Decomposition of a signal (e.g., image or video) into multiple semantic components has been an effective research topic for various image/video processing applications, such as image/video denoising, enhancement, and inpainting. In this paper, we present a survey of signal decomposition frameworks based on the uses of sparsity and morphological diversity in signal mixtures and its applications in multimedia. First, we analyze existing MCA (morphological component analysis) based image decomposition frameworks with their applications and explore the potential limitations of these approaches for image denoising. Then, we discuss our recently proposed self-learning based image decomposition framework with its applications to several image/video denoising tasks, including single image rain streak removal, denoising, deblocking, joint super-resolution and deblocking for a highly compressed image/video. By advancing sparse representation and morphological diversity of image signals, the proposed framework first learns an over-complete dictionary from the high frequency part of an input image for reconstruction purposes. An unsupervised or supervised clustering technique is applied to the dictionary atoms for identifying the morphological component corresponding to the noise pattern of interest (e.g., rain streaks, blocking artifacts, or Gaussian noises). Different from prior learning-based approaches, our method does not need to collect training data in advance and no image priors are required. Our experimental results have confirmed the effectiveness and robustness of the proposed framework, which has been shown to outperform state-of-the-art approaches.