Null space and resolution in dynamic computerized tomography

The 2D Radon transform (1) can be considered as mapping

$$\begin{eqnarray*}&&{ \mathcal R }\ :{L}_{2}({V}_{1}(0))\to {L}_{2}(Z)\\ &&{ \mathcal R }f({\boldsymbol{\theta }},s)={\displaystyle \int }_{{V}_{1}(0)}f(x)\;\delta (s-{x}^{T}{\boldsymbol{\theta }})\;{\rm{d}}x\end{eqnarray*}$$

with the cylinder $Z:= {S}^{1}\times {\mathbb{R}}.$ Without loss of generality, it is assumed that the investigated object described by the x-ray attenuation coefficient f is compactly supported in the unit ball ${V}_{1}(0)\subset {{\mathbb{R}}}^{2},$ [21].

The aim is to determine the unknown function f from the measured Radon data, i.e. to solve the inverse problem

$$\begin{eqnarray*}&&{ \mathcal R }f=g.\end{eqnarray*}$$

A data set $g\in {L}_{2}(Z)$ is called consistent if it is in the range of ${ \mathcal R },$ denoted by $\mathrm{Im}({ \mathcal R }).$ The following Helgason–Ludwig-consistency conditions hold for the Radon transform, [6, 12].

Theorem 2.1. It holds $g\in \bar{\mathrm{Im}({ \mathcal R })}$ if and only if

• (i)  
  $g({\boldsymbol{\theta }},s)=0$ for $| s| \geqslant 1,$
• (ii)  
  g is even,
• (iii)  
  ${\displaystyle \int }_{{\mathbb{R}}}{s}^{n}\;g({\boldsymbol{\theta }},s)\;{\rm{d}}s={p}_{n}({\boldsymbol{\theta }})$ with a trigonometric polynomial p_n of degree $\leqslant n.$

In [16, 17], the singular value decomposition of the Radon transform was derived in arbitrary dimensions. Denote $w(s):= {(1-{s}^{2})}^{-1/2}.$

Theorem 2.2. The singular value decomposition of the Radon transform

$$\begin{eqnarray*}&&{ \mathcal R }\ :{L}_{2}({V}_{1}(0))\to {L}_{2}(Z,w)\end{eqnarray*}$$

is given by

$$\begin{eqnarray*}&&\{({\sigma }_{{nl}},{v}_{{nl}},{u}_{{nl}})\},n\geqslant 0,l\in {\mathbb{Z}},| l| \leqslant n,n+l\in 2{\mathbb{Z}},\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\sigma }_{{nl}}={\sigma }_{n}=2\sqrt{\displaystyle \frac{\pi }{n+1}},\\ \;{v}_{{nl}}(x)=\left\{\begin{array}{ll}\sqrt{\displaystyle \frac{n+1}{\pi }}\;{| x| }^{| l| }\;{P}_{(n-| l| )/2}^{(0,| l| )}(2{| x| }^{2}-1)\;{Y}_{l}\left(\displaystyle \frac{x}{| x| }\right) & \mathrm{for}\ | x| \leqslant 1,\\ 0 & \mathrm{for}\ | x| \gt 1,\end{array}\right.\end{eqnarray*}$$

and

$$\begin{eqnarray*}{u}_{{nl}}({\boldsymbol{\theta }},s)=\left\{\begin{array}{ll}{\pi }^{-1}\;w{(s)}^{-1}\;{U}_{n}(s)\;{Y}_{l}({\boldsymbol{\theta }}) & \mathrm{for}\ | s| \leqslant 1\\ 0 & \mathrm{for}\ | s| \gt 1\end{array}\right.\end{eqnarray*}$$

with the Jacobi-polynomials ${P}_{n}^{(\alpha ,\beta )},$ the Chebychev-polynomials of the second kind U_n and the spherical harmonics Y_l of degree l in ${{\mathbb{R}}}^{2}.$

In practical applications, only a finite number of projections can be measured. In the stationary setting, it is known that the Radon transform with complete projections for finitely many directions has a nontrivial null space, so f is not uniquely determined [16]. Complete projections means that $g({{\boldsymbol{\theta }}}_{j},s)$ is known for $s\in {\mathbb{R}}.$ Thus, this is refered to as the semi-discrete case. Let

$$\begin{eqnarray*}&&{{ \mathcal N }}_{p}({ \mathcal R }):= \{f\in {L}_{2}({V}_{1}(0))\ :{ \mathcal R }f({{\boldsymbol{\theta }}}_{j},s)=0\ \mathrm{for}\ \mathrm{almost}\ \mathrm{all}\ s\in {\mathbb{R}}\ \mathrm{and}\ j=1,...,p\}\end{eqnarray*}$$

denote this null space and let $A:= \{{{\boldsymbol{\theta }}}_{1},...,{{\boldsymbol{\theta }}}_{p}\}\subset {S}^{1}$ not be contained in an algebraic variety of degree $\lt p,$ i.e. there is no $q\in {{ \mathcal P }}_{l},$ $l\lt p,q\not\equiv 0$ with $q({\boldsymbol{\theta }})=0$ for all ${\boldsymbol{\theta }}\in A,$ see [16]. Here

$$\begin{eqnarray}&&{{ \mathcal P }}_{l}:= \mathrm{span}\{\mathrm{cos}(k\varphi ),\mathrm{sin}(k\varphi )\ :0\leqslant k\leqslant l,k+l\ \mathrm{even}\}\end{eqnarray} \tag{ 2 }$$

is the set of spherical harmonics of degree $\lt l$ which are even for l even, and odd for l odd. Please note that in the two-dimensional case, a set of p distinct source positions fulfils this condition, and that the angles of ${{\boldsymbol{\theta }}}_{1},...,{{\boldsymbol{\theta }}}_{p}$ are not necessarily equispaced, see [21].

Then, for $f\in {{ \mathcal N }}_{p}({ \mathcal R }),$ it holds

$$\begin{eqnarray}&&{ \mathcal R }f({\boldsymbol{\theta }},s)=w{(s)}^{-1}\displaystyle \sum _{n\geqslant p}{U}_{n}(s)\;{q}_{n}({\boldsymbol{\theta }})\end{eqnarray} \tag{ 3 }$$

with a trigonometric polynomial q_n of degree $\leqslant n,$ see [15, 16]. This means, for fixed ${\boldsymbol{\theta }}\in {S}^{1},$ the series expansion of ${ \mathcal R }f({\boldsymbol{\theta }},\cdot )$ in Chebychev-polynomials U_n contains only polynomials of degree p and larger. An explicit characterization of the elements in the null space, the so-called ghosts, is given in [17], showing that they are highly oscillating functions by studying their frequency distribution. Especially, this yields that only details of size $2\pi /p$ and larger can be reliably reconstructed from Radon data measured from p directions.

Due to the rotation of the radiation source, the data acquisition in computerized tomography takes a certain amount of time. The position of the x-ray source is described by ${\boldsymbol{\theta }}\in {S}^{1},$ and therefore, each unit vector ${\boldsymbol{\theta }}\in {S}^{1}$ can be uniquely described by a time instance t, via the parametrization

$$\begin{eqnarray*}&&{\boldsymbol{\theta }}(t)=\left(\begin{array}{c}\mathrm{cos}(t\phi )\\ \mathrm{sin}(t\phi )\end{array}\right),\end{eqnarray*}$$

where ϕ is the rotation angle of the source [8, 9].

