  X-ray Bragg coherent diffraction imaging has been demonstrated as a powerful
three-dimensional (3D) microscopy approach for the investigation of
sub-micrometer-scale crystalline particles. It is based on the measurement of a
series of coherent diffraction intensity patterns that are numerically inverted
to retrieve an image of the spatial distribution of relative phase and
amplitude of the Bragg structure factor of the scatterer. This 3D information,
which is collected through an angular rotation of the sample, is necessarily
obtained in a non-orthogonal frame in Fourier space that must be eventually
reconciled. To deal with this, the currently favored approach (detailed in Part
I) is to perform the entire inversion in conjugate non-orthogonal real and
Fourier space frames, and to transform the 3D sample image into an orthogonal
frame as a post-processing step for result analysis. In this article, a direct
follow-up of Part I, we demonstrate two different transformation strategies
that enable the entire inversion procedure of the measured data set to be
performed in an orthogonal frame. The new approaches described here build
mathematical and numerical frameworks that apply to the cases of evenly and
non-evenly sampled data along the direction of sample rotation (the rocking
curve). The value of these methods is that they rely on and incorporate
significantly more information about the experimental geometry into the design
of the phase retrieval Fourier transformation than the strategy presented in
Part I. Two important outcomes are 1) that the resulting sample image is
correctly interpreted in a shear-free frame, and 2) physically realistic
constraints of BCDI phase retrieval that are difficult to implement with
current methods are easily incorporated. Computing scripts are also given to
aid readers in the implementation of the proposed formalisms.
