A Multi-Dimensional Transfer Function Approach to Photoacoustic Signal Analysis by Natalie Baddour

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Journal of the Franklin Institute 345 (2008) 792–818 www.elsevier.com/locate/jfranklin

A multi-dimensional transfer function approach to photo-acoustic signal analysis Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N5 Received 4 September 2007; received in revised form 2 February 2008; accepted 20 April 2008

Abstract Photo-acoustic signal generation has shown potential for medical tomography. This paper aims to present a consistent and uniﬁed approach to the mathematical modelling of the photo-acoustic problem, using a transfer function approach. A generalized version of the Fourier slice theorem is presented and proved. Reconstruction algorithms can be developed based on speciﬁc cases of this general theorem. Closed-form solutions to special cases are given in Cartesian, cylindrical and spherical polar coordinates. These can be used to simulate the forward problem and as test cases for any reconstruction algorithms. r 2008 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Modelling; Photo-acoustic; Transfer function; Fourier; Imaging

1. Introduction Photo-acoustic signal generation is a new technique, which has demonstrated great potential for non-invasive medical tomography. With this technique, a short-pulsed laser source is used to irradiate the sample. The energy absorbed produces a small temperature rise, which induces a pressure inside the sample through thermal expansion. This pressure acts as an acoustic source and generates further acoustic waves, which can be detected by ultrasound transducers positioned outside the sample. Since there is a large difference in optical absorption between blood and surrounding tissue, the laser irradiation induces an Tel.: +1 613 562 5800; fax: +1 613 562 5177.

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ultrasound wave on the inhomogeneities within the investigated volume. Hence, the acquired photothermo-acoustic signals carry information about the optical absorption property of the tissue. This approach is thus suitable for the imaging of the micro-vascular system or for tissue characterization, with contrast similar to that of pure optical imaging and spatial resolution similar to that of pure ultrasonic imaging. It therefore combines the advantages of two imaging modalities in a single modality. The issue of the strong scattering of light in media like biological tissue is overcome and the ability of acoustic waves to travel long distances without significant distortion or attenuation is also exploited. Photo-acoustic detection has shown concrete promise of imaging in turbid media at depths potentially through the full thickness of skin [1,2]. Several groups of researchers have made significant progress towards achieving these imaging goals [2–6]. However, a number of different approaches have been developed by each group. This paper thus aims to present a unifying framework for the mathematical theory of photoacoustic imaging, using a transfer function/Green’s function approach. A photo-acoustic Fourier theorem is presented and proved. Related reconstruction results are given in [4,7,8], which can be shown to be special cases of the general results shown here. Those papers do not use multi-dimensional Fourier transforms and thus cannot demonstrate that these results can all be interpreted in a concise way as specific cases of a generalized Fourier theorem. This paper aims to present the theorem in a clear and general way, with no assumption regarding the detection geometry. This theorem is extremely important as it forms the basis for most reconstruction algorithms. The representations in this paper are embedded in time-dependent, spatially three-dimensional (3D) descriptions. Different coordinate systems are considered. Section 2 presents the governing equations for this problem. Section 3 considered the solution in terms of Cartesian coordinates and presents several special cases, which can be solved in closed form and which are useful for elucidating the main features of the proposed approach. Section 4 formally introduces the Green’s function, transfer function approach to the problem and Section 5 derives the relevant functions in spherical polar coordinates. Section 6 develops the photo-acoustic Fourier theorem and is the main result of this paper. Section 7 considers the application of this theorem to the special case of spherically symmetric heterogeneity functions. Section 8 discusses the disadvantage of using short Dirac–delta pulses as input pulses and Section 9 considers the development of the proposed approach in terms of cylindrical polar coordinates and presents an example. Section 10 summarizes the main mathematical results that have been developed in the paper and Section 11 concludes the paper. 2. Governing equations The physical principle behind this imaging modality is the photo-acoustic effect. This entails the generation of an acoustic wave as a result of the absorption of light pulse. While optical energy can be converted to mechanical energy through various pathways, it is often the case that thermal expansion is the dominant mechanism. In pulsed photo-acoustic tomography, the pulse duration is so short that the thermal conduction time is greater than the thermo-acoustic transit time and the effect of thermal conduction can be ignored [1]. The equation describing the thermo-acoustic wave propagation with a thermal expansion source term is given by [1,2,6,7,9] r2 pð~ r; tÞ

