The inverse problem for the local geodesic ray transform

Abstract

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

Appendix: Local X-ray transform for a general family of curves by Hanming Zhou (H.Z.’s address is Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA, and e-mail address is hzhou@math.washington.edu, New contact information: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK. Email: hz318@dpmms.cam.ac.uk)

In this appendix, we extend the result of the local invertibility of the geodesic ray transform proved in the present paper to the X-ray transform for a general family of curves.

Given a Riemannian manifold $(\tilde{X}, g)$ of dimension $\ge 3$, we consider smooth curves $\gamma$ on $\tilde{X}$, $|\dot{\gamma }|\ne 0$, that satisfy the following equation

$$\begin{aligned} \nabla _{\dot{\gamma }}\dot{\gamma }=G(\gamma , \dot{\gamma }), \end{aligned}$$

(4.1)

where $\nabla$ is the Levi-Civita connection, $G(z, v)\in T_z\tilde{X}$ is smooth on $T\tilde{X}$. $\gamma =\gamma _{z,v}$ depends smoothly on $(z,v)=(\gamma (0),\dot{\gamma }(0))$. We call the collection of such smooth curves on $\tilde{X}$, denoted by $\mathcal {G}$, a general family of curves. For the sake of simplicity, we assume $\gamma \in \mathcal {G}$ are parameterized by arclength (one can always reparametrize the curve to make this happen, and we will see later that our method also works for curves with non-constant speed). Note that if $G\equiv 0$, $\mathcal {G}$ is the set of usual geodesics. The X-ray transform of a smooth function $f\in C^{\infty }(\tilde{X})$ for a general family of curves is

$$\begin{aligned} (If)(\gamma )=\int f(\gamma (t))\, dt,\quad \gamma \in \mathcal {G}. \end{aligned}$$

Let X be a domain in $\tilde{X}$ with boundary defining function $\rho$, and $p\in \partial X$ a boundary point. We say that X (or $\partial X$) is convex (concave) at p with respect to $\mathcal {G}$ if for any $\gamma \in \mathcal {G}$ with $\gamma (0)=p$, $\dot{\gamma }(0)=v\in T_p(\partial X)$, we have $\frac{d^2}{dt^2}\rho (\gamma (t))|_{t=0}\le 0\,(\ge 0)$. If the inequality is always strict, then we say X is strictly convex (strictly concave) at p with respect to $\mathcal {G}$. It is easy to see that the geometric meaning of our definition is similar to the usual convexity with respect to the metric (geodesics).

Now assume X is strictly convex at $p\in \partial X$ with respect to $\mathcal {G}$. Similar to the settings in Sect. 3.1, we obtain a smooth function x whose level sets are strictly concave with respect to $\mathcal {G}$ from the super-level sets of x. This means that the arguments of Sects. 3.1, 3.2 and 3.3 also work for our case. In particular, if for $\gamma =\gamma _{x,y,{\uplambda },\omega }\in \mathcal {G}$, $\alpha (x,y,{\uplambda },\omega ):=\frac{1}{2}\frac{d^2}{dt^2}x(\gamma (t))|_{t=0}$, then $\alpha (x,y,0,\omega )>0$, i.e. $\alpha$ is positive on the tangent planes of the level sets of x. It was shown in Sect. 3.1 that $\alpha (x,y,0,\omega )$ defines a positive definite quadratic form for the usual geodesics, however for a general family of curves, it no longer has such special structure.

We consider the operator $A_\digamma =x^{-1}e^{-\digamma /x}Ae^{\digamma /x}$. By the proof of Proposition 3.3, we see that $A_\digamma \in \Psi ^{-1,0}_{sc}(x\ge 0)$ for $\digamma >0$. In particular, the Schwartz kernel of $A_\digamma$ at the scattering front face $x=0$, $\tilde{K}(y,X,Y)$, is given in Lemma 3.5. To show the invertibility of $A_\digamma$ we need to verify the ellipticity of its boundary principal symbol. The proof of Lemma 3.6 relies on the quadratic structure of $\alpha (x,y,0,\omega )$ which is not available for general curves, so we analyze the principal symbol in a different way, and our method works for a general family of curves.

