In this paper we study the relaxation with respect to the L1 norm of integral functionals of the type F(u), where F(u) is the integral on a bounded open set of Rn Omega of an integrand f(x, u,∇u) and u ∈ W1,1(Omega; Sd−1), where Sd−1 denotes the unit sphere in Rd, n and d being any positive integers, and f satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that if, in addition, f is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional admits an integral representation on BV (Omega; Sd−1), via a suface energy density K on the jump set S(u), defined by a suitable Dirichlet-type problem (for the complete integral formula look at the abstract in the pdf of the paper)