1 q2 b q r; tÞ pð~ r; tÞ ¼ s Hð~ c2s qt2 C p qt

(1)

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where p is the pressure of the acoustic wave, Cp is the specific heat, H is the heating function defined as the thermal energy deposited by the energy source per unit time and volume, bs is the coefficient of thermal volume expansion and cs is the speed of sound. The light incident on the sample must be pulsed or modulated in order for photo-acoustic waves to be generated. The heating function can be written as the product of a spatial absorption function and a temporal illumination function of the RF source Hð~ r; tÞ ¼ I o fð~ rÞZðtÞ

(2)

where Io is a scaling factor proportional to the incident radiation intensity and f describes the optical absorption properties of the medium. It is the inhomogeneity in optical absorption of the medium that is the inhomogeneity whose image is sought and the aim of photo-acoustic imaging is to reconstruct f from pressure measurements made at the surface. This will be referred to as the inheterogeneity function since it represents the absorbed optical energy and is thus a function of the actual optical absorption inhomogeneity of the medium and any inhomogeneity of the illuminating beam. It is noted that this stands in contrast to ultrasonic tomography where it would be the inhomogeneity in the speed of sound in the medium that would be the quantity being imaged. The speed of sound is considered constant in Eq. (1). The function Z(t) describes the shape of the irradiating pulse and is a non-negative function whose integration over time equals the pulse energy. The validity of Eq. (2) requires the conditions of thermal and stress confinement to be met. Thermal confinement implies that heat diffusion is negligible during the excitation pulse Z(t). Let tp denote the length of the excitation pulse, then the condition of thermal confinement will be met if tp 5tth ðLp =4aÞ, where tth is the time scale for heat dissipation by thermal conduction, Lp is the characteristic linear dimension of the tissue volume being heated or the size of the absorbing structure and a is the thermal diffusivity of the material [2]. Additionally, the condition of stress confinement is met if the length of the excitation pulse is smaller than the time for the stress to transit the heated region so that tp 5ts ðLp =cs Þ. Under typical values for most soft tissues, tth 40 ms and ts 100 ns, so that stress confinement is usually the more stringent condition. Under the condition of stress confinement, a high thermo-elastic pressure can build up rapidly [2]. 2.1. Governing equations in the time and space frequency domain Taking the temporal Fourier transform of Eq. (1), we obtain r2 pðr; oÞ þ k2s pðr; oÞ ¼

bs io Hðr; oÞ Cp

(3)

where k2s ¼ o2 =c2s is the acoustic wave number and o is the temporal Fourier frequency variable. It is noted for the sake of completeness that the one-dimensional (1D) version of the Fourier transform that will be used is Z Z 1 1 1 ~ f ðoÞeiot do Fðf ðtÞÞ ¼ f~ðoÞ ¼ f ðtÞe iot dt3f ðtÞ ¼ (4) 2p 1 1 where the tilde denotes the Fourier transform, but is often dropped if it is clear from the argument that it is the transform of the function, instead of the function itself, that is being used. This deﬁnition can be simply extended to include multi-dimensional Fourier

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transforms over the spatial variables so that for example Z 1 f ðx; y; zÞe iðox xþoy yþoz zÞ dx dy dz. F3D ðf ðx; y; zÞÞ ¼ f~ðox ; oy ; oz Þ ¼

795

(5)

As for standard diffraction tomography theory, we assume a background medium infinite in extent and an inhomogeneity structure of finite extent. The previously given Eq. (3) for the resulting pressure field is valid for all points outside the inhomogeneity and for arbitrary source–detector configurations. This is the same geometry assumption made for standard acoustic diffraction tomography, as well as for diffuse photonic wave tomography, thus allowing for straightforward comparisons. 3. Cartesian coordinates Cartesian coordinates are most convenient where planes are part of the inherent system geometry, for example if the acoustic wave is measured by a plane of detectors, or if the illuminating wave is a plane wave. Consider the Fourier transform of the detected wave measured in the z ¼ zd plane. Since detection in one of the z planes is being considered, we proceed by taking the spatial Fourier transform of both sides of the governing equation (3), however, only with respect to the x and y variables leaving the z variable untransformed. This gives d2 b io pðox ; oy ; z; oÞ g2s pðox ; oy ; z; oÞ ¼ s Hðox ; oy ; z; oÞ Cp dz2

