{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T14:59:35Z","timestamp":1791385175667,"version":"4.3.3"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T00:00:00Z","timestamp":1791331200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T00:00:00Z","timestamp":1791331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National Institutes of Health","award":["R01GM132651"],"award-info":[{"award-number":["R01GM132651"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2026,11]]},"abstract":"Abstract<\/jats:title>\n \n Symmetry is a fundamental property of biological systems, yet it is commonly reduced to discrete, subjective classifications that fail to capture the continuous and graded nature of symmetry in real organisms. We present a high-throughput, user-friendly computational framework for quantifying and visualizing biological symmetry as a continuous variable using\n transformation information<\/jats:italic>\n (\n \n \n $$\\mathcal{T}\\mathcal{I}$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ), an information theoretic measure that quantifies the divergence between an object and its transformed self. We establish key mathematical properties of\n \n \n $$\\mathcal{T}\\mathcal{I}$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , including invariance, equivariance, and Lipschitz continuity, that guarantee robustness to transformations such as translation, rotation, reflection, and rescaling, enabling reliable symmetry quantification across heterogeneous image datasets without manual landmarking or image alignment. Building on this theoretical foundation, we introduce a computational pipeline that automates symmetry center detection,\n \n \n $$\\mathcal{T}\\mathcal{I}$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n curve computation, and visualization of the symmetry data using functional principal component analysis (fPCA). We demonstrate the utility of this framework on two biological datasets: a large collection of angiosperm flower images, where we investigate rotational and reflectional symmetries, and a dataset of simulated fern images, where we investigate fractal self-similarity under scaling and translation. Our analysis of the floral dataset provides novel evidence of convergent evolution. In both datasets, fPCA of the resulting\n \n \n $$\\mathcal{T}\\mathcal{I}$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n profiles reveals that symmetry is best characterized as a continuum rather than a set of discrete classes, and there is subtle morphological variation that is not captured by traditional categorical approaches. Together, these results suggest that\n \n \n $$\\mathcal{T}\\mathcal{I}$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n provides a mathematically principled and biologically informative foundation for integrating symmetry into quantitative evolutionary, ecological, and developmental models.\n <\/jats:p>","DOI":"10.1007\/s11538-026-01754-9","type":"journal-article","created":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T14:33:05Z","timestamp":1791383585000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A High Throughput and User-Friendly Computational Tool for Quantifying and Visualizing Symmetry Variation in Image Datasets"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-3827-9771","authenticated-orcid":false,"given":"Liam D.","family":"O\u2019Brien","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8812-7724","authenticated-orcid":false,"given":"Punit","family":"Gandhi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6859-4659","authenticated-orcid":false,"given":"Maria-Veronica","family":"Ciocanel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6055-2868","authenticated-orcid":false,"given":"Adriana T.","family":"Dawes","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,7]]},"reference":[{"key":"1754_CR1","unstructured":"Barnsley MF (1993) Fractals Everywhere, 2nd edn., pp. 86\u201387. 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