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Research Statement | Brian Sherson

My research area is Inverse Problems in Tomography, but also hold interest in distribution theory and the propagation of singularities by operators on distribution spaces. Tomography deals with the reconstruction of a function from its line integrals, which has applications to medical imaging [1]. In my doctoral thesis [?], I explored the Broken Ray Transform, in which we reconstruct a function from its integrals over broken rays — a broken ray being the union of two rays sharing a common endpoint. A motivation for studying the Broken Ray transform is based on the fact that light is attenuated not only by absorption, but also by scattering.

Since a broken ray can be parametrized by the location of the vertex, its incoming and outgoing directions, the space of broken rays in $\mathbb{R}^n$ can be conceived as a $3n-2$-dimensional manifold. Reconstructing functions from all possible broken rays is an over-determined problem as it is shown by Florescu, Markel, and Schotland in 2010 that a function on $\mathbb{R}^2$ can be reconstructed from a $2$-dimensional submanifold of broken rays, those with fixed initial and terminal directions [2]. My thesis introduces the Polar Broken Ray Transform, in which we consider broken rays that start at an origin, and all have a fixed breaking angle (the angle between the two rays).

Part of my research also involved studying how singularities are propagated by the Broken Ray Transform (restricted to 2-dimensional manifolds), as well as their inversions whenever possible. The object of study in this endeavor is called the Wavefront Set, which takes the idea of singular support even further by describing how a function’s (or distribution’s) Fourier transform fails to decay rapidly [3]. We then describe the relationships between the wavefront sets of distributions and its Broken Ray Transform.

## Background

### Broken Ray Transform

Florescu, Markel, and Schotland studied the Broken Ray Transform with fixed initial and terminal directions given by unit vectors $\vec{\mathbf{v}}_1 = \langle 1, 0 \rangle$ and $\vec{\mathbf{v}}_2 = \langle \cos\theta, \sin\theta \rangle$, with $0<\theta<\frac{\pi}{2}$ fixed, by$\mathcal{B}f\left(x;\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}\right)=\int_{0}^{\infty}f\left(x-t\vec{\mathbf{v}}_{1}\right)\,dt+\int_{0}^{\infty}\mu_{\mathrm{t}}\left(x+t\vec{\mathbf{v}}_{2}\right)\,dt,$where $f$ is a function supported in the strip $[0,L]\times\mathbb{R}$. This naturally extends to what we call the full Broken Ray Transform by taking $f$ to be a function with compact support in $\mathbb{R}^n$, and allowing $\vec{\mathbf{v}}_1$ and $\vec{\mathbf{v}}_2$ to vary in $S^{n-1}$, provided that $\vec{\mathbf{v}}_1 \ne \pm \vec{\mathbf{v}}_2$.

In the case that $\vec{\mathbf{v}}_1$ and $\vec{\mathbf{v}}_2$ are fixed, an inversion formula is given by$f\left(x\right)=\int_{0}^{\infty}\mathcal{D}_{\vec{\mathbf{v}}_{1}}\mathcal{D}_{\vec{\mathbf{v}}_{2}}\mathcal{B}f\left(x-s\left(\vec{\mathbf{v}}_{2}-\vec{\mathbf{v}}_{1}\right)\right)\,ds.$Here, $\mathcal{D}_{\vec{\mathbf{v}}_{j}}$ refers to a directional derivative for $j=1,2$.

When considered in the classical sense, the above inversion formula can only be verified when $f$ is $\mathcal{C}^2$. However, when considered in a distributional sense, this inversion formula remains valid, contigient upon carefully giving a distributional meaning to the integration.

Of interest is the observation that the above inversion formula is not the only one, and as such, there are actually infinitely many. This proves useful when investigating the propagation of singularities of the Broken Ray transform and any of its inversions, since we can conclude more information than can be done with just one inversion formula.

### Polar Broken Ray Transform

The Polar Broken Ray Transform came about in an attempt to find a different 2-dimensional submanifold of broken rays on which we can restrict the general Broken Ray Transform to, and still maintain invertibility for functions defined on $\mathbb{R}^2$. For a fixed angle $0<\theta<\frac{\pi}{2}$, let $A$ denote the corresponding (counter-clockwise) rotation matrix, and $f$ be a function with compact support bounded away from the origin. The Polar Broken Ray Transform of $f$ is defined as$\mathcal{Q} f\left(s\vec{\sigma}\right)=\int_{0}^{s}f\left(t\vec{\sigma}\right)\,dt+\int_{0}^{\infty}f\left(s\vec{\sigma}+tA\vec{\sigma}\right)\,dt,\quad s>0,\;\vec{\sigma}\in S^1.$ I then find that the Polar Broken Ray is injective on the space of continuous functions with compact support bounded away from the origin, however, the inversion formula is at best an infinite series.

