North-Holland Nonlinear total variation based noise removal algorithms*

A constrained optimization type of numerical algorithm for removing no sefrom images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradient-projection method. This amounts to olving a time dependent partial differential equation on manifold determined by the constraints. As t--- ~ 0o the solution converges to a steady state which is t e denoised image. The numerical algorithm is simple and relatively fast. The results appearto be state-of-the-art forvery noisy images. The method is n ninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set. 1. In t roduct ion The presence of noise in images is unavoid-able. It may be introduced by the image forma-tion process, image recording, image transmis-sion, etc. These random distortions make it dif-ficult to perform any required picture processing. For example, the feature oriented enhancement introduced in refs. [6,7] is very effective in re-storing blurry images, but it can be "frozen " by an oscillatory noise component. Even a small amount of noise is harmful when high accuracy is required, e.g. as in subcell (subpixel) image analysis. In practice, to estimate a true signal in noise, the most frequently used methods are based on the least squares criteria. The rationale comes from the statistical argument that the least squares estimation is the best over an entir