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# The derivative-free Fourier shell identity for photoacoustics

- Natalie Baddour
^{1}Email authorView ORCID ID profile

**Received:**2 December 2015**Accepted:**11 September 2016**Published:**19 September 2016

## Abstract

In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data. The Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion techniques. A similar relationship, referred to as the ‘Fourier shell identity’ has been previously derived for photoacoustic applications. However, this identity relates the pressure wavefield data function and its normal derivative measured on an arbitrary enclosing aperture to the three-dimensional Fourier transform of the enclosed object evaluated on a sphere. Since the normal derivative of pressure is not normally measured, the applicability of the formulation is limited in this form. In this paper, alternative derivations of the Fourier shell identity in 1D, 2D polar and 3D spherical polar coordinates are presented. The presented formulations do not require the normal derivative of pressure, thereby lending the formulas directly adaptable for Fourier based absorber reconstructions.

## Keywords

- Photoacoustics
- Frequency domain photoacoustics
- Mathematical physics
- Tomography
- Fourier-based tomography

## Background

Photoacoustics is the generation of acoustic waves as a consequence of the absorption of light energy by an absorbing material and the subsequent thermo-elastic expansion of the material. The method combines the spatial resolution of ultrasound with the contrast of optical absorption for deep imaging in biological tissues (Xu and Wang 2006; Telenkov et al. 2009; Telenkov and Mandelis 2009), and has therefore shown great promise for biomedical imaging applications.

In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data (Slaney and Kak 1988) and is the basis for Fourier-based inversion techniques (Chandra et al. 2014). A similar relationship for Photoacoustic Tomography (PAT) would also be very useful. Anastasio et al. (2007) derived the “Fourier shell identity”, a mathematical relationship between the pressure wavefield data function, its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional (3D) Fourier transform of the optical absorption distribution evaluated on concentric spheres. This relationship can be regarded as the PAT analog of the classic Fourier slice theorem from X-ray tomography. It provides a mapping that relates the temporal Fourier component of the pressure data and its normal derivative to a specified spatial Fourier component of the source object function. The difficulty with Anastasio et al’s derivation is that knowledge of the normal derivative of pressure is required—a value that is not normally obtained through experiments.

In this paper, the Fourier shell identity in 1, 2 and 3 dimensions, such that the normal derivative is not required, are derived and presented. This provides a direct relationship between the temporal Fourier components of the pressure data and the Fourier components of the object function. The Fourier shell identity as derived in this paper requires no additional information than that typically acquired in experiments and thus can be directly used for absorber reconstructions.

## Photoacoustic governing equations

*H*is the energy per unit volume and time deposited by the optical radiation beam, and \( p\left( {\vec{r},t} \right) \) is the pressure of the acoustic wave, a function of space (\( \vec{r} \)) and time,

*t*. As is common, it is assumed that

*H*is a separable function of space and time, so that \( H\left( {\vec{r},t} \right) = A\left( {\vec{r}} \right)I(t) \). In this work, the temporal Fourier transform is employed with the angular frequency, non-unitary convention for forward and inverse transforms, as given by

*f*for Fourier transformation. The same formulation can be used to define spatial Fourier transforms, where an overhat notation is used to denote a spatial Fourier transform where the spatial variable \( \vec{r} \) is transformed to the spatial frequency variable \( \vec{\omega } \), so for example \( f\left( {\vec{r}} \right) \) is spatially Fourier transformed to \( \hat{f}\left( {\vec{\omega }} \right) \). Taking the temporal Fourier transform of (1), it then follows that

## Fourier shell identity

In practical applications, the normal derivative of the pressure wavefield will not be measured so the Fourier shell identity as given in (4) is not immediately useful. Equation (4) permits determination of concentric ‘shells’ of Fourier components of \( \hat{A}\left( {\vec{\omega } = k{\hat{\mathbf{s}}}} \right) \) from knowledge of \( \tilde{p}\left( {\vec{r}_{0}^{\prime } ,k} \right) \) and its derivative along the \( {\mathbf{\hat{n}^{\prime}}} \). Because \( {\hat{\mathbf{s}}} \) can be chosen to specify any direction, \( \hat{A}\left( {\vec{\omega } = k{\hat{\mathbf{s}}}} \right) \) specifies the Fourier components of \( \hat{A}\left( {\vec{\omega }} \right) \) that reside on a spherical surface of radius |k|, whose center is at the origin.

It is shown in the proceeding sections how Eq. (4) can be made specific for the 1D, 2D polar and 3D spherical polar cases. The identity is also re-derived in those three cases so that the derivative of pressure does not appear in the formulation.

