Seeing Living Atoms, One Electron at a Time: the expectation-maximization algorithm with Poisson statistics for analyzing counting frames of direct electron detectors in electron cryomicroscopy
In single particle electron cryomicroscopy, we image the same fragile biological molecule many times in 2D, and average them together to get a 3D voxelized scalar density map. This is often formulated mathematically as an ill posed inverse problem. An imaging operator encodes how the imaging happens, the signal is the underlying model to be estimated, and the noise is often modelled as additive gaussian.
However, low dose imaging is better modelled by a Poisson statistic, where each real space pixel has its own lambda from signal and noise. Of course, the 1D Gaussian distribution is a good approximation to the 1D Poisson distribution, even at very low Poisson lambdas, perhaps even 5. But what happens at an even lower dose? Experimentalists choose to operate the microscope at low dose because biological samples are fragile to the energy from the illuminating electron beam. New direct electron detectors count electrons as fast as 1000s of frames per second very accurately, and can get to Poison lambdas around 0.01, meaning an average of one count in every hundred pixels - very low dose. In practice, data processing algorithms work on summed counting frames, termed “movie frames”. Summing yields gaussian stats, averages out noise, but slightly blurs signal. I am interested in how we could avoid the information loss from summing together counting frames into movie frames.
In this talk I investigate 2D image alignment. To what extent do Gaussian stats break down at low dose, and can we overcome this by re-working algorithms under the assumption of Poisson stats? We will consider the 2D expectation-maximization algorithm with Poisson stats, and how Bayesian update equations can be derived analytically. Using Poisson simulated data based on an empirical cryoEM map, I show that Gaussian stats break down at surprisingly low dose. I will show how the expectation-maximization algorithm assuming Gaussian stats can fail on low dose data by over emphasizing high intensity pixels, which blurs out the image.
My current approach models rotations and translation in real space, but through convolution other imaging effects can be included that are more commonly handled in Fourier space, such as the detectors transducer behaviour (detect quantum efficiency) and the magnetic lens’ contrast transfer function. A Poisson expectation-maximization could be done patch wise for a larger field of views with non-uniform effects over the whole micrograph. In the future I hope to work with developers of popular cryoEM data processing software to help implement this approach in a numerically robust and efficient way, and do further experiments with experimental counting frame data from different direct electron detectors.