This paper describes a graph-spectral method for 3D surface integration. The algorithm takes as its input a 2D field of surface normal estimates, delivered, for instance, by a shape-from-shading or shape-from-texture procedure. The method borrows ideas from routing theory. We exploit the well-known fact that the leading eigenvector of a Markov chain transition probability matrix is the steady-state random walk on the equivalent weighted graph. We use this property to find the minimum total curvature path through the available surface normals. To do this we construct a transition probability matrix whose elements are related to the differences in surface normal direction. By threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration. The height increments along the path are simply related to the traversed path length and the slope of the local tangent plane. The method is evaluated on data delivered by a shape-from-shading algorithm.