## Analysis in 1D

### Fourier shell identity in 1D

Equation (5) is the 1D equivalent of Eq. (4), the Fourier shell identity. In Eq. (5), the overhat indicates a spatial Fourier transform in the 1D spatial variable, *z*, meaning that the spatial variable *z* transforms to the spatial frequency variable \( \omega_{z} \). Equation (5) at first appears different from Anastasio’s formulation in Eq. (4). The source of this apparent difference is that in it is necessary to enclose the source function and to do so in 1D, the “detecting surface” must be located at *both*
\( z = \pm z_{0} \). This follows because a detector at only one of \( z = \pm z_{0} \) will not enclose the source.

### Multivariable Fourier analysis

*z*only, \( \vec{r} = \left( z \right) \). Taking the temporal Fourier transform of (1) (denoted with an over-tilde) so that time,

*t*, transforms to the temporal frequency variable \( \omega \), and then further taking a spatial Fourier transform (denoted with an overhat) in the spatial variable,

*z*, so that spatial variable

*z*transforms to the spatial frequency variable \( \omega_{z} \), leads to

### The Fourier shell theorem in 1D

*Assuming that*\( \hat{A}\left( {\omega_{z} } \right) \)

*has no poles*and remains bounded, then the application of the inverse spatial Fourier transform (2) to (6), while making use of the identity in (7) gives

Hence, the success in being able to reconstruct \( A\left( z \right) \) is partly in the ability to gather sufficient (*k* intensive) information about \( \hat{A}\left( k \right) \) to enable the inverse spatial Fourier transform to be computed.

Equation (10) requires measurements of \( \tilde{p}\left( {z_{0} ,\omega } \right) \) and \( \tilde{I}\left( \omega \right) \) at a sufficiently wide bandwidth to enable the Fourier integral to be computed or approximated. In Eq. (10), the \( z_{0} /c_{s} \) term in the exponential represents the time taken for the signal to travel from the absorber to the detector and that time is an indication of the spatial location of the absorber. The term in front of the integral is simply a scaling term. Therefore, Eq. (10) states that the absorber profile in space can be found by finding the inverse temporal Fourier transform \( \bar{A}\left( t \right) = {\mathbb{F}}^{ - 1} \left\{ {\tilde{p}\left( {z_{0} ,\omega } \right)/\tilde{I}\left( \omega \right)|\omega \to t} \right\} \) via well known numerical techniques for calculating Fourier transforms and then scaling \( A\left( z \right) = \bar{A}\left( {c_{s} t} \right) \) to find the shape of the absorber function as a function of space.

### Comparison with the Fourier shell identity

Now, Eq. (5) is the “Fourier shell identity” as put forward by Anastasio et al. (2007), where the temporal Fourier component of pressure values \( \tilde{p}(z_{0} ,k) \) and its derivative are related to the values of a specified Fourier component, \( \hat{A}\left( k \right) \). The problem with the formulation in (5) is the requirement for measurements of the derivative of \( \tilde{p}(z_{0} ,k) \), which does not occur in practice. However, it can be shown that the formulation presented in this paper, as given in Eq. (8), leads to the same result in Eq. (5).

Hence, the version of the Fourier shell theorem in 1D as presented in Eq. (8) satisfies the Fourier shell identity in 1D as derived by Anastasio et al. (2007) and shown in Eq. (5). However, the formulation of Eq. (8) provides a useful alternative since no knowledge or measurement of the derivative of pressure is required.

## Analysis in 2D polar coordinates

### 2D polar coordinates

The process of finding \( f_{n} (r) \) from \( f(r,\theta ) \) can be thought of as a “forward Fourier series transform” where the continuous \( \theta \) variable is transformed to the discrete (although infinite) \( n \) variable. The reverse transform is the process of performing the summation over the discrete *n* variable, as given in Eq. (13), to recover the original function as a continuous function of \( \theta \). The use of the Discrete Fourier Transform (DFT) via fast algorithms such as the FFT (Cooley and Tukey 1965) to numerically compute the continuous, infinite expressions of (13) and (14) has received much attention in the literature. There is a large body of work on how well the continuous Fourier transform or series can be approximated with the discrete FFT, for example in Epstein (2005).

