Figuresto 7–9. be interpreted There is thus as easilya trade-off as possible between by the the amount specialist of doctor. noise removed Figures7 –and9 show the results shown in Figures 7–9. There is thus a trade-off between the amount of noise removed and the quality of preservingobtained inthe the contours case of in such the images. image. the quality of preserving the contours in the image.

(a) (b) (a) (b) Figure 7. (a) Initial image with low contrast; (b) improved image—CT cranial layer image pro- Figure 7. (acessedFigure) Initial with7. image (a) modified Initial with lowimage Wiener contrast; with filter. low (b) improvedcontrast; ( image—CTb) improved cranial image—CT layer image cranial processed layer image with modified pro- Wiener filter. cessed with modified Wiener filter.

(a) (a)

(b) (c) (b) (c) Figure 8. (aFigure) Initial 8.image (a) Initial with image low contrast; with low (b contrast;) improved (b) image—CTimproved image—CT cranial layer cranial image layer processed image with pro- modified Wiener Figure 8. (a) Initial image with low contrast; (b) improved image—CT cranial layer image pro- filter; (c) CTcessed cranial with layer modified image Wiener processed filter; with (c) CT the cranial combined layer algorithmimage processed between with the the modified combined Wiener algo- filter and the rithmcessed between with modified the modified Wiener Wiener filter; ( filterc) CT and cranial the layernonlinear image cellular processed network. with the combined algo- nonlinear cellularrithm between network. the modified Wiener filter and the nonlinear cellular network.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 23 Appl. Sci. 2021, 11, 3895 20 of 23

(a) (b)

Figure Figure9. (a) Initial 9. (a) image Initial imagewith low with contrast; low contrast; (b) improved (b) improved image—CT image—CT cranial cranial layer image layer image pro- processed cessed withwith thethe modiﬁedmodified WienerWiener ﬁlterfilter andand thethe nonlinearnonlinear cellular cellular network. network.

It shouldAs canbe noted be seen, that the in proposedthe results method presented provides in Figures robust 8 and results 9, the in eliminating noise reduction noise from is accompaniedimages, as by well an as improvement improving image of the contrast contours and contourof the processed detection. image, The method leading implemented to a microscopichas the analysis, advantage highlighting of a small computationvery well the time contours because of thetumor nonlinear formations cellular from network dis- used eased tissueshas a single to healthy layer, andtissues. the operatorsSuch an estimate used are is of useful size 3 for× 3,the which medical allows act, the especially calculations for to be surgery.performed Additionally, in a single the proposed step. The method, method canin addition be extended to the to contour more complex enhancement processing prop- that can erty, highlightsbe described the bydeconvolution differential equations, applications considering for removing the relations or reducing (26, 27) the÷ blurring(32). of energy areas inFollowing images by the highlighting simulations their performed, vascularization it is found elements that both (Figures methods 8 and lead 9). to significant Imagingimprovements techniques both are in especially terms of theuseful amount for interpreting of noise removed biomedical from CT/MRI the image, images. but also The improvementin terms of theof the quality quality of theof the contours tomogr inaphic the processedimages aims images. at obtaining At the a samesuperior time, the visibilitycombined of the image use of components, the two methods the adaptive that are increase the subject of the of contrast, this article to isbetested, interpreted the results as easilyobtained as possible being by superior the specialist to those doctor. obtained In the by usingcase of each CT ofimages, them. when the contrast is satisfactoryThe ( interest𝜆 = 0.05) in, such it is filtersnecessary is determined to preprocess by the th facte image that muchusing ofthe the Wiener existing filter informa- proposedtion in in the the article images and is provided nonlinear by cellular contours, network and. Thus, most ofimage the noise analysis removal after methodsedge de- cause tectionthem in preprocessed to fade. In the images present can case, be facilitated. at the cost Based of removing on the apresented smaller amount results, of the noise va- in the lidity contour-containingof the proposed method regions, for itimprovemen can be bettert of preserved, medical images as shown can by be the found results by shownthe in simultaneousFigures7 use–9. Thereof the ismodified thus a trade-off Wiener betweenfilter described the amount in the of article noise and removed cellular and neural the quality networks,of preserving for concrete the applications contours in thein CT image. medical imaging. It should be noted that in the results presented in Figures8 and9, the noise reduction 4. Conclusionsis 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 The article presents two powerful procedures for improving images obtained by soft diseased tissues to healthy tissues. Such an estimate is useful for the medical act, especially truncation of wavelet coefficients correlated with nonlinear cellular network. The first for surgery. Additionally, the proposed method, in addition to the contour enhancement methodproperty, consists highlights in using the the invariant deconvolution wavelet applications transform in for translation, removing orand reducing the second the blurringin designingof energy Wiener areas filters in imagesin the wavelet by highlighting domain that their use vascularization as an estimate elements the image (Figures obtained8 and 9). by soft truncationImaging of techniques the wavelet are coefficients especially.useful For both for methods, interpreting the biomedical theoretical CT/MRIsubstanti- images. ation, Theimplementation, improvement and of thetesting quality are ofpresented. the tomographic Thus, both images methods aims are at obtaining characterized a superior based visibilityon simulations of the imageperformed components, on many theimag adaptivees. New increase filters are of proposed: the contrast, one to based be interpreted on the useas of easily the translation as possible invariant by the specialist wavelet tr doctor.ansform, In which the case mediates of CT images, the results when obtained the contrast by repeatingis satisfactory this procedure (λ = 0.05 for) ,several it is necessary sets of wavelet to preprocess coefficients the image and method using the based Wiener on filter Wienerproposed filtering inin thethe articlewavelet and domain. nonlinear The cellular Wiener network. filters in Thus,the wavelet image domain analysis were after edge called detectionmultiple Wiener in preprocessed filter, mediated images Wiener can be filter facilitated. and iter Basedated Wiener on the presentedfilter. Although results, the close, validitythe results of of the the proposed methods method from Wiener for improvement wavelet filters of medical differ depending images can on be the found com- by the putationalsimultaneous effort involved. use of theThe modified largest computational Wiener filter described effort requires in the the article multiple and cellular Wiener neural filter, andnetworks, the lowest, for concrete the mediated applications Wiener in filter. CT medical imaging.

