Publications about Phase Field

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• 1 Article
• 4 PhD Thesis and Habilitations
• 8 Conference articles

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Article

1 - Extended Phase Field Higher-Order Active Contour Models for Networks. T. Peng and I. H. Jermyn and V. Prinet and J. Zerubia. International Journal of Computer Vision, 88(1): pages 111-128, May 2010. Keywords : Active contour, Phase Field, Shape prior, Parameter analysis, remote sensing, Road network extraction. @ARTICLE{Peng09, author = {Peng, T. and Jermyn, I. H. and Prinet, V. and Zerubia, J.}, title = { Extended Phase Field Higher-Order Active Contour Models for Networks}, year = {2010}, month = {May}, journal = {International Journal of Computer Vision}, volume = {88}, number = {1}, pages = { 111-128}, url = {http://www.springerlink.com/content/d3641g2227316w58/}, keyword = {Active contour, Phase Field, Shape prior, Parameter analysis, remote sensing, Road network extraction} } Abstract : This paper addresses the segmentation from an image of entities that have the form of a ‘network’, i.e. the region in the image corresponding to the entity is composed of branches joining together at junctions, e.g. road or vascular networks. We present new phase field higher-order active contour (HOAC) prior models for network regions, and apply them to the segmentation of road networks from very high resolution satellite images. This is a hard problem for two reasons. First, the images are complex, with much ‘noise’ in the road region due to cars, road markings, etc., while the background is very varied, containing many features that are locally similar to roads. Second, network regions are complex to model, because they may have arbitrary topology. In particular, we address a limitation of a previous model in which network branch width was constrained to be similar to maximum network branch radius of curvature, thereby providing a poor model of networks with straight narrow branches or highly curved, wide branches. We solve this problem by introducing first an additional nonlinear nonlocal HOAC term, and then an additional linear nonlocal HOAC term to improve the computational speed. Both terms allow separate control of branch width and branch curvature, and furnish better prolongation for the same width, but the linear term has several advantages: it is more efficient, and it is able to model multiple widths simultaneously. To cope with the difficulty of parameter selection for these models, we perform a stability analysis of a long bar with a given width, and hence show how to choose the parameters of the energy functions. After adding a likelihood energy, we use both models to extract the road network quasi-automatically from pieces of a QuickBird image, and compare the results to other models in the literature. The state-of-the-art results obtained demonstrate the superiority of our new models, the importance of strong prior knowledge in general, and of the new terms in particular.

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4 PhD Thesis and Habilitations

