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As a diffeomorphic transformation should have positive Jacobian determinant\u00a0$$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> everywhere, the number of pixels\u00a0(2D) or voxels\u00a0(3D) with $$\\vert J\\vert <0$$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is commonly approximated using a central difference, but this strategy can yield positive $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2019s for transformations that are clearly not diffeomorphic\u2014even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2019s must be positive to ensure that the entire domain is invertible and free of folding at the pixel level. For a 3D transformation, ten unique finite differences approximations of $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2019s are required to be positive. Our proposed digital diffeomorphism<\/jats:italic> criteria solves several errors inherent in the central difference approximation of $$\\vert J\\vert $$<\/jats:tex-math>\n \n |<\/mml:mo>\n J<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https:\/\/github.com\/yihao6\/digital_diffeomorphism<\/jats:ext-link>.<\/jats:p>","DOI":"10.1007\/s11263-024-02047-1","type":"journal-article","created":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T17:01:36Z","timestamp":1713459696000},"page":"3678-3688","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["On Finite Difference Jacobian Computation in\u00a0Deformable\u00a0Image \u00a0Registration"],"prefix":"10.1007","volume":"132","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3187-9903","authenticated-orcid":false,"given":"Yihao","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junyu","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuwen","family":"Wei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aaron","family":"Carass","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jerry","family":"Prince","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,4,18]]},"reference":[{"issue":"1","key":"2047_CR1","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.media.2007.06.004","volume":"12","author":"BB Avants","year":"2008","unstructured":"Avants, B. 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