no code implementations • 20 May 2024 • Alexandre Cafaro, Amaury Leroy, Guillaume Beldjoudi, Pauline Maury, Charlotte Robert, Eric Deutsch, Vincent Grégoire, Vincent Lepetit, Nikos Paragios
Earlier methods that work by deforming this volume to match the projections typically fail when the number of projections is very low as the alignment becomes underconstrained.
no code implementations • CVPR 2024 • Marius Schmidt-Mengin, Alexis Benichoux, Shibeshih Belachew, Nikos Komodakis, Nikos Paragios
We apply it to weakly supervised medical image segmentation by training the 2D encoder to output high values for slices containing the regions of interest.
1 code implementation • 5 Oct 2023 • Othmane Laousy, Alexandre Araujo, Guillaume Chassagnon, Nikos Paragios, Marie-Pierre Revel, Maria Vakalopoulou
In this paper, we present for the first time a certified segmentation baseline for medical imaging based on randomized smoothing and diffusion models.
1 code implementation • 18 Jun 2023 • Luuk H. Boulogne, Julian Lorenz, Daniel Kienzle, Robin Schon, Katja Ludwig, Rainer Lienhart, Simon Jegou, Guang Li, Cong Chen, Qi Wang, Derik Shi, Mayug Maniparambil, Dominik Muller, Silvan Mertes, Niklas Schroter, Fabio Hellmann, Miriam Elia, Ine Dirks, Matias Nicolas Bossa, Abel Diaz Berenguer, Tanmoy Mukherjee, Jef Vandemeulebroucke, Hichem Sahli, Nikos Deligiannis, Panagiotis Gonidakis, Ngoc Dung Huynh, Imran Razzak, Reda Bouadjenek, Mario Verdicchio, Pasquale Borrelli, Marco Aiello, James A. Meakin, Alexander Lemm, Christoph Russ, Razvan Ionasec, Nikos Paragios, Bram van Ginneken, Marie-Pierre Revel Dubois
STOIC2021 consisted of a Qualification phase, where participants developed challenge solutions using 2000 publicly available CT scans, and a Final phase, where participants submitted their training methodologies with which solutions were trained on CT scans of 9724 subjects.
1 code implementation • 26 Aug 2022 • Joseph Boyd, Irène Villa, Marie-Christine Mathieu, Eric Deutsch, Nikos Paragios, Maria Vakalopoulou, Stergios Christodoulidis
We present a use case on whole slide images, where an IHC stain provides an experimentally generated signal for metastatic cells.
no code implementations • 13 Dec 2021 • Bhakti Baheti, Satrajit Chakrabarty, Hamed Akbari, Michel Bilello, Benedikt Wiestler, Julian Schwarting, Evan Calabrese, Jeffrey Rudie, Syed Abidi, Mina Mousa, Javier Villanueva-Meyer, Brandon K. K. Fields, Florian Kofler, Russell Takeshi Shinohara, Juan Eugenio Iglesias, Tony C. W. Mok, Albert C. S. Chung, Marek Wodzinski, Artur Jurgas, Niccolo Marini, Manfredo Atzori, Henning Muller, Christoph Grobroehmer, Hanna Siebert, Lasse Hansen, Mattias P. Heinrich, Luca Canalini, Jan Klein, Annika Gerken, Stefan Heldmann, Alessa Hering, Horst K. Hahn, Mingyuan Meng, Lei Bi, Dagan Feng, Jinman Kim, Ramy A. Zeineldin, Mohamed E. Karar, Franziska Mathis-Ullrich, Oliver Burgert, Javid Abderezaei, Aymeric Pionteck, Agamdeep Chopra, Mehmet Kurt, Kewei Yan, Yonghong Yan, Zhe Tang, Jianqiang Ma, Sahar Almahfouz Nasser, Nikhil Cherian Kurian, Mohit Meena, Saqib Shamsi, Amit Sethi, Nicholas J. Tustison, Brian B. Avants, Philip Cook, James C. Gee, Lin Tian, Hastings Greer, Marc Niethammer, Andrew Hoopes, Malte Hoffmann, Adrian V. Dalca, Stergios Christodoulidis, Theo Estiene, Maria Vakalopoulou, Nikos Paragios, Daniel S. Marcus, Christos Davatzikos, Aristeidis Sotiras, Bjoern Menze, Spyridon Bakas, Diana Waldmannstetter
Registration of longitudinal brain MRI scans containing pathologies is challenging due to dramatic changes in tissue appearance.
no code implementations • 23 Nov 2021 • Théo Estienne, Maria Vakalopoulou, Enzo Battistella, Theophraste Henry, Marvin Lerousseau, Amaury Leroy, Nikos Paragios, Eric Deutsch
Deep learning overtook this limitation by taking advantage of GPU calculation and the learning process.