1 - Phase fields for network extraction from images. A. El Ghoul. PhD Thesis, Universite de Nice - Sophia-Antipolis, September 2010. Keywords : Shape prior, Higher-order actif contours, Phase Field, Stability analysis, Directed networks, river extraction. @PHDTHESIS{elghoul10c, author = {El Ghoul, A.}, title = {Phase fields for network extraction from images}, year = {2010}, month = {September}, school = {Universite de Nice - Sophia-Antipolis}, url = {http://tel.archives-ouvertes.fr/docs/00/55/01/34/PDF/ThesisMunuscript2010_EL_GHOUL.pdf}, keyword = {Shape prior, Higher-order actif contours, Phase Field, Stability analysis, Directed networks, river extraction} } Résumé : Cette thèse décrit la construction d'un modèle de réseaux non-directionnels (e.g. réseaux routiers), fondé sur les contours actifs d'ordre supérieur (CAOSs) et les champs de phase développés récemment, et introduit une nouvelle famille des CAOSs des champs de phase pour des réseaux directionnels (e.g. réseaux hydrographiques en imagerie de télédétection, vaisseaux sanguins en imagerie médicale). Dans la première partie de cette thèse, nous nous intéressons à l'analyse de stabilité d'une énergie de type CAOSs aboutissant à un ‘diagramme de phase'. Les résultats, qui sont confirmés par des expériences numériques, permettent une bonne sélection des valeurs des paramètres pour la modélisation de réseaux non-directionnels. Au contraire des réseaux routiers, les réseaux hydrographiques sont directionnels, i.e. ils contiennent un ‘flux' monodimensionnel circulant dans chaque branche. Cela implique des propriétés géométriques spécifiques des branches et particulièrement des jonctions, propriétés qu'il est utile de traduire dans un modèle, pour l'extraction de réseaux. Nous développons donc un modèle de champ de phase non-local de réseaux directionnels, qui, en plus du champ de phase scalaire décrivant une région par une fonction caractéristique lisse et qui interagit non-localement afin que des configurations de réseaux linéiques soient favorisées, introduit un champ vectoriel représentant le ‘flux' dans les branches du réseau. Ce champ vectoriel est contraint d'être nul à l'extérieur, et de magnitude égale à 1 à l'intérieur du réseau ; circulant dans le sens longitudinal des branches du réseau ; et de divergence très faible. Cela prolonge les branches du réseau ; contrôle la variation de largeur tout au long une branche ; et forme des jonctions non-symétriques telles que la somme des largeurs entrantes soit approximativement égale à celle des largeurs sortantes. En conjonction avec une nouvelle fonction d'interaction pour le champ de phase scalaire, le modèle assure aussi une vaste gamme de valeurs des largeurs stables des branches. Ce nouveau modèle a été appliqué au problème d'extraction de réseaux hydrographiques à partir d'images satellitaires très haute résolution. Abstract : This thesis describes the construction of an undirected network (e.g. road network) model, based on the recently developed higher-order active contours (HOACs) and phase fields, and introduces a new family of phase field HOACs for directed networks (e.g. hydrographic networks in remote sensing imagery, vascular networks in medical imagery). In the first part of this thesis, we focus on the stability analysis of a HOAC energy leading to a ‘phase diagram'. The results, which are confirmed by numerical experiments, enable the selection of parameter values for the modelling of undirected networks. Hydrographic networks, unlike road networks, are directed, i.e. they carry a unidirectional flow in each branch. This leads to specific geometric properties of the branches and particularly of the junctions, that it is useful to capture in a model, for network extraction purposes. We thus develop a nonlocal phase field model of directed networks, which, in addition to a scalar field representing a region by its smoothed characteristic function, and interacting nonlocally so as to favour network configurations, contains a vector field representing the ‘flow' through the network branches. The vector field is strongly encouraged to be zero outside, and of unit magnitude inside the network; and to have zero divergence. This prolongs network branches; controls width variation along a branch; and produces asymmetric junctions for which total incoming branch width approximately equals total outgoing branch width. In conjunction with a new interaction function for the scalar field, it also allows a broad range of stable branch widths. The new proposed model is applied to the problem of hydrographic network extraction from VHR satellite images, and it outperforms the undirected network model.