*n*, defined by the integral (Chirikjian and Kyatkin 2000)

*n*th order Bessel function. In 2D coordinates, the overhat symbol \( \hat{f} \) is used to denote a Hankel transform. The superscript \( \left( n \right) \) is used as a reminder of the order of the Hankel transform—in this case an

*n*th order Hankel transform. If \( n > - 1/2 \), the transform is self-reciprocating and the inversion formula is given by

*n*th order Hankel transform to the 2D Laplacian \( \nabla_{n}^{2} \) of any function of

*r*only, \( {\mathbb{H}}_{n} \left\{ {\nabla_{n}^{2} g(r)} \right\} \):

*n*th order Hankel transform. In other words, the

*n*th order Hankel transform of the

*n*th order Laplacian of any function \( g\left( r \right) \) is the product of \( - \rho^{2} \) and the

*n*th order Hankel transform of \( g\left( r \right) \). As is familiar from Fourier theory, derivatives transform to products under a suitable choice of integral transform. It is noted that the use of the Laplacian \( \nabla_{n}^{2} \) necessitates an

*n*th order Hankel transform to give the desired simple result in (20).

### Fourier shell identity in 2D

In Eq. (22), \( \hat{A}_{n}^{\left( n \right)} \left( k \right) \) is the *n*th order Hankel transform of \( A_{n} \left( r \right) \) evaluated at \( \rho = k \) and \( r = r_{0} \) is the location where a measurement is made. The tilde refers to a temporal Fourier transform, the overhat refers to a forward Hankel transform and the superscript (*n*) indicates the order of the transform. Equation (22) is the 2D statement of Eq. (4), the Fourier shell identity, where it is noted that the value of \( \tilde{p}_{n} (r_{0} ,k) \) and its derivative in the radial direction are required.

### Multivariable Fourier analysis

*n*th order Hankel transform. Specifically, \( \tilde{p}\left( {r,\theta ,k} \right) \) transforms to \( \tilde{p}_{n} \left( {r,k} \right) \) via (14), which in turn transforms to \( \hat{\tilde{p}}_{n}^{\left( n \right)} \left( {\rho ,k} \right) \) via (15), an

*n*th order Hankel transform of the

*n*th term in the Fourier series for \( \tilde{p}\left( {r,\theta ,k} \right) \). Symbolically, \( \tilde{p}\left( {r,\theta ,k} \right) \to \tilde{p}_{n} \left( {r,k} \right) \to \hat{\tilde{p}}_{n}^{\left( n \right)} \left( {\rho ,k} \right) \). The same applies to the absorber function so that \( A\left( {r,\theta } \right) \to A_{n} \left( r \right) \to \hat{A}_{n}^{\left( n \right)} \left( \rho \right) \), and furthermore the Laplacian transforms under the same set of operations as \( \nabla^{2} \to \nabla_{n}^{2} \to - \rho^{2} \). The overhat with superscript \( \left( n \right) \) denotes an

*n*th order Hankel transform, and the subscript of

*n*indicates the

*n*th term of the Fourier series. Taking transforms and then cleaning up gives

### Fourier shell theorem in 2D polar coordinates

Here, \( \phi \) is an analytic function defined on the positive real line that remains bounded as *x* goes to infinity (has no poles), \( H_{n}^{(2)} \left( x \right) \) is a Hankel function of order *n.* Given the definition of the Fourier transform that is being currently used, the result in (24) satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.

*k*intensive) information about \( \hat{A}_{n}^{\left( n \right)} \left( k \right) \) to enable the inverse Hankel transform to be computed via

*n*) to enable \( A\left( {\vec{r}} \right) = A\left( {r,\theta } \right) \) to be reconstructed from (or an approximation to)

### Comparison with the Fourier shell identity

This is exactly the same expression as on the right hand side of (22), which implies that the results derived in the previous section and those derived by Anastasio et al. (2007) are identical. However, the formulation proposed here, namely Eq. (25) provides a useful alternative since no knowledge or measurement of the derivative of pressure is required.

## Analysis in 3D spherical polar coordinates

### 3D spherical polar coordinates

*m*, and \( P_{l}^{m} \) is an associated Legendre function. The coefficients \( f_{l}^{m} \left( r \right) \) are sometimes referred to as the

*spherical Fourier transform*of \( f\left( {r,\psi ,\theta } \right) \) (Telenkov et al. 2009). The process of finding \( f_{l}^{m} \left( r \right) \) from \( f\left( {r,\psi ,\theta } \right) \) can be thought of as a “forward spherical Fourier series transform” where the continuous \( \left( {\psi ,\theta } \right) \) variables are transformed to the discrete (although infinite) \( m,l \) variables. The reverse transform is the process of performing the summation over the discrete \( m,l \) variables, as given in (34), to recover the original continuous function.

*n*. The spherical Hankel transform is particularly useful for problems involving spherical symmetry.

