  We prove a homogenization result for Mumford-Shah-type energies associated to
a brittle composite material with weak inclusions distributed periodically at a
scale ${\varepsilon}>0$. The matrix and the inclusions in the material have the
same elastic moduli but very different toughness moduli, with the ratio of the
toughness modulus in the matrix and in the inclusions being
$1/\beta_{\varepsilon}$, with $\beta_{\varepsilon}>0$ small. We show that the
high-contrast behaviour of the composite leads to the emergence of interesting
effects in the limit: The volume and surface energy densities interact by
$\Gamma$-convergence, and the limit volume energy is not a quadratic form in
the critical scaling $\beta_{\varepsilon} = {\varepsilon}$, unlike the
${\varepsilon}$-energies, and unlike the extremal limit cases.
