{"doi":"10.1007/s11263-024-02047-1","title":"On Finite Difference Jacobian Computation in Deformable Image  Registration","abstract":"Abstract Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> everywhere, the number of pixels (2D) or voxels (3D) with $$\\vert J\\vert &lt;0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> is commonly approximated using a central difference, but this strategy can yield positive $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> ’s for transformations that are clearly not diffeomorphic—even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> ’s 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 $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> ’s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of $$\\vert J\\vert $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>J</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism .","journal":"International Journal of Computer Vision","year":2024,"id":426410,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":17,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9532,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2024-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":993446,"name":"Junyu Chen","orcid":"0000-0002-3407-1005","position":1,"is_corresponding":false},{"id":952164,"name":"Shuwen Wei","orcid":"0000-0001-8679-9615","position":2,"is_corresponding":false},{"id":249457,"name":"Aaron Carass","orcid":"0000-0003-4939-5085","position":3,"is_corresponding":false},{"id":249467,"name":"Jerry L. Prince","orcid":"0000-0002-6553-0876","position":4,"is_corresponding":false},{"id":636217,"name":"Yihao Liu","orcid":"0000-0003-3187-9903","position":0,"is_corresponding":true}],"reference_count":33,"raw_metadata":null,"created_at":"2026-07-19T01:58:37.924718Z","pmid":"40026410","pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}