Advances in Graph-Cut Optimization: Multi-Surface Models, Label Costs, and Hierarchical Costs

Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of low-level inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing better energies and better algorithms for energies. This dissertation presents work along the same line, specifically new energies and algorithms based on graph cuts. We present three distinct contributions. First we consider biomedical segmentation where the object of interest comprises multiple distinct regions of uncertain shape (e.g. blood vessels, airways, bone tissue). We show that this common yet difficult scenario can be modeled as an energy over multiple interacting surfaces, and can be globally optimized by a single graph cut. Second, we introduce multi-label energies with label costs and provide algorithms to minimize them. We show how label costs are useful for clustering and robust estimation problems in vision. Third, we characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm with improved approximation guarantees. Hierarchical costs are natural for modeling an array of difficult problems, e.g. segmentation with hierarchical context, simultaneous estimation of motions and homographies, or detecting hierarchies of patterns

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Linear Programming Tools and Approximation Algorithms for Combinatorial Optimization

We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio. Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques. The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem. The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon

Strategic algorithms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 193-201).Classical algorithms from theoretical computer science arise time and again in practice. However,a practical situations typically do not fit precisely into the traditional theoretical models. Additional necessary components are, for example, uncertainty and economic incentives. Therefore, modem algorithm design is calling for more interdisciplinary approaches, as well as for deeper theoretical understanding, so that the algorithms can apply to more realistic settings and complex systems. Consider, for instance, the classical shortest path algorithm, which, given a graph with specified edge weights, seeks the path minimizing the total weight from a source to a destination. In practice, the edge weights are often uncertain and it is not even clear what we mean by shortest path anymore: is it the path that minimizes the expected weight? Or its variance, or some another metric? With a risk-averse objective function that takes into account both mean and standard deviation, we run into nonconvex optimization challenges that require new theory beyond classical shortest path algorithm design. Yet another shortest path application, routing of packets in the Internet, needs to further incorporate economic incentives to reflect the various business relationships among the Internet Service Providers that affect the choice of packet routes. Strategic Algorithms are algorithms that integrate optimization, uncertainty and economic modeling into algorithm design, with the goal of bringing about new theoretical developments and solving practical applications arising in complex computational-economic systems.(cont.) In short, this thesis contributes new algorithms and their underlying theory at the interface of optimization, uncertainty and economics. Although the interplay of these disciplines is present in various forms in our work, for the sake of presentation we have divided the material into three categories: 1. In Part I we investigate algorithms at the intersection of Optimization and Uncertainty. The key conceptual contribution in this part is discovering a novel connection between stochastic and nonconvex optimization. Traditional algorithm design has not taken into account the risk inherent in stochastic optimization problems. We consider natural objectives that incorporate risk, which tum out equivalent to certain nonconvex problems from the realm of continuous optimization. As a result, our work advances the state of art in both stochastic and in nonconvex optimization, presenting new complexity results and proposing general purpose efficient approximation algorithms, some of which have shown promising practical performance and have been implemented in a real traffic prediction and navigation system. 2. Part II proposes new algorithm and mechanism design at the intersection of Uncertainty and Economics. In Part I we postulate that the random variables in our models come from given distributions. However, determining those distributions or their parameters is a challenging and fundamental problem in itself. A tool from Economics that has recently gained momentum for measuring the probability distribution of a random variable is an information or prediction market. Such markets, most popularly known for predicting the outcomes of political elections or other events of interest, have shown remarkable accuracy in practice, though at the same time have left open the theoretical and strategic analysis of current implementations, as well as the need for new and improved designs which handle more complex outcome spaces (probability distribution functions) as opposed to binary or n-ary valued distributions. The contributions of this part include a unified strategic analysis of different prediction market designs that have been implemented in practice.(cont.) We also offer new market designs for handling exponentially large outcome spaces stemming from ranking or permutation-type outcomes, together with algorithmic and complexity analysis. 3. In Part III we consider the interplay of optimization and economics in the context of network routing. This part is motivated by the network of autonomous systems in the Internet where each portion of the network is controlled by an Internet service provider, namely by a self-interested economic agent. The business incentives do not exist merely in addition to the computer protocols governing the network. Although they are not currently integrated in those protocols and are decided largely via private contracting and negotiations, these economic considerations are a principal factor that determines how packets are routed. And vice versa, the demand and flow of network traffic fundamentally affect provider contracts and prices. The contributions of this part are the design and analysis of economic mechanisms for network routing. The mechanisms are based on first- and second-price auctions (the so-called Vickrey-Clarke-Groves, or VCG mechanisms). We first analyze the equilibria and prices resulting from these mechanisms. We then investigate the compatibility of the better understood VCG-mechanisms with the current inter-domain routing protocols, and we demonstrate the critical importance of correct modeling and how it affects the complexity and algorithms necessary to implement the economic mechanisms.by Evdokia Velinova Nikolova.Ph.D