(6)

where g2s ¼ ox2+oy2 k2s and ox, oy, oz are the spatial Fourier frequency variables. It can readily be seen that Eq. (6) is simply a single-dimensional ordinary differential equation in the variable z, with Green’s function given by [10] Gðox ; oy ; z; oÞ ¼

e gs jzj . 2gs

The solution to Eq. (6) is thus given by Z 1 1 gs jz zs j bs io e Hðox ; oy ; zs ; oÞ dzs . pðox ; oy ; z; oÞ ¼ Cp 1 2gs With the deﬁnition of H given in Eq. (2), the foregoing equation becomes Z bs ioI o ZðoÞ 1 gs jz zs j e fðox ; oy ; zs Þ dzs . pðox ; oy ; z; oÞ ¼ 2gs C p 1

(7)

(8)

(9)

It should be noted that gs could have two different forms, depending on the relative sizes of the spatial wave number and the temporal wave number. In fact qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 > for o2x þ o2y 4k2s < gsr ¼ o2x þ o2y k2s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10) gs ¼ > : igsi ¼ i k2s o2x o2y for o2x þ o2y pk2s For ox2+oy24k2s , the exponential in the Green’s function is a decaying exponential which corresponds with strongly damped evanescent waves. However, for ox2+oy2pk2s , then the exponent in the Green’s function is imaginary and the exponential term is

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oscillatory, thus propagating the resulting waves. For a detector plane far enough away from the inhomogeneity, it is only these waves that will be detected [11]. The physical interpretation of the choice ox2+oy2pk2s is that in order to ensure a solution that produces plane waves in the z direction, then there are necessarily restrictions on the wave numbers in the x and y directions and ox2+oy2pk2s is exactly the required restriction. The interpretation of this from a tomographic standpoint is that if we are choosing to probe a sample using only plane waves in the z direction, then increasing the frequency wave vector in the z direction comes at the expense of the frequency wave vectors in the lateral (x and y) dimensions. Thus, any gain in resolution in the z direction necessarily comes at the expense of a loss in resolution in the lateral dimension. Choosing ox2+oy2pk2s in order to ensure propagating waves, the pressure becomes 8 bs oI o ZðoÞ R 1 igsi ðz zs Þ > > fðox ; oy ; zs Þ dzs ; z4zs > 1 e < 2gsi C p (11) pðox ; oy ; z; oÞ ¼ > bs oI o ZðoÞ R 1 igsi ðzs zÞ > e fðo ; o ; z Þ dz ; zoz > x y s s s : 2g C 1 si p where the negative root in Eq. (10) has been utilized and Eq. (11) is only valid for ox2+oy2pk2s . Recognizing the deﬁnition of the Fourier integral, this can be written as 8 bs oI o ZðoÞ igsi z > > rÞgjoz ¼gsi ; z4zs > < 2gsi C p e F3D ffð~ (12) pðox ; oy ; z; oÞ ¼ b oI o ZðoÞ igsi z > > e F3D ffð~ rÞgjoz ¼ gsi ; zozs > s : 2gsi C p Here, F3D denotes the 3D spatial Fourier transform. Eq. (12) makes the statement that for z4zs, the 2D Fourier transform of the pressure detected on the z plane is proportional to the full 3D Fourier transform of the inhomogeneity evaluated on oz ¼ gsi. Similarly, for zozs, the spatial 2D Fourier transform of the pressure detected on the z plane is proportional to the full 3D Fourier transform of the inhomogeneity evaluated on oz ¼ gsi. We note that had the negative root been chosen from Eq. (10), then the signs would change but the essence of the situation would remain the same. In other words, for z4zs, we would detect the 3D transform on oz ¼ gsi and oz ¼ gsi for zozs. It should be noticed that oz ¼ 7gsi are the top and bottom hemispheres of the full sphere in 3D Fourier space, namely the surface that satisﬁes oz2 ¼ g2si ¼ k2s ox2 oy2oz2+ox2+oy2 ¼ k2s . This is a sphere in 3D spatial Fourier space that is centered at the origin and has radius ks. For the case zs ¼ 0, which corresponds to the system heterogeneity being located at the origin, then Eq. (12) can be compactly stated as pðox ; oy ; z; oÞ ¼

bs I o io Gðox ; oy ; z; oÞZðoÞF3D ffð~ rÞgjoz ¼ gsi Cp

(13)

where G(ox, oy, z, o) is the 1D Green’s function as defined in Eq. (7) and we recall that the + or signs are taken depending on whether measurements are made in transmission or reflection. If we further define G total ðox ; oy ; z; oÞ ¼

bs I o io Gðox ; oy ; z; oÞ Cp

(14)

