Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Statistical Models for Co-occurrence Data

Modeling and predicting co-occurrences of events is a fundamental problem of unsupervised learning. In this contribution we develop a statistical framework for analyzing co-occurrence data in a general setting where elementary observations are joint occurrences of pairs of abstract objects from two finite sets. The main challenge for statistical models in this context is to overcome the inherent data sparseness and to estimate the probabilities for pairs which were rarely observed or even unobserved in a given sample set. Moreover, it is often of considerable interest to extract grouping structure or to find a hierarchical data organization. A novel family of mixture models is proposed which explain the observed data by a finite number of shared aspects or clusters. This provides a common framework for statistical inference and structure discovery and also includes several recently proposed models as special cases. Adopting the maximum likelihood principle, EM algorithms are derived to fit the model parameters. We develop improved versions of EM which largely avoid overfitting problems and overcome the inherent locality of EM--based optimization. Among the broad variety of possible applications, e.g., in information retrieval, natural language processing, data mining, and computer vision, we have chosen document retrieval, the statistical analysis of noun/adjective co-occurrence and the unsupervised segmentation of textured images to test and evaluate the proposed algorithms

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

Rich probabilistic models for semantic labeling

Das Ziel dieser Monographie ist es die Methoden und Anwendungen des semantischen Labelings zu erforschen. Unsere Beiträge zu diesem sich rasch entwickelten Thema sind bestimmte Aspekte der Modellierung und der Inferenz in probabilistischen Modellen und ihre Anwendungen in den interdisziplinären Bereichen der Computer Vision sowie medizinischer Bildverarbeitung und Fernerkundung

Gradient Based Mrf Learning For Image Restoration And Segmentation

The undirected graphical model or Markov Random Field (MRF) is one of the more popular models used in computer vision and is the type of model with which this work is concerned. Models based on these methods have proven to be particularly useful in low-level vision systems and have led to state-of-the-art results for MRF-based systems. The research presented will describe a new discriminative training algorithm and its implementation. The MRF model will be trained by optimizing its parameters so that the minimum energy solution of the model is as similar as possible to the ground-truth. While previous work has relied on time-consuming iterative approximations or stochastic approximations, this work will demonstrate how implicit differentiation can be used to analytically differentiate the overall training loss with respect to the MRF parameters. This framework leads to an efficient, flexible learning algorithm that can be applied to a number of different models. The effectiveness of the proposed learning method will then be demonstrated by learning the parameters of two related models applied to the task of denoising images. The experimental results will demonstrate that the proposed learning algorithm is comparable and, at times, better than previous training methods applied to the same tasks. A new segmentation model will also be introduced and trained using the proposed learning method. The proposed segmentation model is based on an energy minimization framework that is iii novel in how it incorporates priors on the size of the segments in a way that is straightforward to implement. While other methods, such as normalized cuts, tend to produce segmentations of similar sizes, this method is able to overcome that problem and produce more realistic segmentations

Spatial Fuzzy clustering with simultaneous estimation of Markov random field parameters and class

Projecte final de carrera fet en col.laboració amb Medical Imaging Research Center. Illinois Institute of Technolog