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Modern industrial systems and robotic complexes use machine vision systems to increase productivity and automate processes. Deep learning systems are used to assist the operator and make decisions. The creation of industrial systems within the framework of "Industry 4.0" and "Industry 5.0" is impossible without precise control of the position of objects in space. Implementing ma-chine vision systems allows you to control the position of robot elements, objects, and humans. Information about objects and their constituent parts allows predicting their movement based on the accumulated knowledge and the embedded algorithm of the program. The analysis of the position of a material point in space and the forecast of its movement under the conditions of without apriori information about the process can be carried out both using neural networks (a large amount of knowledge is needed for training) and using computational methods (exact conditions are required). On the other hand, the fixed data is affected by noise. The appearance of noise is due to: not the ideality of the sensor (the CCD matrix does not consist of pure groups, but contains impurities in the composition); to the influence of external factors (electrical interference, inter-pixel inter-action, etc.); uneven accumulation, and drainage of the charge of the CCD of the matrix; destruction of the pixel; inaccuracy during subsequent digitization of data (quantization noise), etc. Methods of data accumulation and averaging, neural networks, or computational methods can be used to eliminate the noise component. The article discusses a data filtering method based on the analysis of vector spaces and the formation of an optimal solution according to the combined criterion that minimizes the distance between the maximum approximation, monotony, and minimum deviation of the difference in the norms of data vectors. The article contains 4 theorems and their proofs. In work, practical examples of processing data were obtained for various practical applications, including forecasting, searching for object boundaries in images, and data filtering. On a set of test sequences, the results obtained were compared with an example of data processing by approximation functions, which showed the coincidence of the results with an accuracy of 100%. For the case of the presence of a noise component, the accuracy is more than 94%, with a noise standard deviation of 25% of the signal power. It should be noted, that the proposed filtering method can reduce the effect of the noise component in the absence of a priori information about the belonging of a pure signal to any parametric class of functions. In the presence of the specified information, the approximation of the implementation by a function from the corresponding class may be more accurate.
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