1 code implementation • 7 Sep 2021 • Joseph Boyd, Mykola Liashuha, Eric Deutsch, Nikos Paragios, Stergios Christodoulidis, Maria Vakalopoulou
In this study, we propose a novel generative framework that can learn powerful representations for such tiles by learning to plausibly expand their visual field.
no code implementations • 27 Jul 2021 • Othmane Laousy, Guillaume Chassagnon, Edouard Oyallon, Nikos Paragios, Marie-Pierre Revel, Maria Vakalopoulou
In this paper, we propose a deep reinforcement learning method for accurate localization of the L3 CT slice.
no code implementations • 23 Jul 2021 • Théo Estienne, Maria Vakalopoulou, Stergios Christodoulidis, Enzo Battistella, Théophraste Henry, Marvin Lerousseau, Amaury Leroy, Guillaume Chassagnon, Marie-Pierre Revel, Nikos Paragios, Eric Deutsch
Explainability of deep neural networks is one of the most challenging and interesting problems in the field.
no code implementations • MICCAI Workshop COMPAY 2021 • Amaury Leroy, Kumar Shreshtha, Marvin Lerousseau, Théophraste Henry, Théo Estienne, Marion Classe, Nikos Paragios, Vincent Grégoire, Eric Deutsch
The pathological analysis of biopsy specimens is essential to cancer diagnosis, treatment selection and prognosis.
no code implementations • 10 May 2021 • Marvin Lerousseau, Marion Classe, Enzo Battistella, Théo Estienne, Théophraste Henry, Amaury Leroy, Roger Sun, Maria Vakalopoulou, Jean-Yves Scoazec, Eric Deutsch, Nikos Paragios
The vast majority of semantic segmentation approaches rely on pixel-level annotations that are tedious and time consuming to obtain and suffer from significant inter and intra-expert variability.
1 code implementation • MICCAI Workshop COMPAY 2021 • Marvin Lerousseau, Maria Vakalopoulou, Eric Deutsch, Nikos Paragios
Multiple instance learning (MIL) is the preferred approach for whole slide image classification.
no code implementations • 11 Feb 2021 • Enzo Battistella, Maria Vakalopoulou, Roger Sun, Théo Estienne, Marvin Lerousseau, Sergey Nikolaev, Emilie Alvarez Andres, Alexandre Carré, Stéphane Niyoteka, Charlotte Robert, Nikos Paragios, Eric Deutsch
Precision medicine is a paradigm shift in healthcare relying heavily on genomics data.
1 code implementation • 30 Oct 2020 • Theophraste Henry, Alexandre Carre, Marvin Lerousseau, Theo Estienne, Charlotte Robert, Nikos Paragios, Eric Deutsch
Brain tumor segmentation is a critical task for patient's disease management.
2 code implementations • 21 Oct 2020 • Théo Estienne, Maria Vakalopoulou, Enzo Battistella, Alexandre Carré, Théophraste Henry, Marvin Lerousseau, Charlotte Robert, Nikos Paragios, Eric Deutsch
Image registration is one of the most challenging problems in medical image analysis.
no code implementations • 3 Sep 2020 • Marvin Lerousseau, Eric Deutsh, Nikos Paragios
validation) balanced-accuracy, kappa and f1 of 0. 913, 0. 897 and 0. 951 (resp.
