Egil Bae

A study on continuous max-flow and min-cut approaches

We propose and study novel max-flow models in the continuous setting, which directly map the discrete graph-based max-flow problem to its continuous optimization formulation. We show such a continuous max-flow model leads to an equivalent min-cut problem in a natural way, as the corresponding dual model. In this regard, we revisit basic conceptions used in discrete max-flow / min-cut models and give their new explanations from a variational perspective. We also propose corresponding continuous max-flow and min-cut models constrained by priori supervised information and apply them to interactive image segmentation/labeling problems. We prove that the proposed continuous max-flow and min-cut models, with or without supervised constraints, give rise to a series of global binary solutions λ∗(x) ∊ {0,1}, which globally solves the original nonconvex image partitioning problems. In addition, we propose novel and reliable multiplier-based max-flow algorithms. Their convergence is guaranteed by classical optimization theories. Experiments on image segmentation, unsupervised and supervised, validate the effectiveness of the discussed continuous max-flow and min-cut models and suggested max-flow based algorithms.

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding duality-based approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.

Graph Cut Optimization for the Piecewise Constant Level Set Method Applied to Multiphase Image Segmentation

The piecewise constant level set method (PCLSM) has recently emerged as a variant of the level set method for variational interphase problems. Traditionally, the Euler-Lagrange equations are solved by some iterative numerical method for PDEs. Normally the speed is slow. In this work, we focus on the piecewise constant level set method (PCLSM) applied to the multiphase Mumford-Shah model for image segmentation. Instead of solving the Euler-Lagrange equations of the resulting minimization problem, we propose an efficient combinatorial optimization technique, based on graph cuts. Because of a simplification of the length term in the energy induced by the PCLSM, the minimization problem is not NP hard. Numerical experiments on image segmentation demonstrate that the new approach is very superior in terms of efficiency, while maintaining the same quality.

Graph Cuts for Curvature Based Image Denoising

Minimization of total variation (TV) is a well-known method for image denoising. Recently, the relationship between TV minimization problems and binary MRF models has been much explored. This has resulted in some very efficient combinatorial optimization algorithms for the TV minimization problem in the discrete setting via graph cuts. To overcome limitations, such as staircasing effects, of the relatively simple TV model, variational models based upon higher order derivatives have been proposed. The Eulers elastica model is one such higher order model of central importance, which minimizes the curvature of all level lines in the image. Traditional numerical methods for minimizing the energy in such higher order models are complicated and computationally complex. In this paper, we will present an efficient minimization algorithm based upon graph cuts for minimizing the energy in the Eulers elastica model, by simplifying the problem to that of solving a sequence of easy graph representable problems. This sequence has connections to the gradient flow of the energy function, and converges to a minimum point. The numerical experiments show that our new approach is more effective in maintaining smooth visual results while preserving sharp features better than TV models.

Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation

The Mumford-Shah model is an important variational image segmentation model. A popular multiphase level set approach, the Chan-Vese model, was developed for this model by representing the phases by several overlapping level set functions. Recently, exactly the same model was also formulated by using binary level set functions. In both approaches, the gradient descent equations had to be solved numerically, a procedure which is slow and has the potential of getting stuck in a local minima. In this work, we develop an efficient and global minimization method for the binary level set representation of the multiphase Chan-Vese model based on graph cuts. If the average intensity values of the different phases are sufficiently evenly distributed, the discretized energy function becomes submodular. Otherwise, a novel method for minimizing nonsubmodular functions is proposed with particular emphasis on this energy function.

Partial differential equations for interpolation and compression of surfaces

Partial differential equations (PDEs) have recently shown to be very promising for image interpolation and compression. Inspired from this work, we present a PDE based approach to interpolation of surfaces from scattered point sets using the geometric diffusion equation. Triangulated surfaces are considered in the discrete setting, and the geometric diffusion equation is discretized by the finite element method directly on the triangular mesh. Furthermore, a PDE based method for lossy compression of triangulated surfaces is presented. The idea is to store only a few relevant vertex coordinates in the encoding step. In the decoding step, the remaining vertices are reconstructed by solving the geometric diffusion equation. Finally, two modified reconstruction methods are proposed that are shown to improve the compression quality for both images and surfaces. These reconstruction methods approximate instead of interpolating, and have links to Hopscotch methods for the numerical solution of PDEs. Experiments are presented illustrating that results of high quality can be obtained using simple geometric diffusion without any information on surface normals.

Reconstructing Open Surfaces via Graph-Cuts

A novel graph-cuts-based method is proposed for reconstructing open surfaces from unordered point sets. Through a Boolean operation on the crust around the data set, the open surface problem is translated to a watertight surface problem within a restricted region. Integrating the variational model, Delaunay-based tetrahedral mesh and multiphase technique, the proposed method can reconstruct open surfaces robustly and effectively. Furthermore, a surface reconstruction method with domain decomposition is presented, which is based on the new open surface reconstruction method. This method can handle more general surfaces, such as nonorientable surfaces. The algorithm is designed in a parallel-friendly way and necessary measures are taken to eliminate cracks and conflicts between the subdomains. Numerical examples are included to demonstrate the robustness and effectiveness of the proposed method on watertight, open orientable, open nonorientable surfaces and combinations of such.

A continuous max-flow approach to minimal partitions with label cost prior

This paper investigates a convex relaxation approach for minimum description length (MDL) based image partitioning or labeling, which proposes an energy functional regularized by the spatial smoothness prior joint with a penalty for the total number of appearences or labels, the so-called label cost prior. As common in recent studies of convex relaxation approaches, the total-variation term is applied to encode the spatial regularity of partition boundaries and the auxiliary label cost term is penalized by the sum of convex infinity norms of the labeling functions. We study the proposed such convex MDL based image partition model under a novel continuous flow maximization perspective, where we show that the label cost prior amounts to a relaxation of the flow conservation condition which is crucial to study the classical duality of max-flow and min-cut! To the best of our knowledge, it is new to demonstrate such connections between the relaxation of flow conservation and the penalty of the total number of active appearences. In addition, we show that the proposed continuous max-flow formulation also leads to a fast and reliable max-flow based algorithm to address the challenging convex optimization problem, which significantly outperforms the previous approach by direct convex programming, in terms of speed, computation load and handling large-scale images. Its numerical scheme can by easily implemented and accelerated by the advanced computation framework, e.g. GPU.