We note from the above formulation that
the smoothness constraint is applied globally to the entire scene.
While ensuring a smooth surface, this constraint is inadequate near
discontinuities.
Discontinuities thus present serious difficulties to the above formulation.
Most often, discontinuities are blurred by
minimizing E without a proper choice of 
.
Different methods have been proposed to relax the global smoothness constraint in various ways so as to preserve visual discontinuities. The basic philosophy is to find ways to quantify discontinuities either directly or indirectly, and then use this information to apply different degrees of smoothness to different parts of the image [2], [3], [4].
For example in [3], two ways are proposed to identify the discontinuities, namely local validation by bending moment method and the variational continuity control. The smoothness constraint is then relaxed at those discontinuities.
In these two methods for discontinuities estimation, the discontinuities are analyzed using single resolution methods. As pointed out in [7], multiresolution analysis performs better than single resolution in discriminating between noise and desired features.