On an integral geometric formula

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bsV}{{\boldsymbol{V}}}$ $\DeclareMathOperator{\Graffr}{\mathbf{Graff}^c}$  $\newcommand{\be}{\boldsymbol{e}}$ $\newcommand{\bv}{\boldsymbol{v}}$   $\DeclareMathOperator{\Grr}{\mathbf{Gr}^c}$ $\newcommand{\Gr}{\mathbf{Gr}}$ $\newcommand{\Graff}{\mathbf{Graff}}$

Suppose that $\bsV$ is a finite dimensional real Euclidean  space, $M\subset \bsV$  is a smooth compact submanifold of dimension $m$ and codimension $r$ and we set


$$N:=\dim \bsV=m+r.$$

  For any  nonnegative integer $c\leq \dim \bsV$ we denote by $\Graff^c(\bsV)$ the  Grassmannian  of  affine subspaces of $\bsV$ of codimension $c$,   by $\Gr^c(\bsV)$ the Grassmannian of  codimension $c$ vector subspaces of $\bsV$. We set $\Gr_k(\bsV):=\Gr^{N-k}(\bsV)$.

The   codimension $c$ Radon transform of a smooth function $f: M\to  \bR$  is a function

$$\widehat{f}:\Graff^c(\bsV)\to\bR ,$$

such that $\newcommand{\eH}{\mathfrak{H}}$

$$\widehat{f}(S) =\int_{S\cap M} f(x) d\eH^{m-c}(x),  \;\; \forall S\in \Graff^c(M), \label{r}\tag{R}$$

where $d\eH^{m-c}$ denotes the $(m-c)$-dimensional Hausdorff measure.   If $c\leq \dim M$ then a generic   affine plane $S\in\Graff^c(\bsV)$ intersects  $M$ transversally in which case the Hausdorff measure in ($\ref{r}$) is the usual Lebesgue measure induced my the  natural Riemann metric on $S\cap M$.

I want  to explain how to  recover the integral of $f$ over $M$ from its Radon transform.



Observe that we have an incidence set $\newcommand{\eI}{\mathscr{I}}$

$$\eI^c(\bsV) :=\Bigl\{ (\bv, S)\in \bsV\times \Graffr(\bsV);\;\; \bv\in S\;\Bigr\}$$

equipped with  natural projections

$$\bsV\stackrel{\lambda}{\leftarrow}\eI^c(\bsV)\stackrel{\rho}{\to}\Graffr(\bsV).\label{F}\tag{F}$$


For any subset $X\subset \bsV$ we define

$$\eI^c(X):=\lambda^{-1}(M)\subset \eI^r(X),\;\; \Graffr(X)=\rho\Bigl(\;\eI^r(X)\;\Bigr).$$

Note that

$$\Graffr(X)=\Bigl\{ S\in \Graffr(\bsV);\;\; S\cap X\neq \emptyset\;\Bigr\}$$

and for any $\bv\in\bsV$ we have

$$\lambda^{-1}(\bv) =\bigl\{ \bv+S;\;\;S\in \Grr(\bsV)\;\bigr\}=\Graffr(\bv)\subset \Graffr(\bsV).$$


Observe that  $\eI^c(V)\to \bsV$ is a smooth fiber bundle  with fiber  $\Gr^r(\bsV)$. In particular,  $\eI^c(M)\to M$ is the bundle obtained  by restricting to the submanifold $M$.  Its fiber is also $\Gr^c(\bsV)$.

At this point I need to recall some  basic facts described in great detail in Sections 9.1.2, 9.1.3 of  Lectures on the Geometry of Manifolds.

 The Grassmannain $\Gr^c(\bsV)$ is equipped with a canonical $O(\bsV)$-invariant metric  with volume  density $|d\gamma^c_\bsV|$ with total volume $\newcommand{\sbinom}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}$

$$\int_{\Gr^c(\bsV)} |d\gamma_\bsV^c(L)|=\sbinom{N}{c},$$

where $\sbinom{N}{c}$ is defined  in   equation (9.1.66) of the Lectures.