1 code implementation • 16 Jul 2020 • Mihir Sahasrabudhe, Stergios Christodoulidis, Roberto Salgado, Stefan Michiels, Sherene Loi, Fabrice André, Nikos Paragios, Maria Vakalopoulou
We show that the identification of the magnification level for tiles can generate a preliminary self-supervision signal to locate nuclei.
no code implementations • 20 Apr 2020 • Guillaume Chassagnon, Maria Vakalopoulou, Enzo Battistella, Stergios Christodoulidis, Trieu-Nghi Hoang-Thi, Severine Dangeard, Eric Deutsch, Fabrice Andre, Enora Guillo, Nara Halm, Stefany El Hajj, Florian Bompard, Sophie Neveu, Chahinez Hani, Ines Saab, Alienor Campredon, Hasmik Koulakian, Souhail Bennani, Gael Freche, Aurelien Lombard, Laure Fournier, Hippolyte Monnier, Teodor Grand, Jules Gregory, Antoine Khalil, Elyas Mahdjoub, Pierre-Yves Brillet, Stephane Tran Ba, Valerie Bousson, Marie-Pierre Revel, Nikos Paragios
Chest computed tomography (CT) is widely used for the management of Coronavirus disease 2019 (COVID-19) pneumonia because of its availability and rapidity.
1 code implementation • 10 Apr 2020 • Marvin Lerousseau, Maria Vakalopoulou, Marion Classe, Julien Adam, Enzo Battistella, Alexandre Carré, Théo Estienne, Théophraste Henry, Eric Deutsch, Nikos Paragios
In this paper, we propose a weakly supervised framework for whole slide imaging segmentation that relies on standard clinical annotations, available in most medical systems.
1 code implementation • 20 Mar 2020 • Théo Estienne, Marvin Lerousseau, Maria Vakalopoulou, Emilie Alvarez Andres, Enzo Battistella, Alexandre Carré, Siddhartha Chandra, Stergios Christodoulidis, Mihir Sahasrabudhe, Roger Sun, Charlotte Robert, Hugues Talbot, Nikos Paragios, Eric Deutsch
Image registration and segmentation are the two most studied problems in medical image analysis.
1 code implementation • 10 Oct 2019 • Théo Estienne, Maria Vakalopoulou, Stergios Christodoulidis, Enzo Battistella, Marvin Lerousseau, Alexandre Carre, Guillaume Klausner, Roger Sun, Charlotte Robert, Stavroula Mougiakakou, Nikos Paragios, Eric Deutsch
We evaluated the proposed architecture using the publicly available OASIS 3 dataset, measuring the dice coefficient metric for both registration and segmentation tasks.
1 code implementation • 5 Nov 2018 • Spyridon Bakas, Mauricio Reyes, Andras Jakab, Stefan Bauer, Markus Rempfler, Alessandro Crimi, Russell Takeshi Shinohara, Christoph Berger, Sung Min Ha, Martin Rozycki, Marcel Prastawa, Esther Alberts, Jana Lipkova, John Freymann, Justin Kirby, Michel Bilello, Hassan Fathallah-Shaykh, Roland Wiest, Jan Kirschke, Benedikt Wiestler, Rivka Colen, Aikaterini Kotrotsou, Pamela Lamontagne, Daniel Marcus, Mikhail Milchenko, Arash Nazeri, Marc-Andre Weber, Abhishek Mahajan, Ujjwal Baid, Elizabeth Gerstner, Dongjin Kwon, Gagan Acharya, Manu Agarwal, Mahbubul Alam, Alberto Albiol, Antonio Albiol, Francisco J. Albiol, Varghese Alex, Nigel Allinson, Pedro H. A. Amorim, Abhijit Amrutkar, Ganesh Anand, Simon Andermatt, Tal Arbel, Pablo Arbelaez, Aaron Avery, Muneeza Azmat, Pranjal B., W Bai, Subhashis Banerjee, Bill Barth, Thomas Batchelder, Kayhan Batmanghelich, Enzo Battistella, Andrew Beers, Mikhail