What is the object in the above image?

Is it a vase, or two faces? The decision problem is ill-posed since the solution is not unique.
On the other hand, if the problem is regularized by stating that "the object has higher intensity (i.e. is more white) than the background"
then the decision task is easy.

Formal examples of ill-posed problems, such as numerical differentiation, can also be given. See the article by Terzopoulos
(under ill-posed problems in vision) from the literature list.