Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem (original) (raw)
Related papers
Continuous Multiclass Labeling Approaches and Algorithms
Computing Research Repository, 2011
We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity -- one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent Douglas-Rachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other first-order methods, the approach shows competitive performance on synthetical and real-world images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1%--5% of the global optimum for the combinatorial image labeling problem.
Convex Multi-class Image Labeling by Simplex-Constrained Total Variation
2009
Multi-class labeling is one of the core problems in image analysis. We show how this combinatorial problem can be approximately solved using tools from convex optimization. We suggest a novel functional based on a multidimensional total variation formulation, allowing for a broad range of data terms. Optimization is carried out in the operator splitting framework using Douglas-Rachford Splitting. In this connection, we compare two methods to solve the Rudin-Osher-Fatemi type subproblems and demonstrate the performance of our approach on single- and multichannel images.
2009
We introduce a linearly weighted variant of the total variation for vector fields in order to formulate regularizers for multi-class labeling problems with non-trivial interclass distances. We characterize the possible distances, show that Euclidean distances can be exactly represented, and review some methods to approximate non-Euclidean distances in order to define novel total variation based regularizers. We show that the convex relaxed problem can be efficiently optimized to a prescribed accuracy with optimality certificates using Nesterov's method, and evaluate and compare our approach on several synthetical and real-world examples.
What is optimized in tight convex relaxations for multi-label problems?
2012
Abstract In this work we present a unified view on Markov random fields and recently proposed continuous tight convex relaxations for multi-label assignment in the image plane. These relaxations are far less biased towards the grid geometry than Markov random fields. It turns out that the continuous methods are non-linear extensions of the local polytope MRF relaxation. In view of this result a better understanding of these tight convex relaxations in the discrete setting is obtained.
Minimal partitions and image classification using a gradient-free perimeter approximation
Inverse Problems and Imaging, 2014
In this paper we propose a new optimal partition algorithm and show applications to multilabel image classification problems. Possibly noisy and blurred greyscale and color images can be processed, with or without automatic update of the labels. Regularization is performed by a non standard approximation of the total interface length, which involves a system of uncoupled linear partial differential equations and shows Γ-converge properties in the set of characteristic functions. These good mathematical properties are recovered in the numerical convergence scheme.
Energy Minimization under Constraints on Label Counts
Lecture Notes in Computer Science, 2010
Many computer vision problems such as object segmentation or reconstruction can be formulated in terms of labeling a set of pixels or voxels. In certain scenarios, we may know the number of pixels or voxels which can be assigned to a particular label. For instance, in the reconstruction problem, we may know size of the object to be reconstructed. Such label count constraints are extremely powerful and have recently been shown to result in good solutions for many vision problems. Traditional energy minimization algorithms used in vision cannot handle label count constraints. This paper proposes a novel algorithm for minimizing energy functions under constraints on the number of variables which can be assigned to a particular label. Our algorithm is deterministic in nature and outputs εapproximate solutions for all possible counts of labels. We also develop a variant of the above algorithm which is much faster, produces solutions under almost all label count constraints, and can be applied to all submodular quadratic pseudoboolean functions. We evaluate the algorithm on the two-label (foreground/background) image segmentation problem and compare its performance with the stateof-the-art parametric maximum flow and max-sum diffusion based algorithms. Experimental results show that our method is practical and is able to generate impressive segmentation results in reasonable time.
Approximation schemes for partitioning: convex decomposition and surface approximation
Symposium on Discrete Algorithms, 2015
Recently, Adamaszek and Wiese [1, 2] presented a quasipolynomial time approximation scheme (QPTAS) for the problem of computing a maximum weight independent set for certain families of planar objects. This major advance on the problem was based on their proof that a certain type of separator exists for any independent set. Subsequently, Har-Peled [22] simplified and generalized their result. Mustafa et al. [36] also described a simplification, and somewhat surprisingly, showed that QPTAS's can be obtained for certain, albeit special, type of covering problems. Building on these developments, we revisit two NPhard geometric partitioning problems-convex decomposition and surface approximation. Partitioning problems combine the features of packing and covering. In particular, since the optimal solution does form a packing, the separator theorems are potentially applicable. Nevertheless, the two partitioning problems we study bring up additional difficulties that are worth examining in the context of the wider applicability of the separator methodology. We show how these issues can be handled in presenting quasi-polynomial time algorithms for these two problems with improved approximation guarantees.
Lecture Notes in Computer Science, 2004
In this paper we present a new fast approximation algorithm for the Uniform Metric Labeling Problem. This is an important classification problem that occur in many applications which consider the assignment of objects into labels, in a way that is consistent with some observed data that includes the relationship between the objects. The known approximation algorithms are based on solutions of large linear programs and are impractical for moderated and large size instances. We present an 8 log n-approximation algorithm analyzed by a primal-dual technique which, although has factor greater than the previous algorithms, can be applied to large sized instances. We obtained experimental results on computational generated and image processing instances with the new algorithm and two others LP-based approximation algorithms. For these instances our algorithm present a considerable gain of computational time and the error ratio, when possible to compare, was less than 2% from the optimum.
Large-Scale Integer Programs in Image Analysis
Operations Research, 2002
An important problem in image analysis is to segment an image into regions with di erent class-labels. This is releveant in applications in medicine and cartography. In a proper statistical framework this problem may be viewed as a discrete optimization problem. We present t w o integer linear programming formulations of the problem and study some properties of these models and associated polytopes. Di erent algorithms for solving these problems are suggested and compared on some realistic data. In particular, a Lagrangian algorithm is shown to have a v ery promising performance. The algorithm is based on the technique of cost splitting and uses the fact that certain relaxed problems may be solved as shortest path problems.
Efficient Energy Minimization for Enforcing Label Statistics
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000
Energy minimization algorithms, such as graph cuts, enable the computation of the MAP solution under certain probabilistic models such as Markov random fields. However, for many computer vision problems, the MAP solution under the model is not the ground truth solution. In many problem scenarios, the system has access to certain statistics of the ground truth. For instance, in image segmentation, the area and boundary length of the object may be known. In these cases, we want to estimate the most probable solution that is consistent with such statistics, i.e., satisfies certain equality or inequality constraints.