1 Introduction

A sample is illuminated by a parallel beam of coherent X-rays; the intensity of the diffracted pattern (hologram) is registered at a plane detector for determination of distribution of attenuation and refractivity of the object. This type of imaging is implemented at third generation synchrotron radiation sources due to their high degree of coherence. This method guarantees improved contrast compared to purely absorption-based radiography but involves phase retrieval problem. The linearized version of this problem is known as the contrast transfer function model (CTF [9]). For narrow beams the Helmholtz equation is replaced by the paraxial approximation. Pogany et al. [12], Cloetens et al. [4], Paganin et al. [10], Gureyev et al. [5] applied reconstruction algorithms for medium with zero absorption by replacing the Fresnel propagator by its quadratic term in the frequency plane (the dashed parabola on Fig. 1). Bronnikov’s method [1, 2] is based on Radon’s and Lorentz’s inversion formulas. See Burvall et al. [3] for similar algorithms. Methods of complex analysis are applied in for stating uniqueness [6] and stability of inversion [7]. The exact estimate of the norm of the inversion shows exponential growth for large Fresnel numbers.

In this paper an analytic reconstruction method of arbitrary refraction index is proposed. Our method is exact in frame of CTF model for optically weak objects and requires measurements of intensity of a single hologram. The key point is interpolation in the Fourier domain from a nonuniform sampling of data. The asymptotic density of the sampling is minimal for any natural Fresnel number which depends on size of the support of the object. The interpolation operator looks as the classical Whittecker-Shannon-Kotelnikov interpolation. An earlier version is given in [11].

2 Setup and Linearization

A three-dimensional object of interest is subjected by a plane wave beam of perfectly coherent radiation of high frequency with no polarization. Intensity of radiation is the measurable quantity

$$\begin{aligned} I\left( \psi \right) =\left| P \left( \exp \left( i\psi \right) \right) \right| ^{2}, \end{aligned}$$

where\(\ P\) is the Fresnel propagator which is the paraxial approximation of the Helmholtz propagator. Integral

$$\begin{aligned} \psi \left( x\right) =\mathrm {k}\int \left( 1-\mathrm {n}\left( x,z\right) \right) \mathrm {d}z=\mathrm {k}\int \left( \delta \left( x,z\right) -i\beta \left( x,z\right) \right) \mathrm {d}z \end{aligned}$$

is well defined on the object plane \(\mathbb {R}^{2}\), where\(\ \mathrm {k}\) is the space frequency, and z is the coordinate along the central ray. The function \(\mathrm {n}=1-\delta +i\beta \) is the refraction index of object where \(\delta \ \)and\(\ \beta \) are real and assumed small. This yields

$$\begin{aligned} P \left( \exp \left( i\psi \right) \right) =1+iP \left( \psi \right) +O\left( \left\| \psi \right\| ^{2}\right) \end{aligned}$$

and

$$\begin{aligned} I\left( \psi \right) =1+2T\left( \psi \right) +O\left( \left\| \psi \right\| ^{2}\right) \end{aligned}$$

for an appropriate norm. CTF operator \(T\left( \psi \right) =-{\text {Im}}{} P \left( \psi \right) \) is the linearization of \(I\left( \psi \right) /2\) at \(\psi =0\). By [7] the norm of the operator \(T^{-1}\) is estimated by \(\exp \left( \mathrm {const}\ \mathfrak {f}\right) \). The Fresnel propagator can be written in the form

$$\begin{aligned} P (\psi )\doteqdot F^{-1}EF(\psi ),\ E\left( \xi \right) =\exp \left( \frac{-id\left| \xi \right| ^{2}}{2\mathrm {k}}\right) , \end{aligned}$$
(1)

where \(z=d\) is the distance from the object plane to the detector, F is Fourier transform on the object plane \(\mathbb {R}^{2}\ \)orthogonal to the central ray and \(\xi =\left( \xi _{1},\xi _{2}\right) \) are coordinates on the frequency plane \(\text {R}^{2}\). We assume that the support of \(\psi \) is contained in disc \(\Omega _{b}=\left\{ x;\left| x\right| \le b/2\right\} \) for some \(b>0\) and call the quotient

$$\begin{aligned} \mathfrak {f}=\frac{\mathrm {k}b^{2}}{2\pi d} \end{aligned}$$
(2)

Fresnel number of the setting.

