Spherical harmonics-based parametric deconvolution of 3D surface images using bending energy minimization.

Numerical deconvolution of 3D fluorescence microscopy data yields sharper images by reversing the known optical aberrations introduced during the acquisition process. When additional prior information such as the topology and smoothness of the imaged object surface is available, the deconvolution can be performed by fitting a parametric surface directly to the image data. In this work, we incorporate such additional information into the deconvolution process and focus on a parametric shape description suitable for the study of organelles, cells and tissues. Such membrane-bound closed biological surfaces are often topologically equivalent to the sphere and can be parameterized as series expansions in spherical harmonic functions (SH). Because image data are noisy and the SH-parameterization is prone to the formation of high curvatures even at low expansion orders, the parametric deconvolution problem is ill-posed and must be regularized. We use the shape bending energy as a regularizing (smoothing) function, and determine the regularization parameter graphically with the help of the L-curve method. We demonstrate the complete deconvolution scheme, including the initial image segmentation, the calculation of a good starting surface and the construction of the L-curve, using real and synthetic image data.