Semi-algebraic geometry of common lines

Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded g...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Dynerman, David [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2014

Schlagwörter:	Cryo-EM Common lines Semi-algebraic geometry

Anmerkung:	© Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (

Übergeordnetes Werk:	Enthalten in: Research in the mathematical sciences - New York, NY [u.a.] : Springer, 2014, 1(2014), 1 vom: 02. Dez.
Übergeordnetes Werk:	volume:1 ; year:2014 ; number:1 ; day:02 ; month:12

Links:	Volltext

DOI / URN:	10.1186/s40687-014-0014-5

Katalog-ID:	SPR037353624

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allfields	10.1186/s40687-014-0014-5 doi (DE-627)SPR037353624 (SPR)s40687-014-0014-5-e DE-627 ger DE-627 rakwb eng Dynerman, David verfasserin aut Semi-algebraic geometry of common lines 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213 Enthalten in Research in the mathematical sciences New York, NY [u.a.] : Springer, 2014 1(2014), 1 vom: 02. Dez. (DE-627)815914725 (DE-600)2806676-5 2197-9847 nnns volume:1 year:2014 number:1 day:02 month:12 https://dx.doi.org/10.1186/s40687-014-0014-5 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 1 2014 1 02 12
spelling	10.1186/s40687-014-0014-5 doi (DE-627)SPR037353624 (SPR)s40687-014-0014-5-e DE-627 ger DE-627 rakwb eng Dynerman, David verfasserin aut Semi-algebraic geometry of common lines 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213 Enthalten in Research in the mathematical sciences New York, NY [u.a.] : Springer, 2014 1(2014), 1 vom: 02. Dez. (DE-627)815914725 (DE-600)2806676-5 2197-9847 nnns volume:1 year:2014 number:1 day:02 month:12 https://dx.doi.org/10.1186/s40687-014-0014-5 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 1 2014 1 02 12
allfields_unstemmed	10.1186/s40687-014-0014-5 doi (DE-627)SPR037353624 (SPR)s40687-014-0014-5-e DE-627 ger DE-627 rakwb eng Dynerman, David verfasserin aut Semi-algebraic geometry of common lines 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213 Enthalten in Research in the mathematical sciences New York, NY [u.a.] : Springer, 2014 1(2014), 1 vom: 02. Dez. (DE-627)815914725 (DE-600)2806676-5 2197-9847 nnns volume:1 year:2014 number:1 day:02 month:12 https://dx.doi.org/10.1186/s40687-014-0014-5 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 1 2014 1 02 12
allfieldsGer	10.1186/s40687-014-0014-5 doi (DE-627)SPR037353624 (SPR)s40687-014-0014-5-e DE-627 ger DE-627 rakwb eng Dynerman, David verfasserin aut Semi-algebraic geometry of common lines 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213 Enthalten in Research in the mathematical sciences New York, NY [u.a.] : Springer, 2014 1(2014), 1 vom: 02. Dez. (DE-627)815914725 (DE-600)2806676-5 2197-9847 nnns volume:1 year:2014 number:1 day:02 month:12 https://dx.doi.org/10.1186/s40687-014-0014-5 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 1 2014 1 02 12
allfieldsSound	10.1186/s40687-014-0014-5 doi (DE-627)SPR037353624 (SPR)s40687-014-0014-5-e DE-627 ger DE-627 rakwb eng Dynerman, David verfasserin aut Semi-algebraic geometry of common lines 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213 Enthalten in Research in the mathematical sciences New York, NY [u.a.] : Springer, 2014 1(2014), 1 vom: 02. Dez. (DE-627)815914725 (DE-600)2806676-5 2197-9847 nnns volume:1 year:2014 number:1 day:02 month:12 https://dx.doi.org/10.1186/s40687-014-0014-5 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 AR 1 2014 1 02 12
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author	Dynerman, David
spellingShingle	Dynerman, David misc Cryo-EM misc Common lines misc Semi-algebraic geometry Semi-algebraic geometry of common lines
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topic_title	Semi-algebraic geometry of common lines Cryo-EM (dpeaa)DE-He213 Common lines (dpeaa)DE-He213 Semi-algebraic geometry (dpeaa)DE-He213
topic	misc Cryo-EM misc Common lines misc Semi-algebraic geometry
topic_unstemmed	misc Cryo-EM misc Common lines misc Semi-algebraic geometry
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title	Semi-algebraic geometry of common lines
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title_full	Semi-algebraic geometry of common lines
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title_sort	semi-algebraic geometry of common lines
title_auth	Semi-algebraic geometry of common lines
abstract	Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (
abstractGer	Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (
abstract_unstemmed	Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. Conclusions The polynomials we derive provide the desired intrinsic characterization of common lines. We discuss possible applications of these polynomials to reconstruction algorithms that are robust to the high levels of noise present in cryo-electron images. © Dynerman; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (
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title_short	Semi-algebraic geometry of common lines
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up_date	2024-07-03T22:23:06.141Z
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Purpose Cryo-electron microscopy is a technique in structural biology for determining the 3D structure of macromolecules. A key step in this process is detecting common lines of intersection between unknown embedded image planes. We wish to characterize such common lines in terms of the unembedded geometric data detected in experiments. Methods We use techniques from spherical geometry, real algebraic geometry, and linear algebra. Results We show that common lines are the solutions to a system of polynomial equalities and inequalities, i.e., they form a semi-algebraic set. These polynomials are low degree, and we explicitly derive them in this paper. 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