Mapping of Continuously Heterogeneous Volumes

This work generalizes the problem of reconstructing a molecule to the problem of reconstructing a “hyper-molecule” which is a higher dimensional objects that describe the variability of the molecule. see this report.

The idea is implemented in the following prototype and applied to simulated data.

True Model

The ground truth, a continuously heterogeneous “molecule” (represented in this simulation by a “cat”). Each value of “t” in the title represents a different state in the continuum of states in which the molecule can be found. The following video shows the object morphs between states and spun around.


Ground truth model




The measurements are images, each image is a projection of one unknown state of the molecule in an unknown direction, with high levels of noise. In this simulation we are given 50,000 such images, each of 65×65 pixels. The SNR in this example is 1/16.

Figure: a clean projection image, and the same image with noise.


Reconstruction (preliminary results)


Reconstruction results, a heterogeneous object


The cat is not dancing, this is a nice example of alignment ambiguity in reconstruction, see this report.




Simultaneous Alignment and Classification

This project also treats heterogeneity, but in a different approach. This work is a representation theory approach to simultaneous alignment and classification, with applications in the heterogeneity problem in cryo-EM.
We generalize multireference alignment and non unique games (optimization over compact groups with pairwise cost functions) to the case of alignment and classification, which can be written as the product of compact groups.

While the classification x alignment ideas here and the mapping/reconstruction ideas above have some similar motivations, they apply to very different approaches to the cryo-EM problem.