Let S ⊂ R2 be the union of two convex sets with smooth boundary. We connect the levelsets of the minimizers uλ of

(ROF) 1 2

ku − χSk 2

L2 + λ kukT V

to the minimizers of a (simpler) set-minimization problem in order to obtain a geometrical characterization of the levelsets of uλ. Moreover, we calculate explicit minimizers of (ROF), when S is the union of two nonintersecting circles/squares or a starshaped (nonconvex) set using simple morphological operators. Finally I give an idea how to obtain the dual BV-norm of χS.