The medical diagnosis is strongly based on different medical imaging modalities. Consequently, the amount of digital image data to be stored or transmitted is huge. Hence, efficient and low-complexity compression becomes necessary. Medical image compression "intuitively" requires ensuring the quality of the diagnostic image while reducing the bit rate. Regarding the modalities, ultrasound is one of the most widely used one. However, it has a drawback of visual quality caused by the speckle noise. This work aims to achieve speckle noise reduction together with lossy compression in ultrasound medical images, where the losses are mainly the undesirable speckle noise. Furthermore, the proposed compression scheme is efficiently applied to Magnetic Resonance Images(MRI). In this thesis we propose three main contributions: a multi-output filter for speckle reduction of medical ultrasound images, an image quality metric for medical context (ultrasound and MRI) and a scheme of compression for medical images. We perform a study on the usability of opinion-unaware no-reference natural image quality metrics in the context of medical images: filtered ultrasound and compressed Magnetic Resonance Images (MRI), as result of the study we propose a metric adapted to medical images. Then, we propose a multi-output speckle reduction filter based on MMD (MOF-MMD), which makes it possible to obtain three filtered images that help the radiologist’s diagnostic needs. Indeed, the diagnostic task is based on different characteristics of the image. Since the end-user of the medical images is the radiologist, we believe that it is more reasonable to evaluate the algorithm using human observers. Thus, we conduct a subjective experiment involving three radiologists with different years of experience. The correlations between the subjective scores and the objective metrics outputs and the impact of the radiologists years of experience are presented and analyzed. The last contribution consists on developing a lossy coding scheme based on the MMD applied to ultrasound and MR Images.