Fig. 1
figure 1

Scheme of the phase contrast imaging

Full size image

We can write \(\psi =\varphi -i\mu \) where

$$\begin{aligned} \varphi \left( x\right) =\mathrm {k}\int \delta \left( x,z\right) \mathrm {d}z,\ \mu \left( x\right) =\mathrm {k}\int \beta \left( x,z\right) \mathrm {d}z \end{aligned}$$

are phase shift and absorption integral on the ray through a point x on the object plane, respectively and supports of \(\varphi \) and \(\mu \) are contained in disc \(\left\{ y:\left| y\right| \le 1/2\right\} \) where \(y_{1}=x_{1}/b,y_{2}=x_{2}/b\) are normalized object coordinates. The Fourier transform on the object plane is defined by

$$\begin{aligned} \hat{a}\left( \eta \right) =F_{y\mapsto \eta }\left( a\right) =\int \int \exp \left( -2\pi i\left( y_{1}\eta _{1}+y_{2}\eta _{2}\right) \right) a\left( y_{1},y_{2}\right) \mathrm {d}y_{1}\mathrm {d}y_{2} \end{aligned}$$

where \(\eta _{i}=b\xi _{i},\ i=1,2\) are the normalized coordinates on the frequency plane. Equation

$$\begin{aligned} \hat{T}\left( \psi \right) =\sin \left( \frac{\pi \left| \eta \right| ^{2}}{\mathfrak {f}}\right) \hat{\varphi }\left( \eta \right) -\cos \left( \frac{\pi \left| \eta \right| ^{2}}{\mathfrak {f}}\right) \hat{\mu }\left( \eta \right) \end{aligned}$$
(3)

follows from (1) and (2).

3 Reconstruction of Phase and Attenuation Integrals

Theorem 3.1

For any natural Fresnel number, compactly supported functions \(\varphi ,\mu \in L_{2}\left( R^{2}\right) \) can be found from (3) by means of interpolation of\(\ \hat{\varphi }\) and \(\hat{\mu }\) on each line through the origin in the frequency plane R\(^{2}\). The sampling points have minimal density.

Proof

We focus first on reconstruction of the phase shift \(\varphi .\ \)Let PW denote the (Paley-Wiener) space of functions \(g\in L_{2}\left( \mathbb {R}\right) \) such that \(F\left( g\right) \) is supported in \(\left[ -1/2,1/2\right] .\)

Definition

We shall say that a function Z defined on the complex plane is of sine-type if it is entire, zeros of Z are real, simple and separated that is there exists \(c>0\) such that

$$\begin{aligned} \left| \lambda -\mu \right| \ge c \end{aligned}$$
(4)

for arbitrary zeros \(\lambda \),\(\ \mu \ne \lambda ,\ \)and there are constants A, B, h such that

$$\begin{aligned} A\exp \left( \pi \left| {\text {Im}}\lambda \right| \right) \ge \left| Z\left( \lambda \right) \right| \ge B\exp \left( \pi \left| {\text {Im}}\lambda \right| \right) \end{aligned}$$
(5)

for any \(\lambda \) such that \(\left| {\text {Im}}\lambda \right| \ge h.\ \)We call\(\ Z\) generating function for a Fresnel number \(\mathfrak {f}\) if it is of sine-type and each zero \(\lambda \) of \(Z\ \) satisfies

$$\begin{aligned} \lambda ^{2}=\left( l\left( \lambda \right) +\frac{1}{2}\right) \mathfrak {f.} \end{aligned}$$
(6)

where \(l\ (\lambda )\) is an integer. The proof will continue through Sects. 4–8.

4 Lagrange Interpolation in Paley–Wiener Space

The following fact is classical (see for ex.[13]).

Theorem 4.1

Let Z be a generating function for \(\mathfrak {f}\) and \(\Lambda \) be the zero set of \(Z.\ \)An arbitrary function \(f\in PW\) can be interpolated from set \(\Lambda \) of zeros of Z by

$$\begin{aligned} f\left( t\right) =Z\left( t\right) \sum _{\lambda \in \Lambda }\frac{f\left( \lambda \right) }{\left( t-\lambda \right) Z^{\prime }\left( \lambda \right) }, \end{aligned}$$
(7)

where the series converges uniformly on each bounded interval.

