I am trying to implement a diffeomorphic registration of two images using an unsupervised algorithm$^\color{magenta}{\star}$ and one of the steps is a generation of velocity field $\bf z$ which is then exponentiated to produce a deformation field $\boldsymbol{\phi}$. The exponentiation is performed via squaring and scaling steps, which as I understand is a fast method for the matrix exponential. According to my understanding what is happening here is the integration of the velocity field to obtain the deformation field, which involves matrix exponential, as explained here. What I don't understand is why $\boldsymbol{\phi} = \exp({\bf v})$ in the first place. Is the integral of the velocity vector field interpreted as a deformation flow?
$\color{magenta}{\star}$ Adrian V. Dalca, Guha Balakrishnan, John Guttag, Mert R. Sabuncu, Unsupervised learning of probabilistic diffeomorphic registration for images and surfaces, Medical Image Analysis, Volume 57, October 2019. [PUBMED]