The aim of the paper is to provide a recent overview of Kalman filter employment in the non-accelerating inflation rate of unemployment (NAIRU) estimation. The NAIRU plays a key part in an economic system. A certain unemployment rate which is consistent with a stable rate of inflation is one of the conditions for economic system stability. Since the NAIRU cannot be directly observed and measured, it is one of the most fitting problems for the Kalman filter application. The search for original, NAIRU focused and Kalman filter employment studies was performed in three scientific databases: Web of Science, Scopus, and ScienceDirect. A sample of 152 papers was narrowed down to 25 studies, which were described in greater detail regarding the focus, methods, model features, limitations, and other characteristics. A group of studies using a purely statistical approach of decomposing unemployment into a trend and cyclical component was identified. The next group uses the reduced-form approach which is sometimes combined with statistical decomposition. In such cases, the models are usually based on the backward-looking Phillips curve. Nevertheless, the forward-looking, New Keynesian or rarely hybrid New Keynesian variant can also be encountered.

The Kalman filter is a classical algorithm of estimation and control theory. Its use in image processing is not very well known as it is not its typical application area. The paper deals with the presentation and demonstration of selected possibilities of using the Kalman filter in image processing. Particular attention is paid to problems of image noise filtering and blurred image restoration. The contribution presents the reduced update Kalman filter algorithm, that can be used to solve both the tasks. The construction of the image model, which is the necessary first step prior to the application of the algorithm itself, is briefly mentioned too. The described procedures are then implemented in the MATLAB software and the results are presented and discussed in the paper.

Kalman filters are a set of algorithms based on the idea of a filter described by Rudolf Emil Kalman in 1960. Kalman filters are used in various application domains, including localization, object tracking, and navigation. The text provides an overview and discussion of the possibilities of using Kalman filters in indoor localization. The problems of static localization and localization of dynamically moving objects are investigated, and corresponding stochastic models are created. Three algorithms for static localization and one algorithm for dynamic localization are described and demonstrated. All algorithms are implemented in the MATLAB software, and then their performance is tested on Bluetooth Low Energy data from a real indoor environment. The results show that by using Kalman filters, the mean localization error of two meters can be achieved, which is one meter less than in the case of using the standard fingerprinting technique. In general, the presented principles of Kalman filters are applicable in connection with various technologies and data of various nature.

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