Lemma 4.2

[13]If Z is a sine-type function with zero set \(\Lambda \), then for arbitrary \(\varepsilon >0,\) there exists \(\delta >0\) such that for any real \(\alpha \) such that \(\mathrm {dist}\left( \alpha ,\Lambda \right) \ge \varepsilon \), inequality holds

$$\begin{aligned} \left| Z\left( \alpha +i\beta \right) \right| \ge \delta \exp \left( \pi \left| \beta \right| \right) . \end{aligned}$$
(8)

Proof of theorem

For a real \(\beta ,\) we denote \(\Gamma _{\beta }=\left\{ {\text {Im}}\lambda =\beta \right\} \) and have

$$\begin{aligned} \int _{\Gamma _{\beta }}\left| f\left( \lambda \right) \right| ^{2}\mathrm {d}\alpha \le \left\| f\right\| ^{2}\exp \left( 2\pi \left| \beta \right| \right) . \end{aligned}$$
(9)

To check this inequality we write \(f=\hat{g}\) for a function \(g\in PW,\) and have

$$\begin{aligned} f\left( \lambda \right) =F_{\xi \mapsto \alpha }\left( g\left( \xi \right) \exp \left( 2\pi \beta \xi \right) \right) \end{aligned}$$

where \(\lambda =\xi +i\beta .\) It follows that

$$\begin{aligned} \int _{\Gamma _{\beta }}\left| f\left( \lambda \right) \right| ^{2}\mathrm {d}\alpha \le \int \left| g\left( \xi \right) \exp \left( 2\pi \beta \xi \right) \right| ^{2}\mathrm {d}\xi \le \exp \left( 2\pi \left| \beta \right| \right) \left\| g\right\| ^{2} \end{aligned}$$

since g is supported on \(\left[ -1,1\right] .\) Now (9) follows from Plancherel Theorem. By (5), the Cauchy inequality and (9) we conclude that for any real t,

$$\begin{aligned}&B^{2}\left( \int _{\Gamma _{\beta }}\left| \frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z\left( \lambda \right) }\right| \mathrm {d}\lambda \right) ^{2}\le \exp \left( -2\pi \left| \beta \right| \right) \left( \int _{\Gamma _{\beta }}\left| \frac{f\left( \lambda \right) }{\lambda -t}\right| \mathrm {d}\lambda \right) ^{2}\\&\quad \le \exp \left( -2\pi \left| \beta \right| \right) \int _{\Gamma _{\beta }}\frac{\mathrm {d}\lambda }{\left| \lambda -t\right| ^{2}}\int _{\Gamma _{\beta }}\left| f\left( \lambda \right) \right| ^{2}\mathrm {d}\lambda \le \frac{\pi }{\left| \beta \right| }\left\| f\right\| ^{2}, \end{aligned}$$

which yields

$$\begin{aligned} \int _{\Gamma _{\beta }}\left| \frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z\left( \lambda \right) }\right| \mathrm {d}\lambda \le \frac{1}{B}\left( \frac{\pi }{\beta }\right) ^{1/2}\left\| f\right\| . \end{aligned}$$

For arbitrary \(\alpha >0,\) we can replace here \(\Gamma _{\beta }\) by \(\Lambda _{\beta ,\alpha }=\left\{ \left| {\text {Re}}\lambda \right| \le \alpha ,\ \left| {\text {Im}}\lambda \right| =\beta \right\} \) for 2 times the same right-hand side. This implies

$$\begin{aligned} \int _{\Lambda _{\beta ,\alpha }}\left| \frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z\left( \lambda \right) }\right| \mathrm {d}\lambda \rightarrow 0 \end{aligned}$$

as \(\beta \rightarrow \infty \) uniformly with respect to t and \(\alpha .\) Set \(\Lambda _{\alpha ,\beta }=\left\{ {\text {Re}}\lambda =\pm \alpha ,\ \left| {\text {Im}}\lambda \right| \le \beta \right\} \) and choose a sequence \(\alpha =\alpha \left( k\right) \rightarrow \infty \) such that \(\alpha \left( k\right) \ \)and \(-\alpha \left( k\right) \) are kept at distance \(\ge c/4\) from \(\Lambda \) for any k where c is the constant in (4). Then

$$\begin{aligned} \left| \int _{\Lambda _{\alpha \left( k\right) ,\beta }}\frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z\left( \lambda \right) }\mathrm {d}\lambda \right| \rightarrow 0 \end{aligned}$$