A Markov Random Field Based Approach to 3D Mosaicing and Registration Applied to Ultrasound Simulation

A novel Markov Random Field (MRF) based method for the mosaicing of 3D ultrasound volumes is presented in this dissertation. The motivation for this work is the production of training volumes for an affordable ultrasound simulator, which offers a low-cost/portable training solution for new users of diagnostic ultrasound, by providing the scanning experience essential for developing the necessary psycho-motor skills. It also has the potential for introducing ultrasound instruction into medical education curriculums. The interest in ultrasound training stems in part from the widespread adoption of point-of-care scanners, i.e. low cost portable ultrasound scanning systems in the medical community. This work develops a novel approach for producing 3D composite image volumes and validates the approach using clinically acquired fetal images from the obstetrics department at the University of Massachusetts Medical School (UMMS). Results using the Visible Human Female dataset as well as an abdominal trauma phantom are also presented. The process is broken down into five distinct steps, which include individual 3D volume acquisition, rigid registration, calculation of a mosaicing function, group-wise non-rigid registration, and finally blending. Each of these steps, common in medical image processing, has been investigated in the context of ultrasound mosaicing and has resulted in improved algorithms. Rigid and non-rigid registration methods are analyzed in a probabilistic framework and their sensitivity to ultrasound shadowing artifacts is studied. The group-wise non-rigid registration problem is initially formulated as a maximum likelihood estimation, where the joint probability density function is comprised of the partially overlapping ultrasound image volumes. This expression is simplified using a block-matching methodology and the resulting discrete registration energy is shown to be equivalent to a Markov Random Field. Graph based methods common in computer vision are then used for optimization, resulting in a set of transformations that bring the overlapping volumes into alignment. This optimization is parallelized using a fusion approach, where the registration problem is divided into 8 independent sub-problems whose solutions are fused together at the end of each iteration. This method provided a speedup factor of 3.91 over the single threaded approach with no noticeable reduction in accuracy during our simulations. Furthermore, the registration problem is simplified by introducing a mosaicing function, which partitions the composite volume into regions filled with data from unique partially overlapping source volumes. This mosaicing functions attempts to minimize intensity and gradient differences between adjacent sources in the composite volume. Experimental results to demonstrate the performance of the group-wise registration algorithm are also presented. This algorithm is initially tested on deformed abdominal image volumes generated using a finite element model of the Visible Human Female to show the accuracy of its calculated displacement fields. In addition, the algorithm is evaluated using real ultrasound data from an abdominal phantom. Finally, composite obstetrics image volumes are constructed using clinical scans of pregnant subjects, where fetal movement makes registration/mosaicing especially difficult. Our solution to blending, which is the final step of the mosaicing process, is also discussed. The trainee will have a better experience if the volume boundaries are visually seamless, and this usually requires some blending prior to stitching. Also, regions of the volume where no data was collected during scanning should have an ultrasound-like appearance before being displayed in the simulator. This ensures the trainee\u27s visual experience isn\u27t degraded by unrealistic images. A discrete Poisson approach has been adapted to accomplish these tasks. Following this, we will describe how a 4D fetal heart image volume can be constructed from swept 2D ultrasound. A 4D probe, such as the Philips X6-1 xMATRIX Array, would make this task simpler as it can acquire 3D ultrasound volumes of the fetal heart in real-time; However, probes such as these aren\u27t widespread yet. Once the theory has been introduced, we will describe the clinical component of this dissertation. For the purpose of acquiring actual clinical ultrasound data, from which training datasets were produced, 11 pregnant subjects were scanned by experienced sonographers at the UMMS following an approved IRB protocol. First, we will discuss the software/hardware configuration that was used to conduct these scans, which included some custom mechanical design. With the data collected using this arrangement we generated seamless 3D fetal mosaics, that is, the training datasets, loaded them into our ultrasound training simulator, and then subsequently had them evaluated by the sonographers at the UMMS for accuracy. These mosaics were constructed from the raw scan data using the techniques previously introduced. Specific training objectives were established based on the input from our collaborators in the obstetrics sonography group. Important fetal measurements are reviewed, which form the basis for training in obstetrics ultrasound. Finally clinical images demonstrating the sonographer making fetal measurements in practice, which were acquired directly by the Philips iU22 ultrasound machine from one of our 11 subjects, are compared with screenshots of corresponding images produced by our simulator