*n*th order spherical Hankel transform to the 3D Laplacian \( \nabla_{n}^{2} \) of any function of

*r*only, \( {\mathbb{S}}_{n} \left\{ {\nabla_{n}^{2} g(r)} \right\} \):

*n*th order spherical Bessel function gives

In other words, the *n*th order spherical Hankel transform of the *n*th order 3D Laplacian of \( g\left( r \right) \) is the product of \( - \rho^{2} \) and the *n*th order spherical Hankel transform of *g*. As seen before, derivatives transform to products under the appropriate choice of transform.

### Fourier shell theorem in 3D spherical polar coordinates

Equation (44) is 3D statement of Eq. (4), the Fourier shell identity. In (44), the tilde refers to a temporal Fourier transform, the overhat refers to a forward spherical Hankel transform, and the \( \left( l \right) \) superscript refers to order of the spherical Hankel transform.

### Multivariable Fourier analysis

*n*th order spherical Hankel transform of Eq. (45). Specifically, \( \tilde{p}\left( {r,\psi ,\theta ,k} \right) \) transforms to \( \tilde{p}_{l}^{m} \left( {r,k} \right) \) via (35), which in turn transforms to \( \hat{\tilde{p}}_{l}^{m\;\left( l \right)} \left( {\rho ,k} \right) \) via (37), an

*l*th order spherical Hankel transform of the \( _{l}^{m} \) th term in the spherical harmonic series for \( \tilde{p}\left( {r,\psi ,\theta ,k} \right) \). Symbolically, \( \tilde{p}\left( {r,\psi ,\theta ,k} \right) \to \tilde{p}_{l}^{m} \left( {r,k} \right) \to \hat{\tilde{p}}_{l}^{m\;\left( l \right)} \left( {\rho ,k} \right) \). The same applies to \( A\left( {r,\psi ,\theta } \right) \to A_{l}^{m} \left( r \right) \to \hat{A}_{l}^{m\;\left( l \right)} \left( \rho \right) \) and the Laplacian transforms under the same set of operations as \( \nabla^{2} \to \nabla_{l}^{2} \to - \rho^{2} \). The overhat with superscript \( \left( l \right) \) denotes an

*l*th order spherical Hankel transform, and the \( _{l}^{m} \) indicates the \( _{l}^{m} \) th term of the spherical hamornic series. Taking transforms and then cleaning up Eq. (45) gives

### Fourier shell theorem in 3D polar coordinates

Here, \( \phi \) is an analytic function defined on the positive real line that remains bounded as *x* goes to infinity (has no poles), \( h_{n}^{(2)} \left( x \right) \) is a spherical Hankel function of order *n.* Given the definition of the Fourier transform that is being currently used, the presented result satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.

*k*intensive) information about \( \hat{A}_{l}^{m\;\left( l \right)} \left( k \right) \) to enable the inverse spherical Hankel transform to be computed via

*m, l*) to enable \( A\left( {\vec{r}} \right) = A\left( {r,\psi ,\theta } \right) \) to be reconstructed from

### Comparison with the Fourier shell identity

Equation (56) is exactly the same expression as on the right hand side of (44), which implies that the results derived in the previous section and those derived by Anastasio et al. (2007) are identical. However, the formulation of the Fourier shell theorem proposed here, namely Eq. (48) provides a useful alternative since no knowledge or measurement of the derivative of pressure is required.

## Summary and conclusions

In this paper, re-derived and derivative-free formulations of the Fourier shell identity were presented in 1D, 2D polar and 3D spherical polar coordinates. These were shown to be equivalent to the previously derived Fourier shell identity (Anastasio et al. 2007), however knowledge of the derivative of pressure is not required, hence lending the formulas directly applicable to Fourier-based absorber reconstruction schemes. The formulas are restated here as a summary.

In Eq. (57), the \( \sim \) refers to a temporal Fourier transform and the overhat refers to a 1D spatial Fourier transform.

In Eq. (58), the *n* subscript refers to the *n*th Fourier coefficient in the Fourier series. The overhat refers to a forward Hankel transform, the superscript (*n*) refers to the order of the Hankel transform and \( H_{n}^{(2)} \left( x \right) \) is a Hankel function of order *n*.

In Eq. (59), the \( _{l}^{m} \) subscript/superscript refer to the \( _{l}^{m} \) term in the spherical harmonic series expansion, the overhat refers to a forward spherical Hankel transform, the \( \left( l \right) \) superscript refers to order of the spherical Hankel transform and \( h_{n}^{(2)} \left( x \right) \) is a spherical Hankel function of order *n*.

## Declarations

### Acknowledgements

This research was financially supported by the Natural Sciences and Engineering Research Council of Canada.

### Competing interests

The author declares that she has no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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