  Tomography has made a revolutionary impact on diverse fields, ranging from
macro-/mesoscopic scale studies in biology, radiology, plasma physics to the
characterization of 3D atomic structure in material science. The fundamental of
tomography is to reconstruct a 3D object from a set of 2D projections. To solve
the tomography problem, many algorithms have been developed. Among them are
methods using transformation technique such as computed tomography (CT) based
on Radon transform and Generalized Fourier iterative reconstruction (GENFIRE)
based on Fourier slice theorem (FST), and direct methods such as Simultaneous
Iterative Reconstruction Technique (SIRT) and Simultaneous Algebraic
Reconstruction Technique (SART) using gradient descent and algebra technique.
In this paper, we propose a hybrid gradient descent to solve the tomography
problem by combining Fourier slice theorem and calculus of variations. By using
simulated and experimental data, we show that the state-of-art RESIRE can
produce more superior results than previous methods; the reconstructed objects
have higher quality and smaller relative errors. More importantly, RESIRE can
deal with partially blocked projections rigorously where only part of
projection information are provided while other methods fail. We anticipate
RESIRE will not only improve the reconstruction quality in all existing
tomographic applications, but also expand tomography method to a broad class of
functional thin films. We expect RESIRE to find a broad applications across
diverse disciplines.
