## 1 Introduction
It was demonstrated that SSDA could achieve performance comparable to the Bayes least squares with Gaussian scale mixture (BLS-GSM) method [41] and K-SVD [42]. A neural network framework named QIS-SPFT (QIS Serial-Parallel Fusion Transformer), which integrates CNN and transformer components, was employed to suppress Poisson (photon shot) noise during the QIS (quanta image sensor) imaging process was proposed in [43]. QIS-SPFT demonstrated superior performance in terms of PSNR compared to other filtering techniques, such as MLE, TD-BM3D, and QIS-Net. In this paper, we focus on reducing the effects of speckle noise and additive Gaussian noise on digital images in the frequency domain using a complex-valued neural network. Our idea is to perform image filtering by processing overlapping image fragments (patches) similarly how it was performed in [44; 45]. However, while in that work a neural network is used to process patches taken from the spatial domain, we will focus on processing patches represented in the frequency domain using a multilayer neural network with multi-valued neurons (MLMVN). The idea behind this processing can be considered somewhat similar to the idea behind the BM3D filter. This filter performs processing by grouping similar image fragments into 3D blocks and processing them as a single unit. The idea behind this is that if patches are similar, then the noise affecting them should also be similar and can be filtered in a similar way. In our case, we would like to employ MLMVN as an intelligent processor that performs frequency domain convolution of image patches with adaptive convolutional kernels generated from the learning process. The idea of using MLMVN as a framework for this task is based on the nature of image representation in the frequency domain, which relies on complex numbers. In turn, MLMVN is a type of neural network that shares the same architecture as the classical MLP but at the same time is based on multi-valued neurons (MVNs). MVNs, which are comprehensively described in [46], are artificial neurons whose weights, inputs, and outputs are complex numbers. This fact makes them a perfect processing unit for a vast number of problems where complex numbers are essential. This includes image processing in the frequency domain. MLMVN, along with its derivative-free backpropagation learning algorithm, was introduced in [47]. A number of studies have been conducted that build upon the concepts of MVNs and MLMVN [48; 49; 50; 51; 52]. Since their introduction, MVNs and MLMVN have demonstrated successful practical applications in various areas [53; 54; 55; 56; 57; 58; 59]. The structure of this paper is as follows. The second section describes the process of noise filtering based on the use of a set of overlapping patches in the frequency domain. The respective algorithm is described, and the process of finding convolutional filter kernels in the frequency domain using MLMVN is presented. Then the training set design is described. In Section 3, experimental results on filtering additive Gaussian noise and speckle noise using the MLMVN are presented. A detailed description of the learning process is provided, followed by a comparison of the filtering results obtained using our approach with those of some well-known popular filtering algorithms. Additionally, an analysis of the convolutional kernels generated from the learning process is conducted.