The escalating data rate demands of future wireless communications necessitate the deployment of extremely large aperture arrays (ELAAs) in communication systems. Acquiring accurate channel state information is crucial to execute effective precoding for such systems, in which the near-field curvature effects on the channel must be considered. Current channel estimation algorithms are generally restricted to the spherical wavefront channel (SWC), which is appropriate for isotropic scatterers, point sources, and planar reflecting surfaces. However, in practical scenarios involving curved reflecting surfaces, the reflected waves exhibit anisotropic rather than spherical wavefronts, significantly degrading the accuracy of conventional SWC-based algorithms. To tackle this challenge, we first derive a parameterized model for the anisotropic wavefront channel (AWC). Based on this model, we then propose the matching-free acquisition of channels with anisotropic wavefronts (MACAW) algorithm. Unlike conventional dictionary-based matching pursuit techniques, MACAW recovers channel parameters through fast-Fourier-transform-based frequency analysis. This approach enables precise channel estimation in AWC scenarios while maintaining a significantly lower computational complexity than existing methods. Simulation results illustrate how physical characteristics of the propagation environment influence the degree of wavefront anisotropy, and demonstrate the effectiveness of the proposed algorithm.