Appl. Sci. 2021, 11, 3895 21 of 23

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 Appl. Sci. 2021, 11, x FOR PEER REVIEWon simulations performed on many images. New filters are proposed: one based21 on of the23

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 WienerThe filteringexperimental in the results wavelet presented domain. for The testing Wiener nonlinear filters in cellular the wavelet network domain methods were forcalled image multiple processing Wiener were filter, obtained mediated using Wiener the Matlab filter development and iterated Wienerenvironment. filter. Although The pro- posedclose, algorithm the results was of thetested, methods in a semiparallel from Wiener variant, wavelet on Field filters Programmable differ depending Gate onArray the (FPGA)computational platform effort for digital involved. emulation The largestof the nonlinear computational cellular effort network, requires using the FPGA multiple Xil- inxWiener Series-7. filter, The and proposed the lowest, variant, the mediated of semipa Wienerrallel implementation, filter. leads to obtaining an The experimental results presented for testing nonlinear cellular network methods optimal ratio between the processing speed and the necessary hardware resources. Thus, for image processing were obtained using the Matlab development environment. The the computation time depends only on the time constant of the hardware structure and proposed algorithm was tested, in a semiparallel variant, on Field Programmable Gate the maximum number of steps of the algorithm (equal to the maximum displacement), Array (FPGA) platform for digital emulation of the nonlinear cellular network, using FPGA being independent of the dimensions of the processed images (Figure 10). Xilinx Series-7. The proposed variant, of semiparallel implementation, leads to obtaining The obtained results showed that the unit tact frequency is only to a small extent an optimal ratio between the processing speed and the necessary hardware resources. Thus, dependent on the size of the template image, practically the tact frequency of 400 MHz the computation time depends only on the time constant of the hardware structure and the can be reached. The calculation time thus estimated may be a few orders of magnitude maximum number of steps of the algorithm (equal to the maximum displacement), being smaller than in the case of serial implementation. independent of the dimensions of the processed images (Figure 10).

FigureFigure 10. 10. GeneralGeneral resource resource requirem requirementsents for for implementation. implementation.