### Wavefront Sets

A known result is that a function is $\mathcal{C}^\infty$ if and only if its Fourier Transform is rapidly decaying (that is, decays faster than any negative power of $\left\Vert\vec{\xi}\right\Vert$.) Likewise, a distribution with compact support is $\mathcal{C}^\infty$ if its Fourier Transform satisfies the aforementioned decay condition [3].

The wavefront set was coined by Lars Hörmander, and describes how a distributution fails to be $\mathcal{C}^\infty$ [3]. The Wavefront is defined precisely as follows:

Let $u\in\mathscr{D}'\left(X\right)$. For $x_{0}\in X$ and nonzero $\vec{\xi}_{0}\in\mathbb{R}^{n}$, we say $\left(x_{0},\vec{\xi}_{0}\right)\in WF\left(u\right)$ if for every $\phi\in\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ with $\phi\left(x_{0}\right)\ne0$ and open conic (closed under positive scaling) neighborhood $\Gamma$ of $\vec{\xi}_{0}$, there exists a positive integer $N$ such that the estimate$\left|\widehat{\phi u}\left(\vec{\xi}\right)\right|\le C\left(1+\Vert\vec{\xi}\Vert^{2}\right)^{-N/2},\quad\vec{\xi}\in\Gamma$ fails for all constants $C$.

This defines the wavefront set as a subset of $X \times (\mathbb{R}^n\backslash 0)$, and is properly seen as a subset of the cotangent bundle of $X$.

Example: Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary. Then $\left(x_0, \vec{\xi}_0\right) \in WF\left(\mathbf{1}_U\right)$ if and only if $\vec{\xi}_0$ is a normal vector to $\partial U$ at the point $x_0 \in U$.

### Propagation of Singularities

We use wavefront sets to describe the propagation of singularities of an operator on distribution spaces. In particular, we compare the wavefront sets of a distribution and its transform. In my thesis, I found these propagation of singularities results:

Theorem: Given a distribution $u$ with support conditions that allow its Broken Ray Transform to be defined. Then\begin{align*} WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right)\backslash WF\left(u\right)&\subseteq\left\{ \left(x+t\vec{\mathbf{v}}_{1},\vec{\xi}\right)\,\middle|\,\left(x,\vec{\xi}\right)\in WF\left(u\right),\vec{\xi}\in\vec{\mathbf{v}}_{1}^{\perp},t>0\right\} \\&\hspace{1em}\cup\;\left\{ \left(x-t\vec{\mathbf{v}}_{2},\vec{\xi}\right)\,\middle|\,\left(x,\vec{\xi}\right)\in WF\left(u\right),\vec{\xi}\in\vec{\mathbf{v}}_{2}^{\perp},t>0\right\} , \end{align*} and \begin{align*} WF\left(u\right)\backslash WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right)&\subseteq\left\{ \left(x+t\left(\vec{\mathbf{v}}_{2}-\vec{\mathbf{v}}_{1}\right),\vec{\xi}\right)\,\middle|\,\left(x,\vec{\xi}\right)\in WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right),\vec{\xi}\in\left(\vec{\mathbf{v}}_{2}-\vec{\mathbf{v}}_{1}\right)^{\perp},t>0\right\} \\&\hspace{1em}\cap\;\left\{ \left(x+t\left(\vec{\mathbf{v}}_{1}-\vec{\mathbf{v}}_{2}\right),\vec{\xi}\right)\,\middle|\,\left(x,\vec{\xi}\right)\in WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right),\vec{\xi}\in\left(\vec{\mathbf{v}}_{2}-\vec{\mathbf{v}}_{1}\right)^{\perp},t>0\right\}. \end{align*}

Moreover, $WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right)\backslash WF\left(u\right)$ consists of the interiors of line segments and rays that are parallel to either $\vec{\mathbf{v}}_{1}$ or $\vec{\mathbf{v}}_{2}$, such that the endpoints lie in $WF\left(u\right)$, while $WF\left(u\right)\backslash WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right)$ consists of the interiors of line segments parallel to $\vec{\mathbf{v}}_{1}-\vec{\mathbf{v}}_{2}$, having endpoints in $WF\left(\mathcal{B}_{\vec{\mathbf{v}}_{1},\vec{\mathbf{v}}_{2}}u\right)$.