A Markov Random Field Based Approach to 3D Mosaicing and Registration Applied to Ultrasound Simulation

A novel Markov Random Field (MRF) based method for the mosaicing of 3D ultrasound volumes is presented in this dissertation. The motivation for this work is the production of training volumes for an affordable ultrasound simulator, which offers a low-cost/portable training solution for new users of diagnostic ultrasound, by providing the scanning experience essential for developing the necessary psycho-motor skills. It also has the potential for introducing ultrasound instruction into medical education curriculums. The interest in ultrasound training stems in part from the widespread adoption of point-of-care scanners, i.e. low cost portable ultrasound scanning systems in the medical community. This work develops a novel approach for producing 3D composite image volumes and validates the approach using clinically acquired fetal images from the obstetrics department at the University of Massachusetts Medical School (UMMS). Results using the Visible Human Female dataset as well as an abdominal trauma phantom are also presented. The process is broken down into five distinct steps, which include individual 3D volume acquisition, rigid registration, calculation of a mosaicing function, group-wise non-rigid registration, and finally blending. Each of these steps, common in medical image processing, has been investigated in the context of ultrasound mosaicing and has resulted in improved algorithms. Rigid and non-rigid registration methods are analyzed in a probabilistic framework and their sensitivity to ultrasound shadowing artifacts is studied. The group-wise non-rigid registration problem is initially formulated as a maximum likelihood estimation, where the joint probability density function is comprised of the partially overlapping ultrasound image volumes. This expression is simplified using a block-matching methodology and the resulting discrete registration energy is shown to be equivalent to a Markov Random Field. Graph based methods common in computer vision are then used for optimization, resulting in a set of transformations that bring the overlapping volumes into alignment. This optimization is parallelized using a fusion approach, where the registration problem is divided into 8 independent sub-problems whose solutions are fused together at the end of each iteration. This method provided a speedup factor of 3.91 over the single threaded approach with no noticeable reduction in accuracy during our simulations. Furthermore, the registration problem is simplified by introducing a mosaicing function, which partitions the composite volume into regions filled with data from unique partially overlapping source volumes. This mosaicing functions attempts to minimize intensity and gradient differences between adjacent sources in the composite volume. Experimental results to demonstrate the performance of the group-wise registration algorithm are also presented. This algorithm is initially tested on deformed abdominal image volumes generated using a finite element model of the Visible Human Female to show the accuracy of its calculated displacement fields. In addition, the algorithm is evaluated using real ultrasound data from an abdominal phantom. Finally, composite obstetrics image volumes are constructed using clinical scans of pregnant subjects, where fetal movement makes registration/mosaicing especially difficult. Our solution to blending, which is the final step of the mosaicing process, is also discussed. The trainee will have a better experience if the volume boundaries are visually seamless, and this usually requires some blending prior to stitching. Also, regions of the volume where no data was collected during scanning should have an ultrasound-like appearance before being displayed in the simulator. This ensures the trainee\u27s visual experience isn\u27t degraded by unrealistic images. A discrete Poisson approach has been adapted to accomplish these tasks. Following this, we will describe how a 4D fetal heart image volume can be constructed from swept 2D ultrasound. A 4D probe, such as the Philips X6-1 xMATRIX Array, would make this task simpler as it can acquire 3D ultrasound volumes of the fetal heart in real-time; However, probes such as these aren\u27t widespread yet. Once the theory has been introduced, we will describe the clinical component of this dissertation. For the purpose of acquiring actual clinical ultrasound data, from which training datasets were produced, 11 pregnant subjects were scanned by experienced sonographers at the UMMS following an approved IRB protocol. First, we will discuss the software/hardware configuration that was used to conduct these scans, which included some custom mechanical design. With the data collected using this arrangement we generated seamless 3D fetal mosaics, that is, the training datasets, loaded them into our ultrasound training simulator, and then subsequently had them evaluated by the sonographers at the UMMS for accuracy. These mosaics were constructed from the raw scan data using the techniques previously introduced. Specific training objectives were established based on the input from our collaborators in the obstetrics sonography group. Important fetal measurements are reviewed, which form the basis for training in obstetrics ultrasound. Finally clinical images demonstrating the sonographer making fetal measurements in practice, which were acquired directly by the Philips iU22 ultrasound machine from one of our 11 subjects, are compared with screenshots of corresponding images produced by our simulator