as \(k\rightarrow \infty \) uniformly for \(\beta \ge 0.\) This property follows from (8) if we take \(\varepsilon =c/4.\) Sum

$$\begin{aligned} \int _{\Lambda _{\alpha \left( k\right) ,\beta }}+\int _{\Lambda _{\beta ,\alpha \left( k\right) }}\frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z\left( \lambda \right) }\mathrm {d}\lambda \end{aligned}$$
(10)

is the integral over the perimeter of the rectangle of size \(2\alpha \times 2\beta .\) If \(k\rightarrow \infty \) and \(\beta \rightarrow \infty ,\) sum (10) tends to zero. If the orientation of the boundary is counterclockwise the sum can be calculated by the Residue theorem for poles \(\lambda =t,\ \left| t\right| <\alpha \) and \(\lambda \in \Lambda \), \(\left| \lambda \right| <\alpha .\) This yields

$$\begin{aligned} \mathrm {res}_{t}+\sum _{\lambda \in \Lambda ,\left| \lambda \right|<\alpha \left( k\right) }\mathrm {res}_{\lambda }=\frac{f\left( t\right) }{Z\left( t\right) }+\sum _{\lambda \in \Lambda ,\left| \lambda \right| <\alpha \left( k\right) }\frac{f\left( \lambda \right) }{\left( \lambda -t\right) Z^{\prime }\left( \lambda \right) }\rightarrow 0 \end{aligned}$$

as \(k\rightarrow \infty .\) The limit of the partial sum vanishes and (7) follows. \(\square \)

5 Generating Functions

Theorem 5.1

For any natural \(\mathfrak {f},\) there exists an even generating function \(Z_{\mathfrak {f}}\) whose zeros have asymptotic density 1.

Proof for odd \(\mathfrak {f}=2p+1\). Check that product \(Z_{\mathfrak {f}}=Y_{\mathfrak {f}}R_{\mathfrak {f}}\) is an even generating function where

$$\begin{aligned} Y_{\mathfrak {f}}\left( \lambda \right)&=\cos \left( \frac{\pi }{\mathfrak {f}}\sqrt{\sigma _{0}\left( \lambda \right) }\right) \prod _{q=1}^{p}\left( \cos \left( \frac{2\pi }{\mathfrak {f}}\sqrt{\sigma _{q}\left( \lambda \right) }\right) +\cos \left( \frac{2\pi }{\mathfrak {f}}q\right) \right) ,\\ \sigma _{q}\left( \lambda \right)&=\lambda ^{2}-\frac{\mathfrak {f}}{2}+\left( \frac{\mathfrak {f}}{2}-q\right) ^{2},\ q=0,...,p \end{aligned}$$

and

$$\begin{aligned} R_{\mathfrak {f}}\left( \lambda \right) =\prod _{r=1}^{\mathfrak {f}-1} \frac{\lambda ^{2}-\left( 2r-1/2\right) \mathfrak {f}}{\lambda ^{2} -\mathfrak {f}/2}. \end{aligned}$$

We have \(\left| R_{\mathfrak {f}}\left( \lambda \right) \right| \rightarrow 1\ \)as \(\lambda \rightarrow \infty \), function \(Y_{\mathfrak {f} }\left( \lambda \right) \ \)fulfils (5) and has only real zeros \(\lambda _{q,k}\ \)such that

$$\begin{aligned} \lambda _{q,k}^{2}=\left( l+\frac{1}{2}\right) \mathfrak {f},\ l=k^{2} \mathfrak {f}+k\left( \mathfrak {f}-2q\right) ,\ q=0,...,\mathfrak {f} -1,\ k=0,1,2,.... \end{aligned}$$
(11)

It is easy to check that the sequence of zeros has asymptotic density 1. It follows that \(Z_{\mathfrak {f}}\) fulfils (4). All zeros of \(Y_{\mathfrak {f}}\) are simple except zeros \(\lambda _{q,0}=\pm \sqrt{\mathfrak {f/}2},\) \(q=0,...,\mathfrak {f}-1\) which has multiplicity \(\mathfrak {f}\). This zero has multiplicity 1 in\(\ Z_{\mathfrak {f}}\) since of factor \(R_{\mathfrak {f}}\ \)which we call the correction factor. This factor has zeros \(\rho _{r}=\pm \sqrt{\left( 2r-1/2\right) \mathfrak {f}},\) \(r=1,...,\mathfrak {f}-1\) which also fulfil (6). Check that none of these zeros is a zero of \(Y_{\mathfrak {f}}.\) Suppose the opposite, let \(\lambda _{q,k}^{2}=\rho _{r}^{2}\) for some r and q. Then

$$\begin{aligned} 2r-1=k^{2}\mathfrak {f}+k\left( \mathfrak {f}-2q\right) =\left( k^{2}+k\right) \mathfrak {f}-2qk \end{aligned}$$

for an integer k which is not possible since the right hand side is even for any q, k and \(\mathfrak {f}.\) This completes the proof of Theorem 5.1 for odd Fresnel numbers. \(\square \)