Occasionally, throughout the paper, we have to depict a unit vector in terms of its angular phase. To keep the notation simple, ${\boldsymbol{\theta }}$ denotes the unit vector and θ its respective angle, so it holds the correlation ${\boldsymbol{\theta }}={(\mathrm{cos}\theta ,\mathrm{sin}\theta )}^{T}.$

In the following, let $f\in {L}_{2}({V}_{1}(0))$ describe the state of the investigated object at the beginning of the scanning. Its state at the moment of measuring the data ${g}_{{\boldsymbol{\theta }}}$ is then assumed to be given by $f\;\circ \;{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}$ with diffeomorphic motion functions ${{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}\ :{{\mathbb{R}}}^{2}\to {{\mathbb{R}}}^{2}.$ This is a common model for the dynamic behaviour in tomography, see e.g. [2, 8, 14]. Within this setting, the mathematical model of dynamic computerized tomography is given by

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}\ :{L}_{2}({V}_{1}(0))\to {L}_{2}(Z)\\ &&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)={\displaystyle \int }_{{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}({V}_{1}(0))}f({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}x)\;\delta (s-{x}^{T}{\boldsymbol{\theta }})\;{\rm{d}}x.\end{eqnarray*}$$

A detailed derivation of the dynamic operator ${{ \mathcal R }}_{{\rm{\Gamma }}}$ is given in [10] in the general context of linear inverse problems.

The task of reconstruction methods in the dynamic case is to determine both the motion functions and the reference function f from the measured data. In this paper, we want to study the effect of the dynamic behaviour on the achievable resolution, independent of a reconstruction procedure. Therefore, considering the exact motion functions ${{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}$ for our mathematical analysis is justified.

We now discuss some properties of the underlying motion model. Without loss of generality, we can make the assumption that the trajectory of a particle x,

$$\begin{eqnarray}&&{\mathrm{tr}}_{x}\ :{S}^{1}\to {{\mathbb{R}}}^{2},\quad {\mathrm{tr}}_{x}({\boldsymbol{\theta }})={{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}x,\end{eqnarray} \tag{ 4 }$$

is smooth, since in applications, only a discrete number of projections is measured.

The classical Radon transform ${ \mathcal R }$ is a symmetric operator in the sense that

$$\begin{eqnarray}&&{ \mathcal R }f(-{\boldsymbol{\theta }},-s)={ \mathcal R }f({\boldsymbol{\theta }},s),\end{eqnarray} \tag{ 5 }$$

see also theorem 2.1. In contrast, functions in the range of the dynamic operator ${{ \mathcal R }}_{{\rm{\Gamma }}}$ are in general not even functions of the cylinder variables. However, the data acquisition scheme in 2D parallel scanning geometry makes use of the symmetry of the Radon transform (5), i.e. the data are collected for unit vectors ${\boldsymbol{\theta }}$ covering only the half sphere. The same acquisition process is applied in the dynamic case, i.e. we can assume ${{\rm{\Gamma }}}_{-{\boldsymbol{\theta }}}={{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}$ without affecting the measured data, [8]. This yields

$$\begin{eqnarray}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f(-{\boldsymbol{\theta }},-s)={{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s).\end{eqnarray} \tag{ 6 }$$

Changes of the object during the scanning lead in general to inconsistent data sets, i.e. $g\notin \mathrm{Im}({ \mathcal R }).$ However, in the noise-free case, it is an element in the range of the dynamic operator ${{ \mathcal R }}_{{\rm{\Gamma }}}.$ For $g\in \mathrm{Im}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ we prove a condition similar to the property in theorem 2.1, iii), which is then a necessary consistency condition for ${{ \mathcal R }}_{{\rm{\Gamma }}}.$ To keep the notation simple, we denote

$$\begin{eqnarray*}&&{{ \mathcal T }}_{n}:= \{p\ :[0,2\pi )\to {\mathbb{R}},\mathrm{being}\ {\rm{a}}\ \mathrm{trigonometric}\ \mathrm{polynomial}\ \mathrm{of}\ \mathrm{degree}\leqslant n\}.\end{eqnarray*}$$

Theorem 3.1. Let the inverse motion functions be given by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}x=\left(\begin{array}{c}{h}_{1}(\theta ,x)\\ {h}_{2}(\theta ,x)\end{array}\right),\ \mathrm{with}\ {h}_{i}(\cdot ,x)\in {{ \mathcal T }}_{L},i=1,2,\mathrm{for}\ \mathrm{every}\ x.\end{eqnarray} \tag{ 7 }$$

Then, for $f\in {L}_{2}({V}_{1}(0)),$ it holds

$$\begin{eqnarray*}&&{\displaystyle \int }_{{\mathbb{R}}}{s}^{n}\;{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{\rm{d}}s\ \in {{ \mathcal T }}_{(L+1)n+2L}.\end{eqnarray*}$$

Proof. Using the transformation of coordinates $x:= {{\rm{\Gamma }}}_{\theta }^{-1}z,$ we obtain

$$\begin{eqnarray*}{\displaystyle \int }_{{\mathbb{R}}}{s}^{n}\;{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{\rm{d}}s & = & {\displaystyle \int }_{{\mathbb{R}}}{\displaystyle \int }_{{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}({V}_{1}(0))}f({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}x)\;\delta (s-{x}^{T}{\boldsymbol{\theta }})\;{s}^{n}\;{\rm{d}}x\;{\rm{d}}s\\ & = & {\displaystyle \int }_{{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}({V}_{1}(0))}f({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}x)\;{({x}^{T}{\boldsymbol{\theta }})}^{n}\;{\rm{d}}x\\ & = & {\displaystyle \int }_{{V}_{1}(0)}f(z)\;| \mathrm{det}D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z| {\left({({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z)}^{T}{\boldsymbol{\theta }}\right)}^{n}\;{\rm{d}}z\\ & =: & p(\theta ).\end{eqnarray*}$$

Using the representation (7) of the inverse motion functions ${{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1},$ it holds

$$\begin{eqnarray*}&&{({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z)}^{T}{\boldsymbol{\theta }}={h}_{1}(\theta ,z)\;\mathrm{cos}\theta +{h}_{2}(\theta ,z)\mathrm{sin}\theta \in {{ \mathcal T }}_{L+1}.\end{eqnarray*}$$

Thus, the mapping $\theta \mapsto {\left({({{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z)}^{T}{\boldsymbol{\theta }}\right)}^{n}$ is an element in ${{ \mathcal T }}_{(L+1)n}.$ The components of the Jacobian $D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z$ are given by

$$\begin{eqnarray*}&&{\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{i,j}=\displaystyle \frac{\partial {h}_{i}(\theta ,z)}{\partial {z}_{j}},\quad i,j=1,2.\end{eqnarray*}$$

Since the differentiation takes place with respect to the spatial coordinates only, we obtain

$$\begin{eqnarray*}&&{\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{i,j}\in {{ \mathcal T }}_{L},\end{eqnarray*}$$

and thus

$$\begin{eqnarray*}&&\mathrm{det}D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z={\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{\mathrm{1,1}}\cdot {\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{\mathrm{2,2}}-{\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{\mathrm{1,2}}\cdot {\left(D{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}z\right)}_{\mathrm{2,1}}\in {{ \mathcal T }}_{2L}.\end{eqnarray*}$$

Altogether, it holds $p\in {{ \mathcal T }}_{(L+1)n+2L}.$

This result will play an important role within the characterization of the null space of the semi-discrete dynamic operator. Especially, it is not restricted to affine deformations and holds for all motion models satisfying (7). In applications, this condition (7) can be realized by choosing a suitable underlying motion model. Thus, it is not a restrictive condition.

