This paper proposes a functional that assigns low `energy' to sets of subsets of the image domain consisting of a number of possibly overlapping near-circular regions of approximately a given radius: a `gas of circles'. The model can be used as a prior for object extraction whenever the objects conform to the `gas of circles' geometry, e.g. cells in biological images. Configurations are represented by a multi-layer phase field. Each layer has an associated function, regions being defined by thresholding. Intra-layer interactions assign low energy to configurations consisting of non-overlapping near-circular regions, while overlapping regions are represented in separate layers. Inter-layer interactions penalize overlaps. Here we present a theoretical and experimental analysis of the model.

1 aMolnar, Csaba1 aKato, Zoltan1 aJermyn, Ian1 aEklundh, Jan-Olof1 aOhta, Yuichi1 aTanimoto, Steven uhttps://www.inf.u-szeged.hu/en/publication/a-multi-layer-phase-field-model-for-extracting-multiple-near-circular-objects01458nas a2200121 4500008004100000020002200041245011700063210006900180260004600249300001400295520093900309856008801248 2011 eng d a978-3-642-23686-000aA Multi-Layer 'Gas of Circles' Markov Random Field Model for the Extraction of Overlapping Near-Circular Objects0 aMultiLayer Gas of Circles Markov Random Field Model for the Extr aGhent, BelgiumbSpringer-VerlagcAug 2011 a171 - 1823 a

We propose a multi-layer binary Markov random field (MRF) model that assigns high probability to object configurations in the image domain consisting of an unknown number of possibly touching or overlapping near-circular objects of approximately a given size. Each layer has an associated binary field that specifies a region corresponding to objects. Overlapping objects are represented by regions in different layers. Within each layer, long-range interactions favor connected components of approximately circular shape, while regions in different layers that overlap are penalized. Used as a prior coupled with a suitable data likelihood, the model can be used for object extraction from images, e.g. cells in biological images or densely-packed tree crowns in remote sensing images. We present a theoretical and experimental analysis of the model, and demonstrate its performance on various synthetic and biomedical images.

uhttp://www.inf.u-szeged.hu/ipcg/publications/Year/2011.complete.xml#Nemeth-etal201100655nas a2200169 4500008004100000020001400041245010700055210006900162260001200231300001400243490000700257100001900264700001600283700001700299700002100316856014800337 2009 eng d a0031-320300aA higher-order active contour model of a 'gas of circles' and its application to tree crown extraction0 ahigherorder active contour model of a gas of circles and its app c2009/// a699 - 7090 v421 aHorvath, Peter1 aJermyn, Ian1 aKato, Zoltan1 aZerubia, Josiane uhttps://www.inf.u-szeged.hu/en/publication/a-higher-order-active-contour-model-of-a-gas-of-circles-and-its-application-to-tree-crown-extraction00402nas a2200097 4500008004100000245007200041210007200113260003300185300001000218856007600228 2009 hun d00aKör alakú objektumok szegmentálása Markov mező segítségével0 aKör alakú objektumok szegmentálása Markov mező segítségével aBudapestbAkaprintcJan 2009 a1 - 9 uhttp://vision.sztaki.hu/~kepaf/kepaf2009_CD/files/116-4-MRFCircle08.pdf01108nas a2200157 4500008004100000020002300041245006800064210006500132260003300197300001600230520053700246100002300783700001700806700001600823856011100839 2009 eng d a978-1-4244-5653-6 00aA Markov random field model for extracting near-circular shapes0 aMarkov random field model for extracting nearcircular shapes aCairo, EgyptbIEEEcNov 2009 a1073 - 10763 aWe propose a binary Markov Random Field (MRF) model that assigns high probability to regions in the image domain consisting of an unknown number of circles of a given radius. We construct the model by discretizing the 'gas of circles' phase field model in a principled way, thereby creating an 'equivalent'MRF. The behaviour of the resultingMRF model is analyzed, and the performance of the new model is demonstrated on various synthetic images as well as on the problem of tree crown detection in aerial images. ©2009 IEEE.