Now observe that  we have a natural projection $\pi: \Graff^c(\bsV)\to \Gr^c(\bsV)$ that associates  to each affine  plane its translate through the origin.    A plane $S\in\Graff^c(\bsV)$ intersects the orthogonal complement  of $\pi(S)$ in a unique point $C(S)=S\cap \pi(S)^\perp$.   We obtain a an embeding

$$\Gr^c(\bsV)\ni S\mapsto \bigl(\;C(S), \pi(S)\;\bigr)\in \bsV\times\Gr^c(\bsV),\;\;C(S)\perp \pi(S),$$
$\newcommand{\eQ}{\mathfrak{Q}}$
and we  will regard  $\Graff^c(\bsV)$ as a submanifold of $\bsV\times \Gr^c(\bsV)$.  As such,     it becomes  the total space of  a vector bundle $\eQ_c\to\Gr^c(\bsV)$, in fact a subbundle of the trivial bundle $\bsV\times  \Gr^c(\bsV)\to\Gr^c(\bsV)$.   The orthogonal complement  $\eQ_c^\perp$ of this bundle is the tautological vector bundle $\newcommand{\eU}{\mathscr{U}}$  ${\eU}^c\to\Gr^c(\bsV)$.  In particular

$$\dim\Gr^c(\bsV)= c(N-c)+  c.$$

Along $\Graff^c(\bsV)$ we have a canonical vector bundle,  the  vertical bundle $VT\Graff^c(\bsV)\subset T\Graff^c(\bsV)$   consisting of the kernels of $d\pi$, i.e., vectors tangent to the fibers of $\pi$. The  vertical bundle is equipped with a natural density  $|d\bv|_c$ which when restricted to a fiber of $\pi^{-1}(L)$  induces the natural volume form on the fiber $L^\perp$ viewed as a vector subspace of $\bsV$. As in Section 9.1.3 of the Lectures we define a product  density $|d\tilde{\gamma}^c|=|d\tilde{\gamma}_\bsV^c|$ on $\Graff^c(\bsV)$,

$$|d\tilde{\gamma}_\bsV^c|= |d\bv|_c\times \pi^*|d\gamma_\bsV^{c}|$$

Alternatively, the vector bundle $\eQ_c$, as a subbundle of the trivial bundle $\bsV\times \Gr^c(\bsV)\to\Gr^c(\bsV)$  is equipped with a natural metric connection. The  horizontal subbundle   $HT\eQ_c\subset T\eQ_c$  is isomorphic to $\pi^* T\Gr^c(\bsV)$ and thus comes equipped with a natural metric.    The  vertical subbundle $VT\eQ_c=VT\Graff^c(\bsV)$ is also equipped with a  natural  metric and in this fashion we obtain a metric on $\Graff^c(\bsV)=\eQ_c$. The density $|d\tilde{\gamma}^c_\bsV|$  is the volume density defined by this metric.


Suppose now that $c\leq m=\dim M$.   We denote by $\Graff^c_*(M)$ the subset of $\Graff^c(M)$ consisting of affine planes that intersect $M$ transversally.    This is an open subset of $\Graff^c(M)$.  The condition $c\leq m$ implies that this set is nonempty.  (For $c=1$ this follows from the fact that the restriction to $M$ of a generic linear function is a Morse function. Then look at iterated slicing by hyperplanes.)

Set

$$\eI^c_*(M)= \rho^{-1}\bigl(\;\Graff^c_*(M)\;\bigr)\subset \eI_M$$

The fiber of  $\rho:\eI_*^c(M)\to \Graff^c_*(M)$ over $S\in \Graff^c_*(M)$ is the submanifold $S\cap M$ which is equipped with a metric density.  We obtain a density on $\eI^c_*(M)$

$$|d\nu^c_M|= |dV_{S\cap M}|\times \rho^*|d\tilde{\gamma}^c|. \tag{$\nu^c$}\label{nu}$$

If $f: M\to\bR$ is a smooth function, then

$$\int_{\eI^c_*(M)}\lambda^*(f) |d\nu^c_M|=\int_{\Graff^c_*(M)}\left(\int_{S\cap M} f|dV_{S\cap M}\right) |d\tilde{\gamma}^c(S)|. \label{1}\tag{1}$$

For any  vector subspace $U\subset \bsV)$ we denote by $\Gr^c(\bsV)_U$ the set consisiting of subspaces $L\in\Gr^c(\bsV)$ that intersect $U$ transversely.


