In this paper, we give a decomposition of the gradient measure Du of an arbitrary function of bounded variation u into a linear combination of atoms mu = D-chi F, where F is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each mu, there exists a cube Q such that supp mu subset of Q, mu(Q) = 0, |mu|(Q) 0|t(1/2)p(t) & lowast; mu(x)|