A frequency-domain analysis of 3D reconstruction techniques based on defocusing

A technique used for three-dimensional reconstruction of microscopic objects is based on linear processing of a collection of images taken at different focus positions, that is, acquired step by step while the objective moves along the microscope optical axis. As the relationship between the "object" I(x,y,z) and the collection of images (if these latter are enough) is known to be a convolution via the microscope point spread function, a deconvolution procedure yields I(x,y,z). Here I(x,y,z) is the local absorption of the object. Based on the fact that reasonable approximations for the point spread function are available in the literature, the above technique seems to work satisfactorily, and has been applied by various researchers. In the present work, the method is analyzed from a strictly mathematical point of view. It Is shown that being the optical transfer function (OTF), that is, the Fourier transform of the point spread function, identically vanishing in an infinite region of the frequency domain, a deconvolution procedure is not feasible. If the z-coordinate refers to the optical axis, the OTF is actually zero outside a conical region in the frequency domain. It is also shown that what can be actually recovered are projections of the given object. To this end, the so-called slice theorem, or projection theorem, is used. More exactly, we prove that one can obtain all projections of the object within an angle of projection - with respect to the optical axis - not greater than the aperture angle of the image formation system, that is, the microscope. Examples of simulations and of processing of experimental images are reported