Lemma 4.1

For $\digamma >0$ there exists $\chi \in C^{\infty }_c(\mathbb {R})$, $\chi \ge 0, \chi (0)=1$, such that the boundary principal symbol of corresponding $A_\digamma$ is elliptic.

Proof

Similar to the strategy in the proof of Lemma 3.6, we first calculate the boundary principal symbol for the case $\chi (s)=e^{-s^2/(2\digamma ^{-1}\alpha )}$ with $\digamma >0$ (here we need the positivity of $\alpha$), hence ${\hat{\chi }}(\zeta )=c\sqrt{\digamma ^{-1}\alpha }e^{-\digamma ^{-1}\alpha |\zeta |^2/2}$ for some $c>0$. First, the Fourier transform of ${\tilde{K}}$ in X equals a non-zero multiple of

$$\begin{aligned} |Y|^{2-n}\{(\digamma ^{-1}\alpha _+)^{\frac{1}{2}}e^{-\digamma ^{-1}(\xi ^2+\digamma ^2)\alpha _+|Y|^2/2}+(\digamma ^{-1}\alpha _-)^{\frac{1}{2}}e^{-\digamma ^{-1}(\xi ^2+\digamma ^2)\alpha _-|Y|^2/2}\}, \end{aligned}$$

(4.2)

where $\alpha _+=\alpha (0,y,0,{\hat{Y}}), \, \alpha _-=\alpha (0,y,0,-{\hat{Y}})$.

As mentioned previously, our $\alpha$ may contain terms other than a quadratic form in ${\hat{Y}}$, which means the exponential term in (4.2) is not Gaussian like in Y, thus we use polar coordinates to compute the Fourier transform of (4.2) in Y. We denote $\frac{\digamma ^{-1}(\xi ^2+\digamma ^2)}{2}$ by b, then

$$\begin{aligned} \mathcal {F}_Y(\mathcal {F}_X{\tilde{K}})(y,\xi ,\eta )\simeq & {} \int e^{-i\eta \cdot Y}|Y|^{2-n}(\alpha _+^{1/2}e^{-b\alpha _+|Y|^2}+\alpha _-^{1/2}e^{-b\alpha _-|Y|^2})\,dY\\= & {} \int _0^{+\infty }\int _{{\mathbb {S}}^{n-2}} e^{-i\eta \cdot \hat{Y}|Y|}|Y|^{2-n}(\alpha _+^{1/2}e^{-b\alpha _+|Y|^2}\\&+\,\alpha _-^{1/2}e^{-b\alpha _-|Y|^2})|Y|^{n-2}\,d|Y|d{\hat{Y}}\\= & {} \frac{1}{2}\int _{{\mathbb {R}}}\int _{{\mathbb {S}}^{n-2}} e^{-i\eta \cdot \hat{Y}t}(\alpha _+^{1/2}e^{-b\alpha _+t^2}+\alpha _-^{1/2}e^{-b\alpha _-t^2})\,dt d{\hat{Y}}\\\simeq & {} \frac{1}{2}\int _{{\mathbb {S}}^{n-2}} b^{-1/2}(e^{-|\eta \cdot {\hat{Y}}|^2/4b\alpha _+}+e^{-|\eta \cdot {\hat{Y}}|^2/4b\alpha _-})\,d{\hat{Y}}\\= & {} b^{-1/2}\int _{{\mathbb {S}}^{n-2}} e^{-|\eta \cdot {\hat{Y}}|^2/4b\alpha (0,y,0,{\hat{Y}})}\,d{\hat{Y}}, \end{aligned}$$

here $\simeq$ means equal up to a non-zero multiple.

