In this paper, we propose a new large-deformation nonlinear image registration model in three dimensions, based on nonlinear elastic regularization and unbiased registration. Both the nonlinear elastic and the unbiased functionals are simplified introducing, in the modeling, a second unknown that mimics the Jacobian matrix of the displacement vector field, reducing the minimization to involve linear differential equations. In contrast to recently proposed unbiased fluid registration method, the new model is written in a unified variational form and is minimized using gradient descent on the corresponding Euler-Lagrange equations. As a result, the new unbiased nonlinear elasticity model is computationally more efficient and easier to implement than the unbiased fluid registration. The model was tested using three-dimensional serial MRI images and shown to have some advantages for computational neuroimaging.