An algorithm is described for simultaneously compressing and denoising images. The algorithm is called tree-adapted wavelet shrinkage and compression (TAWS-Comp). TAWS-Comp is a synthesis of an image compression algorithm, adaptively scanned wavelet difference reduction (ASWDR), and a denoising algorithm, tree-adapted wavelet shrinkage (TAWS). As a compression procedure, TAWS-Comp inherits all of the advantages of ASWDR: its ability to achieve a precise bit rate assigned before compressing, its scalability of decompression, and its capability for enhancing regions of interest. Such a full range of features has not been available with previous compressor-plus-denoiser algorithms. As a denoising procedure, TAWS-Comp is nearly as effective as TAWS alone. TAWS has been shown to have good performance, comparable to state of the art denoisers. In many cases, TAWS-Comp matches the performance of TAWS while simultaneously performing compression. TAWS-Comp is compared with other combined compressor/denoisers, in terms of error reduction and compressed bit rates. Simultaneous compression and denoising is needed when images are acquired from a noisy source and storage or transmission capacity is severely limited (as in some video coding applications). An application is described where the features of TAWS-Comp as both compressor and denoiser are exploited.