We want to study resolution properties in the dynamic setting at the case of affine deformations. Thus, throughout the rest of the paper, the motion functions are given by

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}x:= {A}_{{\boldsymbol{\theta }}}x+{b}_{{\boldsymbol{\theta }}},\qquad {\boldsymbol{\theta }}\in {S}^{1}\end{eqnarray*}$$

with invertible matrix ${A}_{{\boldsymbol{\theta }}}\in {{\mathbb{R}}}^{2\times 2}$ and vector ${b}_{{\boldsymbol{\theta }}}\in {{\mathbb{R}}}^{2}.$ Due to the smoothness assumption on the trajectories ${\mathrm{tr}}_{x},$ (4), the entries of ${A}_{{\boldsymbol{\theta }}}$ and ${b}_{{\boldsymbol{\theta }}}$ are considered to be continuously differentiable. Further, we assume that the motion model satisfies

$$\begin{eqnarray}&&h({\boldsymbol{\theta }}):= {\left({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\right)}_{1}\cdot \displaystyle \frac{\partial }{\partial \theta }{\left({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\right)}_{2}-{\left({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\right)}_{2}\cdot \displaystyle \frac{\partial }{\partial \theta }{\left({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\right)}_{1}\ne 0\quad \mathrm{for}\ {\boldsymbol{\theta }}\in {S}^{1},\end{eqnarray} \tag{ 8 }$$

where ${A}_{{\boldsymbol{\theta }}}^{-*}:= {\left({A}_{{\boldsymbol{\theta }}}^{*}\right)}^{-1}$ denotes the inverse of the adjoint matrix ${A}_{{\boldsymbol{\theta }}}^{*}$. This condition means that the dynamic behaviour does not lead to infinitesimally rigid integration curves, see [11], where this condition is analysed in more detail.

Theorem 3.2. Let the object's deformation be given by affine motion functions as described above. Then, the dynamic operator ${{ \mathcal R }}_{{\rm{\Gamma }}}\ :{L}_{2}({V}_{1}(0))\to {L}_{2}(Z)$ is related to the classical Radon transform via

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}={{ \mathcal V }}_{{\rm{\Gamma }}}{ \mathcal R }\end{eqnarray*}$$

with

$$\begin{eqnarray}&&{{ \mathcal V }}_{{\rm{\Gamma }}}g({\boldsymbol{\theta }},s)=| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}{\parallel }^{-1}\;g(T({\boldsymbol{\theta }}),{T}_{{\boldsymbol{\theta }}}(s)),\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&T({\boldsymbol{\theta }})=\displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel },\qquad {T}_{{\boldsymbol{\theta }}}(s)=\displaystyle \frac{s+{b}_{{\boldsymbol{\theta }}}^{T}{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }.\end{eqnarray} \tag{ 10 }$$

The proof of this representation can be found in [10].

In the following, we consider the case where the measured data cover directions distributed over the complete sphere. For our analytical framework, we therefore assume the condition

$$\begin{eqnarray*}&&\left\{\displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }\ :{\boldsymbol{\theta }}\in {S}^{1}\right\}={S}^{1}.\end{eqnarray*}$$

Otherwise, the dynamic behaviour would result in limited-data, where only certain singularities of the object are detectable from the data, which has to be taken into account while analysing the spatial resolution.

Lemma 3.3. The mappings

$$\begin{eqnarray*}&&T\ :{S}^{1}\to {S}^{1},\quad T({\boldsymbol{\theta }})=\displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{T}_{{\boldsymbol{\theta }}}\ :{\mathbb{R}}\to {\mathbb{R}},\quad {T}_{{\boldsymbol{\theta }}}(s)=\displaystyle \frac{s+{b}_{{\boldsymbol{\theta }}}^{T}{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel },\quad {\boldsymbol{\theta }}\in {S}^{1}\end{eqnarray*}$$

are, except on a measure-zero set, diffeomorphic functions. Especially, it holds

$$\begin{eqnarray*}&&| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| =\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}{\parallel }^{-1},\quad | \mathrm{det}{DT}({\boldsymbol{\theta }})| =| h({\boldsymbol{\theta }})| \;\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}{\parallel }^{-2}\end{eqnarray*}$$

with $h({\boldsymbol{\theta }})$ stated in (8).

Proof. The mapping ${T}_{{\boldsymbol{\theta }}}$ is differentiable with respect to s. Since ${A}_{{\boldsymbol{\theta }}}$ is injective, it holds especially

$$\begin{eqnarray*}&&\displaystyle \frac{{\rm{d}}{T}_{{\boldsymbol{\theta }}}}{{\rm{d}}s}(s)=\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}{\parallel }^{-1}\ne 0\quad \mathrm{for}\ \mathrm{all}\ s\in {\mathbb{R}}.\end{eqnarray*}$$

Hence, ${T}_{{\boldsymbol{\theta }}}$ is a diffeomorphism.

To prove the property for the mapping T, we use again the isometric correlation between a unit vector and its phase angle. Especially, T corresponds to

$$\begin{eqnarray*}&&T\ :[0,2\pi )\to [0,2\pi ),\quad T(\theta )=\tau \ \ \mathrm{with}\ \ \left(\begin{array}{c}\mathrm{cos}\tau \\ \mathrm{sin}\tau \end{array}\right)=\displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }.\end{eqnarray*}$$

In case of ${({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}=0,$ it holds either

$$\begin{eqnarray*}&&\displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }=\left(\begin{array}{c}0\\ 1\end{array}\right)\quad \mathrm{or}\quad \displaystyle \frac{{A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel }=\left(\begin{array}{c}0\\ -1\end{array}\right),\end{eqnarray*}$$

such that $\mu \left(\left\{\varphi \in [0,2\pi )\ :{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}=0\right\}\right)=0$ with Lebesgue-measure μ. In case of ${({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}\ne 0,$ it holds

$$\begin{eqnarray*}&&\displaystyle \frac{\mathrm{sin}\tau }{\mathrm{cos}\tau }=\displaystyle \frac{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{2}}{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}}.\end{eqnarray*}$$

Thus, we obtain the representation

$$\begin{eqnarray*}T(\theta )=\left\{\begin{array}{ll}\mathrm{arctan}\left(\displaystyle \frac{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{2}}{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}}\right), & \mathrm{if}\ {({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}\gt 0,\\ \mathrm{arctan}\left(\displaystyle \frac{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{2}}{{({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}}\right)+\pi , & \mathrm{if}\ {({A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }})}_{1}\lt 0.\end{array}\right.\end{eqnarray*}$$

Using the chain rule and the property (8) yields

$$\begin{eqnarray*}&&\mathrm{det}{DT}(\theta )=\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}{\parallel }^{-2}\;h({\boldsymbol{\theta }})\ne 0\quad \mathrm{for}\ \mathrm{all}\ \theta \in [0,2\pi ].\end{eqnarray*}$$

The following theorem states the singular value decomposition of ${{ \mathcal R }}_{{\rm{\Gamma }}}$ with Γ describing affine deformations.

Theorem 3.4. Let ${w}_{{\rm{\Gamma }}}({\boldsymbol{\theta }},s):= | \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{2}\;| h({\boldsymbol{\theta }})| \;| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w({T}_{{\boldsymbol{\theta }}}(s))$ with the weight function $w(s)={(1-{s}^{2})}^{-1/2}$ of the Radon transform and $h({\boldsymbol{\theta }})$ as in (8).