Examples. Case \(\mathfrak {f}=1.\ \)Check that \(Z_{1}\left( \lambda \right) =\cos \left( \pi \sqrt{\lambda ^{2}-1/4}\right) \) is a generating function. Numbers \(\lambda _{k}=\pm \sqrt{k^{2}+k+1/2},\ k=0,1,...\) are all zeros and (6) is fulfilled. This condition is illustrated in Fig. 2 for \(\mathfrak {f}=1\)

Fig. 2
figure 2

Common zeros of \(Z_{1}=\) \(\cos \left( \pi \sqrt{x^{2}-1/4}\right) \) (red) and of \(E\left( x\right) =\cos \pi x^{2}\) (blue) are shown by vertical lines

Full size image

For \(\mathfrak {f}=3,\ \)we have

$$\begin{aligned} Y_{3}\left( \lambda \right)&=\cos \left( \frac{\pi }{3}\sqrt{\lambda ^{2}+\frac{3}{4}}\right) \left( \cos \frac{\pi }{3}-\cos \frac{2\pi }{3} \sqrt{\lambda ^{2}-\frac{5}{4}}\right) ,\\ R_{3}\left( \lambda \right)&=\frac{\left( \lambda ^{2}-9/2\right) \left( \lambda ^{2}-21/2\right) }{\left( \lambda ^{2}-3/2\right) ^{2}}. \end{aligned}$$

Positive zeros of \(Z_{3}=Y_{3}R_{3}\ \)are shown in Fig. 3

Fig. 3
figure 3

Positive zeros of \(Z_{3}\)

Full size image

where zeros of \(R_{3}\ \)is shown by red. Case \(\mathfrak {f}=5\) is shown in Fig. 4

Fig. 4
figure 4

Positive zeros of \(Z_{5}\)

Full size image

where zeros of \(R_{5}\) are shown by red.

6 Even Fresnel numbers

For arbitrary even Fresnel number \(\mathfrak {f,}\) we set

$$\begin{aligned} Z_{\mathfrak {f}}=\prod _{q=1}^{\mathfrak {f}/2}Z_{\mathfrak {f,q}}, \end{aligned}$$

where

$$\begin{aligned} Z_{\mathfrak {f,q}}\left( \lambda \right)&=\frac{\lambda ^{2}-\left( 2q-3/2\right) \mathfrak {f}}{\lambda ^{2}-\mathfrak {f/}2}\left( \cos \frac{2\pi }{\mathfrak {f}}\sqrt{\tau _{q}\left( \lambda \right) }+\cos \frac{2\pi }{\mathfrak {f}}\left( q-\frac{1}{2}\right) \right) ,\nonumber \\ \tau _{q}\left( \lambda \right)&=\lambda ^{2}-\frac{\mathfrak {f}}{2} +\frac{1}{4}\left( \mathfrak {f}-2q+1\right) ^{2}. \end{aligned}$$
(12)

Positive zeros of the first and of the second factors are\(\ \mu _{q} =\sqrt{\left( 2q-3/2\right) \mathfrak {f}}\) and

$$\begin{aligned} \lambda _{q,k}=\sqrt{\left( l\left( q,k\right) +\frac{1}{2}\right) \mathfrak {f}},\ l\left( q,k\right) =k^{2}\mathfrak {f}+k\left( \mathfrak {f}-2q+1\right) ,\ q=0,...,\mathfrak {f}-1,\ k\in \mathbb {Z}, \end{aligned}$$
(13)

respectively. Note that for any q,  the second factor of (12) vanishes at \(\lambda _{q,0}=\sqrt{\mathfrak {f/}2}\ \)but the denominator in the first factor cancels this zero. The first factor of \(Z_{\mathfrak {f},1}\) equals 1, the second factor has a simple zero at \(\sqrt{\mathfrak {f/}2}\) and so does \(Z_{\mathfrak {f},1}\). Function \(Z_{\mathfrak {f}}\ \)is entire and has only simple zeros \(\lambda _{k}\ \)of \(Z_{\mathfrak {f}}\) which fulfil condition (6) Indeed, \(l\left( j\right) =l\left( q,k\right) \ \)if \(\lambda _{j}=\lambda _{q,k}\) and \(l\left( j\right) =q-1\) if \(\lambda _{j}=\mu _{q}\). Condition (6) is illustrated in Fig. 5 for \(\mathfrak {f}=2.\)

