Spherical Radon transforms with smoothly varying radii

Abstract

We present an analysis of a novel spherical Radon transform, $R$, which defines the integrals of a function, $f$, in $\mathbb{R}^n$ over spheres with arbitrary center ($\mathbf{y}$) and radii, $r(\mathbf{y})$, which vary smoothly with $\mathbf{y}$. We first establish sufficient and necessary conditions on $r$ and $\text{supp}(f)$ so that $R$ satisfies the Bolker condition, and further conditions which allow $f$ to be recovered stably from $Rf$. We then apply this theory to a number of example applications in Compton Scatter Tomography (CST) and Ultrasound Reflection Tomography (URT). For each application considered, we also provide injectivity proofs and explicit inversion formulae, some of which are based on the generalized theory presented by Palamodov ("Palamodov, V. P. (2012). A uniform reconstruction formula in integral geometry. Inverse Problems, 28(6), 065014."). We then combine our microlocal theory and injectivity results to prove stability estimates for our transforms. In addition, to validate our theory, we provide simulated image reconstructions.

Figures (10)

• Figure 1: Example CST scanner design. Photons scatter on circular arcs bound by the source $\mathbf{s}$ and detector $\mathbf{d}$.
• Figure 2: Circular CST scanner design. For ${\mathbf{y}}\neq \mathbf{0}$, ${\mathbf{z}} = \frac{{\mathbf{y}}}{\left|{\mathbf{y}}\right|}$.
• Figure 3: Plot of circle centers, ${\mathbf{y}}$ that $R^*$ integrates over for varying ${\mathbf{x}}$. Here, $n=2$. The colored dots are the ${\mathbf{x}}$ coordinates of the points listed in the figure, and the corresponding colored curves are the set of ${\mathbf{y}}$ such that $S({\mathbf{y}})$ intersects ${\mathbf{x}}$.
• Figure 4: Spherical ultrasound scanner.
• Figure 5: In ${{\mathbb R}^2}$, the white circle is the boundary of $D$, and the set in yellow is $S_D$--the set of circle centers ${\mathbf{y}} \in Y'$ for which $S({\mathbf{y}})\cap D \neq \emptyset$. Here $r=1.25$ and $d = 0.25$.

Theorems & Definitions (78)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3: hormanderI
• Definition 2.4: hormanderIII
• Definition 2.5: hormanderIII and hormander
• Definition 2.6
• Theorem 2.7: Hörmander-Sato Lemma
• Definition 2.8
• Definition 2.9: Bolker condition