We now want to integrate $\lambda^*(f)$ along the fibers of $\lambda :\eI^c_*(M)\to M$.  For any  vector subspace $U\subset \bsV)$ we denote by $\Gr^c(\bsV)_U$ the set consisting of subspaces $L\in\Gr^c(\bsV)$ that intersect $U$ transversely.

The fiber of this map over a point $x\in M$ is an open  subset of $x+\Gr^c(\bsV)_{T_xM}\subset \Graff^c(\bsV)$ with negligible complement.    The density $|d\nu^c_M|$ on $\eI^c_*(M)$ induces  a density

$$|d\nu^c_x|=|d\nu^M|/\lambda^*|dV_M|$$


on each fiber $\lambda^{-1}(x)$ and we deduce

$$\int_{\eI^c_*(M)} \lambda^* f|d\nu^c(M)|= \int_M\left(\int_{\lambda^{-1}(x)}|d\nu^c_x|\right) f(x)|dV_N(x)|. \label{2}\tag{2}$$

The density $|d\nu^c(x)|$ is  the restriction of a density $|d\bar{\nu}^c_x|$ on $\Gr^c(\bsV)_{T_xM}$. In fact, a reasoning similar to the one   in the proof  of Lemma 9.3.21 in the Lectures implies that  for any $U\in\Gr_m(\bsV)$ there exists a canonical density $|d\bar{\nu}^c_U|$ on $\Gr^c(\bsV)_U$ such that

$$T_*|d\nu^c_U|=|d\bar{\nu^c}_{T(U)}|,\;\;\forall T\in O(\bsV),\;\;U\in \Gr_m(\bsV), \label{3}\tag{3}$$


$$|d\bar{\nu}^c_x|=|d\bar{\nu}^c_{T_xM}|,\;\;\forall x\in M. \label{4}\tag{4}$$


 Using ($\ref{3}$) ($\ref{4}$) in ($\ref{2}$) we deduce that there  exists a constant $Z=Z(N,m,c)$ that depends only on $N,m,c$ such that

$$Z(N,m,c)=\int_{\nu^{-1}(x)} |d\bar{\nu}^c_x|,\;\;forall x\in M.$$

Using this in ($\ref{2}$)  we conclude from ($\ref{1}$) that

$$Z(N,m,c)\int_{M}f(x)\; |dV_M(x)| =\int_{\Graff^c(\bsV)}\left(\int_{S\cap M} f(x)|dV_{S\cap M}|\right) |d\tilde{\gamma}^c_\bsV(S)|.\label{5}\tag{5}$$




To find the constant $Z(N,m,c)$ we choose $M$ and $f$ judiciously.  We let $M=\Sigma^m$, the unit $m$-dimensional sphere contained in some $(m+1)$-dimensional subspace of $\bsV$.  Then, we let $f\equiv 1$. We deduce from ($\ref{5}$) that

$$Z(N,m,c)=\frac{1}{{\rm vol}\;(\Sigma^m)} \int_{\Graff^c(\bsV)} {\rm vol}\,(S\cap \Sigma^m)\;|d\tilde{\gamma}^c_\bsV(S)|.\label{6}\tag{6}$$


Using the Crofton formula in Theorem 9.3.34 in the Lectures  in the special case $p=m-c$ we deduce


$$Z(N,m,c)=\sbinom{m}{c}.$$

Remark. 1        Consider the Radon transform


$$C_0^\infty(\bsV)\ni f\mapsto  \widehat{f}\in C^\infty\bigl(\;\Graff^c(\bsV)\;\bigr), \;\; \widehat{f}(S)=\int_S f(x)|dV_S(x)|,\;\;\forall S\in \Graff^c(\bsV).$$

Observe that $\widehat{f}$ has compact support.   Indeed, if  the support of $f$ is contained in a ball of radius $R$, then for any affine plane $S\in \Graff^c(\bsV)$ such that ${\rm dist}\,(0,S)>R$ we have $\widehat{f}(S)=0$.

