We present an inference scheme that maps relative lateral tensions along the cell wall based on the cell outline coordinates. The full tension-inference approach presented here includes the primary scheme that maps the tension distributions on the discretized cell outlines and their optimizations to smooth polynomials. We have studied the stability of the primary scheme against small noise analytically and have shown that the scheme is more stable to noise when discretization of the cell outline is coarser. We have found agreement quantitatively between the error bounds predicted by the analysis and the error computed from numerical experiments, when the scheme is applied to synthetic cell outlines from a computational model of hyper-elastic thin-shell deformation. Then, we develop an optimization method that effectively restores the spatial resolution of the primary scheme by converting the discrete tension distributions to smooth polynomials. In the end, we apply the full scheme to map tension distributions in moss protonemal cells.