Institut für Informatik III der Rheinischen Friedrich-Wilhelms-Universität Bonn On Local Region Models and the Statistical Interpretation of the Piecewise Smooth Mumford-Shah Functional

This report provides additional results on local region statistics and accompanies the work in [2]. In particular, we derive the Euler-Lagrange equations of the local Gaussian region model including variance as well as the local nonparametric model. Moreover, we show some experimental results on contour tracking with local region statistics. 1 Euler-Lagrange Equations for Local Region Statistics Here we show that for energies based on local region statistics, in contrast to their first-order approximations, one can compute precise shape gradients. In order to minimize the Mumford-Shah functional, one usually alternates the optimization of the smooth region approximations given a preliminary partitioning and, vice-versa, the optimization of the partitioning given the smooth approximations. In fact, this is not a gradient descent but a coordinate descent. In case of local region statistics, the smooth region approximations are available in an analytic form by means of convolution expressions. Thus, their dependence on the partitioning can be considered when computing the Euler-Lagrange equations of the functional. This is not possible for the Mumford-Shah energy, since here the smooth approximation can only be computed numerically. Considering the dependence of the statistics on the partitioning, generally leads to additional secondary terms that respond to the changes of the distributions by moving points from one region to another. Precise shape gradients including secondary terms are not unusual, though they are rarely implemented. A general work on this topic is [1]. In [6] secondary terms are computed for a homogeneous model with nonparametric densities, in [5] the secondary terms are given for the homogeneous Laplace distribution. In the latter work, it is shown empirically that these secondary terms have very little effect in case of the homogeneous Laplace distribution and can be neglected. In [9] it was shown that in case of the homogeneous Gaussian region model the secondary terms in fact turn out to be zero. In [7] the secondary terms for a local Gaussian distribution with fixed variance are derived, and they are brought in a form that allows for an efficient implementation. In case of local region models, implementation of these terms can have positive effects regarding the convergence of the energy [8]. In the following, we derive the Euler-Lagrange equations for the local Gaussian model including variance as well as the local nonparametric model. Contours are represented as zero-level lines of a number of level set functions Φi : Ω→ R. The number of regions N is assumed to be fixed. The Euler-Lagrange equations we derive are subject to the side conditions: ⋃ i Ωi = Ω and Ωi ∩ Ωj = ∅ ∀i 6= j, i.e., there must be no vacuum or region overlap. How to integrate such side conditions efficiently in case of