SORFAC-CT (Cylindrical-Topology Self-Organizing Reference-Free Alignment and Classification; pronounced sôr-fakt or sôr-fak-si-ti) is an efficient method of reference-free rotational image alignment. SORFAC-CT circumvents the dependence of subjective user- or computer-generated external or internal reference images used by other reference-based techniques. Alignment is performed using a Kohonen self-organizing map (SOM), configured on a cylindrical array of artificial neurons. Although alternative alignment protocols often depend on an expert to adjust many ad hoc parameters, SORFAC-CT instead achieves the minimization of one objectively calculated target function by varying only two free parameters. Because SOMs preserve the topological properties of the training (dataset) images, the relative in-plane rotational orientations of dataset images are obtained directly from the array’s intrinsic cylindrical coordinate system by noting each mapped dataset image’s respective azimuthal angle coordinate placement around the cylinder. Dataset images are not rotated into alignment with internally or externally generated reference images. Instead, SORFAC-CT starts with a cylindrical array of proto-models consisting of random pixel values. The proto-models gradually morph to become the rotational class models by an unsupervised process. The dataset images are then mapped to the class models according to greatest similarity; the alignment angles directly read off the cylinder. This was tested on datasets including square, rectangular and hexagonal geometrical shapes in six different sizes, to introduce heterogeneity. It was also tested using noisy 2-dimensional projections of the AQP-1 x-ray model. The results were near-perfect alignments that could be improved by increasing cylinder circumference, until the resolution limits of the dataset images are reached.

As explained in the Materials and Methods, these datasets were generated using IMAGIC software. 

SORFAC or SORFAC-CT may be used on these three sets of test images. The full-featured software is freely available upon request to the author, and the source code is available too.