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then Eq. (13) can further be written as pðox ; oy ; z; oÞ ¼ G total ðox ; oy ; z; oÞZðoÞ;

F3D ffð~ rÞgjoz ¼ gsi

(15)

As previously stated, Eq. (12) states that the 2D Fourier transform of the pressure P detected on a plane is proportional to the 3D Fourier transform of the inhomogeneity function on a sphere in 3D Fourier space. In other words, detection of the pressure on a plane is akin to extracting a spherical ‘slice’ of the 3D Fourier transform of the inhomogeneity. It can be seen from Eq. (15) that Gtotal(ox, oy, z, o) behaves as a transfer function between the input pulse, the spherical slice of 3D Fourier transform of the inhomogeneity function and the pressure measured on the same variables as in the argument of the transfer function, namely p(ox, oy, z, o). If the spatial frequency variables are rescaled to become dimensionless such that cs oy cs ox cs oz o0x ¼ ; o0y ¼ ; o0z ¼ o o o then the 3D spherical slice of the 3D Fourier transform of the inhomogeneity function is a sphere of radius 1 in dimensionless spatial frequency space. With this view of the dimensionless spatial frequency variables, then it is always the same area of the Fourier plane that is being considered, however, changing the input (temporal) frequency, o, changes the measurement scale in the Fourier plane. The temporal frequencies, o, that appear are those that are present in the input pulse. It should be obvious that based on this analysis the ‘optimal’ choice of input pulse is a true delta function pulse. If the time dependence of the input pulse is strictly a delta function, its temporal Fourier transform is 1 for all frequencies and Z(o) ¼ 1, indicates that all frequencies are present. Since all frequencies are present, the sphere in 3D spatial frequency space will eventually assume all possible radii and thus information about the entire 3D Fourier transform of the inhomogeneity function will be obtained. The opposite situation is true if the input pulse is chosen to be a sinusoid, say sinðo0 tÞ. In this case, there will only be temporal frequency present, namely o ¼ o0 and information regarding the inhomogeneity function will be obtained strictly on the 3D sphere of radius o0 =cs . Since this sphere is centered at the origin, we note that rotating the sphere about the origin obtains no new information about the nature of the 3D transform of the object function away from that sphere. This is in contrast with standard ultrasonic diffraction tomography where the sphere in spatial Fourier space has center at (0, 0, ks). In this case, the obtained measurements at the same frequency but different views give a rotation of the 3D spherical slice in Fourier space. Since this spherical slice is not centered at the origin, rotating it serves to obtain further coverage of the Fourier space. For ultrasonic tomography, if projections from all angles but one input frequency are given, their Fourier transforms will completely cover a low-pass version of the Fourier transform of the object function, with the frequency of the low-pass ﬁlter being determined by the frequency of the incident ultrasound radiation. Thus, the object function can be found from its 3D spatial Fourier transform by assembling the Fourier transforms at various views and obtaining enough coverage in spatial Fourier space to correctly back-transform to the space domain. 3.1. Special cases Some special cases will now be considered.

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3.1.1. Special case: one-dimensional case In 1D, Eq. (11) becomes 8 bs cs I o ZðoÞ iks z ~ o > > f e ; > < 2C p cs pðz; oÞ ¼ bs cs I o ZðoÞ iks z ~ o > > > f e ; : 2C p cs

z4zs (16) zozs

where f~ denotes the 1D Fourier transform of f. Inverse transforming with respect to time yields 8 bs cs I o R 1 o ioðtþðz=cs ÞÞ > ~ > do; z4zs > < 4pC p 1 ZðoÞf cs e (17) pðz; tÞ ¼ bs cs I o R 1 o ioðt ðz=cs ÞÞ > > ~ > ZðoÞf do: zozs e : cs 4pC p 1 The factors of eiks z ¼ eioz=cs and e iks z ¼ e ðioz=cs Þ simply become time delays of ðz=cs Þ upon transformation to the time domain. The pressure response is simply the time-delayed ~ ~ inverse Fourier transform of ZðoÞfðo=c s Þ or ZðoÞfð o=cs Þ. Since this is a product in the Fourier domain, the result in the time-domain is one of convolution between the input pulse and the object function. Clearly, if the input pulse is a delta function so that Z(o) ¼ 1 then the measured response will almost directly measure the object function itself. 3.1.2. Special case: measurements in transmission for one-dimensional case The schematic for measurements in transmission and reﬂection is shown in Fig. 1. For measurements in transmission at zd4zs, we can immediately write from Eq. (16) 2C p pðzd ; oÞe iks zd o f~ . (18) ¼ cs bs cs I o ZðoÞ

Fig. 1. Conceptual diagram for measurements in transmission and reﬂection for the 1D planar detection scheme.