Belyaev, Martin Bendszus, Eze Benson, Jose Bernal, Halandur Nagaraja Bharath, George Biros, Sotirios Bisdas, James Brown, Mariano Cabezas, Shilei Cao, Jorge M. Cardoso, Eric N Carver, Adrià Casamitjana, Laura Silvana Castillo, Marcel Catà, Philippe Cattin, Albert Cerigues, Vinicius S. Chagas, Siddhartha Chandra, Yi-Ju Chang, Shiyu Chang, Ken Chang, Joseph Chazalon, Shengcong Chen, Wei Chen, Jefferson W. Chen, Zhaolin Chen, Kun Cheng, Ahana Roy Choudhury, Roger Chylla, Albert Clérigues, Steven Colleman, Ramiro German Rodriguez Colmeiro, Marc Combalia, Anthony Costa, Xiaomeng Cui, Zhenzhen Dai, Lutao Dai, Laura Alexandra Daza, Eric Deutsch, Changxing Ding, Chao Dong, Shidu Dong, Wojciech Dudzik, Zach Eaton-Rosen, Gary Egan, Guilherme Escudero, Théo Estienne, Richard Everson, Jonathan Fabrizio, Yong Fan, Longwei Fang, Xue Feng, Enzo Ferrante, Lucas Fidon, Martin Fischer, Andrew P. French, Naomi Fridman, Huan Fu, David Fuentes, Yaozong Gao, Evan Gates, David Gering, Amir Gholami, Willi Gierke, Ben Glocker, Mingming Gong, Sandra González-Villá, T. Grosges, Yuanfang Guan, Sheng Guo, Sudeep Gupta, Woo-Sup Han, Il Song Han, Konstantin Harmuth, Huiguang He, Aura Hernández-Sabaté, Evelyn Herrmann, Naveen Himthani, Winston Hsu, Cheyu Hsu, Xiaojun Hu, Xiaobin Hu, Yan Hu, Yifan Hu, Rui Hua, Teng-Yi Huang, Weilin Huang, Sabine Van Huffel, Quan Huo, Vivek HV, Khan M. Iftekharuddin, Fabian Isensee, Mobarakol Islam, Aaron S. Jackson, Sachin R. Jambawalikar, Andrew Jesson, Weijian Jian, Peter Jin, V Jeya Maria Jose, Alain Jungo, B Kainz, Konstantinos Kamnitsas, Po-Yu Kao, Ayush Karnawat, Thomas Kellermeier, Adel Kermi, Kurt Keutzer, Mohamed Tarek Khadir, Mahendra Khened, Philipp Kickingereder, Geena Kim, Nik King, Haley Knapp, Urspeter Knecht, Lisa Kohli, Deren Kong, Xiangmao Kong, Simon Koppers, Avinash Kori, Ganapathy Krishnamurthi, Egor Krivov, Piyush Kumar, Kaisar Kushibar, Dmitrii Lachinov, Tryphon Lambrou, Joon Lee, Chengen Lee, Yuehchou Lee, M Lee, Szidonia Lefkovits, Laszlo Lefkovits, James Levitt, Tengfei Li, Hongwei Li, Hongyang Li, Xiaochuan Li, Yuexiang Li, Heng Li, Zhenye Li, Xiaoyu Li, Zeju Li, Xiaogang Li, Wenqi Li, Zheng-Shen Lin, Fengming Lin, Pietro Lio, Chang Liu, Boqiang Liu, Xiang Liu, Mingyuan Liu, Ju Liu, Luyan Liu, Xavier Llado, Marc Moreno Lopez, Pablo Ribalta Lorenzo, Zhentai Lu, Lin Luo, Zhigang Luo, Jun Ma, Kai Ma, Thomas Mackie, Anant Madabushi, Issam Mahmoudi, Klaus H. Maier-Hein, Pradipta Maji, CP Mammen, Andreas Mang, B. S. Manjunath, Michal Marcinkiewicz, S McDonagh, Stephen McKenna, Richard McKinley, Miriam Mehl, Sachin Mehta, Raghav Mehta, Raphael Meier, Christoph Meinel, Dorit Merhof, Craig Meyer, Robert Miller, Sushmita Mitra, Aliasgar Moiyadi, David Molina-Garcia, Miguel A. B. Monteiro, Grzegorz Mrukwa, Andriy Myronenko, Jakub Nalepa, Thuyen Ngo, Dong Nie, Holly Ning, Chen Niu, Nicholas K Nuechterlein, Eric Oermann, Arlindo Oliveira, Diego D. C. Oliveira, Arnau Oliver, Alexander F. I. Osman, Yu-Nian Ou, Sebastien Ourselin, Nikos Paragios, Moo Sung Park, Brad Paschke, J. Gregory Pauloski, Kamlesh Pawar, Nick Pawlowski, Linmin Pei, Suting Peng, Silvio M. Pereira, Julian Perez-Beteta, Victor M. Perez-Garcia, Simon