Partial differential equations (PDEs) and variational equipment have been brought into photo processing approximately fifteen years in the past. due to the fact that then, extensive learn has been conducted. The targets of this ebook are to offer a number of photo research functions, the best arithmetic concerned and the way to discretize them. therefore, this e-book is meant for 2 audiences. the 1st is the mathematical group via displaying the contribution of arithmetic to this area. it's also the party to focus on a few unsolved theoretical questions. the second one is the pc imaginative and prescient neighborhood via proposing a transparent, self-contained and international evaluation of the maths excited by snapshot procesing difficulties. This paintings will function an invaluable resource of reference and notion for fellow researchers in utilized arithmetic and laptop imaginative and prescient, in addition to being a foundation for complex classes inside those fields. through the 4 years because the booklet of the 1st version, there was significant development within the diversity of picture processing purposes lined by way of the PDE framework. the most ambitions of the second one version are to replace the 1st variation by way of giving a coherent account of a few of the hot demanding functions, and to replace the present fabric. furthermore, this publication presents the reader with the chance to make his personal simulations with a minimum attempt. To this finish, programming instruments are made to be had, on the way to enable the reader to enforce and try simply a few classical approaches.