2 - New higher-order active contour models, shape priors, and multiscale analysis: their application to road network extraction from very high resolution satellite images. T. Peng. PhD Thesis, Universite de Nice Sophia Antipolis, November 2008. Keywords : Higher-order active contour, Phase Field, Prior, Multiresolution, Road network, Very high resolution. Copyright : @PHDTHESIS{Peng08d, author = {Peng, T.}, title = {New higher-order active contour models, shape priors, and multiscale analysis: their application to road network extraction from very high resolution satellite images}, year = {2008}, month = {November}, school = {Universite de Nice Sophia Antipolis}, pdf = {http://tel.archives-ouvertes.fr/tel-00349768/fr/}, keyword = {Higher-order active contour, Phase Field, Prior, Multiresolution, Road network, Very high resolution} } Résumé : L'objectif de cette thèse est de développer et de valider des approches robustes d'extraction semi-automatique de réseaux routiers en zone urbaine dense à partir d'images satellitaires optiques à très haute résolution (THR). Nos modèles sont fondés sur une modélisation par champs de phase des contours actifs d'ordre supérieur (CAOS). Le probléme est difficile pour deux raisons principales : les images THR sont intrinsèquement complexes, et certaines zones des réseaux peuvent prendre une topologie arbitraire. Pour remédier à la complexité de l'information contenue dans les images THR, nous proposons une modélisation statistique multi-résolution des données ainsi qu'un modèle multi-résolution contraint a priori. Ces derniers permettent l'intégration des résultats de segmentation de résolution brute et de résolution fine. De plus, dans le cadre particulier de la mise à jour de réseaux routiers, nous présentons un modèle de forme a priori spécifique, dérivé d'une ancienne carte numérique issue d'un SIG. Ce terme spécifique a priori équilibre l'effet de la connaissance a priori générique apportée par le modèle de CAOS, qui décrit la forme géométrique générale des réseaux routiers. Cependant, le modèle classique de CAOS souffre d'une limitation importante : la largeur des branches du réseau est contrainte à d'être similaire au maximum du rayon de courbure des branches du réseau, fournissant ainsi un modèle non satisfaisant dans le cas de réseaux aux branches droites et étroites ou aux branches fortement incurvées et larges. Nous résolvons ce problème en proposant deux nouveaux modèles : l'un contenant un terme additionnel, nonlocal, non-linéaire de CAOS, et l'autre contenant un terme additionnel, nonlocal, linéaire de CAOS. Ces deux termes permettent le contrôle séparé de la largeur et de la courbure des branches, et fournissent une meilleure prolongation pour une même largeur. Le terme linéaire a plusieurs avantages : d'une part il se calcule plus efficacement, d'autre part il peut modéliser plusieurs largeurs de branche simultanément. Afin de remédier à la difficulté du choix des paramètres de ces modèles, nous analysons les conditions de stabilité pour une longue barre d'une largeur donnée décrite par ces énergies, et montrons ainsi comment choisir rigoureusement les paramètres des fonctions d'énergie. Des expériences sur des images satellitaires THR et la comparaison avec d'autres modèles démontrent la supériorité de nos modèles. Abstract : The objective of this thesis is to develop and validate robust approaches for the semi-automatic extraction of road networks in dense urban areas from very high resolution (VHR) optical satellite images. Our models are based on the recently developed higher-order active contour (HOAC) phase field framework. The problem is difficult for two main reasons: VHR images are intrinsically complex and network regions may have arbitrary topology. To tackle the complexity of the information contained in VHR images, we propose a multiresolution statistical data model and a multiresolution constrained prior model. They enable the integration of segmentation results from coarse resolution and fine resolution. Subsequently, for the particular case of road map updating, we present a specific shape prior model derived from an outdated GIS digital map. This specific prior term balances the effect of the generic prior knowledge carried by the HOAC model, which describes the geometric shape of road networks in general. However, the classical HOAC model suffers from a severe limitation: network branch width is constrained to be similar to maximum network branch radius of curvature, thereby providing a poor model of networks with straight narrow branches or highly curved, wide branches. We solve this problem by introducing two new models: one with an additional nonlinear nonlocal HOAC term, and one with an additional linear nonlocal HOAC term. Both terms allow separate control of branch width and branch curvature, and furnish better prolongation for the same width, but the linear term has several advantages: it is more efficient from a computational standpoint, and it is able to model multiple widths simultaneously. To cope with the difficulty of parameter selection of these models, we analyze the stability conditions for a long bar with a given width described by these energies, and hence show how to choose rigorously the parameters of the energy functions. Experiments on VHR satellite images and comparisons with other approaches demonstrate the superiority of our models.

3 - The 'Gas of circles' model and its application to tree crown extraction. P. Horvath. PhD Thesis, Universite de Szeged, Universite de Nice Sophia Antipolis, December 2007. Keywords : geometric prior, Contours actifs d'ordre supérieur, Phase Field, Gas of circles. @PHDTHESIS{horvath_these, author = {Horvath, P.}, title = {The 'Gas of circles' model and its application to tree crown extraction}, year = {2007}, month = {December}, school = {Universite de Szeged, Universite de Nice Sophia Antipolis}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_horvath_these.pdf}, keyword = {geometric prior, Contours actifs d'ordre supérieur, Phase Field, Gas of circles} }

4 - Contours actifs d'ordre supérieur et leur application à la détection de linéiques dans des images de télédétection. M. Rochery. PhD Thesis, Universite de Nice Sophia Antipolis, Sophia Antipolis, September 2005. Keywords : Active contour, Higher-order, Phase Field, Line networks, Road network. @PHDTHESIS{rochery_these, author = {Rochery, M.}, title = {Contours actifs d'ordre supérieur et leur application à la détection de linéiques dans des images de télédétection}, year = {2005}, month = {September}, school = {Universite de Nice Sophia Antipolis}, address = {Sophia Antipolis}, pdf = {http://hal.inria.fr/docs/00/04/86/28/PDF/tel-00010631.pdf}, keyword = {Active contour, Higher-order, Phase Field, Line networks, Road network} }