The singular value decomposition of the operator

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}\ :{L}_{2}({V}_{1}(0))\to {L}_{2}(Z,{w}_{{\rm{\Gamma }}})\end{eqnarray*}$$

is given by

$$\begin{eqnarray*}&&\left\{({\sigma }_{n},{v}_{{nl}},{u}_{{nl}}^{{\rm{\Gamma }}})\in {{\mathbb{R}}}_{\geqslant 0}\times {L}_{2}({V}_{1}(0))\times {L}_{2}(Z,{w}_{{\rm{\Gamma }}}),n\in {\mathbb{N}},l\in {\mathbb{Z}},| l| \leqslant n,n+l\in 2{\mathbb{Z}}\right\}\end{eqnarray*}$$

with

$$\begin{eqnarray*}&&{\sigma }_{n}=2\sqrt{\displaystyle \frac{\pi }{n+1}},\\ \;{v}_{n,l}(x)=\left\{\begin{array}{ll}\sqrt{\displaystyle \frac{n+1}{\pi }}\;\parallel x{\parallel }^{| l| }{P}_{(n-| l| )/2}^{(0,| l| )}(2\parallel x{\parallel }^{2}-1)\;{Y}_{l}\left(\displaystyle \frac{x}{\parallel x\parallel }\right), & \mathrm{for}\ \parallel x\parallel \leqslant 1,\\ 0 & \mathrm{for}\ \parallel x\parallel \gt 1,\end{array}\right.\end{eqnarray*}$$

and

$$\begin{eqnarray*}{u}_{{nl}}^{{\rm{\Gamma }}}({\boldsymbol{\theta }},s)=\left\{\begin{array}{ll}| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;{\pi }^{-1}\;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{Y}_{l}(T({\boldsymbol{\theta }})), & s\in {I}_{{\boldsymbol{\theta }}},\\ 0 & s\notin {I}_{{\boldsymbol{\theta }}},\end{array}\right.\end{eqnarray*}$$

where ${I}_{{\boldsymbol{\theta }}}:= {T}_{{\boldsymbol{\theta }}}^{-1}([-1,1]).$ As in the singular value decomposition of the Radon transform, P denotes the Jacobi polynomials, U_n the Chebychev-polynomials of the second kind and Y_l the spherical harmonics.

We prove this statement in the appendix, using a result derived in [10] in the general context of dynamic linear inverse problems.

Remark. Since the singular values of ${{ \mathcal R }}_{{\rm{\Gamma }}}$ coincide with the ones of the classical Radon transform, this shows that the degree of the ill-posedness is not changed by a dynamic behaviour with affine motion functions. This was already shown in [8] in terms of sobolev space estimates for ${{ \mathcal R }}_{{\rm{\Gamma }}}.$

In addition, the singular value decomposition of ${{ \mathcal R }}_{{\rm{\Gamma }}}$ provides with the Picard criterion a sufficient consistency condition for $g\in {L}_{2}(Z,{w}_{{\rm{\Gamma }}})$ to be in $\mathrm{Im}({{ \mathcal R }}_{{\rm{\Gamma }}}).$

Based on the singular value decomposition, we deduce a representation of ${{ \mathcal R }}_{{\rm{\Gamma }}}f,$ which is later used to characterize the null space of the dynamic operator. Within the proof, the following lemma is required.

Lemma 3.5. For ${{ \mathcal R }}_{{\rm{\Gamma }}}f\in {L}_{2}(Z,{w}_{{\rm{\Gamma }}}),$ it holds

$$\begin{eqnarray*}&&{\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s=\displaystyle \sum _{| l| \leqslant n,l+n\in 2{\mathbb{Z}}}{\sigma }_{n}\;\langle f,{v}_{n,l}\rangle \;{\pi }^{-1}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;{Y}_{l}(T({\boldsymbol{\theta }})).\end{eqnarray*}$$

Proof. For reasons of simplicity, we shortly write ${\displaystyle \sum }_{m}{\displaystyle \sum }_{l}$ for the sum ${\displaystyle \sum }_{m}{\displaystyle \sum }_{| l| \leqslant m,l+m\in 2{\mathbb{Z}}}.$ Using the singular value decomposition of ${{ \mathcal R }}_{{\rm{\Gamma }}},$ along with the coordinate transform $\tilde{s}:= {T}_{{\boldsymbol{\theta }}}(s),$ it holds

$$\begin{eqnarray*}&&{\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\\ &&\quad ={\displaystyle \int }_{{T}_{{\boldsymbol{\theta }}}^{-1}([-1,1])}\displaystyle \sum _{m}\displaystyle \sum _{l}{\sigma }_{m}\;\langle f,{v}_{{ml}}\rangle \;{u}_{{ml}}^{{\rm{\Gamma }}}({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\\ &&\quad =\displaystyle \sum _{m}\displaystyle \sum _{l}{\sigma }_{m}\;\langle f,{v}_{{ml}}\rangle {\pi }^{-1}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;{Y}_{l}(T({\boldsymbol{\theta }}))\\ &&\quad \times {\displaystyle \int }_{{T}_{{\boldsymbol{\theta }}}^{-1}([-1,1])}| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;{U}_{m}({T}_{{\boldsymbol{\theta }}}(s))\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\\ &&\quad =\displaystyle \sum _{m}\displaystyle \sum _{l}{\sigma }_{m}\;\langle f,{v}_{{ml}}\rangle {\pi }^{-1}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;{Y}_{l}(T({\boldsymbol{\theta }}))\\ &&\quad \times {\displaystyle \int }_{[-1,1]}w{(\tilde{s})}^{-1}\;{U}_{m}(\tilde{s})\;{U}_{n}(\tilde{s})\;{\rm{d}}\tilde{s}.\end{eqnarray*}$$

Since $w{(s)}^{-1}=\sqrt{1-{s}^{2}},$ the assertion follows with the fact that the Chebyshev-polynomials form an orthonormal set on ${L}_{2}([-1,1],{w}^{-1}).$

For $f\in {L}_{2}({V}_{1}(0)),$ we denote in the following

$$\begin{eqnarray}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}):= {\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s.\end{eqnarray} \tag{ 11 }$$

Theorem 3.6. Let ${{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}$ satisfy condition (7), i.e. ${{\rm{\Gamma }}}_{{\boldsymbol{\theta }}}^{-1}$ is given by trigonometric polynomials of degree $L\in {\mathbb{N}}.$ Then, for $f\in {L}_{2}({V}_{1}(0)),$ it holds

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)=| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;\displaystyle \sum _{n\in {\mathbb{N}}}{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}),\end{eqnarray*}$$

and ${q}_{n}^{{\rm{\Gamma }},f}\in {{ \mathcal T }}_{(L+1)n+2L}$ with $((L+1)n+2L+1)$ coefficients.

Proof. Together with the representation of the singular functions ${u}_{{nl}}^{{\rm{\Gamma }}}$ and lemma 3.5, the singular value decomposition yields

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)=\displaystyle \sum _{n\in {\mathbb{N}}}\displaystyle \sum _{| l| \lt n,l+n\in 2{\mathbb{Z}}}{\sigma }_{n}\;\langle f,{v}_{{nl}}\rangle \;{u}_{{nl}}^{{\rm{\Gamma }}}(\theta ,s)\\ &&\quad =| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\\ &&\quad \times \displaystyle \sum _{n}{U}_{n}({\boldsymbol{\theta }},s)\;\displaystyle \sum _{l}{\sigma }_{n}\langle f,{v}_{{nl}}\rangle \;{\pi }^{-1}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;{Y}_{l}(T({\boldsymbol{\theta }}))\\ &&\quad =| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;\displaystyle \sum _{n}{U}_{n}({\boldsymbol{\theta }},s){\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s.\end{eqnarray*}$$

For fixed ${\boldsymbol{\theta }}\in {S}^{1},\;$ ${U}_{n}({T}_{{\boldsymbol{\theta }}}(s))$ is a polynomial in s of degree n. Thus, according to the consistency condition 3.1 for the dynamic operator, we have

$$\begin{eqnarray*}&&{\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\ \ \in {{ \mathcal T }}_{(L+1)n+2L}.\end{eqnarray*}$$

Further, we obtain

$$\begin{eqnarray*}{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}) & = & {\displaystyle \int }_{{\mathbb{R}}}{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\\ & = & {\displaystyle \int }_{{\mathbb{R}}}{ \mathcal R }f(T({\boldsymbol{\theta }}),{T}_{{\boldsymbol{\theta }}}(s))\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{\rm{d}}s\\ & = & {\displaystyle \int }_{{\mathbb{R}}}\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \;{ \mathcal R }f(T({\boldsymbol{\theta }}),s)\;{U}_{n}(s){\rm{d}}s.\end{eqnarray*}$$