Fig. 5
figure 5

Common zeros of \(Z_{2}=\) \(\cos \left( \pi \sqrt{x^{2}-3/4}\right) \) and of \(\cos \frac{1}{2}\pi x^{2}\) are shown by vertical lines

Full size image

7 Reconstruction of Phase Shift

Resume to the proof of Theorem 3.1 Let \(Z_{f}\) be an even generating function for a natural Fresnel number \(\mathfrak {f}\) and \(\Lambda =\left\{ \lambda _{k},\ k\in \mathbb {Z}\right\} \) be the zero set of \(Z_{\mathfrak {f}}\ \)shown in (11) and in (13). Let \(\theta \) be an arbitrary unit vector in the frequency plane. By (6) for any zero\(\ \lambda _{j},\) vector \(\eta =\pm \lambda _{k}\theta \) satisfies \(\left| \eta \right| ^{2}=\left( l\left( \lambda _{k}\right) +1/2\right) \mathfrak {f}\). It follows that \(\cos \left( \pi \left| \lambda _{k} \theta \right| ^{2}/\mathfrak {f}\right) =0\) which implies \(\hat{\varphi }\left( \lambda _{k}\theta \right) =\left( -1\right) ^{l\left( \lambda _{k}\right) }\hat{T}\left( \psi ;\lambda _{k}\theta \right) .\) By the slice theorem for any unit vector \(\theta ,\)

$$\begin{aligned} \hat{\varphi }\left( t\theta \right) =\int \exp \left( -2\pi ipt\right) \mathbf {R}\varphi \left( p,\theta \right) \mathrm {d}p,\ t\in \mathbb {R}, \end{aligned}$$

where \(\mathbf {R}\) means the plane Radon transform and the function \(\mathbf {R}\varphi \left( \cdot ,\theta \right) \in L_{2}\left( R\right) \) is supported in \(\left[ -1/2,1/2\right] .\) Therefore \(f_{\theta }\in PW\) where \(f_{\theta }\left( \lambda \right) =\hat{\varphi }\left( \lambda \theta \right) \) for an arbitrary unit vector \(\theta .\) Theorem 4.1 can be applied to \(f_{\theta }\) and function \(\hat{\varphi }\) can be reconstructed on the whole frequency plane by

$$\begin{aligned} \hat{\varphi }\left( \lambda \theta \right) =Z\left( \lambda \right) \sum _{k}\frac{\left( -1\right) ^{l\left( \lambda _{k}\right) }}{\left( \lambda -\lambda _{k}\right) Z_{\mathfrak {f}}^{\prime }\left( \lambda _{k}\right) }\hat{T}\left( \psi ;\lambda _{k}\theta \right) \end{aligned}$$
(14)

This equation follows from (3) since

$$\begin{aligned} \sin \left( \frac{\pi \left| \lambda _{k}\right| ^{2}}{\mathfrak {f} }\right) =\left( -1\right) ^{l\left( \lambda _{k}\right) } \end{aligned}$$

for any k. Equation (14) provides reconstruction of \(\hat{\varphi }\) on all dual plane when \(\theta \) runs over a half-circle. This completes the proof for phase shift if we apply the inverse Fourier transform to\(\ \hat{\varphi }.\ \square \)

8 Reconstruction of Attenuation Integrals

The same method can be applies for reconstruction of\(\ \hat{\mu }.\)

Lemma 8.1

For any odd \(\mathfrak {f,}\) there exists an interpolation formula that holds for an arbitrary function \(g=\hat{f}\) such that\(\ f\in L_{2}\left[ -1/2,1/2\right] \) whose knots \(\lambda _{k}\) have asymptotic density 1 and satisfy

$$\begin{aligned} \left| \lambda _{k}\right| ^{2}=l\left( k\right) \mathfrak {f} \end{aligned}$$

where \(l\left( k\right) \) is an integer for any k.