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8 Conference articles

1 - A theoretical and numerical study of a phase field higher-order active contour model of directed networks. A. El Ghoul and I. H. Jermyn and J. Zerubia. In The Tenth Asian Conference on Computer Vision (ACCV), Queenstown, New Zealand, November 2010. Keywords : Phase Field, Shape prior, Directed networks, Stability analysis, river extraction, remote sensing. Copyright : Springer-Verlag GmbH Berlin Heidelberg @INPROCEEDINGS{Elghoul10b, author = {El Ghoul, A. and Jermyn, I. H. and Zerubia, J.}, title = {A theoretical and numerical study of a phase field higher-order active contour model of directed networks}, year = {2010}, month = {November}, booktitle = {The Tenth Asian Conference on Computer Vision (ACCV)}, address = {Queenstown, New Zealand}, pdf = {http://hal.archives-ouvertes.fr/inria-00522443/fr/}, keyword = {Phase Field, Shape prior, Directed networks, Stability analysis, river extraction, remote sensing} } Abstract : We address the problem of quasi-automatic extraction of directed networks, which have characteristic geometric features, from images. To include the necessary prior knowledge about these geometric features, we use a phase field higher-order active contour model of directed networks. The model has a large number of unphysical parameters (weights of energy terms), and can favour different geometric structures for different parameter values. To overcome this problem, we perform a stability analysis of a long, straight bar in order to find parameter ranges that favour networks. The resulting constraints necessary to produce stable networks eliminate some parameters, replace others by physical parameters such as network branch width, and place lower and upper bounds on the values of the rest.We validate the theoretical analysis via numerical experiments, and then apply the model to the problem of hydrographic network extraction from multi-spectral VHR satellite images.

2 - Segmentation of networks from VHR remote sensing images using a directed phase field HOAC model. A. El Ghoul and I. H. Jermyn and J. Zerubia. In Proc. ISPRS Technical Commission III Symposium on Photogrammetry Computer Vision and Image Analysis (PCV), Paris, France, September 2010. Keywords : Phase Field, Shape prior, Directed networks, Road network extraction, river extraction, remote sensing. Copyright : ISPRS @INPROCEEDINGS{Elghoul10a, author = {El Ghoul, A. and Jermyn, I. H. and Zerubia, J.}, title = {Segmentation of networks from VHR remote sensing images using a directed phase field HOAC model}, year = {2010}, month = {September}, booktitle = {Proc. ISPRS Technical Commission III Symposium on Photogrammetry Computer Vision and Image Analysis (PCV)}, address = {Paris, France}, pdf = {https://hal.inria.fr/inria-00491017}, keyword = {Phase Field, Shape prior, Directed networks, Road network extraction, river extraction, remote sensing} } Abstract : We propose a new algorithm for network segmentation from VHR remote sensing images. The algorithm performs this task quasi-automatically, that is, with no human intervention except to fix some parameters. The task is made difficult by the amount of prior knowledge about network region geometry needed to perform the task, knowledge that is usually provided by a human being. To include such prior knowledge, we make use of methodological advances in region modelling: a phase field higher-order active contour of directed networks is used as the prior model for region geometry. By adjoining an approximately conserved flow to a phase field model encouraging network shapes (i.e. regions composed of branches meeting at junctions), the model favours network regions in which different branches may have very different widths, but in which width change along a branch is slow; in which branches do not come to an end, hence tending to close gaps in the network; and in which junctions show approximate ‘conservation of width’. We also introduce image models for network and background, which are validated using maximum likelihood segmentation against other possibilities. We then test the full model on VHR optical and multispectral satellite images.

3 - A phase field higher-order active contour model of directed networks. A. El Ghoul and I. H. Jermyn and J. Zerubia. In 2nd IEEE Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment, at ICCV, Kyoto, Japan, September 2009. Keywords : Geometric prior, Shape, Higher-order actif contours, Phase Field, Directed networks. Copyright : ©2009 IEEE. @INPROCEEDINGS{ElGhoul09b, author = {El Ghoul, A. and Jermyn, I. H. and Zerubia, J.}, title = {A phase field higher-order active contour model of directed networks}, year = {2009}, month = {September}, booktitle = {2nd IEEE Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment, at ICCV}, address = {Kyoto, Japan}, url = {https://hal.inria.fr/inria-00409910}, pdf = {http://hal.inria.fr/docs/00/40/99/10/PDF/nordia09aymenelghoul.pdf}, keyword = {Geometric prior, Shape, Higher-order actif contours, Phase Field, Directed networks} } Abstract : The segmentation of directed networks is an important problem in many domains, e.g. medical imaging (vascular networks) and remote sensing (river networks). Directed networks carry a unidirectional flow in each branch, which leads to characteristic geometric properties. In this paper, we present a nonlocal phase field model of directed networks. In addition to a scalar field representing a region by its smoothed characteristic function and interacting nonlocally so as to favour network configurations, the model contains a vector field representing the ‘flow’ through the network branches. The vector field is strongly encouraged to be zero outside, and of unit magnitude inside the region; and to have zero divergence. This prolongs network branches; controls width variation along a branch; and produces asymmetric junctions for which total incoming branch width approximately equals total outgoing branch width. In conjunction with a new interaction function, it also allows a broad range of stable branch widths. We analyse the energy to constrain the parameters, and show geometric experiments confirming the above behaviour. We also show a segmentation result on a synthetic river image.

