Local electron tomography using angular variations of surface tangents: Stomo version 2 Original Research Article

In a recent publication, we investigated the prospect of measuring the outer three-dimensional (3D) shapes of nano-scale atom probe specimens from tilt-series of images collected in the transmission electron microscope. For this purpose alone, an algorithm and simplified reconstruction theory were developed to circumvent issues that arise in commercial “back-projection” computations in this context. In our approach, we give up the difficult task of computing the complete 3D continuum structure and instead seek only the 3D morphology of internal and external scattering interfaces. These interfaces can be described as embedded 2D surfaces projected onto each image in a tilt series. Curves and other features in the images are interpreted as inscribed sets of tangent lines, which intersect the scattering interfaces at unknown locations along the direction of the incident electron beam. Smooth angular variations of the tangent line abscissa are used to compute the surface tangent intersections and hence the 3D morphology as a “point cloud”. We have published the explicit details of our alternative algorithm along with the source code entitled “stomo_version_1”. For this work, we have further modified the code to efficiently handle rectangular image sets, perform much faster tangent-line “edge detection” and smoother tilt-axis image alignment using simple bi-linear interpolation. We have also adapted the algorithm to detect tangent lines as “ridges”, based upon 2nd order partial derivatives of the image intensity; the magnitude and orientation of which is described by a Hessian matrix. Ridges are more appropriate descriptors for tangent-line curves in phase contrast images outlined by Fresnel fringes or absorption contrast data from fine-scale objects. Improved accuracy, efficiency and speed for “stomo_version_2” is demonstrated in this paper using both high resolution electron tomography data of a nano-sized atom probe tip and simulated absorption-contrast images.