With the symmetry ${{\rm{\Gamma }}}_{-{\boldsymbol{\theta }}}={{\rm{\Gamma }}}_{{\boldsymbol{\theta }}},$ it holds $T(-{\boldsymbol{\theta }})=-T({\boldsymbol{\theta }}),$ and, using the symmetry property of the Radon transform and the fact that U_n is even (odd) for n even (odd), we obtain

$$\begin{eqnarray*}{q}_{n}^{{\rm{\Gamma }},f}(-{\boldsymbol{\theta }})=\left\{\begin{array}{ll}{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}) & \mathrm{if}\ n\ \mathrm{even},\\ -{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}) & \mathrm{if}\ n\ \mathrm{odd}.\end{array}\right.\end{eqnarray*}$$

This property yields the representation

$$\begin{eqnarray*}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})=\displaystyle \sum _{l=0}^{\lfloor ((L+1)n+2L)/2\rfloor }({d}_{1\;l}^{n}\mathrm{cos}({\alpha }_{l}^{n}\theta )+{d}_{2\;l}^{n}\mathrm{sin}({\alpha }_{l}^{n}\theta )),\quad {d}_{1,l}^{n},{d}_{2,l}^{n}\in {\mathbb{R}}\end{eqnarray*}$$

with ${\alpha }_{l}^{n}=\left\{\begin{array}{ll}2l & \mathrm{if}\ n\ \mathrm{even},\\ 2l+1 & \mathrm{if}\ n\ \mathrm{odd}.\end{array}\right.$

In the following, we study the elements in the null space of the semi-discrete dynamic operator ${{ \mathcal R }}_{{\rm{\Gamma }}},$

$$\begin{eqnarray*}&&{{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}})=\{f\in {L}_{2}({V}_{1}(0))\ :{{ \mathcal R }}_{{\rm{\Gamma }}}f({{\boldsymbol{\theta }}}_{j},s)=0\ \mathrm{for}\ \mathrm{almost}\ \mathrm{all}\ s\in {\mathbb{R}}\ \mathrm{and}\ j=1,...,p\},\end{eqnarray*}$$

with p distinct source positions ${{\boldsymbol{\theta }}}_{1},...,{{\boldsymbol{\theta }}}_{p}\in {S}^{1}.$ As in the static case, $\{{{\boldsymbol{\theta }}}_{1},...,{{\boldsymbol{\theta }}}_{p}\}$ is then not contained in an algebraic variety of degree $\lt p,$ i.e. there is no $q\in {{ \mathcal P }}_{l},l\lt p,q\not\equiv 0$ with $q({{\boldsymbol{\theta }}}_{j})=0$ for all $j=1,...,p,$ see (2), [21].

Now, we first derive a series expansion for ${{ \mathcal R }}_{{\rm{\Gamma }}}f$ with $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ which holds independently of any sampling scheme. In a next step, the a priori information of the specific source positions is taken into account. Finally, based on these characterizations of ${{ \mathcal R }}_{{\rm{\Gamma }}}f,$ we state an explicit representation of the ghosts in the dynamic case.

Theorem 4.1. For $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ it holds

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)=| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;\displaystyle \sum _{n\geqslant \frac{p-2L}{L+1}}{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\end{eqnarray*}$$

with ${q}_{n}^{{\rm{\Gamma }},f}$ as in (11) and ${q}_{n}^{{\rm{\Gamma }},f}({{\boldsymbol{\theta }}}_{j})=0$ for $j=1,...,p.$

Proof. For $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ it holds ${{ \mathcal R }}_{{\rm{\Gamma }}}f({{\boldsymbol{\theta }}}_{j},s)=0\ \mathrm{for}\ \mathrm{almost}\ \mathrm{all}\ s\in {\mathbb{R}}\ \mathrm{and}\ j\quad =\quad 1,...,p.$ Due to theorem 3.6, this is equivalent to

$$\begin{eqnarray*}&&| \mathrm{det}{{DT}}_{{{\boldsymbol{\theta }}}_{j}}(s)| \;w{({T}_{{{\boldsymbol{\theta }}}_{j}}(s))}^{-1}\;\displaystyle \sum _{n\in {\mathbb{N}}}{U}_{n}({T}_{{{\boldsymbol{\theta }}}_{j}}(s))\;{q}_{n}^{{\rm{\Gamma }},f}({{\boldsymbol{\theta }}}_{j})=0\end{eqnarray*}$$

for almost all $s\in {\mathbb{R}}$ and $j=1,...,p.$

For fixed ${\boldsymbol{\theta }},$ the transformed Chebyshev-polynomials ${({U}_{n}({T}_{{\boldsymbol{\theta }}}(s)))}_{n}$ form an orthonormal system on ${L}_{2}\left({T}_{{\boldsymbol{\theta }}}^{-1}([-1,1]),| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\right),$ which can be directly verified by substituting ${T}_{{\boldsymbol{\theta }}}(s).$ Hence, it holds

$$\begin{eqnarray}&&{q}_{n}^{{\rm{\Gamma }},f}({{\boldsymbol{\theta }}}_{j})=0\quad \mathrm{for}\ \mathrm{all}\ n\in {\mathbb{N}}\ \mathrm{and}\ j=1,...,p.\end{eqnarray} \tag{ 12 }$$

Due to theorem 3.6, it is ${q}_{n}^{{\rm{\Gamma }},f}\in {{ \mathcal T }}_{(L+1)n+2L}$ with $(L+1)n\quad +\quad 2L+1$ coefficients. Hence, for each n, the above condition (12) corresponds to a system of p linear homogenous equations with $((L+1)n+2L+1)$ unknowns. If $(L+1)n\quad +\quad 2L\lt p,$ or equivalently

$$\begin{eqnarray*}&&n\lt \displaystyle \frac{p-2L}{L+1},\end{eqnarray*}$$

the matrix has full rank since $\{{{\boldsymbol{\theta }}}_{1},...,{{\boldsymbol{\theta }}}_{p}\}$ is not contained in an algebraic variety of degree $\lt p,$ and it follows ${q}_{n}^{{\rm{\Gamma }},f}\equiv 0.$

Since ${T}_{{\boldsymbol{\theta }}}(s)$ is a polynomial in s of degree 1, theorem 4.1 yields that ${{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},\cdot )$ has a series expansion in polynomials of degree $\frac{p-2L}{L+1}$ and larger. However, in the static case, the series expansion of ${ \mathcal R }$ starts with polynomials of degree p, see (3). For $L\gt 0,$ i.e. the truly dynamic case, it holds the estimate

$$\begin{eqnarray*}&&\displaystyle \frac{p-2L}{L+1}\lt p-2\;L\lt p.\end{eqnarray*}$$

This comparison shows that, in the dynamic case, the series expansion of the imaging operator can contain lower order terms than in the stationary case.

Our provided series expansion, especially the lower bound, describes the worst-case scenario for any deformation of degree L and for any discretization of the projection space S¹. By taking into account the effect of a given motion model on a specific sampling scheme, one can obtain more detailed information. Therefore, we denote

$$\begin{eqnarray*}&&{\rm{\Xi }}:= \{T({{\boldsymbol{\theta }}}_{1}),...,T({{\boldsymbol{\theta }}}_{p})\}\end{eqnarray*}$$

and $\tilde{p}:= | {\rm{\Xi }}| .$ The set Ξ comprises the directions from which the reference object is actually scanned, and especially, it holds $\tilde{p}\leqslant p.$ In applications, it is possible to determine the number $\tilde{p}$ directly from the data by detecting e.g. redundancies in the measured data set.

