Fuzzy Connectedness Image Segmentation in Graph Cut Formulation: A Linear-Time Algorithm and a Comparative Analysis

A deep theoretical analysis of the graph cut image segmentation framework presented in this paper simultaneously translates into important contributions in several directions. The most important practical contribution of this work is a full theoretical description, and implementation, of a novel powerful segmentation algorithm, GC(max). The output of GC(max) coincides with a version of a segmentation algorithm known as Iterative Relative Fuzzy Connectedness, IRFC. However, GC(max) is considerably faster than the classic IRFC algorithm, which we prove theoretically and show experimentally. Specifically, we prove that, in the worst case scenario, the GC(max) algorithm runs in linear time with respect to the variable M=|C|+|Z|, where |C| is the image scene size and |Z| is the size of the allowable range, Z, of the associated weight/affinity function. For most implementations, Z is identical to the set of allowable image intensity values, and its size can be treated as small with respect to |C|, meaning that O(M)=O(|C|). In such a situation, GC(max) runs in linear time with respect to the image size |C|. We show that the output of GC(max) constitutes a solution of a graph cut energy minimization problem, in which the energy is defined as the a"" (a) norm ayenF (P) ayen(a) of the map F (P) that associates, with every element e from the boundary of an object P, its weight w(e). This formulation brings IRFC algorithms to the realm of the graph cut energy minimizers, with energy functions ayenF (P) ayen (q) for qa[1,a]. Of these, the best known minimization problem is for the energy ayenF (P) ayen(1), which is solved by the classic min-cut/max-flow algorithm, referred to often as the Graph Cut algorithm. We notice that a minimization problem for ayenF (P) ayen (q) , qa[1,a), is identical to that for ayenF (P) ayen(1), when the original weight function w is replaced by w (q) . Thus, any algorithm GC(sum) solving the ayenF (P) ayen(1) minimization problem, solves also one for ayenF (P) ayen (q) with qa[1,a), so just two algorithms, GC(sum) and GC(max), are enough to solve all ayenF (P) ayen (q) -minimization problems. We also show that, for any fixed weight assignment, the solutions of the ayenF (P) ayen (q) -minimization problems converge to a solution of the ayenF (P) ayen(a)-minimization problem (ayenF (P) ayen(a)=lim (q -> a)ayenF (P) ayen (q) is not enough to deduce that). An experimental comparison of the performance of GC(max) and GC(sum) algorithms is included. This concentrates on comparing the actual (as opposed to provable worst scenario) algorithms' running time, as well as the influence of the choice of the seeds on the output.

Méthodes de segmentation du ventricule gauche basée sur l'algorithme graph cut pour les images par résonance magnétique et échocardiographiques

L’échocardiographie et l’imagerie par résonance magnétique sont toutes deux des techniques non invasives utilisées en clinique afin de diagnostiquer ou faire le suivi de maladies cardiaques. La première mesure un délai entre l’émission et la réception d’ultrasons traversant le corps, tandis que l’autre mesure un signal électromagnétique généré par des protons d’hydrogène présents dans le corps humain. Les résultats des acquisitions de ces deux modalités d’imagerie sont fondamentalement différents, mais contiennent dans les deux cas de l’information sur les structures du coeur humain. La segmentation du ventricule gauche consiste à délimiter les parois internes du muscle cardiaque, le myocarde, afin d’en calculer différentes métriques cliniques utiles au diagnostic et au suivi de différentes maladies cardiaques, telle la quantité de sang qui circule à chaque battement de coeur. Suite à un infarctus ou autre condition, les performances ainsi que la forme du coeur en sont affectées. L’imagerie du ventricule gauche est utilisée afin d’aider les cardiologues à poser les bons diagnostics. Cependant, dessiner les tracés manuels du ventricule gauche requiert un temps non négligeable aux cardiologues experts, d’où l’intérêt pour une méthode de segmentation automatisée fiable et rapide. Ce mémoire porte sur la segmentation du ventricule gauche. La plupart des méthodes existantes sont spécifiques à une seule modalité d’imagerie. Celle proposée dans ce document permet de traiter rapidement des acquisitions provenant de deux modalités avec une précision de segmentation équivalente au tracé manuel d’un expert. Pour y parvenir, elle opère dans un espace anatomique, induisant ainsi une forme a priori implicite. L’algorithme de Graph Cut, combiné avec des stratégies telles les cartes probabilistes et les enveloppes convexes régionales, parvient à générer des résultats qui équivalent (ou qui, pour la majorité des cas, surpassent) l’état de l’art ii Sommaire au moment de la rédaction de ce mémoire. La performance de la méthode proposée, quant à l’état de l’art, a été démontrée lors d’un concours international. Elle est également validée exhaustivement via trois bases de données complètes en se comparant aux tracés manuels de deux experts et des tracés automatisés du logiciel Syngovia. Cette recherche est un projet collaboratif avec l’Université de Bourgogne, en France.

