It has been known for many years that images have sparse or almost sparse representations in some transform domains such as wavelets. In fact, this is the main idea behind modern image-compression techniques. The benefits of having sparse representations are not limited to compression; they also play an important role for denoising and deconvolution. The delicate point here is to acknowledge that the degree of sparsity of the representations is a property defined for the digital images, which are samples of a continuous-space object. As we increase the imaging resolution, we observe that the digital images become sparser and sparser. This fact suggests that the sparsity or compressibility of digital images is a characteristic inherited from the continuous-space object. However, it is not trivial to generalize the sparsity/compressibility concepts to models defined in the the continuous world. The aim of this research is to develop a stochastic framework for continuous-space signals that, in some sense, exhibit a sparse/compressible structure. The new framework is expected to better represent the sparse nature of the images. This might also lead to the discovery of new tools in image processing. |
