Biomedical Imaging — Overview

Notes on biomedical imaging: the physics of imaging modalities and the math of reconstructing images from measurements.

Topics to cover

• MRI: spin physics, k-space, pulse sequences
• CT: X-ray attenuation and the Radon transform
• Ultrasound and optical imaging
• Reconstruction, denoising, and deep-learning approaches

CT and the Radon transform

CT reconstruction inverts the Radon transform, the set of line integrals of the attenuation $f(x, y)$ through the body:

$$R f(\theta, s) = \int_{-\infty}^{\infty} f\big(s\cos\theta - t\sin\theta,\; s\sin\theta + t\cos\theta\big)\, dt$$

X-ray attenuation along a path follows the Beer–Lambert law:

$$I = I_0 \exp\!\left( -\int \mu(x)\, dx \right)$$

where $\mu(x)$ is the linear attenuation coefficient.

MRI signal

The measured MRI signal is the Fourier transform of the transverse magnetization, sampled in k-space:

$$s(\mathbf{k}) = \int m(\mathbf{r})\, e^{-i 2\pi \mathbf{k}\cdot\mathbf{r}}\, d\mathbf{r}.$$

Next steps

Add pages such as mri.md, ct.md, or reconstruction.md to this folder.