Piecewise-constant Mumford–Shah problem (${\min }_{\nabla u=0}\{\Vert u-g{\Vert }_2^2+\beta {\mathcal{H}}^1(J_u)\}$)

The infinite gradient-weight limit restricts to SBV space image signals with $\nabla u=0$ almost everywhere. Their energy is fidelity plus jump length, represented by constants on a Caccioppoli partition. Internal perimeter is counted once by $\frac{1}{2}{\sum }_i\mathrm{Per}(E_i;\Omega )$; adjacent equal-valued regions can be merged.

When grey values are fixed but region labels vary, a smooth interface satisfies $\beta \kappa =(c_j-g{)}^2-(c_i-g{)}^2$. The curvature uses the normal pointing out of region $i$, positive on an outward-oriented circle. It balances the change in data cost against the first variation of interface length. Fixing a full spatial image signal instead fixes its jumps and leaves no such boundary-relocation freedom.

For a fixed positive-area region $E_i$, its least-squares grey value is $c_i=|E_i{|}^{-1}{\int }_{E_i}g$. The minimum fidelity equals $\int g^2-{\sum }_i({\int }_{E_i}g{)}^2/|E_i|$. This eliminates the grey values before optimizing the partition geometry.