Prerequisites & Notation

Before You Begin

This chapter compares the RF imaging forward model with three major medical imaging modalities. The prerequisites below ensure the reader can draw the parallels that motivate the chapter.

Matched-filter imaging, backpropagation, and the PSF as the autocorrelation of the sensing operator (Chapter 13) (Review ch13)
Self-check: Can you write $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ and explain why the PSF width determines the resolution limit?
Diffraction tomography, Fourier coverage on Ewald arcs, and the Fourier Diffraction Theorem (Chapter 15) (Review ch15)
Self-check: Can you sketch the Ewald sphere construction and explain how multi-frequency and multi-view measurements fill k-space?
Learned OAMP and deep unfolding for inverse problems, including the model-based deep learning paradigm (Chapter 18) (Review ch18)
Self-check: Can you explain how algorithm unrolling converts OAMP iterations into a learnable computational graph?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathcal{R}f(\theta, s)$	Radon transform of $f$ at angle $\theta$ , offset $s$	s01
$\tilde{f}(k_x, k_y)$	2D Fourier transform of the object function $f$	s01
$\hat{f}_{\mathrm{FBP}}$	Filtered back-projection reconstruction	s01
$\Omega_k$	k-space sampling pattern (MRI or diffraction tomography)	s02
$\mathbf{F}_{\Omega}$	Undersampled Fourier encoding matrix (rows indexed by $\Omega$ )	s02
$\mathbf{S}_c$	Coil sensitivity map for receive coil $c$ (parallel MRI)	s02
$\Psi$	Sparsifying transform (wavelet, total variation, learned)	s02
$\tau_{ij}$	Round-trip time delay for transducer element $i$ , focus point $j$	s03
$\mathcal{A}_{\mathrm{CT}}$	CT forward operator (Radon transform + discretization)	s01
$\mathcal{A}_{\mathrm{MRI}}$	MRI forward operator ( $\mathbf{F}_{\Omega}$ or $\mathbf{F}_{\Omega}\mathbf{S}$ )	s02
$\mathcal{A}_{\mathrm{US}}$	Ultrasound forward operator (delay-and-sum model)	s03

← Ch 26 Computed Tomography: The Canonical Inverse Problem