# Recent insights on the Uniqueness Problem of Diffeomorphisms determined by Prescribed Jacobian Determinant and Curl

arXiv: https://arxiv.org/abs/2207.13053 · HTML: https://mlsanity.com/paper/2207.13053/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2022.

## Built on by (most cited first, 1 of 1)

- [VPREG: An Optimal Control Formulation for Diffeomorphic Image Registration Based on the Variational Principle Grid Generation Method](https://mlsanity.com/paper/2510.13109/) (2025)

## Abstract (arXiv)

Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP seem to satisfy properties of a Lie group. Hence, it is conjectured that a unique diffeomorphism can be assured by its JD and curl (Uniqueness Conjecture). In this paper, (1) an observation based on VP is derived that a counter example to the Conjecture, if exists, should satisfy a particular property; (2) from the observation, an experimental strategy is formulated to numerically test whether a given diffeomorphism is a valid counter example to the conjecture; (3) a proof of an intermediate step to the conjecture is provided and referred to as the semi-general case, which argues that, given two diffeomorphisms, pmbφ and pmbpsi, if they are close to the identity map, pmbid, then pmbφ is identical pmbpsi.