Mathematical Framework

CryoLike is a GPU-accelerated software package for efficiently computing image-to-structure likelihood in cryo-electron microscopy (cryo-EM). It is built upon a mathematical framework using Fourier-Bessel representations, and draws on prior work in cross-correlation computation for ab initio reconstruction [Rangan, Greengard 2023]. Cryolike extends this method to support full likelihood-based comparisons in a user-friendly Python interface and enables scalable image-to-structure likelihood evaluation across large image datasets. It allows researchers to assess how well candidate structures explain cryo-EM data without the overhead of full reconstruction pipelines.

The CryoLike workflow includes modules for template generation, particle image conversion, cross-correlation computation, and log-likelihood evaluation, all optimized for modern GPUs.

Main objective

Given a cryo-EM image, and a 3D structure or map, CryoLike computes the image-to-structure likelihood. Representing the image and templates (2D projections of the 3D structure) in a Fourier-Bessel basis set, it searches for the optimal cross-correlation between the image and templates, i.e. finds the pose with highest cross-correlation to the image. CryoLike then approximates the image-to-structure likelihood by using the image-to-template likelihood evaluated at the optimal pose obtained by finding the best cross-correlation.

Cross-Correlation in Fourier Space

At its core, the cross-correlation \(\mathcal{C}(\theta; f, g)\) between two functions \(f(\psi)\) and \(g(\psi)\) is defined as:

\[\mathcal{C}(\theta; f, g) = \int_{\Omega} f^*(\psi) \cdot g(\psi + \theta) \, d\psi,\]

where \(f^*\) denotes the complex conjugate and \(\Omega\) is the domain of integration.

Using the convolution theorem, this cross-correlation can be evaluated via the inverse Fourier transform of the element-wise product of the Fourier transforms:

\[\mathcal{C}(\cdot; f, g) = \mathcal{F}^{-1}\left[ \mathcal{F}\left[f\right]^* \odot \mathcal{F}\left[g\right] \right],\]

where \(\mathcal{F}\) represents the 1D Fourier transform and \(\odot\) denotes element-wise multiplication.

CryoLike leverages this principle to compute cross-correlations in the angular coordinate of 2D Fourier space. Specifically, for an image \(\tilde{I}(k, \psi)\) and a template \(\tilde{T}_{\phi}(k, \psi)\), the frequency-space cross-correlation \(\mathcal{C}_{\text{freq}}(\theta; \tilde{T}, \tilde{I})\) is calculated by combining radial integration with angular convolution. In practice, this involves transforming both the image and template into a Fourier-Bessel basis via a 1D angular Fourier transform, multiplying their coefficients, and then applying an inverse transform to obtain the final cross-correlation with respect to in-plane rotation \(\theta\).

Image-to-Template Likelihood

This section outlines a simplified formulation of the main CryoLike output: the image-to-template likelihood. For the sake of simplicity, we formulate it in physical space but it can be easily extended to Fourier space. A full derivation of both formulations is provided in the paper.

We assume a Gaussian white-noise model in physical space. Each cryo-EM image \(I(\mathbf{x})\) is modeled as a scaled projection template \(T_{\phi}(\mathbf{x})\) at pose \(\phi\), with image-specific intensity \(\alpha\), a constant offset \(\beta\), and additive noise:

\[I(\mathbf{x}) \sim \alpha T_{\phi}(\mathbf{x}) + \beta \mathbb{1}(\mathbf{x}) + \epsilon(\mathbf{x}),\]

where \(\epsilon(\mathbf{x})\) is drawn from a zero-mean Gaussian distribution with constant pixel variance \(\lambda^2\).

The likelihood of observing image \(I\) given the template and parameters is:

\[P(I \mid T_{\phi}, \alpha, \beta, \lambda) = \frac{1}{(2\pi)^{N/2} \lambda^N} \exp\left\{ -\frac{\ell_{\text{phys}}(I, T_{\phi}, \alpha, \beta)}{\lambda^2} \right\},\]

where \(\ell_{\text{phys}}\) is the squared L2-norm between the modeled and observed image:

\[\ell_{\text{phys}}(I, T_{\phi}, \alpha, \beta) = \int_{\Omega} \left| \alpha T_{\phi}(\mathbf{x}) + \beta \mathbb{1}(\mathbf{x}) - I(\mathbf{x}) \right|^2 d\mathbf{x},\]

where \(\Omega\) is the image space.