1 aBlaskovics, Tamás1 aKato, Zoltan1 aJermyn, Ian uhttps://www.inf.u-szeged.hu/en/publication/a-markov-random-field-model-for-extracting-near-circular-shapes00593nas a2200145 4500008003900000245008600039210006900125260002500194100001900219700001600238700002000254700002300274700002300297856012700320 2007 d00aA 'gas of Circles' Phase Field Model and its Application to Tree Crown Extraction0 agas of Circles Phase Field Model and its Application to Tree Cro aPoznan, Polandc20071 aHorvath, Peter1 aJermyn, Ian1 aDomanski, Marek1 aStasinski, Ryszard1 aBartkowiak, Maciej uhttps://www.inf.u-szeged.hu/en/publication/a-gas-of-circles-phase-field-model-and-its-application-to-tree-crown-extraction00446nas a2200097 4500008004100000245009900041210007600140260007300216300001400289856004500303 2007 eng d00aKör alakú objektumok szegmentálása magasabb rendű aktív kontúr modellek segítségével0 aKör alakú objektumok szegmentálása magasabb rendű aktív kontúr m aDebrecenbKépfeldolgozók és Alakfelismerők TársaságacJan 2007 a133 - 140 uhttps://www.inf.u-szeged.hu/en/node/127500655nas a2200169 4500008003900000245009500039210006900134260002600203300001200229490000900241100001900250700001600269700002600285700001900311700001900330856013600349 2007 d00aA New Phase Field Model of a 'gas of Circles' for Tree Crown Extraction from Aerial Images0 aNew Phase Field Model of a gas of Circles for Tree Crown Extract aVienna, Austriac2007 a702-7090 v46731 aHorvath, Peter1 aJermyn, Ian1 aKropatsch, Walter, G.1 aKampel, Martin1 aHanbury, Allan uhttps://www.inf.u-szeged.hu/en/publication/a-new-phase-field-model-of-a-gas-of-circles-for-tree-crown-extraction-from-aerial-images01332nas a2200109 4500008004100000245005900041210005600100260001800156300001400174520098900188856004501177 2006 eng d00aA higher-order active contour model for tree detection0 ahigherorder active contour model for tree detection bIEEEc2006/// a130 - 1333 aWe present a model of a 'gas of circles', the ensemble of regions in the image domain consisting of an unknown number of circles with approximately fixed radius and short range repulsive interactions, and apply it to the extraction of tree crowns from aerial images. The method uses the recently introduced 'higher order active contours' (HOACs), which incorporate long-range interactions between contour points, and thereby include prior geometric information without using a template shape. This makes them ideal when looking for multiple instances of an entity in an image. We study an existing HOAC model for networks, and show via a stability calculation that circles stable to perturbations are possible for constrained parameter sets. Combining this prior energy with a data term, we show results on aerial imagery that demonstrate the effectiveness of the method and the need for prior geometric knowledge. The model has many other potential applications. © 2006 IEEE.

uhttps://www.inf.u-szeged.hu/en/node/123301538nas a2200169 4500008003900000245005900039210005600098260003300154300001400187490000600201520098600207100001901193700001601212700001701228700002101245856010201266 2006 d00aA Higher-Order Active Contour Model for Tree Detection0 aHigherOrder Active Contour Model for Tree Detection aHong Kong, ChinabIAPRc2006 a130–1330 v23 aWe present a model of a 'gas of circles', the ensemble of regions in the image domain consisting of an unknown number of circles with approximately fixed radius and short range repulsive interactions, and apply it to the extraction of tree crowns from aerial images. The method uses the recently introduced 'higher order active contours' (HOACs), which incorporate long-range interactions between contour points, and thereby include prior geometric information without using a template shape. This makes them ideal when looking for multiple instances of an entity in an image. We study an existing HOAC model for networks, and show via a stability calculation that circles stable to perturbations are possible for constrained parameter sets. Combining this prior energy with a data term, we show results on aerial imagery that demonstrate the effectiveness of the method and the need for prior geometric knowledge. The model has many other potential applications. ` `