It depends on your application and the images you have to compress. If you have a specific type of images, you can develop specific algorithms with a higher compression rate (e.g., Hotelling). If you want to compress a broad sample of images, you should probably focus more your attention to generic algorithms (JPEG 2000 gives you a good palette of methods). The second part of your question is referring to no change to image visualization. You need to be more specific. Who will observe the decompressed images and for which task? If the task is to recognize some patterns moving on a background, you could accept to compress more the background with some visible differences compred to the original image, as soon as the pattern recognition is not affected. In such a situation, you can implement a lossy compression scheme. If you want to preserve any tiny gray level because your images will be used for a diagnostic and you do not know a priori what i suseful or not for this diagnostic, you will prefer implementing a lossless compression scheme. More recently, researchers are working on near-lossless compression algorithms that lead to unnoticeable differences in image visualization for human observers while some small gray level differences between original and decompressed images are allowed enabling higher compresion rates.Observers can be human observers or mathematical observers (e.g., quantification algorithms using decompressed images as input). For mathematical observers, what is important is that results obtained from decompresed images are identical or similar to results obtained from original images. In summary, the choice of an image compression algorithm depends from the images you are processing, from the observers who will read the images and from the task they will have to realize.
Jpeg2000 is a full defined standard algorithm for image compression. reference [1] showed this approach. it's compression and PSNR have optimum value. if you use compression for communication, you can use some reconstruction algorithm to achieve appropriated resolution in receiver (such as method of [2]).
[1] Christopoulos, Charilaos, Athanassios Skodras, and Touradj Ebrahimi. "The JPEG2000 still image coding system: an overview." Consumer Electronics, IEEE Transactions on 46.4 (2000): 1103-1127.
[2] J. Yang et al. Image super-resolution via sparse representation. IEEE Transactions on Image Processing, Vol 19, Issue 11, pp2861-2873, 2010