The advances in computer technologies and the Internet have made rapid strides and breakthroughs in the field of data communication, which nowadays, is easily accessed. Unfortunately, this easy access offers almost endless opportunities for pirating copyrighted and confidential medical imagery. Many methods are proposed in the literature, however, most of them lack in robustness and perceptibility, and are prone to attacks. Hence, this study proposes a robust image steganographic approach that combines Redundant Integer Wavelet Transform (RIWT), Discrete Wavelet Transforms (DCT) and Singular Value Decomposition (SVD) and the logistic chaotic map. RIWT being a shift invariant, reversibility and robustness were achieved in this proposed technique. Better level of imperceptibility was achieved using SVD and DCT, with embedding carried out on singular values. Extra security was provided using the logistic chaotic map for encryption of secret medical images, which also enhanced the robustness of the technique. The effectiveness of our proposed scheme was compared with similar schemes available in the literature using common parameters such as imperceptibility, robustness and resistance to several geometric transformation attacks. This technique proved superior to other existing methods. The UCID benchmarking database was used during validation. © 2019 Elsevier Ltd