4 - Inflection point model under phase field higher-order active contours for network extraction from VHR satellite images. A. El Ghoul and I. H. Jermyn and J. Zerubia. In Proc. European Signal Processing Conference (EUSIPCO), Glasgow, Scotland, August 2009. Keywords : Geometric prior, Shape, Higher-order active contour, Phase Field, remote sensing. Copyright : EURASIP @INPROCEEDINGS{ElGhoul09a, author = {El Ghoul, A. and Jermyn, I. H. and Zerubia, J.}, title = {Inflection point model under phase field higher-order active contours for network extraction from VHR satellite images}, year = {2009}, month = {August}, booktitle = {Proc. European Signal Processing Conference (EUSIPCO)}, address = {Glasgow, Scotland}, url = {http://hal.inria.fr/inria-00390446/fr/}, pdf = {http://hal.inria.fr/docs/00/39/04/46/PDF/eusipco09aymenelghoul.pdf}, keyword = {Geometric prior, Shape, Higher-order active contour, Phase Field, remote sensing} } Abstract : The segmentation of networks is important in several imaging domains, and models incorporating prior shape knowledge are often essential for the automatic performance of this task. We incorporate such knowledge via phase fields and higher-order active contours (HOACs). In this paper: we introduce an improved prior model, the phase field HOAC ‘inflection point’ model of a network; we present an improved data term for the segmentation of road networks; we confirm the robustness of the resulting model to choice of gradient descent initialization; and we illustrate these points via road network extraction results on VHR satellite images.

5 - An extended phase field higher-order active contour model for networks and its application to road network extraction from VHR satellite images. T. Peng and I. H. Jermyn and V. Prinet and J. Zerubia. In Proc. European Conference on Computer Vision (ECCV), Marseille, France, October 2008. Keywords : Dense urban area, Phase Field, Road network, Variational methods, Very high resolution. Copyright : @INPROCEEDINGS{Peng08c, author = {Peng, T. and Jermyn, I. H. and Prinet, V. and Zerubia, J.}, title = {An extended phase field higher-order active contour model for networks and its application to road network extraction from VHR satellite images}, year = {2008}, month = {October}, booktitle = {Proc. European Conference on Computer Vision (ECCV)}, address = {Marseille, France}, pdf = {http://link.springer.com/chapter/10.1007%2F978-3-540-88690-7_38}, keyword = {Dense urban area, Phase Field, Road network, Variational methods, Very high resolution} } Abstract : This paper addresses the segmentation from an image of entities that have the form of a 'network', i.e. the region in the image corresponding to the entity is composed of branches joining together at junctions, e.g. road or vascular networks. We present a new phase field higher-order active contour (HOAC) prior model for network regions, and apply it to the segmentation of road networks from very high resolution satellite images. This is a hard problem for two reasons. First, the images are complex, with much 'noise' in the road region due to cars, road markings, etc., while the background is very varied, containing many features that are locally similar to roads. Second, network regions are complex to model, because they may have arbitrary topology. In particular, we address a severe limitation of a previous model in which network branch width was constrained to be similar to maximum network branch radius of curvature, thereby providing a poor model of networks with straight narrow branches or highly curved, wide branches. To solve this problem, we propose a new HOAC prior energy term, and reformulate it as a nonlocal phase field energy. We analyse the stability of the new model, and find that in addition to solving the above problem by separating the interactions between points on the same and opposite sides of a network branch, the new model permits the modelling of two widths simultaneously. The analysis also fixes some of the model parameters in terms of network width(s). After adding a likelihood energy, we use the model to extract the road network quasi-automatically from pieces of a QuickBird image, and compare the results to other models in the literature. The results demonstrate the superiority of the new model, the importance of strong prior knowledge in general, and of the new term in particular.

