Executive Summary : | The Total Variation (TV)-q norms are widely used in scientific applications such as signal and image restoration.These are L^q norms with fractional norm values. Fractional regularization approaches are used in various scientific applications to solve ill-posed problems. Most fractional values used are in the range [1-2]. The value beyond 2 results in over smoothing the results and wiping out the detailed structures present in the data as the non-smooth functions are approximated with smooth continuous functions. On the other hand the values close to 1 preserves the non-smooth variations in the data but tens to perform piecewise approximation resulting in staircase like effect in 1D signals and patch like effects in images. However, the q values near to one promotes sparsity in the output leading to reduced dimensions of the output data. The reduced dimensions are useful in many applications such as compressed sensing where the data dimension is expected to be considerably smaller. However, the piecewise approximation leads to visual discrepancies in the output and causes misinterpretations of the data during the analysis phase. Many a times these patch-like structures are mistaken as edges during the analysis phase. The local structure of the variational regularization models are one of the major reasons for this patch like effect in the output. This can considerably be solved with the help of non-local variational models. Further the q-values smaller than 1 are not studies rigorously to analyze their application potential in the scientific world. Though they produce considerably sparse solutions, they are not real norms unlike the q values which are greater than 1. They form good candidate norms for sparse solutions which are useful in various scientific applications. However, since they do not possess the properties of norms they are difficult to be analyzed theoretically. In this study we intent to explore the possibility of exploiting the fractional norms specifically the ones for which the q values are smaller than 1, for possible usage in various applications. The piecewise approximation of the input needs to be addressed using the non-local variational framework. Further, the restoration capacity of the model shall be improved by using the retinex theory while designing the model. Various noise distributions of the input needs to be analyzed adaptively and the model needs to fine tune to address such discrepancies. The model thus designed can cater to various image restoration requirements in various imaging applications. A deep learning based approach will be employed to design the proposed variational framework. |