Theorem 4.2. Let $\tilde{p}=| {\rm{\Xi }}|$ as introduced above. Then, for $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ it holds

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)=| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\end{eqnarray*}$$

with ${q}_{n}^{{\rm{\Gamma }},f}$ as in (11) and ${q}_{n}^{{\rm{\Gamma }},f}({{\boldsymbol{\theta }}}_{j})=0$ for $j=1,...,p.$

Proof. For $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ theorem 4.1 yields the representation

$$\begin{eqnarray*}&&{{ \mathcal R }}_{{\rm{\Gamma }}}f({\boldsymbol{\theta }},s)=| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;\displaystyle \sum _{n\geqslant \frac{p-2L}{L+1}}{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}),\end{eqnarray*}$$

where ${q}_{n}^{{\rm{\Gamma }},f}$ is the polynomial as in (11) with property

$$\begin{eqnarray}&&{q}_{n}^{{\rm{\Gamma }},f}({{\boldsymbol{\theta }}}_{j})=0\quad \mathrm{for}\ j=1,...,p.\end{eqnarray} \tag{ 13 }$$

Using lemma 3.5, ${q}_{n}^{{\rm{\Gamma }},f}$ has the representation

$$\begin{eqnarray*}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})={| \mathrm{det}{A}_{{\boldsymbol{\theta }}}| }^{-1}\;\displaystyle \sum _{| l| \leqslant n,l+n\in 2{\mathbb{Z}}}{c}_{{nl}}\;{Y}_{l}(T({\boldsymbol{\theta }})),\end{eqnarray*}$$

with ${c}_{{nl}}={\sigma }_{n}\langle f,{v}_{{nl}}\rangle {\pi }^{-1}\in {\mathbb{R}}.$ Since ${| \mathrm{det}{A}_{{{\boldsymbol{\theta }}}_{j}}| }^{-1}\ne 0$ for $j=1,...,p,$ (13) is equivalent to the condition

$$\begin{eqnarray*}&&\displaystyle \sum _{| l| \leqslant n,l+n\in 2{\mathbb{Z}}}{c}_{{nl}}\;{Y}_{l}({\xi }_{j})=0\quad \mathrm{for}\ \mathrm{all}\ {\xi }_{j}\in {\rm{\Xi }}.\end{eqnarray*}$$

This corresponds to a system of p equations and $n+1$ unknowns. Since ${\xi }_{j}\in {\rm{\Xi }}$ with $| {\rm{\Xi }}| =\tilde{p},$ the respective matrix has rank $\mathrm{min}(\tilde{p},n+1).$ Thus, for $n\lt \tilde{p},$ we have $\sum {c}_{{nl}}{Y}_{l}(\xi )=0\ \forall \ \xi \in {S}^{1},$ and so

$$\begin{eqnarray*}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\equiv 0\quad \mathrm{for}\ n\lt \tilde{p},\end{eqnarray*}$$

leading to the asserted representation of ${{ \mathcal R }}_{{\rm{\Gamma }}}f.$

Remark. Please note that theorem 4.2 and its proof hold even in the non-symmetric case, i.e. when measurements are taken over the complete sphere and when ${{\rm{\Gamma }}}_{-\theta }={{\rm{\Gamma }}}_{\theta }$ is not necessarily fulfilled for all θ, see (6).

From theorem 4.2 and the respective proof, we obtain the following representation of the functions in the null space.

Corollary 4.3. For $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}),$ it holds the series expansion

$$\begin{eqnarray*}&&f(x)=\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}\displaystyle \sum _{| l| \leqslant n,l+n\in 2{\mathbb{Z}}}{c}_{{nl}}{v}_{{nl}}(x)\end{eqnarray*}$$

with coefficients ${c}_{{nl}}\in {\mathbb{R}}.$

Proof. Let $f(x)={\displaystyle \sum }_{n}{\displaystyle \sum }_{l}{c}_{{nl}}{v}_{{nl}}$ be the series expansion of f in the orthonormal set ${({v}_{{nl}})}_{{nl}},$ with ${c}_{{nl}}=\langle f,{v}_{{nl}}\rangle .$ According to theorem 4.2 and its proof, we have for $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}})$

$$\begin{eqnarray*}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\equiv 0\quad \mathrm{for}\ n\lt \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\},\end{eqnarray*}$$

where ${q}_{n}^{{\rm{\Gamma }},f}$ has the representation

$$\begin{eqnarray*}&&{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})=\displaystyle \sum _{| l| \leqslant n,l+n\in 2{\mathbb{Z}}}{\sigma }_{n}\;{c}_{{nl}}\;{\pi }^{-1}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;{Y}_{l}(T({\boldsymbol{\theta }})).\end{eqnarray*}$$

Thus, with ${q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\equiv 0,$ it holds also

$$\begin{eqnarray*}&&{\sigma }_{n}^{-1}\;| \mathrm{det}{A}_{{T}^{-1}({\boldsymbol{\theta }})}| \;\pi \;{q}_{n}^{{\rm{\Gamma }},f}({T}^{-1}({\boldsymbol{\theta }}))\equiv 0\quad \mathrm{for}\ n\lt \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}.\end{eqnarray*}$$

This yields the condition

$$\begin{eqnarray*}&&\displaystyle \sum _{l}{c}_{{nl}}\;{Y}_{l}({\boldsymbol{\theta }})\equiv 0\end{eqnarray*}$$

for the coefficients in the series expansion of f. Since ${({Y}_{l})}_{l\in {\mathbb{N}}}$ is an orthonormal system on ${L}_{2}({S}^{1}),$ this yields

$$\begin{eqnarray*}&&{c}_{{nl}}=0,\quad \mathrm{for}\ l\in {\mathbb{N}},n\lt \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\},\end{eqnarray*}$$

and hence the asserted representation of f.

To interpret the previous results in terms of resolution, we characterize the elements of ${{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}})$ in the Fourier space.

Theorem 4.4. Let $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}}).$ Then, its Fourier transform has the representation

$$\begin{eqnarray*}&&\widehat{f}(\sigma {\boldsymbol{\theta }})=\displaystyle \frac{1}{2}\;| \mathrm{det}{A}_{{T}^{-1}({\boldsymbol{\theta }})}| \;\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}{(-i)}^{n}\;(n+1)\;{\sigma }^{-1}{J}_{n+1}(\sigma )\;{q}_{n}^{{\rm{\Gamma }},f}({T}^{-1}({\boldsymbol{\theta }}))\end{eqnarray*}$$

with the Bessel-functions J_n of the first kind.

Proof. With the series expansion of ${{ \mathcal R }}_{{\rm{\Gamma }}}f$ as given in theorem 4.2, it holds for $f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}})$

$$\begin{eqnarray*}&&\widehat{{{ \mathcal R }}_{{\rm{\Gamma }}}f}({\boldsymbol{\theta }},\sigma )=\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}{ \mathcal F }{H}_{{\boldsymbol{\theta }}}(\sigma )\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }})\end{eqnarray*}$$

with the function

$$\begin{eqnarray*}&&{H}_{{\boldsymbol{\theta }}}(s):= | \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s)){\chi }_{{T}_{{\boldsymbol{\theta }}}^{-1}([-1,1])}(s).\end{eqnarray*}$$

