In a combined experimental and numerical study, we determine the details of the pinning of domain walls at constrictions in permalloy nanostructures. Using high spatial-resolution (<10nm) electron holography, we image the spin structure of geometrically confined head-to-head domain walls at constrictions. Low-temperature magnetoresistance measurements are used to systematically ascertain the domain-wall depinning fields in constrictions down to 35 nm width. The depinning fields increase from 60 to 335 Oe with decreasing constriction width and depend on the wall spin structure. The energy barrier to depin the wall from the constriction is quantitatively determined and comparison with the depinning field strength allows us to gauge the energy barrier height of the pinning potential.