On Second Order Operators and Quadratic Operators

In pattern recognition, computer vision, and image processing, many approaches are based on second order operators. Well-known examples are second order networks, the 3D structure tensor for motion estimation, and the Harris corner detector. A subset of second order operators are quadratic operators. It is lesser known that every second order operator can be written as a weighted quadratic operator. The contribution of this paper is to provide a constructive proof for this equivalence, i.e., we propose an algorithm for converting an arbitrary second order operator into a quadratic operator. We apply the method to several examples from image processing and machine learning. The advantages of the alternative implementation by quadratic operators is three-fold: first, the reduced data dependency leads to faster parallel algorithms suitable for GPU implementations. Second, the underlying linear operators allow new insights into the theory of the respective second order operators. Finally, replacing second order networks with sums of squares of linear networks reduces significantly the computational burden and the number of required learning samples.