Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos (Q1108527)

The author proposes a new method for the three-dimensional reconstruction of a function \(d(x_ 1,x_ 2,x_ 3)\) with compact support describing the distribution density of the matter of a particle from its planar projections in unknown directions up to a rotation or a reflection. Let the center of mass be the origin, \(\omega\) a rotation and \(d_{\omega}(\vec x)\) the distribution function of the rotated particle. One must recover d from the projections \(d_ i(x_ 1,x_ 2):=\int d_{\omega_ i}(\vec x)dx_ 3\) where \(\omega_ 1\) are unknown. First the author discusses a geometric method to recover the relative rotations \(\omega_ 1^{-1}\omega_ 2,...,\omega_ 1^{- 1}\omega_ n\) and shows the stability with respect to experimental errors. A second approach to find the \(\omega_ 1\) uses the moment method and assigns to d(\(\vec x)\) the positive definite quadratic form \(Q_ d(\vec x):=\int d(\vec y)(\vec x,\vec y)^ 2d\vec y.\) Consider the two ellipsoids \(Q_ d(\vec x)=1\) and \(Q_{d_{\omega}}(\vec x)=Q_ d(D_{\omega}\vec x)=1\) obtained by rotating \(Q_ d\) under \(\omega\). From the projections one can find the lengths of its principal axes and determinate \(\omega\) expressed via Euler angles in a unique way. Algorithms of three-dimensional reconstruction from projections in known directions are well known. The result may be applied in tomography.