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This paper presents a computational method to extract optimum rectangular Regions of Interest (RoI) in images with an associated saliency map. Although saliency maps provide an individual relevance measure for each pixel, to find the sub-image (i.e., rectangular region) that contains the set of the most relevant pixels requires an optimisation procedure to define the boundaries of the best RoI. This is achieved by the method devised in the paper, by following an approach based on balancing the amount of relevant information that is included and excluded from the RoI. The results show that such method is capable of finding the most relevant rectangular RoI and thus to extract the optimum sub-images according to the relevance measure given by a generic saliency map. Since the method is not tied to any particular type of images, it finds application in quite different fields, such as salient object extraction and processing in industry and surveillance, image compression using attention modelling, biomedical imaging, etc. [ABSTRACT FROM AUTHOR]