Since my research did not find an inversion formula for the Polar Broken Ray Transform that was useful for a propagation of singularities result, I only obtained a one-sided result as follows:

Theorem: Given a distribution $f$ with compact support bounded away from the origin,\begin{align*} WF\left(\mathcal{Q}f\right)\backslash WF\left(f\right)&\subseteq\left\{ \left(s\vec{\sigma},\vec{\xi}\right)\,\middle|\,\exists t\le s:\left(t\vec{\sigma},\vec{\xi}\right)\in WF\left(f\right),\vec{\xi}\perp\vec{\sigma}\right\} \\&\cup\;\left\{ \left(s\vec{\sigma},\rho_{\sigma-\theta}\vec{\xi}\right)\,\middle|\,\exists t\ge0:\left(r\vec{\theta},\vec{\xi}\right)\in WF\left(f\right),\vec{\xi}\perp A\vec{\sigma}\right\} ,\end{align*} where $r\vec{\theta}=s\vec{\sigma}+tA\vec{\sigma}$, and $\rho_{\sigma-\theta}$ represents rotation by the angle made between $\vec{\theta}$ and $\vec{\sigma}$.

## Possible Areas of Future Research

There are still some results to be obtained for the Polar Broken Ray transform, including finding an inversion formula that can be used to obtain a more complete propagation of singularities result, as well as to implement a faster numerical inversion.

### $\mathcal{H}^s$ Propagation of singularities

Hörmander’s wavefront set is defined in terms of a Fourier transform failing to decay rapidly. As such, the wavefront set does not tell us if a singularity in a distribution comes from a discontinuity in a function or if one of its particular distributional derivatives comes from such. For $N \ge 0$ the Sobolev space $\mathcal{H}^N\left(\mathbb R^n\right)$ is defined as the space of $\mathcal{L}^2$ functions on $\mathbb{R}^n$ having weak partial derivatives of all orders up to $N$. Due to the Fourier transform acting as an isometry on $\mathcal{R}^2\left(\mathbb R^n\right)$, it is seen that a function $f$ is in $\mathcal{H}^N\left(\mathbb R^n\right)$ whenever $\left(1+\Vert\vec{\xi}\Vert^{2}\right)^{N/2}\hat{f}\left({\vec{\xi}}\right)\in\mathcal{L}^2\left(\mathbb{R}^n\right).$ If we replace $N$ with an arbitrary nonnegative real number $s$, we can then take the above estimate as defining the Sobolev space $\mathcal{H}^s$. We can then define an $\mathcal{H}^s$-wavefront set by considering how a distribution fails to be $\mathcal{H}^s$ locally. One possible definition is as follows:

Let $u\in\mathscr{D}'\left(X\right)$, $s \ge 0$. For $x_{0}\in X$ and nonzero $\vec{\xi}_{0}\in\mathbb{R}^{n}$, we say $\left(x_{0},\vec{\xi}_{0}\right)\in WF_s\left(u\right)$ if for every $\phi\in\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ with $\phi\left(x_{0}\right)\ne0$ and open conic neighborhood $\Gamma$ of $\vec{\xi}_{0}$,$\int_\Gamma \left(1+\Vert\vec{\xi}\Vert^{2}\right)^{s}\left|\widehat{\phi u}\left(\vec{\xi}\right)\right|^2 \; d\vec{\xi}= \infty.$

This, of course is not the only possible definition, and is therefore subject to change.

Once we establish a usable definition of the $\mathcal{H}^s$-wavefront set of a distribution, we can proceed to derive similar propagation of singularities results in terms of this new definition.

## Bibliography

1. Adel Faridani and others, Introduction to the mathematics of computed tomography, Inside Out: Inverse Problems and Applications (2003), 1–46.
2. Lucia Florescu, Vadim A. Markel, and John C. Schotland, Inversion formulas for the broken-ray Radon transform, Inverse Problems 27, 025002 (2011) (2010).
3. L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, second ed., Springer Verlag, Berlin, 1990.