Algorithmic Differentiation: Application to Variational Problems in Computer Vision Export

@article{citeulike:1475012, abstract = {Many vision problems can be formulated as minimization of appropriate energy functionals. These energy functionals are usually minimized, based on the calculus of variations (Euler-Lagrange equation). Once the Euler-Lagrange equation has been determined, it needs to be discretized in order to implement it on a digital computer. This is not a trivial task and, is moreover, error-prone. In this paper, we propose a flexible alternative. We discretize the energy functional and, subsequently, apply the mathematical concept of algorithmic differentiation to directly derive algorithms that implement the energy functional's derivatives. This approach has several advantages: First, the computed derivatives are exact with respect to the implementation of the energy functional. Second, it is basically straightforward to compute second-order derivatives and, thus, the Hessian matrix of the energy functional. Third, algorithmic differentiation is a process which can be automated. We demonstrate this novel approach on three representative vision problems (namely, denoising, segmentation, and stereo) and show that state-of-the-art results are obtained with little effort.}, author = {Pock, Thomas and Pock, Michael and Bischof, Horst}, citeulike-article-id = {1475012}, citeulike-linkout-0 = {http://dx.doi.org/10.1109/TPAMI.2007.1044}, citeulike-linkout-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4204161}, doi = {10.1109/TPAMI.2007.1044}, journal = {Pattern Analysis and Machine Intelligence, IEEE Transactions on}, keywords = {automatic-differentiation}, number = {7}, pages = {1180--1193}, posted-at = {2007-07-23 13:17:16}, priority = {2}, title = {Algorithmic Differentiation: Application to Variational Problems in Computer Vision}, url = {http://dx.doi.org/10.1109/TPAMI.2007.1044}, volume = {29}, year = {2007} }

TY - JOUR ID - citeulike:1475012 L3 - citeulike-article-id:1475012 N2 - Many vision problems can be formulated as minimization of appropriate energy functionals. These energy functionals are usually minimized, based on the calculus of variations (Euler-Lagrange equation). Once the Euler-Lagrange equation has been determined, it needs to be discretized in order to implement it on a digital computer. This is not a trivial task and, is moreover, error-prone. In this paper, we propose a flexible alternative. We discretize the energy functional and, subsequently, apply the mathematical concept of algorithmic differentiation to directly derive algorithms that implement the energy functional's derivatives. This approach has several advantages: First, the computed derivatives are exact with respect to the implementation of the energy functional. Second, it is basically straightforward to compute second-order derivatives and, thus, the Hessian matrix of the energy functional. Third, algorithmic differentiation is a process which can be automated. We demonstrate this novel approach on three representative vision problems (namely, denoising, segmentation, and stereo) and show that state-of-the-art results are obtained with little effort. IS - 7 JF - Pattern Analysis and Machine Intelligence, IEEE Transactions on EP - 1193 TI - Algorithmic Differentiation: Application to Variational Problems in Computer Vision VL - 29 SP - 1180 KW - automatic-differentiation AU - Pock, Thomas AU - Pock, Michael AU - Bischof, Horst PY - 2007/// UR - http://dx.doi.org/10.1109/TPAMI.2007.1044 ER -