Total Variation Model Weighted By Gaussian Curvature In Image Restoration

Image restoration, as an important branch of image processing field, has a great effect on the successive work of image analysis and computer vision, such as image segmentation, registration and classification, etc. Given an observed image, how to reconstruct a clear and clean image that can be correctly understood by a human operator or post-processed by other image analysis methods is still an open problem. Recently, many researchers have shown interests in partial differential equation (PDE) methods in image processing. Compared with other conventional approaches, PDE methods have remarkable advantages in both theories and implementation, and it can profoundly benefit from the mature tools of literature on numerical analysis and computational PDEs.This thesis mainly focuses on PDE methods based on nonlinear diffusion in image restoration. Motivated by the fact that Gauss curvature always keeps the value of zero within regions if any principal curvature is zero, we propose a total variance model concerning Gauss curvature for image restoration problem. Details are in the following: where, u₀( x ) is the observed image, and Gu 0 is the Gauss curvature of the surface z = u₀ ( x), g (0) = 0,and g (∞)â†'1,φ1 ( s ) = ( K² /2)ln(1 + ( s/K)²). The first term in E (u ) is a smoothing term; the second term measures the fidelity to the data. In other words, we search for u that best fits the data so that its Gauss curvature is low (so that noise will be removed). The parameterλis a positive weighted constant.Then obtained the Euler-Lagrange equation for computing the minimization problem of the energy function by variance method and then use the Rudin et. al. method that proposed the use of the artificial time marching to solve the Euler-Lagrange equation (chooseλ= 0): where N is the outward normal to ?? .Meanwhile, we proposed the semi-implicit finite differences scheme and proved the stability of the scheme:1. u₀ is the initial condition, i.e. the observed image . 2. Once uⁿ is calculated compute as a solution of the linear discrete equation:3. The boundary conditions: where, denotes the dual operator of , withWe compare the improved model with other nonlinear PDEs by using a volume-based analysis, which looks at the time change of the volume the volume the domain enclosed by the surface and the plan . The improved model can better preserve zero gauss curvature structures and large gradient edge, such as straight edges, curvy edges, corners, ramp and small-scaled features and edges. In the experiment, improved model obtains better result than PM model, so this model can be used for a variety of tasks in image processing and computer vision.