Now we need to study the joint $(\xi ,\eta )$-behavior of $\mathcal {F}_{X,Y}\tilde{K}(y,\xi ,\eta )$. Denote $(\xi ^2+\digamma ^2)^{1/2}$ by $\langle \xi \rangle$, then ${\mathcal {F}}_{X,Y}{\tilde{K}}(y,\xi ,\eta )$ is a constant multiple of

$$\begin{aligned} \langle \xi \rangle ^{-1}\int _ {{\mathbb {S}}^{n-2}} e^{-|\frac{\eta }{\langle \xi \rangle }\cdot {\hat{Y}}|^2/4c\alpha (0,y,0,{\hat{Y}})}\,d{\hat{Y}} \end{aligned}$$

with $c=c(\digamma )>0$. Since $\alpha (0,y,0,{\hat{Y}})>0$, there exist positive $c_1, c_2$ that depend on y and are locally uniform such that $0< c_1\le \alpha \le c_2$.

When $\frac{|\eta |}{\langle \xi \rangle }$ is bounded from above, then $\langle \xi \rangle ^{-1}$ is equivalent to $\langle (\xi ,\eta )\rangle ^{-1}$ in this region in terms of decay rates,

$$\begin{aligned} \int _ {{\mathbb {S}}^{n-2}} e^{-|\frac{\eta }{\langle \xi \rangle }\cdot {\hat{Y}}|^2/4c\alpha (0,y,0,{\hat{Y}})}\,d{\hat{Y}}\ge \int _ {{\mathbb {S}}^{n-2}} e^{-c'|\frac{\eta }{\langle \xi \rangle }|^2}\,d{\hat{Y}}\ge \int _{{\mathbb {S}}^{n-2}}e^{-C'} \,d{\hat{Y}}=C''. \end{aligned}$$

Thus ${\mathcal {F}}_{X,Y}{\tilde{K}}(y,\xi ,\eta )\ge C''\langle \xi \rangle ^{-1}\simeq C''\langle (\xi ,\eta )\rangle ^{-1}$.

When $\frac{|\eta |}{\langle \xi \rangle }$ is bounded from below, in which case $\langle (\xi ,\eta )\rangle ^{-1}$ is equivalent to $|\eta |^{-1}$, we write ${\hat{Y}}=({\hat{Y}}^{\parallel }, {\hat{Y}}^{\perp })$ according to the orthogonal decomposition of $\hat{Y}$ relative to $\frac{\eta }{|\eta |}$, where ${\hat{Y}}^{\parallel }={\hat{Y}}\cdot \frac{\eta }{|\eta |}$, and $d{\hat{Y}}$ is of the form $a({\hat{Y}}^{\parallel })d{\hat{Y}}^{\parallel }d\theta$ with $\theta =\frac{{\hat{Y}}^{\perp }}{|{\hat{Y}}^{\perp }|}, a(0)=1$ then

$$\begin{aligned} \int _ {{\mathbb {S}}^{n-2}} e^{-|\frac{\eta }{\langle \xi \rangle }\cdot {\hat{Y}}|^2/4c\alpha (0,y,0,{\hat{Y}})}\,d{\hat{Y}}\ge & {} \int _ {{\mathbb {S}}^{n-2}} e^{-c'|\frac{\eta }{\langle \xi \rangle }\cdot {\hat{Y}}|^2}\,d{\hat{Y}}\nonumber \\= & {} \int _{{\mathbb {R}}}\int _{{\mathbb {S}}^{n-3}} e^{-c'({\hat{Y}}^{\parallel }\frac{|\eta |}{\langle \xi \rangle })^2} a({\hat{Y}}^{\parallel })\,d\theta d{\hat{Y}}^{\parallel }\nonumber \\= & {} \frac{\langle \xi \rangle }{|\eta |}\int _{{\mathbb {R}}}\{\frac{|\eta |}{\langle \xi \rangle }e^{-c'({\hat{Y}}^{\parallel }\frac{|\eta |}{\langle \xi \rangle })^2}\}a({\hat{Y}}^{\parallel })\,d{\hat{Y}}^{\parallel }\int _{{\mathbb {S}}^{n-3}} d\theta .\nonumber \\ \end{aligned}$$

(4.3)