For the Fourier transform ${ \mathcal F }{H}_{{\boldsymbol{\theta }}},$ we obtain with the substitution $z:= {T}_{{\boldsymbol{\theta }}}(s)$ and the definition of ${T}_{{\boldsymbol{\theta }}},$ (10),

$$\begin{eqnarray*}{ \mathcal F }{H}_{{\boldsymbol{\theta }}}(\sigma ) & = & {(2\pi )}^{-1/2}\;{\displaystyle \int }_{{T}_{{\boldsymbol{\theta }}}^{-1}([-1,1])}| \mathrm{det}{{DT}}_{{\boldsymbol{\theta }}}(s)| \;w{({T}_{{\boldsymbol{\theta }}}(s))}^{-1}\;{U}_{n}({T}_{{\boldsymbol{\theta }}}(s))\;{{\rm{e}}}^{-{\rm{i}}s\sigma }\;{\rm{d}}s\\ & = & {(2\pi )}^{-1/2}\;{\displaystyle \int }_{[-1,1]}w{(s)}^{-1}\;{U}_{n}(s)\;{{\rm{e}}}^{-{\rm{i}}\sigma {T}_{{\boldsymbol{\theta }}}^{-1}(s)}\;{\rm{d}}s\\ & = & {(2\pi )}^{-1/2}\;\mathrm{exp}\left({\rm{i}}\sigma {({A}_{{\boldsymbol{\theta }}}^{-1}{b}_{{\boldsymbol{\theta }}})}^{T}{\boldsymbol{\theta }}\right)\;{\displaystyle \int }_{[-1,1]}w{(s)}^{-1}\;{U}_{n}(s)\;{{\rm{e}}}^{-{\rm{i}}s\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \sigma }\;{\rm{d}}s\\ & = & \sqrt{\pi /2}\;\mathrm{exp}\left({\rm{i}}\sigma {({A}_{{\boldsymbol{\theta }}}^{-1}{b}_{{\boldsymbol{\theta }}})}^{T}{\boldsymbol{\theta }}\right)\;{(-i)}^{n}\;(n+1)\;{(\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \sigma )}^{-1}\;{J}_{n+1}(\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \sigma ).\end{eqnarray*}$$

In the last step, we used that the Bessel-functions J_n fulfill

$$\begin{eqnarray*}&&{ \mathcal F }\left(\sqrt{1-{s}^{2}}\;{U}_{n}(s)\;{\chi }_{[-1,1]}(s)\right)(\sigma )=\sqrt{\pi /2}\;(n+1)\;{(-i)}^{n}{\sigma }^{-1}{J}_{n}(\sigma ),\end{eqnarray*}$$

see formula 7.321, [7]. Altogether, we obtain

$$\begin{eqnarray}&&\widehat{{{ \mathcal R }}_{{\rm{\Gamma }}}f}({\boldsymbol{\theta }},\sigma )=\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}\sqrt{\pi /2}\;\mathrm{exp}\left({\rm{i}}\sigma {({A}_{{\boldsymbol{\theta }}}^{-1}{b}_{{\boldsymbol{\theta }}})}^{T}{\boldsymbol{\theta }}\right)\;{(-i)}^{n}(n+1)\;{(\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \sigma )}^{-1}\;\\ &&\quad {J}_{n+1}(\parallel {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}\parallel \sigma )\;{q}_{n}^{{\rm{\Gamma }},f}({\boldsymbol{\theta }}).\end{eqnarray} \tag{ 14 }$$

The Fourier transform $\widehat{{{ \mathcal R }}_{{\rm{\Gamma }}}f}$ is related to the Fourier transform of f by the Fourier-slice-theorem

$$\begin{eqnarray*}&&\widehat{{{ \mathcal R }}_{{\rm{\Gamma }}}f}({\boldsymbol{\theta }},\sigma )={(2\pi )}^{1/2}\;| \mathrm{det}{A}_{{\boldsymbol{\theta }}}{| }^{-1}\;\mathrm{exp}\left({\rm{i}}\sigma {({A}_{{\boldsymbol{\theta }}}^{-1}{b}_{{\boldsymbol{\theta }}})}^{T}{\boldsymbol{\theta }}\right)\;\widehat{f}(\sigma {A}_{{\boldsymbol{\theta }}}^{-*}{\boldsymbol{\theta }}),\end{eqnarray*}$$

see [8]. It provides the representation of $\widehat{f}$ in polar coordinates

$$\begin{eqnarray*}&&\widehat{f}(\sigma {\boldsymbol{\theta }})=\widehat{f}(\sigma T({T}^{-1}({\boldsymbol{\theta }})))\\ &&\quad =\widehat{f}\left(\displaystyle \frac{\sigma }{\parallel {A}_{{T}^{-1}({\boldsymbol{\theta }})}^{-*}{T}^{-1}({\boldsymbol{\theta }})\parallel }{A}_{{T}^{-1}({\boldsymbol{\theta }})}^{-*}{T}^{-1}({\boldsymbol{\theta }})\right)\\ &&\quad ={(2\pi )}^{-1/2}\;| \mathrm{det}{A}_{{T}^{-1}({\boldsymbol{\theta }})}| \;\widehat{{{ \mathcal R }}_{{\rm{\Gamma }}}f}\left({T}^{-1}({\boldsymbol{\theta }}),\displaystyle \frac{\sigma }{\parallel {A}_{{T}^{-1}({\boldsymbol{\theta }})}^{-*}{T}^{-1}({\boldsymbol{\theta }})\parallel }\right)\\ &&\qquad \quad \mathrm{exp}\left(-{\rm{i}}\sigma \parallel {A}_{{T}^{-1}({\boldsymbol{\theta }})}^{-*}{T}^{-1}({\boldsymbol{\theta }}){\parallel }^{-1}{({A}_{{T}^{-1}({\boldsymbol{\theta }})}^{-1}{b}_{{T}^{-1}({\boldsymbol{\theta }})})}^{T}{T}^{-1}({\boldsymbol{\theta }})\right).\end{eqnarray*}$$

Using (14) yields

$$\begin{eqnarray*}&&\widehat{f}(\sigma {\boldsymbol{\theta }})=\displaystyle \frac{1}{2}\;| \mathrm{det}{A}_{{T}^{-1}({\boldsymbol{\theta }})}| \;\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}{(-i)}^{n}\;(n+1){\sigma }^{-1}\;{J}_{n+1}(\sigma )\;{q}_{n}^{{\rm{\Gamma }},f}({T}^{-1}({\boldsymbol{\theta }})).\end{eqnarray*}$$

With the representation of ${q}_{n}^{{\rm{\Gamma }},f}$ according to lemma 3.5, we obtain immediately

$$\begin{eqnarray*}&&\widehat{f}(\sigma {\boldsymbol{\theta }})=\displaystyle \frac{1}{2}\displaystyle \sum _{n\geqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}{(-i)}^{n}\;(n+1)\;{\sigma }^{-1}{J}_{n+1}(\sigma )\;{q}_{n}({\boldsymbol{\theta }})\end{eqnarray*}$$

with a ${q}_{n}\in \mathrm{span}\{{Y}_{l},| l| \leqslant n,l+n\in 2{\mathbb{Z}}\},{q}_{n}({\boldsymbol{\theta }})=0$ for ${\boldsymbol{\theta }}\in {\rm{\Xi }}.$

Now, we want to study the frequency distribution of these ghosts by comparing the energy of their Fourier transform restricted to a circle ${V}_{c}(0)$ around 0 of radius c with their total energy. Therefore, we denote $M:= \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}$ and

$$\begin{eqnarray*}&&{\mu }_{M}(c):= \mathrm{sup}\left\{\displaystyle \frac{\parallel \widehat{f}{\parallel }_{{L}_{2}({V}_{c}(0))}}{\parallel f{\parallel }_{{L}_{2}({{\mathbb{R}}}^{2})}},f\in {{ \mathcal N }}_{p}({{ \mathcal R }}_{{\rm{\Gamma }}})\right\}.\end{eqnarray*}$$

The asymptotic behaviour in the static case was derived in [17]. The proof therein is based on a representation of $f\in {{ \mathcal N }}_{p}({ \mathcal R })$ in Fourier space as

$$\begin{eqnarray*}&&\widehat{f}(\sigma {\boldsymbol{\theta }})=\displaystyle \sum _{n\geqslant M}{\sigma }^{-1}{i}^{n}{J}_{n+1}\;{q}_{n}({\boldsymbol{\theta }})\end{eqnarray*}$$

with ${q}_{n}\in \mathrm{span}\{{Y}_{l},| l| \leqslant n,l+n\in 2{\mathbb{Z}}\},{q}_{n}({{\boldsymbol{\theta }}}_{j})=0$ for $j=1,...,M$, $M\gt 0.$ Thus, with the same arguments as in [17], we obtain immediately the following behaviour.

