Mathematical Framework
======================

.. _mathframework:
        :title: Math 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 :math:`\mathcal{C}(\theta; f, g)` between
two functions :math:`f(\psi)` and :math:`g(\psi)` is defined as:

.. math::

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

where :math:`f^*` denotes the complex conjugate and :math:`\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:

.. math::

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

where :math:`\mathcal{F}` represents the 1D Fourier transform and
:math:`\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 :math:`\tilde{I}(k, \psi)` and a template
:math:`\tilde{T}_{\phi}(k, \psi)`, the
frequency-space cross-correlation
:math:`\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 :math:`\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 :math:`I(\mathbf{x})` is modeled as a scaled projection
template :math:`T_{\phi}(\mathbf{x})`
at pose :math:`\phi`, with image-specific intensity :math:`\alpha`,
a constant offset :math:`\beta`, and additive noise:

.. math::

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

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

The likelihood of observing image :math:`I` given the template
and parameters is:

.. math::

    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 :math:`\ell_{\text{phys}}` is the squared L2-norm between
the modeled and observed image:

.. math::

    \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 :math:`\Omega` is the image space.

..     missing math section?