{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T22:15:59Z","timestamp":1773958559843,"version":"3.50.1"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T00:00:00Z","timestamp":1662940800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T00:00:00Z","timestamp":1662940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Bergen"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The alpha complex efficiently computes persistent homology of a point cloud <jats:inline-formula><jats:alternatives><jats:tex-math>$$X$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in Euclidean space when the dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>d<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is low. Given a subset <jats:inline-formula><jats:alternatives><jats:tex-math>$$A$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>A<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$X$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, relative \u010cech persistent homology can be computed as the persistent homology of the relative \u010cech complex <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\check{\\mathrm{C}}}(X, A)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mover>\n                      <mml:mi>C<\/mml:mi>\n                      <mml:mo>\u02c7<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. However, this is not computationally feasible for larger point clouds\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$X$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The aim of this note is to present a method for efficient computation of relative \u010cech persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay\u2013\u010cech complex <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\text {Del}\\check{\\mathrm{C}}}(X, A)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mtext>Del<\/mml:mtext>\n                      <mml:mover>\n                        <mml:mi>C<\/mml:mi>\n                        <mml:mo>\u02c7<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> whose homology is the relative \u010cech persistent homology. It is constructed from the Delaunay complex of an embedding of <jats:inline-formula><jats:alternatives><jats:tex-math>$$X$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:tex-math>$$(d+1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional Euclidean space.<\/jats:p>","DOI":"10.1007\/s00454-022-00421-9","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T15:03:01Z","timestamp":1662994981000},"page":"949-963","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Relative Persistent Homology"],"prefix":"10.1007","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9489-1657","authenticated-orcid":false,"given":"Nello","family":"Blaser","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Morten","family":"Brun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,12]]},"reference":[{"issue":"5","key":"421_CR1","doi-asserted-by":"publisher","first-page":"3741","DOI":"10.1090\/tran\/6991","volume":"369","author":"U Bauer","year":"2017","unstructured":"Bauer, U., Edelsbrunner, H.: The Morse theory of \u010cech and Delaunay complexes. 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