Lemma 4.5. Let $\alpha ,\beta \in {\mathbb{R}}.$ It holds

$$\begin{eqnarray*}{\mu }_{M}({c}_{M})\overset{M\to \infty }{\longrightarrow }\left\{\begin{array}{ll}0 & \mathrm{for}\ {c}_{M}\leqslant M-\alpha {M}^{\beta },\alpha \gt 0,\beta \gt \displaystyle \frac{1}{3},\\ 1 & \mathrm{for}\ {c}_{M}\geqslant M+\alpha {M}^{\beta },\alpha \gt 0,\beta \gt \displaystyle \frac{1}{2}.\end{array}\right.\end{eqnarray*}$$

This provides the following interpretation with respect to resolution. The spectrum of the ghosts is, at least asymptotically, concentrated outside a disk around zero with radius $\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}.$ Therefore, a reliable reconstruction is only possible for functions which are essentially M-band limited with $M=\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\},$ i.e. for details corresponding to frequencies

$$\begin{eqnarray*}&&| \xi | \leqslant \mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}.\end{eqnarray*}$$

In accordance to Shannon's sampling theorem, these are the details of size

$$\begin{eqnarray*}&&\displaystyle \frac{2\pi }{\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}}\end{eqnarray*}$$

and larger.

We compare this result in the dynamic case with the respective result from the stationary setting, where data from p directions are required to reliably reconstruct details of size $2\pi /p$ and larger. Since $\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}\leqslant p,$ i.e.

$$\begin{eqnarray*}&&\displaystyle \frac{2\pi }{p}\leqslant \displaystyle \frac{2\pi }{\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}},\end{eqnarray*}$$

we obtain that, in the static case, smaller details can be reliably reconstructed from the same amount of data than in the dynamic setting. To guarantee a similar resolution in dynamic and static case, more data, i.e. a finer sampling, is required.

Our analysis also provides a worst case estimate on how many projections are required to guarantee a spatial resolution of $2\pi /b:$ Using $(p-2L)/(L+1)$ as lower bound for the resolution, i.e. without any knowledge about the sampling scheme, approximately $(L+1)b$ directions are required. If we want to guarantee the same resolution as in the static setting, i.e. for b = p, approximately $L+1$ times as many data have to be collected.

As a first example, we consider the specimen shown in figure 1, performing a rotation during the scanning with

$$\begin{eqnarray*}{A}_{{\boldsymbol{\theta }}}=\left(\begin{array}{cc}\mathrm{cos}(3\theta ) & -\mathrm{sin}(3\theta )\\ \mathrm{sin}(3\theta ) & \mathrm{cos}(3\theta )\end{array}\right).\end{eqnarray*}$$

Thus, the inverse motion functions are given by trigonometric polynomials of degree L = 3. The Radon data are collected from p = 136 projections uniformly distributed over S¹, i.e.

$$\begin{eqnarray*}&&{{\boldsymbol{\theta }}}_{j}=\left(\begin{array}{c}\mathrm{cos}{\theta }_{j}\\ \mathrm{sin}{\theta }_{j}\end{array}\right),\quad {\theta }_{j}=(j-1)\;\displaystyle \frac{2\pi }{p},\quad j=1,...,p.\end{eqnarray*}$$

To study the effect of different numbers of projections, the fixed number of 451 detector points is used for all numerical examples, independent of the number of projections. More precisely, we used the detector points

$$\begin{eqnarray*}&&{s}_{l}={lh},l=-q,...,q,h=\displaystyle \frac{1}{q}\end{eqnarray*}$$

with q = 225. The worst-case estimate guarantees a spatial resolution of at least $2\pi /33.$ To get a sharper estimate, we take also the effect of the motion on the specific sampling scheme into account. With

$$\begin{eqnarray*}&&T({{\boldsymbol{\theta }}}_{j})=\displaystyle \frac{{A}_{{{\boldsymbol{\theta }}}_{j}}^{-*}{\boldsymbol{\theta }}}{\parallel {A}_{{{\boldsymbol{\theta }}}_{j}}^{-*}\theta \parallel }=\left(\begin{array}{c}\mathrm{cos}((j-1)\frac{2\pi }{34})\\ \mathrm{sin}((j-1)\frac{2\pi }{34})\end{array}\right),\quad j=1,...,136,\end{eqnarray*}$$

we obtain

$$\begin{eqnarray*}&&{\rm{\Xi }}=\left\{{\boldsymbol{\theta }}\left((j-1)\frac{2\pi }{34}\right),j=1,...,34\right\},\end{eqnarray*}$$

and hence, $\tilde{p}=| {\rm{\Xi }}| =34.$ Since $\mathrm{max}\left\{\tilde{p},\frac{p-2L}{L+1}\right\}=34,$ the achievable resolution in this dynamic setting is $2\pi /34.$ In comparison, in case of stationary data, the achievable resolution would be $2\pi /136.$ This illustrates that the resolution is actually significantly reduced by the dynamic behaviour of the object, which gets also obvious by comparing the reconstruction results from both dynamic and stationary data, see figures 2 and 3.

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For the reconstruction in the stationary case, the classical filtered backprojection algorithm was used with the filter

$$\begin{eqnarray}&&{\psi }^{\gamma }(s)=\displaystyle \frac{1}{2{\pi }^{2}{\gamma }^{2}}\left(1-\displaystyle \frac{\sqrt{2}s}{\gamma }D\left(\displaystyle \frac{s}{\sqrt{2}\gamma }\right)\right),\end{eqnarray} \tag{ 15 }$$

where D is the Dawson integral

$$\begin{eqnarray*}&&D(s)=\mathrm{exp}\left(-{s}^{2}\right)\;{\displaystyle \int }_{0}^{s}\mathrm{exp}\left({t}^{2}\right)\;{\rm{d}}t,\end{eqnarray*}$$

see [19]. To reconstruct the reference image from dynamic data, we applied the algorithm proposed in [8] which is exact for affine deformations and corresponds to a filtered backprojection method with a respectively modified version of the filter (15).

Next, we address the question how to achieve the same spatial resolution $2\pi /136.$ Using the lower estimate which is independent of the specific dynamics and sampling, approximately $(L+1)136=544$ projections are required to guarantee the desired resolution. With this number of source positions, we obtain

$$\begin{eqnarray*}&&T({{\boldsymbol{\theta }}}_{j})=\left(\begin{array}{c}\mathrm{cos}((j-1)\frac{2\pi }{136})\\ \mathrm{sin}((j-1)\frac{2\pi }{136})\end{array}\right),\quad j=1,...544.\end{eqnarray*}$$

Hence, it is $\tilde{p}=| {\rm{\Xi }}| =136$ which proves that we obtain actually the desired resolution.

However, a comparable resolution could be obtained with far less than 544 projections, see also Figure 4 and 5. Consider e.g. p = 139, for which we obtain

$$\begin{eqnarray}&&T({{\boldsymbol{\theta }}}_{j})=\left(\begin{array}{c}\mathrm{cos}((j-1)\frac{8\pi }{139})\\ \mathrm{sin}((j-1)\frac{8\pi }{139})\end{array}\right),\quad j=1,...139,\end{eqnarray} \tag{ 16 }$$

such that $\tilde{p}=| {\rm{\Xi }}| =139\gg \lceil \frac{p-2L}{L+1}\rceil=34,$ leading to an actual resolution of $2\pi /139,$ which is far better than the predicted resolution $2\pi /34$ by the worst-case estimate. Thus, this example illustrates that the estimate independent of Ξ only provides a coarse estimation on how many projections are required to obtain a desired resolution. Especially for high-order motion functions, a high number of projections is required to guarantee a desired resolution independent of the specific dynamic sampling scheme.