MASCG: Multi-Atlas Segmentation Constrained Graph method for accurate segmentation of hip CT images

This paper addresses the issue of fully automatic segmentation of a hip CT image with the goal to preserve the joint structure for clinical applications in hip disease diagnosis and treatment. For this purpose, we propose a Multi-Atlas Segmentation Constrained Graph (MASCG) method. The MASCG method uses multi-atlas based mesh fusion results to initialize a bone sheetness based multi-label graph cut for an accurate hip CT segmentation which has the inherent advantage of automatic separation of the pelvic region from the bilateral proximal femoral regions. We then introduce a graph cut constrained graph search algorithm to further improve the segmentation accuracy around the bilateral hip joint regions. Taking manual segmentation as the ground truth, we evaluated the present approach on 30 hip CT images (60 hips) with a 15-fold cross validation. When the present approach was compared to manual segmentation, an average surface distance error of 0.30 mm, 0.29 mm, and 0.30 mm was found for the pelvis, the left proximal femur, and the right proximal femur, respectively. A further look at the bilateral hip joint regions demonstrated an average surface distance error of 0.16 mm, 0.21 mm and 0.20 mm for the acetabulum, the left femoral head, and the right femoral head, respectively.

Probabilistic Atlas Based Segmentation Using Affine Moment Descriptors and Graph-Cuts

We show a procedure for constructing a probabilistic atlas based on affine moment descriptors. It uses a normalization procedure over the labeled atlas. The proposed linear registration is defined by closed-form expressions involving only geometric moments. This procedure applies both to atlas construction as atlas-based segmentation. We model the likelihood term for each voxel and each label using parametric or nonparametric distributions and the prior term is determined by applying the vote-rule. The probabilistic atlas is built with the variability of our linear registration. We have two segmentation strategy: a) it applies the proposed affine registration to bring the target image into the coordinate frame of the atlas or b) the probabilistic atlas is non-rigidly aligning with the target image, where the probabilistic atlas is previously aligned to the target image with our affine registration. Finally, we adopt a graph cut - Bayesian framework for implementing the atlas-based segmentation.

A new label fusion method using graph cuts: application to hippocampus segmentation

The aim of this paper is to develop a probabilistic modeling framework for the segmentation of structures of interest from a collection of atlases. Given a subset of registered atlases into the target image for a particular Region of Interest (ROI), a statistical model of appearance and shape is computed for fusing the labels. Segmentations are obtained by minimizing an energy function associated with the proposed model, using a graph-cut technique. We test different label fusion methods on publicly available MR images of human brains.

A fully-automatic caudate nucleus segmentation of brain MRI: application in volumetric analysis of pediatric attention-deficit/hyperactivity disorder

Background Accurate automatic segmentation of the caudate nucleus in magnetic resonance images (MRI) of the brain is of great interest in the analysis of developmental disorders. Segmentation methods based on a single atlas or on multiple atlases have been shown to suitably localize caudate structure. However, the atlas prior information may not represent the structure of interest correctly. It may therefore be useful to introduce a more flexible technique for accurate segmentations. Method We present Cau-dateCut: a new fully-automatic method of segmenting the caudate nucleus in MRI. CaudateCut combines an atlas-based segmentation strategy with the Graph Cut energy-minimization framework. We adapt the Graph Cut model to make it suitable for segmenting small, low-contrast structures, such as the caudate nucleus, by defining new energy function data and boundary potentials. In particular, we exploit information concerning the intensity and geometry, and we add supervised energies based on contextual brain structures. Furthermore, we reinforce boundary detection using a new multi-scale edgeness measure. Results We apply the novel CaudateCut method to the segmentation of the caudate nucleus to a new set of 39 pediatric attention-deficit/hyperactivity disorder (ADHD) patients and 40 control children, as well as to a public database of 18 subjects. We evaluate the quality of the segmentation using several volumetric and voxel by voxel measures. Our results show improved performance in terms of segmentation compared to state-of-the-art approaches, obtaining a mean overlap of 80.75%. Moreover, we present a quantitative volumetric analysis of caudate abnormalities in pediatric ADHD, the results of which show strong correlation with expert manual analysis. Conclusion CaudateCut generates segmentation results that are comparable to gold-standard segmentations and which are reliable in the analysis of differentiating neuroanatomical abnormalities between healthy controls and pediatric ADHD.

