Summary: We present a geometric flow approach to the segmentation of three-dimensional medical images obtained from Magnetic Resonance Imaging (MRI) or Computed Tomography (CT) scan methods, by minimizing a cost function. This energy term is based on the intensity of the original image, and its minimum is found following a gradient descent curve in an infinite-dimensional space of diffeomorphisms (Diff) to preserve topology. The general framework is reminiscent of variational shape optimization methods but remains closer to general developments on the deformable template theory of geometric flows. In our case, the metric that provides the gradient is defined as a right-invariant inner product on the tangent space \(({\mathcal V})\) at the identity of the group of diffeomorphisms, following the general Lie group approach suggested by Arnold [J. Méc. 5, 29–43 (1966)]. To avoid local solutions of the optimization problem and to mitigate the influence of several sources of noise, a finite set of control points is defined on the boundary of the template binary images, yielding a projected gradient descent on Diff.