Applied partial differential variational techniques

We review and present examples for two new image processing methods based on scale space evolution under partial differential operators. These operators arise from minimizing variational objective functions and from a novel approach that respects the discrete nature of real images. The first method considered is a variational approach that provides a closed-form expression for the optimal boundary function. This method gives good results with a 2-norm penalty term for approximation error and a 1-norm smoothness term; in this case there is a strong connection with total variation methods. The second method is a peer group averaging approach (developed by Hewer et al.) that smooths the image without blurring edges. This is accomplished by averaging over window pixels with nearly the same intensity; these pixels are the `peer group´ of the window´s central pixel. The direct correspondence between the object characteristics and the parameters of peer group size and window diameter make the parameter selection easier for peer group averaging than for standard variational methods. These methods are used in combination with PGA providing the initial approximation for the variational descent procedure. This is applied to real image examples and compared to existing methods including a new level-set histogram modification method (developed by Caselles et al. (see Hewlett Packard Technical Report HPL97-58, 1997)) that treats connected level-sets of pixels as individual units