## 4 Discussion  
Existing studies on the use of neural networks in image filtering have proven the effectiveness of this approach. Thus, MLPs, CNNs, AEs, and other types of neural networks have been utilized for image denoising and restoration. These results can be explained by the ability of neural networks to model complex nonlinear functional relationships between inputs and outputs. In this paper, we considered the application of MLMVN as a framework for filtering additive Gaussian noise and multiplicative speckle noise in the frequency domain. The results obtained in our study demonstrate that MLMVN is effective for solving such tasks. MLMVN proved its efficiency in the development of frequency domain convolutional kernels generated from the learning process and processing overlapping fragments (patches) of images. This approach enables the use of a neural network with fewer neurons and parameters than, for example, in [35,44,45] and facilitates parallelism in programmatic implementation. Figure 9: (**a**) Ideal image. (**b**) Noisy image. Speckle noise \(0.3\sigma\). PSNR 27.8073 dB. (**c**) Image filtered using MLMVN. PSNR 32.2338 dB. (**d**) Image filtered using BM3D filter. SD parameter is \(1\sigma\). PSNR 29.1584 dB. (**e**) Ideal image. (**f**) Noisy image. Speckle noise \(0.3\sigma\). PSNR 27.7893 dB. (**g**) Image filtered using MLMVN. PSNR 31.4565 dB. (**h**) Image filtered using BM3D filter. SD parameter is \(1.5\sigma\). PSNR 31.1511 dB. (**i**) Ideal image. (**j**) Noisy image. Speckle noise \(0.3\sigma\). PSNR 25.929 dB. (**k**) Image filtered using MLMVN. PSNR 32.6948 dB. (**l**) Image filtered using BM3D filter. SD parameter is \(1\sigma\). PSNR 33.8770 dB. Our work also demonstrates the effectiveness of the batch learning algorithm in the training process. Noise filtering results obtained using MLMVN were presented and compared with other methods. The performance of MLMVN was found to be comparable to that of the BM3D, Lee, and Frost filters. Moreover, for some small detailed images our approach performed better because it better preserves small details. For additive Gaussian noise our approach works better for more heavily corrupted images. At the same time, our approach shows better results for speckle noise filtering than traditionally used filters. These findings indicate that MLMVN has potential for solving a wide range of image processing problems, especially those naturally suited to frequency domain analysis. We have also shown that multi-valued neurons may successfully employ the complex-valued logistic and the complex-valued hyperbolic tangent activation functions. This expands the MVNs' functionality and makes it possible to utilize them for input/output mappings whose outputs are not necessarily located on the unit circle. The results presented in this paper create a good background for further work on intelligent image processing in the frequency domain using MLMVN. For example, Poisson noise filtering and filtering of speckle noise with both multiplicative and additive components would be attractive subjects for further work. It would also be attractive and useful to discover possibilities of restoration of blurred images using a similar approach, that is, utilizing deconvolution in the frequency domain using MLMVN. Conceptualization, I.A. ; methodology, I.A. and Y.T. ; software, Y.T. ; validation, I.A. and Y.T. ; formal analysis, I.A. and Y.T. ; investigation, I.A. and Y.T. ; resources, I.A. ; data curation, I.A. and Y.T. ; writing--original draft preparation, I.A. and Y.T. ; writing--review and editing, I.A. and Y.T. ; visualization, Y.T. ; supervision, I.A. ; project administration, I.A. All authors have read and agreed to the published version of the manuscript. This research received no external funding.