A tendency to increase the number of acquired remote sensing images and to make their average size larger has been observed. To manage such data, compression is needed, and lossy compression is often preferable. Since lossy compression introduces distortions, this results in worse classification and object detection. Therefore, lossy compression must be controlled, i.e., the introduced distortions must be under a certain limit. The distortions and the limit can be characterized by different metrics (quantitative criteria). Here, we consider the case of using the HaarPSI metric, which has a very high correlation with visual quality and human attention (saliency map), for three-channel optical band images compressed by the better portable graphics (BPG) encoder, one of the best modern compression techniques. We analyze a two-step procedure of providing a desired visual quality and show its peculiarities for the modes 4:4:4, 4:2:2, and 4:2:0 of image compression. We show how the HaarPSI metric relates to other known metrics of image visual quality and thresholds of distortion visibility. It is demonstrated that the two-step procedure provides about three times better accuracy in providing the desired visual quality compared to the fixed setting of parameter Q that controls compression for the BPG encoder. The provided accuracy is close to the reachable limit determined by the integer value setting of the Q parameter. We also briefly analyze the influence of compression on the classification accuracy of real-life remote sensing data.