📂Tomography

Properties of Radon Transform

Properties¹

The Radon transform $\mathcal{R} : L^{2}(\mathbb{R}^{n}) \to L^{2}(\Lambda)$ has the following properties.

Linearity

For $\alpha, \beta \in \mathbb{R}$ and $f, g \in L^{2}(\mathbb{R}^{2})$, the following holds.

$$\mathcal{R} \left( \alpha f + \beta g \right) = \alpha \mathcal{R}f + \beta \mathcal{R}g$$

Shift Invariance

Let $T_{\mathbf{a}}$ be a translation for $\mathbf{a} \in \mathbb{R}^{n}$.

$$T_{\mathbf{a}}f(\mathbf{x}) := f(\mathbf{x}-\mathbf{a}) \text{ for } f\in L^{2}(\mathbb{R}^{n})\quad \text{and} \quad T_{t}g(s,\boldsymbol{\theta}) := g(s-t, \boldsymbol{\theta}) \text{ for } g \in L^{2}(\Lambda)$$

Then, the following holds.

$$\mathcal{R}T_{\mathbf{a}}f (s, \boldsymbol{\theta}) = T_{\mathbf{a} \cdot \boldsymbol{\theta}}\mathcal{R}f(s,\boldsymbol{\theta})$$

Rotation Invariance

Let $n$ be a $A$-dimensional rotation transform.

$$Af(\mathbf{x}) := f(A \mathbf{x}) \text{ for } f\in L^{2}(\mathbb{R}^{n})\quad \text{and} \quad Ag(s,\boldsymbol{\theta}) := g(s,A \boldsymbol{\theta}) \text{ for } g \in L^{2}(\Lambda)$$

Then, the following holds.

$$RAf = ARf$$

Dilation Invariance

Let $D_{r}$ be a dilation for $r>0$.

$$D_{r}f(\mathbf{x}) := f(r \mathbf{x}) \text{ for } f\in L^{2}(\mathbb{R}^{n}) \quad \text{and} \quad D_{r}g(s,\boldsymbol{\theta}) := g(rs, \boldsymbol{\theta}) \text{ for } g \in L^{2}(\Lambda)$$

Then, the following holds.

$$RD_{r}f = \dfrac{1}D_{r}Rf$$

Proof

Shift Invariance

$$\begin{align*} \mathcal{R} T_{\mathbf{a}} f(s, \boldsymbol{\theta}) =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} T_{\mathbf{a}}f(\mathbf{x}) d\mathbf{x} \\ =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} f(\mathbf{x} - \mathbf{a}) d\mathbf{x} \\ =&\ \int\limits_{\mathbf{y} \cdot \boldsymbol{\theta} = s + \mathbf{a} \cdot \boldsymbol{\theta}} f(\mathbf{y}) d\mathbf{y} \\ =&\ \mathcal{R}f(s + \mathbf{a} \cdot \boldsymbol{\theta}, \boldsymbol{\theta}) \\ =&\ T_{\mathbf{a}\cdot \boldsymbol{\theta}}\mathcal{R}f(s , \boldsymbol{\theta}) \end{align*}$$

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Rotation Invariance

$$\begin{align*} \mathcal{R} A f(s, \boldsymbol{\theta}) =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} Af(\mathbf{x}) d\mathbf{x} \\ =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} f(A\mathbf{x}) d\mathbf{x} \\ =&\ \int\limits_{A^{-1}\mathbf{y} \cdot \boldsymbol{\theta} = s} f(\mathbf{y}) d\mathbf{y} \\ =&\ \int\limits_{\mathbf{y} \cdot A\boldsymbol{\theta} = s} f(\mathbf{y}) d\mathbf{y} \\ =&\ \mathcal{R} f(s, A\boldsymbol{\theta}) \\ =&\ A\mathcal{R} f(s, \boldsymbol{\theta}) \end{align*}$$

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Dilation Invariance

$$\begin{align*} \mathcal{R} D_{r} f(s, \boldsymbol{\theta}) =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} D_{r}f(\mathbf{x}) d\mathbf{x} \\ =&\ \int\limits_{\mathbf{x} \cdot \boldsymbol{\theta} = s} f(r\mathbf{x}) d\mathbf{x} \\ =&\ \int\limits_{\frac{1}{r}\mathbf{y} \cdot \boldsymbol{\theta} = s} f(\mathbf{y}) \dfrac{1}{r}d\mathbf{y} \\ =&\ \dfrac{1}{r} \int\limits_{\mathbf{y} \cdot \boldsymbol{\theta} = rs} f(\mathbf{y}) d\mathbf{y} \\ =&\ \dfrac{1}{r} \mathcal{R} f(rs, \boldsymbol{\theta}) \\ =&\ \dfrac{1}{r} D_{r} \mathcal{R} f(s, \boldsymbol{\theta}) \\ \end{align*}$$

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