Multiscale Fields of Patterns

We describe a framework for defining high-order image models that can be used in a variety of applications. The approach involves modeling local patterns in a multiscale representation of an image. Local properties of a coarsened image reflect non-local properties of the original image. In the case of binary images local properties are defined by the binary patterns observed over small neighborhoods around each pixel. With the multiscale representation we capture the frequency of patterns observed at different scales of resolution. This framework leads to expressive priors that depend on a relatively small number of parameters. For inference and learning we use an MCMC method for block sampling with very large blocks. We evaluate the approach with two example applications. One involves contour detection. The other involves binary segmentation.Comment: In NIPS 201

Computing a partition function of a generalized pattern-based energy over a semiring

Valued constraint satisfaction problems with ordered variables (VCSPO) are a special case of Valued CSPs in which variables are totally ordered and soft constraints are imposed on tuples of variables that do not violate the order. We study a restriction of VCSPO, in which soft constraints are imposed on a segment of adjacent variables and a constraint language $\Gamma$ consists of $\{0,1\}$-valued characteristic functions of predicates. This kind of potentials generalizes the so-called pattern-based potentials, which were applied in many tasks of structured prediction. For a constraint language $\Gamma$ we introduce a closure operator, $\overline{\Gamma^{\cap}}\supseteq \Gamma$, and give examples of constraint languages for which $|\overline{\Gamma^{\cap}}|$ is small. If all predicates in $\Gamma$ are cartesian products, we show that the minimization of a generalized pattern-based potential (or, the computation of its partition function) can be made in ${\mathcal O}(|V|\cdot |D|^2 \cdot |\overline{\Gamma^{\cap}}|^2 )$ time, where $V$ is a set of variables, $D$ is a domain set. If, additionally, only non-positive weights of constraints are allowed, the complexity of the minimization task drops to ${\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}| \cdot |D| \cdot \max_{\rho\in \Gamma}\|\rho\|^2 )$ where $\|\rho\|$ is the arity of $\rho\in \Gamma$. For a general language $\Gamma$ and non-positive weights, the minimization task can be carried out in ${\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}|^2)$ time. We argue that in many natural cases $\overline{\Gamma^{\cap}}$ is of moderate size, though in the worst case $|\overline{\Gamma^{\cap}}|$ can blow up and depend exponentially on $\max_{\rho\in \Gamma}\|\rho\|$