This paper studies the relaxation with respect to the \(L^1\) norm of integral functionals of the type
\[ F(y)= \int_\Omega f(x,u,\nabla u)\,dx,\quad u\in W^{1,1}(\Omega; S^{d-1}), \]
where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\), \(S^{d-1}\) denotes the unite sphere in \(\mathbb{R}^d\), \(N\) and \(d\) being any positive integers, and \(f\) satisfies linear growth conditions in the gradient variable. The relaxed functional \(\overline F\) is defined by
\[ \overline F(u):= \inf\Bigl\{\liminf_{n\to+\infty} F(u_n): u_n\in W^{1,1}(\Omega; S^{d-1}),\;u_n\to u\text{ in }L^1\Bigr\}. \]
It is shown that if in addition \(f\) is quasiconvex in the gradient variable and satisfies the above assumptions, then the functional \(\overline F\), on \(\text{BV}(\Omega; S^{d-1})\), can be expressed by
\[ \overline F(u)= \int_\Omega f(x,u,\nabla u)\,dx+ \int_{S(u)} K(x,u^-, u^+, \nu_u)\,dH^{N-1}+ \int_\Omega f^\infty(x,u,dC(u)), \] where the surface energy density on \(K\) is defined by a suitable Dirichlet-type problem. This result applies in particular to the isotropic case \(f(x,y,z)=|z|\).