Analytical Differentiation for Fan-Beam and Cone-Beam CT Geometry

This repository contains the code for computed tomography (CT) reconstruction in fan-beam and cone-beam geometry which is differentiable with respect to its acquisition geometry. This works by computing the analytical partial derivatives of the reconstructed image to the entries of the projection matrices. The backprojector inherits from PyTorch's torch.autograd.Function to automatically assign the analytical derivative to the differentiable graph in a deep learning sense.

The code in this repository is at the core of the results presented in our paper Gradient-based geometry learning for fan-beam CT reconstruction which has been published in Physics in Medicine & Biology. While the paper focuses on fan-beam geometry, this repository additionally contains an analogous implementation for cone-beam geometry.

Usage

An instance of the differentiable fan-beam backprojector can be obtained via:

from backprojector_fan import DifferentiableFanBeamBackprojector
backprojector = DifferentiableFanBeamBackprojector.apply
reconstruction = backprojector(sinogram, projection_matrices, geometry)

Similarly, for the cone-beam backprojector:

from backprojector_cone import DifferentiableConeBeamBackprojector
backprojector = DifferentiableConeBeamBackprojector.apply
reconstruction = backprojector(sinogram, projection_matrices, geometry)

These backprojectors can be incorporated into any PyTorch differentiable graph and gradients for the projection matrices can be obtained.

Example optimization

The script example.py contains code for a PyTorch-based optimization loop which updates the translational components of the projection matrices for fan-beam geometry based on a supervised MSE-loss in image domain. It produces the following output showing how the reconstructed image improves and the target function is minimized:

Gradient check

The script check_gradients.py implements a comparison between analytical gradients and their numerical approximation to demonstrate the correctness of all computations. It contains four functions:

• check_gradients_fan(): plots analytical and numerical gradients for each projection matrix entry for visual comparison in fan-beam geometry
• check_gradients_cone(): plots analytical and numerical gradients for each projection matrix entry for visual comparison in cone-beam geometry
• pytorch_gradcheck_fan(): runs the PyTorch gradient check for fan-beam geometry
• pytorch_gradcheck_cone(): runs the PyTorch gradient check for cone-beam geometry

Technical details

The code uses numba cuda to parallelize the backprojection operation and gradient computations on GPU. Therefore an NVIDIA GPU and a running CUDA installation is required. So far, the code has only been tested on Linux operating system. We provide some example fan-beam and cone-beam data of the Shepp-Logan phantom in the ./data sub-folder. The geometry settings and conventions of this repository are compatible with those in PyroNN.

Citation

If you use this code for your research, please cite our paper:

@article{10.1088/1361-6560/acf90e,
    author={Thies, Mareike and Wagner, Fabian and Maul, Noah and Folle, Lukas and Meier, Manuela and Rohleder, Maximilian and Schneider, Linda-Sophie and Pfaff, Laura and Gu, Mingxuan and Utz, Jonas and Denzinger, Felix and Manhart, Michael Thomas and Maier, Andreas},
    title={Gradient-based geometry learning for fan-beam CT reconstruction},
    journal={Physics in Medicine & Biology},
    url={http://iopscience.iop.org/article/10.1088/1361-6560/acf90e},
    year={2023}
}

If you have any questions about this repository or the paper, feel free to reach out ([email protected]).