FollowingThe obtained the simulations results showed performed, that the it unit was tact found frequency that the isproposed only to amethod small extentleads todependent significant on improvements the size of the templateboth in terms image, of practicallythe amount the of tact noise frequency removed of from 400 MHz the im- can age,be reached. but also Thein terms calculation of the timequality thus of estimatedthe contours may in be the a fewfiltered orders images. of magnitude smaller than in the case of serial implementation. Considering several circular displacements so that we have approximately the same Following the simulations performed, it was found that the proposed method leads to computational complexity, the Wiener filter in the wavelet domain together with the AB significant improvements both in terms of the amount of noise removed from the image, template in the nonlinear cellular network leads to multiresolution results, superior to those but also in terms of the quality of the contours in the filtered images. obtained by using the translation invariant wavelet transform and of classical neural net- Considering several circular displacements so that we have approximately the same works in the case of medical images and satellite imagery. 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 Author Contributions: Conceptualization, C.D. and M.S.R.; methodology, C.D. and M.S.R.; software, C.D.;those validation, obtained C.D., by using M.S.R. the and translation R.A.F.; formal invariant analysis, wavelet C.D.; investigation, transform and M.S.R. of classical and R.A.F.; neural resources,networks C.D., in M.S.R. the case and of R.A.F.; medical data images curation, and C.D., satellite M.S.R. imagery.and R.A.F.; writing—original draft preparation, M.S.R. and R.A.F.; writing—review and editing, C.D., M.S.R. and R.A.F.; visualization, C.D., M.S.R. and R.A.F.; supervision, C.D.; project administration, M.S.R.; funding acquisition, M.S.R. and R.A.F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was supported by a grant of the Romanian Ministry of Research and Innovation, CCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0776/No. 36 Appl. Sci. 2021, 11, 3895 22 of 23

Author Contributions: Conceptualization, C.D. and M.S.R.; methodology, C.D. and M.S.R.; software, C.D.; validation, C.D., M.S.R. and R.A.F.; formal analysis, C.D.; investigation, M.S.R. and R.A.F.; resources, C.D., M.S.R. and R.A.F.; data curation, C.D., M.S.R. and R.A.F.; writing—original draft preparation, M.S.R. and R.A.F.; writing—review and editing, C.D., M.S.R. and R.A.F.; visualization, C.D., M.S.R. and R.A.F.; supervision, C.D.; project administration, M.S.R.; funding acquisition, M.S.R. and R.A.F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was supported by a grant of the Romanian Ministry of Research and Innovation, CCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0776/No. 36 PC- CDI/15.03.2018, within PNCDI III. Acknowledgments to the National Center for Hydrogen and Fuel Cells (CNHPC)—Installations and Special Objectives of National Interest (IOSIN). Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Dumitrescu, C.; Răboacă, M.S.; Manta, I. Image Compression and Noise Reduction through Algorithms in Wavelet Domain. In Proceedings of the IEEE 26th International Symposium for Design and Technology in Electronic Packaging (SIITME) 2020, Pitesti, Romania, 21–24 October 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 215–220. [CrossRef] 2. Gomez, R.; Kawahara, T.; Nakadai, K. Optimized wavelet-domain filtering under noisy and reverberant conditions. APSIPA Trans. Signal Inf. Process. 2015, 4, 4. [CrossRef] 3. Ayat, S.; Manzuri-Shalmani, M.; Dianat, R. An improved wavelet-based speech enhancement by using speech signal features. Comput. Electr. Eng. 2006, 32, 411–425. [CrossRef] 4. Cohen, I. Noise spectrum estimation in adverse environments: Improved minima controlled recursive averaging. IEEE Trans. Speech Audio Process. 2003, 11, 466–475. [CrossRef] 5. Boll, S. Suppression of acoustic noise in speech using spectral subtraction. IEEE Trans. Acoust. Speech Signal Process. 1979, 27, 113–120. [CrossRef] 6. Babul, M.; Hamidul, T.; Monsur, R. Wiener Filter in Wavelet Domain for Mel-LPC based Noisy Speech Recognition. Int. J. Comput. Appl. 2018, 181, 1–4. [CrossRef] 7. Mallat, S. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 1989, 11, 674–693. [CrossRef] 8. Mihcak, M.K.; Kozintsev, I.; Ramchandran, K. Spatially adaptive statistical modeling of wavelet image coefficients and its application to denoising. In Proceedings of the 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), Phoenix, AZ, USA, 15–19 March 1999; IEEE: Piscataway, NJ, USA, 1999; Volume 6, pp. 3253–3256. 9. Moulin, P.; Liu, J. Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors. IEEE Trans. Inf. Theory 1999, 45, 909–919. [CrossRef] 10. Simoncelli, E.P.; Adelson, E.H. Noise removal via Bayesian wavelet coring. In Proceedings of the 3rd IEEE International Conference on Image Processing, Lausanne, Switzerland, 19 September 1996; IEEE: Piscataway, NJ, USA, 1996; Volume 1, pp. 379–382. [CrossRef] 11. Wainwright, M.; Simoncelli, E. Scale mixtures of Gaussians and the statistics of natural images. In Advances in Neural Information Processing Systems 12; MIT Press: Cambridge, MA, USA, 2000. 12. Kazubek, M. Wavelet domain image denoising by thresholding and Wiener filtering. IEEE Signal Process. Lett. 2003, 10, 324–326. [CrossRef] 13. Zhong, S.; Cherkassky, V. Image denoising using wavelet thresholding and model selection. In Proceedings of the 2000 International Conference on Image Processing (Cat. No. 00CH37101), Vancouver, BC, Canada, 10–13 September 2000; Volume 3, pp. 262–265. [CrossRef] 14. Donoho, D. De-noising by soft-thresholding. IEEE Trans. Inf. Theory 1995, 41, 613–627. [CrossRef] 15. Zhang, H.; Nosratinia, A.; Wells, R.O. Image denoising via wavelet-domain spatially adaptive FIR Wiener filtering. In Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 00CH37100), Istanbul, Turkey, 5–9 June 2000; IEEE: Piscataway, NJ, USA, 2000; Volume 4, pp. 2179–2182. [CrossRef] 16. Senel, H.; Peters, R.; Dawant, B. Topological median filters. IEEE Trans. Image Process. 2002, 11, 89–104. [CrossRef][PubMed] 17. Kumar, S.; Kumar, P.; Gupta, M.; Nagawat, A.K. Performance Comparison of Median and Wiener Filter in Image De-noising. Int. J. Comput. Appl. 2010, 12, 27–31. [CrossRef] Appl. Sci. 2021, 11, 3895 23 of 23