Proof

or arbitrary odd \(\mathfrak {f}=2p+1,\ \)we take generating function

$$\begin{aligned} W_{\mathfrak {f}}&=\cos \frac{\pi }{\mathfrak {f}}\left( \sqrt{\lambda ^{2}+\frac{\mathfrak {f}^{2}}{4}}\right) \prod _{q=1}^{p}\left( \cos \frac{2\pi }{\mathfrak {f}}\sqrt{\lambda ^{2}+\left( \frac{\mathfrak {f}}{2}-q\right) ^{2}}+\cos \frac{2q\pi }{\mathfrak {f}}\right) \\ \lambda _{0,k}^{2}&=\left( \mathfrak {f}\left( k+\frac{1}{2}\right) \right) ^{2}-\frac{\mathfrak {f}^{2}}{4}=\mathfrak {f}^{2}\left( k^{2}+k\right) ,\\ \lambda _{q,k}^{2}&=l \mathfrak {f},\ l=\left( \mathfrak {f}\left( k^{2}+k\right) -2qk\right) ,\ q=1,...,,\mathfrak {f}-1. \end{aligned}$$

The numbers \(\lambda _{q,k}^{2}\) are different for all q and k except for \(\lambda _{0,0}^{2}=...=\lambda _{\mathfrak {f}-1,0}^{2}=0\). Rational function

$$\begin{aligned} R_{\mathfrak {f}}\left( \lambda \right) =\frac{\prod _{q=1}^{\mathfrak {f} -1}\left( \lambda ^{2}-\left( 2q+1\right) \mathfrak {f}\right) }{\lambda ^{2\mathfrak {f}-2}} \end{aligned}$$

is a correction factor. For any zero \(\kappa \ \)of\(\ R_{\mathfrak {f}},\) we have\(\ \kappa ^{2}\ne \lambda _{q,k}^{2}\ \)since otherwise \(\left( 2q+1\right) =\left( \mathfrak {f}\left( k^{2}+k\right) -2qk\right) \ \)where the right hand side is always even.

Finally, we can apply (7) for function \(g\left( t\right) =\hat{\mu }\left( t\theta \right) \) and arbitrary unit vector \(\theta \). Its values at knots are known from (3). For even \(\mathfrak {f,}\) a generating function can written as a similar product of \(\mathfrak {f}/2\) factors. This completes the proof of Theorem 3.1 for absorption integrals \(\mu \). \(\square \)

9 Convergence and Stability

Theorem 9.1

If \(\Lambda \) is the set of zeros of a function Z of sine-type, then functions

$$\begin{aligned} \frac{Z\left( t\right) }{\left( t-\lambda \right) Z^{\prime }\left( \lambda \right) },\ \lambda \in \Lambda \end{aligned}$$

form a Riesz basis in space PW. This means that \(\left\{ f\left( \lambda \right) ,\lambda \in \Lambda \right\} \in l_{2}\ \)for any function \(f\in PW\) and sequence (7) converges to f in \(L_{2}\left( \mathbb {R} \right) .\) Vice versa, for any sequence \(\left\{ c_{\lambda }\right\} \in l_{2}\), series

$$\begin{aligned} \sum _{\lambda \in \Lambda }c_{\lambda }\frac{Z\left( t\right) }{\left( t-\lambda \right) Z^{\prime }\left( \lambda \right) } \end{aligned}$$
(15)

converges to an element of PW.

A proof is given in [13], Ch.4. It follows that reconstructions in Theorem 3.1 are stable in the sense that arbitrary noise \(\delta _{k}\) in data \(\hat{T}\left( \psi ;\lambda _{k}\theta \right) \) causes an error \(\delta \varphi \) in reconstruction by formula (14) whose \(L_{2}\)-norm satisfies

$$\begin{aligned} \left\| \delta \varphi \right\| ^{2}\le C\sum \left| \delta _{k}\right| ^{2} \end{aligned}$$

with a constant \(C=C\left( Z_{\mathfrak {f}}\right) \) which depends only on the generating function \(Z_{\mathfrak {f}},\) respectively \(W_{\mathfrak {f}}.\)

Pointwise convergence of series (15) is similar to that of Whittecker-Kotelnikov-Shannon series which is the particular case for \(Z\left( t\right) =\sin \pi t.\)