Pezold, Bao Pham, Ashish Phophalia, Gemma Piella, G. N. Pillai, Marie Piraud, Maxim Pisov, Anmol Popli, Michael P. Pound, Reza Pourreza, Prateek Prasanna, Vesna Prkovska, Tony P. Pridmore, Santi Puch, Élodie Puybareau, Buyue Qian, Xu Qiao, Martin Rajchl, Swapnil Rane, Michael Rebsamen, Hongliang Ren, Xuhua Ren, Karthik Revanuru, Mina Rezaei, Oliver Rippel, Luis Carlos Rivera, Charlotte Robert, Bruce Rosen, Daniel Rueckert, Mohammed Safwan, Mostafa Salem, Joaquim Salvi, Irina Sanchez, Irina Sánchez, Heitor M. Santos, Emmett Sartor, Dawid Schellingerhout, Klaudius Scheufele, Matthew R. Scott, Artur A. Scussel, Sara Sedlar, Juan Pablo Serrano-Rubio, N. Jon Shah, Nameetha Shah, Mazhar Shaikh, B. Uma Shankar, Zeina Shboul, Haipeng Shen, Dinggang Shen, Linlin Shen, Haocheng Shen, Varun Shenoy, Feng Shi, Hyung Eun Shin, Hai Shu, Diana Sima, M Sinclair, Orjan Smedby, James M. Snyder, Mohammadreza Soltaninejad, Guidong Song, Mehul Soni, Jean Stawiaski, Shashank Subramanian, Li Sun, Roger Sun, Jiawei Sun, Kay Sun, Yu Sun, Guoxia Sun, Shuang Sun, Yannick R Suter, Laszlo Szilagyi, Sanjay Talbar, DaCheng Tao, Zhongzhao Teng, Siddhesh Thakur, Meenakshi H Thakur, Sameer Tharakan, Pallavi Tiwari, Guillaume Tochon, Tuan Tran, Yuhsiang M. Tsai, Kuan-Lun Tseng, Tran Anh Tuan, Vadim Turlapov, Nicholas Tustison, Maria Vakalopoulou, Sergi Valverde, Rami Vanguri, Evgeny Vasiliev, Jonathan Ventura, Luis Vera, Tom Vercauteren, C. A. Verrastro, Lasitha Vidyaratne, Veronica Vilaplana, Ajeet Vivekanandan, Qian Wang, Chiatse J. Wang, Wei-Chung Wang, Duo Wang, Ruixuan Wang, Yuanyuan Wang, Chunliang Wang, Guotai Wang, Ning Wen, Xin Wen, Leon Weninger, Wolfgang Wick, Shaocheng Wu, Qiang Wu, Yihong Wu, Yong Xia, Yanwu Xu, Xiaowen Xu, Peiyuan Xu, Tsai-Ling Yang, Xiaoping Yang, Hao-Yu Yang, Junlin Yang, Haojin Yang, Guang Yang, Hongdou Yao, Xujiong Ye, Changchang Yin, Brett Young-Moxon, Jinhua Yu, Xiangyu Yue, Songtao Zhang, Angela Zhang, Kun Zhang, Xue-jie Zhang, Lichi Zhang, Xiaoyue Zhang, Yazhuo Zhang, Lei Zhang, Jian-Guo Zhang, Xiang Zhang, Tianhao Zhang, Sicheng Zhao, Yu Zhao, Xiaomei Zhao, Liang Zhao, Yefeng Zheng, Liming Zhong, Chenhong Zhou, Xiaobing Zhou, Fan Zhou, Hongtu Zhu, Jin Zhu, Ying Zhuge, Weiwei Zong, Jayashree Kalpathy-Cramer, Keyvan Farahani, Christos Davatzikos, Koen van Leemput, Bjoern Menze
This study assesses the state-of-the-art machine learning (ML) methods used for brain tumor image analysis in mpMRI scans, during the last seven instances of the International Brain Tumor Segmentation (BraTS) challenge, i. e., 2012-2018.
no code implementations • 24 Sep 2018 • Enzo Ferrante, Puneet K. Dokania, Rafael Marini Silva, Nikos Paragios
Conventional approaches refer to the definition of a similarity criterion that, once endowed with a deformation model and a smoothness constraint, determines the optimal transformation to align two given images.
1 code implementation • 13 Sep 2018 • Stergios Christodoulidis, Mihir Sahasrabudhe, Maria Vakalopoulou, Guillaume Chassagnon, Marie-Pierre Revel, Stavroula Mougiakakou, Nikos Paragios
Image registration and in particular deformable registration methods are pillars of medical imaging.