IFTrace: Video segmentation of deformable objects using the Image Foresting Transform

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A comparison about evolutionary algorithms for optimum-path forest clustering optimization

In this paper we deal with the problem of boosting the Optimum-Path Forest (OPF) clustering approach using evolutionary-based optimization techniques. As the OPF classifier performs an exhaustive search to find out the size of sample's neighborhood that allows it to reach the minimum graph cut as a quality measure, we compared several optimization techniques that can obtain close graph cut values to the ones obtained by brute force. Experiments in two public datasets in the context of unsupervised network intrusion detection have showed the evolutionary optimization techniques can find suitable values for the neighborhood faster than the exhaustive search. Additionally, we have showed that it is not necessary to employ many agents for such task, since the neighborhood size is defined by discrete values, with constrain the set of possible solution to a few ones.

Riverbed: A Novel User-Steered Image Segmentation Method Based on Optimum Boundary Tracking

This paper presents an optimum user-steered boundary tracking approach for image segmentation, which simulates the behavior of water flowing through a riverbed. The riverbed approach was devised using the image foresting transform with a never-exploited connectivity function. We analyze its properties in the derived image graphs and discuss its theoretical relation with other popular methods such as live wire and graph cuts. Several experiments show that riverbed can significantly reduce the number of user interactions (anchor points), as compared to live wire for objects with complex shapes. This paper also includes a discussion about how to combine different methods in order to take advantage of their complementary strengths.

Classical segmentation methods on novel MR imaging: a study of brain tissue segmentation of MP2RAGE vs MPRAGE

MP2RAGE has proven to be a bias-free MR acquisition with excellent contrast between grey and white matter. We investigated the ability of three state-of-the-art algorithms to automatically extract white matter (WM), grey matter (GM) and cerebrospinal fluid (CSF) from MPRAGE and MP2RAGE images: unified Segmentation (S) in SPM82 , its extension New Segment (NS), and an in-house Expectation-Maximization Markov Random Field tissue classification3 (EM-MRF) with Graph Cut (GC) optimization4 . Our goal is to quantify the differences between MPRAGE and MP2RAGE-based brain tissue probability maps.

Segmentação de imagens pela transformada imagem-floresta com faixa de restrição geodésica

Vários métodos tradicionais de segmentação de imagens, como a transformada de watershed de marcado- res e métodos de conexidade fuzzy (Relative Fuzzy Connectedness- RFC, Iterative Relative Fuzzy Connected- ness - IRFC), podem ser implementados de modo eficiente utilizando o método em grafos da Transformada Imagem-Floresta (Image Foresting Transform - IFT). No entanto, a carência de termos de regularização de fronteira em sua formulação fazem com que a borda do objeto segmentado possa ser altamente irregular. Um modo de contornar isto é por meio do uso de restrições de forma do objeto, que favoreçam formas mais regulares, como na recente restrição de convexidade geodésica em estrela (Geodesic Star Convexity - GSC). Neste trabalho, apresentamos uma nova restrição de forma, chamada de Faixa de Restrição Geodésica (Geodesic Band Constraint - GBC), que pode ser incorporada eficientemente em uma sub-classe do fra- mework de corte em grafos generalizado (Generalized Graph Cut - GGC), que inclui métodos pela IFT. É apresentada uma prova da otimalidade do novo algoritmo em termos de um mínimo global de uma função de energia sujeita às novas restrições de borda. A faixa de restrição geodésica nos ajuda a regularizar a borda dos objetos, consequentemente melhorando a segmentação de objetos com formas mais regulares, mantendo o baixo custo computacional da IFT. A GBC pode também ser usada conjuntamente com um mapa de custos pré estabelecido, baseado em um modelo de forma, de modo a direcionar a segmentação a seguir uma dada forma desejada, com grau de liberdade de escala e demais deformações controladas por um parâmetro único. Essa nova restrição também pode ser combinada com a GSC e com as restrições de polaridade de borda sem custo adicional. O método é demonstrado em imagens naturais, sintéticas e médicas, sendo estas provenientes de tomografias computadorizadas e de ressonância magnética.