18. Pan, M.-S.; Tang, J.-T.; Yang, X.-L. A Modified Adaptive Median Filter Method and Its Applications in Medical Images. Biomed. Eng. Appl. Basis Commun. 2010, 22, 489–496. [CrossRef] 19. Fan, W.; Xiao, W. Image denoising based on wavelet thresholding and Wiener filtering in the wavelet domain. J. Eng. 2019, 2019, 6012–6015. [CrossRef] 20. Yi, S.-L.; Chen, Z.-C.; Ling, H.-L. Modified Wiener method in diffusion weighted image denoising. J. Central. South Univ. Technol. 2011, 18, 2001–2008. [CrossRef] 21. Hosseini, H.; Hessar, F.; Marvasti, F.; Marvasti, F. Real-Time Impulse Noise Suppression from Images Using an Efficient Weighted-Average Filtering. IEEE Signal Process. Lett. 2014, 22, 1050–1054. [CrossRef] 22. Wink, A.M.; Roerdink, J.B. Denoising Functional MR Images: A Comparison of Wavelet Denoising and Gaussian Smoothing. IEEE Trans. Med Imaging 2004, 23, 374–387. [CrossRef] 23. Ikeda, M.; Makino, R.; Imai, K. A new evaluation method for image noise reduction and usefulness of the spatially adaptive wavelet thresholding method for CT images. Australas. Phys. Eng. Sci. Med. 2012, 35, 475–483. [CrossRef] 24. Indulekha, N.R.; Sasikumar, M. Medical image denoising using three dimensional discrete wavelet transform and bilateral filter. Int. J. Manag. Appl. Sci. 2015, 1. 25. Zhou, Z.-F.; Shui, P.-L. Contourlet-based image denoising algorithm using directional windows. Electron. Lett. 2007, 43, 92. [CrossRef] 26. Pavithra, R.; Ramya, R.; Alaiyarasi, G. Wavelet Based Non Local Means Algorithm for Efficient Denoising of MRI Images. Int. J. Adv. Res. Comput. Commun. Eng. 2015, 4, 388–392. [CrossRef] 27. Satheesh, S.; Prasad, K. Medical Image Denoising Using Adaptive Threshold Based on Contourlet Transform. Adv. Comput. Int. J. 2011, 2, 52–58. [CrossRef] 28. Al Asadi, A.H.H. Contourlet transform based method for medical image denoising. J. AL-Qadisiyah Comput. Sci. Math. 2015, 7, 146–159. 29. Taujuddin, N.S.A.M.; Ibrahim, R. Enhancement of medical image compression by using threshold predicting wavelet-based algorithm. In Advanced Computer and Communication Engineering Technology; Springer: Cham, Switzerland, 2015; pp. 755–765. [CrossRef] 30. Amjad Ali, S.; Vathsal, S.; Lal Kishore, K. An efficient denoising technique for CT images using window based multi wavelet transformation and thresholding. Eur. J. Sci. Res. 2010, 2, 315–325. 31. Yang, G.; Ye, X.; Slabaugh, G.; Keegan, J.; Mohiaddin, R.; Firmin, D. Combined self-learning based single-image super-resolution and dual-tree complex wavelet transform denoising for medical images. In Medical Imaging 2016: Image Processing; International Society for Optics and Photonics: Bellingham, WA, USA, 2016; Volume 9784, p. 97840L. [CrossRef] 32. Yang, A.-F.; Lu, M.; Teng, S.-H.; Sun, J.-X. Local sparse representation for astronomical image denoising. J. Central. South Univ. 2013, 20, 2720–2727. [CrossRef] 33. Saxena, C.; Kourav, D. Noises and image denoising techniques: A brief survey. Int. J. Emerg. Technol. Adv. Eng. 2014, 4, 878–885. 34. Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image Quality Assessment: From Error Visibility to Structural Similarity. IEEE Trans. Image Process. 2004, 13, 600–612. [CrossRef][PubMed] 35. Shah, A.J.; Makwana, R.; Gupta, S.B. Single Image Super-Resolution via Non Sub-sample Contourlet Transform based Learning and a Gabor Prior. Int. J. Comput. Appl. 2013, 64, 32–38. [CrossRef] 36. Apotsos, A.; Raubenheimer, B.; Elgar, S.; Guza, R. Testing and calibrating parametric wave transformation models on natural beaches. Coast. Eng. 2008, 55, 224–235. [CrossRef] 37. Chen, G.; Bui, T.D.; Krzyzak, A. Image denoising with neighbour dependency and customized wavelet and threshold. Pattern Recognit. 2005, 38, 115–124. [CrossRef] 38. Liu, D.; Wen, B.; Liu, X.; Wang, Z.; Huang, T.S. When image denoising meets high-level vision tasks: A deep learning approach. arXiv 2017, arXiv:1706.04284. 39. Liu, J.; Liu, R.; Wang, Y.; Chen, J.; Yang, Y.; Ma, D. Image denoising searching similar blocks along edge directions. Signal Process. Image Commun. 2017, 57, 33–45. [CrossRef] 40. Ghael, S.; Sayeed, A.M.; Baraniuk, R.G. Improved wavelet denoising via empirical Wiener filtering. In Proceedings of the SPIE Technical Conference on Wavelet Applications in Signal Processing, San Diego, CA, USA, 27 July–1 August 1997. 41. Răboacă, M.S.; Dumitrescu, C.; Filote, C.; Manta, I. A New Adaptive Spatial Filtering Method in the Wavelet Domain for Medical Images. Appl. Sci. 2020, 10, 5693. [CrossRef] 42. Joshi, M.A.; Aravind, H.S. Analysis of Adaptive Wavelet Wiener Filtering for ECG Signals. Int. J. Adv. Res. Electron. Commun. Eng. 2014, 3, 395–398. 43. Chen, B.-Q.; Cui, J.-G.; Xu, Q.; Shu, T.; Liu, H.-L. Coupling denoising algorithm based on discrete wavelet transform and modified median filter for medical image. J. Central. South Univ. 2019, 26, 120–131. [CrossRef] 44. Raboaca, M.S.; Dumitrescu, C.; Manta, I. Aircraft Trajectory Tracking Using Radar Equipment with Fuzzy Logic Algorithm. Mathematics 2020, 8, 207. [CrossRef]