6 - A `Gas of Circles' Phase Field Model and its Application to Tree Crown Extraction. P. Horvath and I. H. Jermyn. In Proc. European Signal Processing Conference (EUSIPCO), Poznan, Poland, September 2007. Keywords : Phase Field, Tree Crown Extraction. @INPROCEEDINGS{Horvath07d, author = {Horvath, P. and Jermyn, I. H.}, title = {A `Gas of Circles' Phase Field Model and its Application to Tree Crown Extraction}, year = {2007}, month = {September}, booktitle = {Proc. European Signal Processing Conference (EUSIPCO)}, address = {Poznan, Poland}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_Horvath07d.pdf}, keyword = {Phase Field, Tree Crown Extraction} } Abstract : The problem of extracting the region in the image domain corresponding to an a priori unknown number of circular objects occurs in several domains. We propose a new model of a `gas of circles', the ensemble of regions in the image domain composed of circles of a given radius. The model uses the phase field reformulation of higher-order active contours (HOACs). Phase fields possess several advantages over contour and level set approaches to region modelling, in particular for HOAC models. The reformulation allows us to benefit from these advantages without losing the strengths of the HOAC framework. Combined with a suitable likelihood energy, and applied to the tree crown extraction problem, the new model shows markedly improved performance, both in quality of results and in computation time, which is two orders of magnitude less than the HOAC level set implementation.

7 - A New Phase Field Model of a `Gas of Circles' for Tree Crown Extraction from Aerial Images. P. Horvath and I. H. Jermyn. In Proc. International Conference on Computer Analysis of Images and Patterns (CAIP), Vienna, Austria, August 2007. Keywords : Phase Field, Tree Crown Extraction. @INPROCEEDINGS{Horvath07b, author = {Horvath, P. and Jermyn, I. H.}, title = {A New Phase Field Model of a `Gas of Circles' for Tree Crown Extraction from Aerial Images}, year = {2007}, month = {August}, booktitle = {Proc. International Conference on Computer Analysis of Images and Patterns (CAIP)}, address = {Vienna, Austria}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_Horvath07b.pdf}, keyword = {Phase Field, Tree Crown Extraction} } Abstract : We describe a model for tree crown extraction from aerial images, a problem of great practical importance for the forestry industry. The novelty lies in the prior model of the region occupied by tree crowns in the image, which is a phase field version of the higher-order active contour inflection point `gas of circles' model. The model combines the strengths of the inflection point model with those of the phase field framework: it removes the `phantom circles' produced by the original `gas of circles' model, while executing two orders of magnitude faster than the contour-based inflection point model. The model has many other areas of application e.g., to imagery in nanotechnology, biology, and physics.

8 - Phase field models and higher-order active contours. M. Rochery and I. H. Jermyn and J. Zerubia. In Proc. IEEE International Conference on Computer Vision (ICCV), Beijing, China, October 2005. Keywords : Active contour, Higher-order, Shape, Line networks, Road network, Phase Field. @INPROCEEDINGS{rochery_iccv05, author = {Rochery, M. and Jermyn, I. H. and Zerubia, J.}, title = {Phase field models and higher-order active contours}, year = {2005}, month = {October}, booktitle = {Proc. IEEE International Conference on Computer Vision (ICCV)}, address = {Beijing, China}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/rochery_iccv05.pdf}, keyword = {Active contour, Higher-order, Shape, Line networks, Road network, Phase Field} } Abstract : The representation and modelling of regions is an important topic in computer vision. In this paper, we represent a region via a level set of a `phase field' function. The function is not constrained, eg to be a distance function; nevertheless, phase field energies equivalent to classical active contour energies can be defined. They represent an advantageous alternative to other methods: a linear representation space; ease of implementation (a PDE with no reinitialization); neutral initialization; greater topological freedom. We extend the basic phase field model with terms that reproduce `higher-order active contour' energies, a powerful way of including prior geometric knowledge in the active contour framework via nonlocal interactions between contour points. In addition to the above advantages, the phase field greatly simplifies the analysis and implementation of the higher-order terms. We define a phase field model that favours regions composed of thin arms meeting at junctions, combine this with image terms, and apply the model to the extraction of line networks from remote sensing images.

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