We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Verify that $\Psi\left(M\left(R_\theta, q\vec{u}_\theta\right)\right) = \Delta\left(q, \vec{u}_\theta\right)$; deduce that $\Psi$ is surjective.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$.
a) Describe the elements of $H$.
b) Show that $H$ is a subgroup of $G$.
c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Establish that $f$ is in $\mathcal{B}_1$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Show that $\hat{f}$ is defined on $\mathbb{R}^2$ with $\hat{f}(q,\theta) = \frac{\pi}{\sqrt{1+q^2}}$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
We set $R(q) = \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta$. Prove that $q \mapsto \frac{R^{\prime}(q)}{q}$ is integrable on $]0, +\infty[$ and that $$f(0,0) = -\frac{1}{\pi} \int_0^{+\infty} \frac{R^{\prime}(q)}{q}\,\mathrm{d}q$$ One may, to compute this last integral, use the change of variable $q = \operatorname{sh}(u)$.

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Is the function $\frac{\partial f}{\partial x}$ in $\mathcal{B}_2$?

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
For $r \in \mathbb{R}^+$, compute $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Prove that the Radon transform of $f$ is defined on $\mathbb{R}^2$ and that $$\forall q \in \mathbb{R}^+,\quad \forall \theta \in \mathbb{R} \quad \hat{f}(q,\theta) = 2\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Deduce that $\forall q \in \mathbb{R}^+,\ \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Justify that for all $q$ and all $\theta$ we have $\hat{f}(-q, \theta+\pi) = \hat{f}(q,\theta)$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that the function $r \mapsto r^2 \bar{f}(r)$ is bounded on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Show that if we further assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$, then $r \mapsto r^4 \bar{f}^{\prime}(r)$ is bounded on $\mathbb{R}$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that $H$ is continuous on $]0, +\infty[$.

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.
Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

We now assume that $f$ satisfies the hypotheses allowing us to define its Radon transform.
Demonstrate that if two lines $\Delta\left(q_1, \vec{u}_{\theta_1}\right)$ and $\Delta\left(q_2, \vec{u}_{\theta_2}\right)$ coincide, then $\hat{f}\left(q_1, \theta_1\right) = \hat{f}\left(q_2, \theta_2\right)$.

We define $\hat{f}^\star$ on $G$ by composing $\hat{f}$ with $\Psi$: we set, for all $g \in G$, $\hat{f}^\star(g) = \hat{f}(\Psi(g))$.
Demonstrate that $\hat{f}^\star$ is $H$-invariant, that is, for all $g \in G$ and $h \in H$, $\hat{f}^\star(gh) = \hat{f}^\star(g)$.

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Propose a rigorous definition of the right-hand side of this equation in the case where $\Delta = \Delta\left(q, \vec{u}_\theta\right)$.

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Explain how the Radon inversion formula allows us in principle to know the density of tissues in the radiographed zone.

We denote $\mathcal{D}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ and with compact support. We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Study the variations of $\varphi$. b) Sketch the graph of $\varphi$. c) Show that $\varphi$ is $\mathcal{C}^{\infty}$. d) Show that $\mathcal{D}$ is a vector space over $\mathbb{R}$ not reduced to $\{0\}$.

Show that the derivative of every element of $\mathcal{D}$ is an element of $\mathcal{D}$.

We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number. b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$.
Show that $$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.