Current methods for processing diffusion MRI (dMRI) to map the connectivity of the human brain require precise delineations of anatomical structures. This requirement has been approached by either segmenting the data in native dMRI space or mapping the structural information from T1-weighted (T1w) images. The characteristic features of diffusion data in terms of signal-to-noise ratio, resolution, as well as the geometrical distortions caused by the inhomogeneity of magnetic susceptibility across tissues hinder both solutions. Unifying the two approaches, we propose regseg, a surface-to-volume nonlinear registration method that segments homogeneous regions within multivariate images by mapping a set of nested reference-surfaces. Accurate surfaces are extracted from a T1w image of the subject, using as target image the bivariate volume comprehending the fractional anisotropy (FA) and the apparent diffusion coefficient (ADC) maps derived from the dMRI dataset. We first verify the accuracy of regseg on a general context using digital phantoms distorted with synthetic and random deformations. Then we establish an evaluation framework using undistorted dMRI data from the Human Connectome Project (HCP) and realistic deformations derived from the inhomogeneity fieldmap corresponding to each subject. We analyze the performance of regseg computing the misregistration error of the surfaces estimated after being mapped with regseg onto 16 datasets from the HCP. The distribution of errors shows a 95% CI of 0.56-0.66mm, that is below the dMRI resolution (1.25mm, isotropic). Finally, we cross-compare the proposed tool against a nonlinear b0-to-T2w registration method, thereby obtaining a significantly lower misregistration error with regseg. The accurate mapping of structural information in dMRI space is fundamental to increase the reliability of network building in connectivity analyses, and to improve the performance of the emerging structure-informed techniques for dMRI data processing.