  There is a growing interest in measuring the cell wall mechanical property at
different locations in single walled cells. We present an inference scheme that
maps relative surface elastic modulus distributions along the cell wall based
on tracking the location of material marker points along the turgid and relaxed
cell wall outline. A primary scheme provides a step-function inference of
surface elastic moduli by computing the tensions and elastic stretches between
material marker points. We perform stability analysis for the primary scheme
against perturbations on the marker-point locations, which may occur due to
image acquisition and processing from experiments. The perturbation analysis
shows that the primary scheme is more stable to noise when the spacing between
the marker points is coarser, and has been confirmed by the numerical
experiments where we apply the primary scheme to synthetic cell outlines from
simulations of hyper-elastic membrane deformation with random noise on the
marker-point locations. To improve the spatial resolution of elastic modulus
distribution of the primary scheme with noise, we propose two optimization
schemes that convert the step-function inferences of elastic moduli into
smooth-curve inferences. The first scheme infers a canonical elastic modulus
distribution based on marker-point locations from multiple cell samples of the
same cell type. The second scheme is a simplified cost-effective version that
infers the elastic moduli based on marker-point locations from a single cell.
The numerical experiments show that the first scheme significantly improves the
inference precision for the underlying canonical elastic modulus distributions
and can even capture some degree of nonlinearity when the underlying elastic
modulus gradients are nonlinear. The second cost-effective scheme can predict
the trend of the elastic modulus gradients consistently.
