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Image compression is a vital component for domains in which the computational resources are usually scarce such as automotive or telemedicine fields. Also, when discussing real-time systems, the large amount of data that must flow through the system can represent a bottleneck. Therefore, the storage of images, alongside the compression, transmission, and decompression procedures, becomes vital. In recent years, many compression techniques that only preserve the quality of the region of interest of an image have been developed, the other parts being either discarded or compressed with major quality loss. This paper proposes a study of relevant papers from the last decade which are focused on the selection of a region of interest of an image and on the compression techniques that can be applied to that area. To better highlight the novelty of the hybrid methods, classical state-of-the-art approaches are also analyzed. The current work will provide an overview of classical and hybrid compression methods alongside a categorization based on compression ratio and other quality factors such as mean-square error and peak signal-to-noise ratio, structural similarity index measure, and so on. This overview can help researchers to develop a better idea of what compression algorithms are used in certain domains and to find out if the presented performance parameters are of interest for the intended purpose. [ABSTRACT FROM AUTHOR]