Variational image processing

A variational model balances agreement with observed data against a regularity or geometric penalty. Examples include total variation denoising, the relaxed graph-area functional, and the Mumford–Shah functional. Convex penalties support unique reconstruction, while segmentation energies can have multiple competing partitions.

The energy ${\int }_{\Omega \setminus K}(u-g{)}^2+\alpha {\int }_{\Omega \setminus K}|\nabla u{|}^2+\beta {\mathcal{H}}^1(K)$ balances data fidelity, within-region smoothness and image edge length. In its relaxed SBV space formulation the image edge set is $J_u$. Clipping to the bounded data range, the SBV compactness theorem and lower semicontinuity prove existence; the full segmentation problem is not strictly convex.

As the jump-length weight tends to infinity with the gradient weight fixed, the limiting problem minimizes $\Vert u-g{\Vert }_2^2+\alpha \Vert \nabla u{\Vert }_2^2$ on $H^1$. It has a unique minimizer by the Lax-Milgram theorem and strict convexity. Without prescribed boundary values, its weak equation is $u-\alpha \Delta u=g$ with natural Neumann boundary conditions.

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.

A minimizer of the Mumford–Shah functional with bounded image signal data has an essentially closed jump set: replacing $J_u$ by its relative closure adds no $(n-1)$-dimensional measure. Its complement supports a Sobolev representative. This regularity theorem connects a relaxed SBV space minimizer to the classical closed-edge formulation; it is not a property of every special bounded-variation function.