Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory

Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Effenberger, Felix [verfasserIn] Weiskopf, Daniel [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2010

Schlagwörter:	Vector field topology Interpolation Barycentric interpolation Linear interpolation Bilinear interpolation Level sets Higher-order singularities Computational group theory Colorings

Übergeordnetes Werk:	Enthalten in: Computing and visualization in science - Berlin : Springer, 1997, 13(2010), 8 vom: Dez., Seite 377-396
Übergeordnetes Werk:	volume:13 ; year:2010 ; number:8 ; month:12 ; pages:377-396

Links:	Volltext

DOI / URN:	10.1007/s00791-011-0152-x

Katalog-ID:	SPR007850727

Internformat


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520			\|a Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed.
650		4	\|a Vector field topology \|7 (dpeaa)DE-He213
650		4	\|a Interpolation \|7 (dpeaa)DE-He213
650		4	\|a Barycentric interpolation \|7 (dpeaa)DE-He213
650		4	\|a Linear interpolation \|7 (dpeaa)DE-He213
650		4	\|a Bilinear interpolation \|7 (dpeaa)DE-He213
650		4	\|a Level sets \|7 (dpeaa)DE-He213
650		4	\|a Higher-order singularities \|7 (dpeaa)DE-He213
650		4	\|a Computational group theory \|7 (dpeaa)DE-He213
650		4	\|a Colorings \|7 (dpeaa)DE-He213
700	1		\|a Weiskopf, Daniel \|e verfasserin \|4 aut
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matchkey_str	article:14330369:2010----::idnadlsiynciiapito2vcofedaelretdp
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allfields	10.1007/s00791-011-0152-x doi (DE-627)SPR007850727 (SPR)s00791-011-0152-x-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213 Weiskopf, Daniel verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://dx.doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 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_267 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_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 13 2010 8 12 377-396
spelling	10.1007/s00791-011-0152-x doi (DE-627)SPR007850727 (SPR)s00791-011-0152-x-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213 Weiskopf, Daniel verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://dx.doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 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_267 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_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 13 2010 8 12 377-396
allfields_unstemmed	10.1007/s00791-011-0152-x doi (DE-627)SPR007850727 (SPR)s00791-011-0152-x-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213 Weiskopf, Daniel verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://dx.doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 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_267 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_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 13 2010 8 12 377-396
allfieldsGer	10.1007/s00791-011-0152-x doi (DE-627)SPR007850727 (SPR)s00791-011-0152-x-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213 Weiskopf, Daniel verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://dx.doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 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_267 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_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 13 2010 8 12 377-396
allfieldsSound	10.1007/s00791-011-0152-x doi (DE-627)SPR007850727 (SPR)s00791-011-0152-x-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213 Weiskopf, Daniel verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://dx.doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 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_267 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_2070 GBV_ILN_2086 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_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 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_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 13 2010 8 12 377-396
language	English
source	Enthalten in Computing and visualization in science 13(2010), 8 vom: Dez., Seite 377-396 volume:13 year:2010 number:8 month:12 pages:377-396
sourceStr	Enthalten in Computing and visualization in science 13(2010), 8 vom: Dez., Seite 377-396 volume:13 year:2010 number:8 month:12 pages:377-396
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Vector field topology Interpolation Barycentric interpolation Linear interpolation Bilinear interpolation Level sets Higher-order singularities Computational group theory Colorings
dewey-raw	570
isfreeaccess_bool	false
container_title	Computing and visualization in science
authorswithroles_txt_mv	Effenberger, Felix @@aut@@ Weiskopf, Daniel @@aut@@
publishDateDaySort_date	2010-12-01T00:00:00Z
hierarchy_top_id	253722012
dewey-sort	3570
id	SPR007850727
language_de	englisch
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author	Effenberger, Felix
spellingShingle	Effenberger, Felix ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Vector field topology misc Interpolation misc Barycentric interpolation misc Linear interpolation misc Bilinear interpolation misc Level sets misc Higher-order singularities misc Computational group theory misc Colorings Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
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topic_title	570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory Vector field topology (dpeaa)DE-He213 Interpolation (dpeaa)DE-He213 Barycentric interpolation (dpeaa)DE-He213 Linear interpolation (dpeaa)DE-He213 Bilinear interpolation (dpeaa)DE-He213 Level sets (dpeaa)DE-He213 Higher-order singularities (dpeaa)DE-He213 Computational group theory (dpeaa)DE-He213 Colorings (dpeaa)DE-He213
topic	ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Vector field topology misc Interpolation misc Barycentric interpolation misc Linear interpolation misc Bilinear interpolation misc Level sets misc Higher-order singularities misc Computational group theory misc Colorings
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title	Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
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title_full	Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
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title_sort	finding and classifying critical points of 2d vector fields: a cell-oriented approach using group theory
title_auth	Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
abstract	Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed.
abstractGer	Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed.
abstract_unstemmed	Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed.
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container_issue	8
title_short	Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
url	https://dx.doi.org/10.1007/s00791-011-0152-x
remote_bool	true
author2	Weiskopf, Daniel
author2Str	Weiskopf, Daniel
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doi_str	10.1007/s00791-011-0152-x
up_date	2024-07-03T15:39:48.718Z
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Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory

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