Pub Date : 2021-01-01DOI: 10.1007/978-3-030-56215-1_8
Maëliss Jallais, D. Wassermann
{"title":"Single Encoding Diffusion MRI: A Probe to Brain Anisotropy","authors":"Maëliss Jallais, D. Wassermann","doi":"10.1007/978-3-030-56215-1_8","DOIUrl":"https://doi.org/10.1007/978-3-030-56215-1_8","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50967686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-56215-1_1
Magnus Herberthson, Evren Özarslan, Carl-Fredrik Westin
Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor Rabcd. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor Rabcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors Rabcd and . In terms of components, such an equivalence means that components Rijkl of the first tensor will transform into the components of the second tensor for some change of the coordinate system.
{"title":"Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions.","authors":"Magnus Herberthson, Evren Özarslan, Carl-Fredrik Westin","doi":"10.1007/978-3-030-56215-1_1","DOIUrl":"https://doi.org/10.1007/978-3-030-56215-1_1","url":null,"abstract":"<p><p>Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor <i>R</i><sub><i>abcd</i></sub>. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor <i>R</i><sub><i>abcd</i></sub> the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors <i>R</i><sub><i>abcd</i></sub> and <math><mrow><msub><mover><mi>R</mi><mo>~</mo></mover><mrow><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mrow></msub></mrow></math>. In terms of components, such an equivalence means that components <i>R</i><sub><i>ijkl</i></sub> of the first tensor will transform into the components <math><mrow><msub><mover><mi>R</mi><mo>~</mo></mover><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub></mrow></math> of the second tensor for some change of the coordinate system.</p>","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"2021 ","pages":"3-22"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10201932/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9887503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-83500-2_14
Dennis Mosbach, K. Schladitz, B. Hamann, H. Hagen
{"title":"Topologically Robust B-spline Reconstruction of Fibers from 3D Images","authors":"Dennis Mosbach, K. Schladitz, B. Hamann, H. Hagen","doi":"10.1007/978-3-030-83500-2_14","DOIUrl":"https://doi.org/10.1007/978-3-030-83500-2_14","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"34 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50980907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-56215-1_11
L. Florack, Rick Sengers, S. Meesters, L. Smolders, A. Fuster
{"title":"Riemann-DTI Geodesic Tractography Revisited","authors":"L. Florack, Rick Sengers, S. Meesters, L. Smolders, A. Fuster","doi":"10.1007/978-3-030-56215-1_11","DOIUrl":"https://doi.org/10.1007/978-3-030-56215-1_11","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"14 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50968017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-56215-1_13
P. Slator, A. Ho, S. Bakalis, L. Jackson, L. Chappell, D. Alexander, J. Hajnal, M. Rutherford, J. Hutter
{"title":"Anisotropy in the Human Placenta in Pregnancies Complicated by Fetal Growth Restriction","authors":"P. Slator, A. Ho, S. Bakalis, L. Jackson, L. Chappell, D. Alexander, J. Hajnal, M. Rutherford, J. Hutter","doi":"10.1007/978-3-030-56215-1_13","DOIUrl":"https://doi.org/10.1007/978-3-030-56215-1_13","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50968115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-83500-2_9
Alexander Straub, G. Karch, F. Sadlo, T. Ertl
{"title":"Implicit Visualization of 2D Vector Field Topology for Periodic Orbit Detection","authors":"Alexander Straub, G. Karch, F. Sadlo, T. Ertl","doi":"10.1007/978-3-030-83500-2_9","DOIUrl":"https://doi.org/10.1007/978-3-030-83500-2_9","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"14 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50981031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-56215-1_7
Daniel Jörgens, M. Descoteaux, R. Moreno
{"title":"Challenges for Tractogram Filtering","authors":"Daniel Jörgens, M. Descoteaux, R. Moreno","doi":"10.1007/978-3-030-56215-1_7","DOIUrl":"https://doi.org/10.1007/978-3-030-56215-1_7","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"152 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50967675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-83500-2_15
Tobias Günther, Irene Baeza Rojo
{"title":"Introduction to Vector Field Topology","authors":"Tobias Günther, Irene Baeza Rojo","doi":"10.1007/978-3-030-83500-2_15","DOIUrl":"https://doi.org/10.1007/978-3-030-83500-2_15","url":null,"abstract":"","PeriodicalId":74126,"journal":{"name":"Mathematics and visualization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50980912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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