This paper suggests a novel image compression scheme, using the discrete wavelet transformation (DWT) and the fuzzy c-means clustering technique. The goal is to achieve higher compression rates by applying different compression thresholds for the wavelet coefficients of each DWT band, in terms of how they are clustered according to their absolute values. This methodology is compared to another one based on preserving texturally important image characteristics, by dividing images into regions of textural significance, employing textural descriptors as criteria and fuzzy clustering methodologies. These descriptors include cooccurrence matrices based measures. While rival image compression methodologies utilizing the DWT apply it to the whole original image, the herein presented novel approaches involve a more sophisticated scheme. That is, different compression ratios are applied to the wavelet coefficients belonging in the different regions of interest, in which either each wavelet domain band of the transformed image or the image itself is clustered, respectively. Regarding the first method, its reconstruction process involves using the inverse DWT on the remaining wavelet coefficients. Concerning the second method, its reconstruction process involves linear combination of the reconstructed regions of interest. An experimental study is conducted to qualitatively assessing both approaches in comparison with the original DWT compression technique, when applied to a set of medical images.