| Detection of edges in piecewise smooth functions is important in many applications such as image processing, computer vision and seismology. The area has therefore attracted much attention over the past decades and several edge detection algorithms have been proposed. Unfortunately, these algorithms are mostly first order and depend on external factors such as the choice of appropriate filters and thresholds. Further, their use is mostly limited to digital images. An early attempt to make edge detection a high order venture from physical data was made by Archibald, Gelb and Yoon in 2005, where the polynomial annihilation edge detector was introduced. The method has several advantages over already existing methods, the most important being that it is applicable to multi-dimensional scattered data and that it is high order, so that the underlying function can contain more variation.;This work is an effort to bring together the high order perspective from the numerical partial differential equations community and the needs of edge detection, as required not only in the field of image processing, but also in a broad range of other applications. These include determining edges in the derivatives of functions for post processing partial differential equations and also determining discontinuitites in greater than three dimensions which arise in stochastic partial differential equations. Various ideas have been used to achieve this, a novel one being the use of Essentially Non-Oscillatory (ENO) techniques. ENO methods are popularly used for solving numerical partial differential equations. In this work, it is shown how the use of ENO helps expand the polynomial annihilation edge detection method for varied applications. ENO based techniques also look promising for adapting the method to noisy data on irregular grids. Efforts have been made to keep all the algorithms as free as possible of outside thresholding. |