2 code implementations • ECCV 2018 • Zhixin Shu, Mihir Sahasrabudhe, Alp Guler, Dimitris Samaras, Nikos Paragios, Iasonas Kokkinos
In this work we introduce Deforming Autoencoders, a generative model for images that disentangles shape from appearance in an unsupervised manner.
Ranked #9 on
Unsupervised Facial Landmark Detection
on MAFL
2 code implementations • CVPR 2018 • D. Khuê Lê-Huu, Nikos Paragios
In this paper, we study a nonconvex continuous relaxation of MAP inference in discrete Markov random fields (MRFs).
no code implementations • CVPR 2017 • Hariprasad Kannan, Nikos Komodakis, Nikos Paragios
Linear programming relaxations are central to {\sc map} inference in discrete Markov Random Fields.
2 code implementations • 19 Jul 2017 • Afshine Amidi, Shervine Amidi, Dimitrios Vlachakis, Vasileios Megalooikonomou, Nikos Paragios, Evangelia I. Zacharaki
During the past decade, with the significant progress of computational power as well as ever-rising data availability, deep learning techniques became increasingly popular due to their excellent performance on computer vision problems.
no code implementations • 19 Jul 2017 • Enzo Ferrante, Puneet K. Dokania, Rafael Marini, Nikos Paragios
We propose a novel weakly supervised discriminative algorithm for learning context specific registration metrics as a linear combination of conventional similarity measures.
no code implementations • 6 Feb 2017 • Enzo Ferrante, Nikos Paragios
During the last decades, the research community of medical imaging has witnessed continuous advances in image registration methods, which pushed the limits of the state-of-the-art and enabled the development of novel medical procedures.
2 code implementations • CVPR 2017 • D. Khuê Lê-Huu, Nikos Paragios
In this paper, we introduce a graph matching method that can account for constraints of arbitrary order, with arbitrary potential functions.
1 code implementation • 19 Aug 2016 • Roque Porchetto, Franco Stramana, Nikos Paragios, Enzo Ferrante
Rigid slice-to-volume registration is a challenging task, which finds application in medical imaging problems like image fusion for image guided surgeries and motion correction for volume reconstruction.
no code implementations • 22 Jul 2016 • Mahsa Shakeri, Enzo Ferrante, Stavros Tsogkas, Sarah Lippe, Samuel Kadoury, Iasonas Kokkinos, Nikos Paragios
We propose a modular and scalable framework for dense coregistration and cosegmentation with two key characteristics: first, we substitute ground truth data with the semantic map output of a classifier; second, we combine this output with population deformable registration to improve both alignment and segmentation.
no code implementations • 5 Feb 2016 • Mahsa Shakeri, Stavros Tsogkas, Enzo Ferrante, Sarah Lippe, Samuel Kadoury, Nikos Paragios, Iasonas Kokkinos
In this paper we propose a deep learning approach for segmenting sub-cortical structures of the human brain in Magnetic Resonance (MR) image data.
no code implementations • 30 Aug 2013 • Pierre-Yves Baudin, Danny Goodman, Puneet Kumar, Noura Azzabou, Pierre G. Carlier, Nikos Paragios, M. Pawan Kumar
However, one of the main drawbacks of using the RW algorithm is that its parameters have to be hand-tuned.
no code implementations • 5 Jun 2013 • Pierre-Yves Baudin, Danny Goodman, Puneet Kumar, Noura Azzabou, Pierre G. Carlier, Nikos Paragios, M. Pawan Kumar
However, one of the main drawbacks of using the RW algorithm is that its parameters have to be hand-tuned.
no code implementations • NeurIPS 2009 • Jean Honorio, Dimitris Samaras, Nikos Paragios, Rita Goldstein, Luis E. Ortiz
Locality information is crucial in datasets where each variable corresponds to a measurement in a manifold (silhouettes, motion trajectories, 2D and 3D images).
1 code implementation • NeurIPS 2008 • Nikos Komodakis, Nikos Paragios, Georgios Tziritas
To deal with the most critical issue in a center-based clustering algorithm (selection of cluster centers), we also introduce the notion of stability of a cluster center, which is a well defined LP-based quantity that plays a key role to our algorithm's success.