Since $\frac{|\eta |}{\langle \xi \rangle }e^{-c'({\hat{Y}}^{\parallel }\frac{|\eta |}{\langle \xi \rangle })^2}\rightarrow \delta _0$ in distributions as $\frac{|\eta |}{\langle \xi \rangle }\rightarrow \infty$, (4.3) is equal to $\frac{\langle \xi \rangle }{|\eta |}\int _{\mathbb {S}^{n-3}} d\theta =2C\frac{\langle \xi \rangle }{|\eta |}$ ($C>0$) modulo terms decaying faster as $\frac{|\eta |}{\langle \xi \rangle }\rightarrow \infty$. In particular, there is $N>0$, such that

$$\begin{aligned} \int _{{\mathbb {R}}}\left\{ \frac{|\eta |}{\langle \xi \rangle }e^{-c'({\hat{Y}}^{\parallel }\frac{|\eta |}{\langle \xi \rangle })^2}\right\} a({\hat{Y}}^{\parallel })\,d{\hat{Y}}^{\parallel }\int _{{\mathbb {S}}^{n-3}} d\theta \ge C \end{aligned}$$

for $\frac{|\eta |}{\langle \xi \rangle }\ge N$. (Notice that the integral on ${\mathbb {S}}^{n-3}$ uses the assumption $n\ge 3$; when $n=3$, $d\theta$ is the point measure.) Thus ${\mathcal {F}}_{X,Y}{\tilde{K}}(y,\xi ,\eta )\ge C\frac{1}{\langle \xi \rangle }\frac{\langle \xi \rangle }{|\eta |}=C|\eta |^{-1}\simeq C\langle (\xi ,\eta )\rangle ^{-1}$.

Therefore we deduce that ${\mathcal {F}}_{X,Y}{\tilde{K}}(y,\xi ,\eta )\ge c\langle (\xi ,\eta )\rangle ^{-1}$ for some $c>0$, i.e the ellipticity claim for the case that $\chi$ is a Gaussian. Then by an approximation argument one obtains some $\chi \in C^{\infty }_c({\mathbb {R}})$ such that the Fourier transform of ${\tilde{K}}$ with this $\chi$ still has lower bounds ${\tilde{c}}\langle (\xi ,\eta )\rangle ^{-1}, {\tilde{c}}>0$, and the Lemma is proved. $\square$

Now the argument in Sect. 3.7 shows the local invertibility of the X-ray transform for a general family of curves, with a stability estimate. We denote the set of O-local curves with respect to $\mathcal {G}$ by $\mathcal {G}_O$.

Theorem 4.2

Assume X is strictly convex at $p\in \partial X$ with respect to a general family of curves $\mathcal {G}$, with $O_p=\{x>0\}\cap \overline{X}$, then the local X-ray transform for $\mathcal {G}_{O_p}$ is injective on $H^s(O_p)$, $s\ge 0$ with the stability estimate

$$\begin{aligned}\Vert f\Vert _{H_\digamma ^{s-1}(O_p)}\le C\Vert If|_{\mathcal G_{O_p}}\Vert _{H^s(\mathcal G_{O_p})}.\end{aligned}$$

Similar to the geodesic case, if X has compact closure and can be foliated by hypersurfaces that are strictly convex with respect to $\mathcal G$, then the corresponding global X-ray transform is injective. Previously, injectivity result of the global X-ray transform for a general family of curves was only known in the real analytic category [4].

Remark 4.3

If we add a non-vanishing weight $w\in C^{\infty }(T{\tilde{X}})$ to the X-ray transform, i.e.

$$\begin{aligned} (I_wf)(\gamma )=\int w(\gamma (t),\dot{\gamma }(t)) f(\gamma (t))\,dt, \end{aligned}$$

a slight modification of the proof of Lemma 4.1 allows one to conclude the local invertibility of $I_w$ for a general family of curves. Notice that given a curve $\gamma$ with $|\dot{\gamma }|\ne 0$, a reparametrization exactly introduces a non-vanishing weight to the integral, thus local invertibility of X-ray transform also holds for general families of curves with non-constant speed.