Consider the dual Radon transform  $\newcommand{\vfi}{\varphi}$


$$C^\infty\bigl(\;\Graff^c(\bsV)\;\bigr)\ni \vfi\mapsto \check{\vfi}\in C^\infty(\bsV),\;\;\check{\vfi}(x)=\int_{\Gr^c(\bsV)} \vfi(x+L)\;|d\gamma^c(L)|,\;\;\forall x\in \bsV.$$



Consider the fundamental double fibration  ($\ref{F}$). Given $f\in C_0^\infty(\bsV)$, $\vfi\in C^\infty_0\bigl(\;\Graff^c(\bsV)\;\bigr)$ we obtain a function 


$$\Phi=\lambda^*(f)\cdot \rho^*(\vfi)\in C_0^\infty(\bsV)$$

Arguing as above, with $M=\bsV$  we  observe that $\Graff^c_*(\bsV)=\Graff^c(\bsV)$ and we obtain as in ($\ref{nu}$) a density  $|\nu^c_\bsV|$ on $\eI^c_*(\bsV)=\eI^c(\bsV)$.   Denote by $\rho_*\Phi |d\nu^c_\bsV|$ the   pushfoward  of the density $\Phi|d\nu^c_\bsV$. It is a density on $\Graff^c(\bsV)$  and we have the  Fubini formula (coarea formula)

$$\int_{\eI^c(\bsV)} \Phi(x,S) |d\nu^c_\bsV(x,S)|=\int_{\Graff^c(\bsV)}\rho_*\Phi|d\ni^c_\bsV|\label{7}\tag{7}$$

Similarly, we obtain

$$\int_{\eI^c(\bsV)} \Phi(x,S) |d\nu^c_\bsV(x,S)|=\int_{\bsV} \lambda_*\Phi |d\nu^c_\bsV|(x).\label{8}\tag{8}$$

From the construction  of $|d\nu^c_\bsV|$ we deduce immediately that

$$\rho_*\Phi|d\nu^c_\bsV|(S)=  \widehat{f}(S) \vfi(S) |d\tilde{\gamma}^c|(S).$$


From the definitions of $|d\nu^c_\bsV|$, $|d\gamma^c_\bsV|$ and  $|d\tilde{\gamma}^c_\bsV|$ it follows easily that

$$\lambda_*\Phi |d\nu^c_\bsV|(x) =  f(x)\check{\vfi}(x)|dx|$$

Using the last equalities in ($\ref{7}$) and ($\ref{8}$)  we deduce

$$\int_{\bsV} f(x)\check{\vfi}(x)=\int_{\Graff^c(\bsV)} \widehat{f}(S)\vfi(S) |d\tilde{\Gamma}^c(S)|. \tag{D}  \label{d}$$

The equality ($\ref{d}$) shows that the operations $f\mapsto \widehat{f}$ and $\vfi\mapsto \widehat{\vfi}$ are indeed dual to each other.   Note also that if we set  $\vfi\equiv 1$ in ($\ref{d}$) then

$$\check{\vfi}(x)={\rm vol}\,\bigl(\;\Gr^c(\bsV)\;\bigr)=\sbinom{N}{c}$$

and in this case we reobtain ($\ref{5}$) in the special case $M=\bsV$.  The  equality ($\ref{d}$) is  important for another reason.

Denote by $C_0^{-\infty}(\bsV)$ the space of generalized functions with compact supports, then we can extend the Radon transform to such objects. If $u\in C_0^{-\infty}(\bsV)$ then we define its   Radon transform $\widehat{u}$ to be the compactly supported  generalized density on $\Graff^c(\bsV)$ defined by the equality

$$\langle \widehat{u},\vfi\rangle=\langle u,\check{\vfi}\rangle ,\;\;\forall \vfi\in C^\infty\bigl(\;\Graff^c(\bsV)\;\bigr).$$

If $M$ is a compact submanifold of $\bsV$, then we get a  Dirac-type  generalized function $\delta_M$ on $\bsV$   defined by integration along $M$ with respect to the volume density on $M$ determined by the induced metric.   Then

$$\langle\widehat{\delta}_M,\vfi\rangle =\int_M  \check{\vfi}(x) |dV_M(x)|,\;\;\forall  C^\infty\bigl(\;\Graff^c(\bsV)\;\bigr).$$

The  generalized function $\widehat{\delta}_M$ is represented by a locally integrable function

$$\widehat{\delta}_M(S) =\eH^{m-c}(M\cap S),\;\;\forall S\in \Graff^c(M).$$