README.md

Compressed Sensing for the Radon transform

This is an implementation of an algorithm to solve the Quadratically Constrained Basis Pursuit (QCBP) problem. The algorithm solves the problem $$\min_{x\in\mathbb{C}^{N\times N}} \Vert\Phi^* x\Vert_{l^1} \quad \text{subject to} \quad \Vert Ax-y\Vert_{l^2}<\|e\|_{l^2},$$ where $x$ denotes a $N\times N$ matrix (representing an image), $A$ denotes the Radon transform, $\Phi$ the Fourier transform, $y$ the measured data (sometimes called a sinogram) and $e$ denotes Gaussian noise.
In words, the above minimization process chooses among all solutions $x$ of $\Vert Ax-y\Vert_{l^2}<\|e\|_{l^2}$ the one whose Fourier transform is approximately sparse.

Contents:

Matlab:

• QCBP_fourier.m Minimization algorithm based on the gradient descent method,
• demo.m Demo script that compares the above method to classical filtered backprojection.

Dependencies:

• Matlab's Image processing toolbox

Python:

• cs_algorithms.py Minimization algorithm based on the gradient descent method,
• demo.py Demo script that compares the above method to classical filtered backprojection.

Dependencies:

• numpy
• matplotlib
• scikit-image

Any comments or queries are welcome at https://frank-roesler.github.io/contact/