A note on the primal-dual method for the semi-metric labeling problem

Recently, Komodakis et al. [6] developed the FastPD algorithm for the semi-metric labeling problem, which extends the expansion move algorithm of Boykov et al. [2]. We present a slightly different derivation of the FastPD method

Minimizing a sum of submodular functions

We consider the problem of minimizing a function represented as a sum of submodular terms. We assume each term allows an efficient computation of {\em exchange capacities}. This holds, for example, for terms depending on a small number of variables, or for certain cardinality-dependent terms. A naive application of submodular minimization algorithms would not exploit the existence of specialized exchange capacity subroutines for individual terms. To overcome this, we cast the problem as a {\em submodular flow} (SF) problem in an auxiliary graph, and show that applying most existing SF algorithms would rely only on these subroutines. We then explore in more detail Iwata's capacity scaling approach for submodular flows (Math. Programming, 76(2):299--308, 1997). In particular, we show how to improve its complexity in the case when the function contains cardinality-dependent terms.Comment: accepted to "Discrete Applied Mathematics

Structural Studies of Decaying Fluid Turbulence: Effect of Initial Conditions

We present results from a systematic numerical study of structural properties of an unforced, incompressible, homogeneous, and isotropic three-dimensional turbulent fluid with an initial energy spectrum that develops a cascade of kinetic energy to large wavenumbers. The results are compared with those from a recently studied set of power-law initial energy spectra [C. Kalelkar and R. Pandit, Phys. Rev. E, {\bf 69}, 046304 (2004)] which do not exhibit such a cascade. Differences are exhibited in plots of vorticity isosurfaces, the temporal evolution of the kinetic energy-dissipation rate, and the rates of production of the mean enstrophy along the principal axes of the strain-rate tensor. A crossover between non-`cascade-type' and `cascade-type' behaviour is shown numerically for a specific set of initial energy spectra.Comment: 9 pages, 27 figures, Accepted for publication in Physical Review

Generalized roof duality and bisubmodular functions

Consider a convex relaxation $\hat f$ of a pseudo-boolean function $f$. We say that the relaxation is {\em totally half-integral} if $\hat f(x)$ is a polyhedral function with half-integral extreme points $x$, and this property is preserved after adding an arbitrary combination of constraints of the form $x_i=x_j$, $x_i=1-x_j$, and $x_i=\gamma$ where \gamma\in\{0, 1, 1/2} is a constant. A well-known example is the {\em roof duality} relaxation for quadratic pseudo-boolean functions $f$. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations $\hat f$ by establishing a one-to-one correspondence with {\em bisubmodular functions}. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality.Comment: 14 pages. Shorter version to appear in NIPS 201

A Faster Approximation Algorithm for the Gibbs Partition Function

We consider the problem of estimating the partition function $Z(\beta)=\sum_x \exp(-\beta(H(x))$ of a Gibbs distribution with a Hamilton $H(\cdot)$, or more precisely the logarithm of the ratio $q=\ln Z(0)/Z(\beta)$. It has been recently shown how to approximate $q$ with high probability assuming the existence of an oracle that produces samples from the Gibbs distribution for a given parameter value in $[0,\beta]$. The current best known approach due to Huber [9] uses $O(q\ln n\cdot[\ln q + \ln \ln n+\varepsilon^{-2}])$ oracle calls on average where $\varepsilon$ is the desired accuracy of approximation and $H(\cdot)$ is assumed to lie in $\{0\}\cup[1,n]$. We improve the complexity to $O(q\ln n\cdot\varepsilon^{-2})$ oracle calls. We also show that the same complexity can be achieved if exact oracles are replaced with approximate sampling oracles that are within $O(\frac{\varepsilon^2}{q\ln n})$ variation distance from exact oracles. Finally, we prove a lower bound of $\Omega(q\cdot \varepsilon^{-2})$ oracle calls under a natural model of computation

The effect of wavefront corrugations on fringe motion in an astronomical interferometer with spatial filters

Numerical simulations of atmospheric turbulence and AO wavefront correction are performed to investigate the timescale for fringe motion in optical interferometers with spatial filters. These simulations focus especially on partial AO correction, where only a finite number of Zernike modes are compensated. The fringe motion is found to depend strongly on both the aperture diameter and the level of AO correction used. In all of the simulations the coherence timescale for interference fringes is found to decrease dramatically when the Strehl ratio provided by the AO correction is <~30%. For AO systems which give perfect compensation of a limited number of Zernike modes, the aperture size which gives the optimum signal for fringe phase tracking is calculated. For AO systems which provide noisy compensation of Zernike modes (but are perfectly piston-neutral), the noise properties of the AO system determine the coherence timescale of the fringes when the Strehl ratio is <~30%.Comment: 11 pages, submitted to Applied Optics 17 August 2004, accepted 2 June 200

Generalized sequential tree-reweighted message passing

This paper addresses the problem of approximate MAP-MRF inference in general graphical models. Following [36], we consider a family of linear programming relaxations of the problem where each relaxation is specified by a set of nested pairs of factors for which the marginalization constraint needs to be enforced. We develop a generalization of the TRW-S algorithm [9] for this problem, where we use a decomposition into junction chains, monotonic w.r.t. some ordering on the nodes. This generalizes the monotonic chains in [9] in a natural way. We also show how to deal with nested factors in an efficient way. Experiments show an improvement over min-sum diffusion, MPLP and subgradient ascent algorithms on a number of computer vision and natural language processing problems

New algorithms for the dual of the convex cost network flow problem with application to computer vision

Motivated by various applications to computer vision, we consider an integer convex optimization problem which is the dual of the convex cost network flow problem. In this paper, we first propose a new primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. The main contribution in this paper is to provide a tight bound for the number of the iterations. We show that the time complexity of the primal algorithm is K ¢ T(n;m) where K is the range of primal variables and T(n;m) is the time needed to compute a minimum cut in a graph with n nodes and m edges. We then propose a primal-dual algorithm for the dual of the convex cost network flow problem. The primal-dual algorithm can be seen as a refined version of the primal algorithm by maintaining dual variables (flow) in addition to primal variables. Although its time complexity is the same as that for the primal algorithm, we can expect a better performance practically. We finally consider an application to a computer vision problem called the panoramic stitching problem. We apply several implementations of our primal-dual algorithm to some instances of the panoramic stitching problem and test their practical performance. We also show that our primal algorithm as well as the proofs can be applied to the L\-convex function minimization problem which is a more general problem than the dual of the convex cost network flow problem