Automatic Abdominal Aortic Aneurysm segmentation in MR images

Abdominal Aortic Aneurism is a disease related to a weakening in the aortic wall that can cause a break in the aorta and the death. The detection of an unusual dilatation of a section of the aorta is an indicative of this disease. However, it is difficult to diagnose because it is necessary image diagnosis using computed tomography or magnetic resonance. An automatic diagnosis system would allow to analyze abdominal magnetic resonance images and to warn doctors if any anomaly is detected. We focus our research in magnetic resonance images because of the absence of ionizing radiation. Although there are proposals to identify this disease in magnetic resonance images, they need an intervention from clinicians to be precise and some of them are computationally hard. In this paper we develop a novel approach to analyze magnetic resonance abdominal images and detect the lumen and the aortic wall. The method combines different algorithms in two stages to improve the detection and the segmentation so it can be applied to similar problems with other type of images or structures. In a first stage, we use a spatial fuzzy C-means algorithm with morphological image analysis to detect and segment the lumen; and subsequently, in a second stage, we apply a graph cut algorithm to segment the aortic wall. The obtained results in the analyzed images are pretty successful obtaining an average of 79% of overlapping between the automatic segmentation provided by our method and the aortic wall identified by a medical specialist. The main impact of the proposed method is that it works in a completely automatic way with a low computational cost, which is of great significance for any expert and intelligent system.

Automatic object segmentation from calibrated images

This paper addresses the problem of automatically obtaining the object/background segmentation of a rigid 3D object observed in a set of images that have been calibrated for camera pose and intrinsics. Such segmentations can be used to obtain a shape representation of a potentially texture-less object by computing a visual hull. We propose an automatic approach where the object to be segmented is identified by the pose of the cameras instead of user input such as 2D bounding rectangles or brush-strokes. The key behind our method is a pairwise MRF framework that combines (a) foreground/background appearance models, (b) epipolar constraints and (c) weak stereo correspondence into a single segmentation cost function that can be efficiently solved by Graph-cuts. The segmentation thus obtained is further improved using silhouette coherency and then used to update the foreground/background appearance models which are fed into the next Graph-cut computation. These two steps are iterated until segmentation convergences. Our method can automatically provide a 3D surface representation even in texture-less scenes where MVS methods might fail. Furthermore, it confers improved performance in images where the object is not readily separable from the background in colour space, an area that previous segmentation approaches have found challenging. © 2011 IEEE.

The tree cut and merge algorithm for estimation of network reliability

This article presents Monte Carlo techniques for estimating network reliability. For highly reliable networks, techniques based on graph evolution models provide very good performance. However, they are known to have significant simulation cost. An existing hybrid scheme (based on partitioning the time space) is available to speed up the simulations; however, there are difficulties with optimizing the important parameter associated with this scheme. To overcome these difficulties, a new hybrid scheme (based on partitioning the edge set) is proposed in this article. The proposed scheme shows orders of magnitude improvement of performance over the existing techniques in certain classes of network. It also provides reliability bounds with little overhead.

Semi-supervised segmentation of ultrasound images based on patch representation and continuous min cut.

Ultrasound segmentation is a challenging problem due to the inherent speckle and some artifacts like shadows, attenuation and signal dropout. Existing methods need to include strong priors like shape priors or analytical intensity models to succeed in the segmentation. However, such priors tend to limit these methods to a specific target or imaging settings, and they are not always applicable to pathological cases. This work introduces a semi-supervised segmentation framework for ultrasound imaging that alleviates the limitation of fully automatic segmentation, that is, it is applicable to any kind of target and imaging settings. Our methodology uses a graph of image patches to represent the ultrasound image and user-assisted initialization with labels, which acts as soft priors. The segmentation problem is formulated as a continuous minimum cut problem and solved with an efficient optimization algorithm. We validate our segmentation framework on clinical ultrasound imaging (prostate, fetus, and tumors of the liver and eye). We obtain high similarity agreement with the ground truth provided by medical expert delineations in all applications (94% DICE values in average) and the proposed algorithm performs favorably with the literature.