The main goal of machine learning is the creation of self-learning algorithms in many areas of human activity. It allows a replacement of a person with artificial intelligence in seeking to expand production. The theory of artificial neural networks, which have already replaced humans in many problems, remains the most well-utilized branch of machine learning. Thus, one must select appropriate neural network architectures, data processing, and advanced applied mathematics tools. A common challenge for these networks is achieving the highest accuracy in a short time. This problem is solved by modifying networks and improving data pre-processing, where accuracy increases along with training time. Bt using optimization methods, one can improve the accuracy without increasing the time. In this review, we consider all existing optimization algorithms that meet in neural networks. We present modifications of optimization algorithms of the first, second, and information-geometric order, which are related to information geometry for Fisher–Rao and Bregman metrics. These optimizers have significantly influenced the development of neural networks through geometric and probabilistic tools. We present applications of all the given optimization algorithms, considering the types of neural networks. After that, we show ways to develop optimization algorithms in further research using modern neural networks. Fractional order, bilevel, and gradient-free optimizers can replace classical gradient-based optimizers. Such approaches are induced in graph, spiking, complex-valued, quantum, and wavelet neural networks. Besides pattern recognition, time series prediction, and object detection, there are many other applications in machine learning: quantum computations, partial differential, and integrodifferential equations, and stochastic processes.

Denoising and compression of 2D and 3D images are important problems in modern medical imaging systems. Discrete wavelet transform (DWT) is used to solve them in practice. We analyze the quantization noise effect in coefficients of DWT filters for 3D medical imaging in this paper. The method for wavelet filters coefficients quantizing is proposed, which allows minimizing resources in hardware implementation by simplifying rounding operations. We develop the method for estimating the maximum error of 3D grayscale and color images DWT with various bits per color (BPC). The dependence of the peak signal-to-noise ratio (PSNR) of the images processing result on wavelet used, the effective bit-width of filters coefficients and BPC is revealed. We derive formulas for determining the minimum bit-width of wavelet filters coefficients that provide a high (PSNR ≥ 40 dB for images with 8 BPC, for example) and maximum (PSNR = ∞ dB) quality of 3D medical imaging by DWT depending on wavelet used. The experiments of 3D tomographic images processing confirmed the accuracy of theoretical analysis. All data are presented in the fixed-point format in the proposed method of 3D medical images DWT. It is making possible efficient, from the point of view of hardware and time resources, the implementation for image denoising and compression on modern devices such as field-programmable gate arrays and application-specific integrated circuits.

The residue number system (RNS) is an unconventional number system which can lead to parallel and fault-tolerant arithmetic operations. However, the complexity of residue-to-binary conversion for large number of moduli reduces the overall RNS performance, and makes it inefficient for nowadays high-performance computation systems. In this paper, we present an improved approximate Chinese remainder theorem (CRT) with the aim of performing efficient residue-to-binary conversion for general RNS moduli sets. To achieve this aim, the required number of fraction bits for accurate residue-to-binary conversion is derived. Besides, a method is proposed to substitute fractional calculations by similar computations based on integer numbers to have a hardware amenable algorithm. The proposed approach results in high-speed and low-area residue-to-binary converters for general RNS moduli sets. Therefore, with this conversion method, high dynamic range residue number systems suitable for cryptography and digital signal processing can be designed.

In this paper, we analyze the noise quantization effects in coefficients of discrete wavelet transform (DWT) filter banks for image processing. We propose the implementation of the DWT method, making it possible to determine the effective bit-width of the filter banks coefficients at which the quantization noise does not significantly affect the image processing results according to the peak signal-to-noise ratio (PSNR). The dependence between the PSNR of the DWT image quality on the wavelet and the bit-width of the wavelet filter coefficients is analyzed. The formulas for determining the minimal bit-width of the filter coefficients at which the processed image achieves high quality (PSNR ≥ 40 dB) are given. The obtained theoretical results were confirmed through the simulation of DWT for a test image using the calculated bit-width values. All considered algorithms operate with fixed-point numbers, which simplifies their hardware implementation on modern devices: field-programmable gate array (FPGA), application-specific integrated circuit (ASIC), etc.

Implants are now the standard method of replacing missing or damaged teeth. Despite the improving technologies for the manufacture of implants and the introduction of new protocols for diagnosing, planning, and performing implant placement operations, the percentage of complications in the early postoperative period remains quite high. In this regard, there is a need to develop new methods for preliminary assessment of the patient’s condition to predict the success of single implant survival. The intensive development of artificial intelligence technologies and the increase in the amount of digital information that is available for analysis make it relevant to develop systems based on neural networks for auxiliary diagnostics and forecasting. Systems based on artificial intelligence in the field of dental implantology can become one of the methods for forming a second opinion based on mathematical decision making and forecasting. The actual clinical evaluation of a particular case and further treatment are carried out by the dentist, and AI-based systems can become an integral part of additional diagnostics. The article proposes an artificial intelligence system for analyzing various patient statistics to predict the success of single implant survival. As the topology of the neural network, the most optimal linear neural network architectures were developed. The one-hot encoding method was used as a preprocessing method for statistical data. The novelty of the proposed system lies in the developed optimal neural network architecture designed to recognize the collected and digitized database of various patient factors based on the description of the case histories. The accuracy of recognition of statistical factors of patients for predicting the success of single implants in the proposed system was 94.48%. The proposed neural network system makes it possible to achieve higher recognition accuracy than similar neural network prediction systems due to the analysis of a large number of statistical factors of patients. The use of the proposed system based on artificial intelligence will allow the implantologist to pay attention to the insignificant factors affecting the quality of the installation and the further survival of the implant, and reduce the percentage of complications at all stages of treatment. However, the developed system is not a medical device and cannot independently diagnose patients. At this point, the neural network system for analyzing the statistical factors of patients can predict a positive or negative outcome of a single dental implant operation and cannot be used as a full-fledged tool for supporting medical decision-making.