The recent years have witnessed rapid strides in the use of cloud computing and its countless applications. A cloud can contain massive volumes of multimedia data in the form of images, video and audio. Cloud computing platforms confront challenges in terms of data confidentiality, message integrity, user authentication and compression. Multimedia data needs plenty of storage capacity. Consequently, there is a need for multimedia data compression to reduce data size. Compression techniques are quite reliable, offering benefits to organizations dealing with metasized data in the cloud. Compressing large quanta of data leads to superior utilization of cloud storage. Compression techniques can compress data used for storage and transmission, yet compression alone is inadequate because multimedia data shared should, of necessity, be secure. Therefore, both multimedia compression and security are mandatory in the cloud. The chief goal of this paper is to propose a new framework, comprising multiscale transforms, public key cryptography and appropriate encoding techniques, that performs joint medical image compression and image encryption in the cloud. Multiscale transforms play a lead role in image compression, and the ones discussed in this paper include wavelet, bandelet, curvelet, ridgelet and contourlet transforms. Wavelet transforms offer robust localization both in terms of time and frequency domains. Bandelet transforms offer natural images geometric regularity to help improve the efficiency of representation. Curvelet transforms handle curve discontinuities well, with ridgelet transforms being the core idea behind curvelets. Contourlet transforms capture smooth contours and edges at any orientation. The Rivest-Shamir-Adleman (RSA) algorithm is used to encrypt images to provide maximum security when they are being transferred. Encoding techniques involved in this paper comprise the Embedded Zerotree Wavelet (EZW), Set Partitioning in Hierarchical Trees (SPIHT), Wavelet Difference Reduction (WDR), and Adaptively Scanned Wavelet Difference Reduction (ASWDR). Performance parameters such as peak signal to noise ratio (PSNR), mean square error (MSE), image quality index and structural similarity index (SSIM) are used for evaluation. It is justified that the proposed framework compresses images securely in the cloud.