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Inverse Fourier transformation of both sides from o-z yields 2C p z zd fðzÞ ¼ Q bs c2s I o cs where QðzÞ ¼ F

pðzd ; oÞ jo ! z . ZðoÞ

799

(19)

(20)

3.1.3. Special case: measurements in reflection for one-dimensional case For measurements in reﬂection at zdozs, we can immediately write from Eq. (16) iks zd ~ o ¼ 2C p pðzd ; oÞe . (21) f cs bs cs I o ZðoÞ Inverse Fourier transformation of both sides from o-z yields 2C p zd z fðzÞ ¼ Q . cs bs c2s I o

(22)

3.1.4. Special case: Dirac– delta input in one-dimensional case Consider the special case of a Dirac–delta source in the 1D case. This does NOT correspond to a point source, but rather to a planar source at z ¼ z0. If f(z) ¼ d(z z0), then in this case Eq. (16) becomes 8 bs cs I o ZðoÞ iks z iks z0 > > e e ; z4zs > < 2C p (23) pðz; oÞ ¼ bs cs I o ZðoÞ iks z iks z0 > > e e : zozs > : 2C p Upon inverse-Fourier transforming back into the time-domain, this becomes 8 bcI ðz zÞ > > s s oZ t 0 ; z4zs > < 2C p cs pðz; tÞ ¼ bs cs I o ðz z0 Þ > > > Z t : zozs : 2C cs p

(24)

Clearly, the time-domain response to a planar source is simply a time-delayed version of the input pulse. The magnitude of the time-delay is given by the time taken for the signal to travel between the source plane and the measurement plane. 4. Green’s function and transfer function approach Taking the full spatial Fourier transform of Eq. (3) and rearranging yields pð~ o; oÞ ¼ pðox ; oy ; oz ; oÞ ¼

C p ðo2x

bs I o io ~ oÞZðoÞ fð~ þ o2y þ o2z k2s Þ

(25)

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where we have used the shorthand notation ð~ oÞ ¼ ðox ; oy ; oz Þ to denote a point in 3D spatial Fourier frequency space. Although the same variables are used to denote a function and its Fourier transform, it should be clear from the arguments, which is being indicated. In cases where it may not be clear, a tilde ( ) will be used to denote the function in spatial Fourier space. For shorthand notation, let ok2 ¼ ox2+oy2+oz2, so that ok is the length of the spatial Fourier vector. Then by spatial inverse transformation of Eq. (25), the pressure function becomes Z 1 b I o ioZðoÞ 1 fð~ oÞ pð~ r; oÞ ¼ s ei~o ~r d~ o. (26) Cp ð2pÞ3 1 ðo2k k2s Þ Using the deﬁnition of the Fourier transform, the above equation can be re-written as Z 1 bs I o ioZðoÞ 1 fð~ oÞ ei~o ~r d~ pð~ r; oÞ ¼ o 3 2 2 Cp ð2pÞ 1 ðok ks Þ Z 1 Z 1 b I o ioZðoÞ 1 ei~o ~r fð~ xÞ e i~o ~x d~ x d~ o. (27) ¼ s Cp ð2pÞ3 1 ðo2k k2s Þ 1 Let us deﬁne the spatial Green’s function as Z 1 i~o ð~r ~xÞ 1 e d~ o. Gð~ rj~ xÞ ¼ Gð~ r ~ xÞ ¼ 3 ð2pÞ 1 o2k k2s

(28)

This is the Green’s function for the Helmholtz equation (3). It then follows that Eq. (27) can be rewritten such that it can be clearly interpreted as a convolution in space Z bs I o ioZðoÞ 1 pð~ r; oÞ ¼ fð~ xÞGð~ r ~ xÞ d~ x (29) Cp 1 The frequency domain equivalent to Eq. (29) is Eq. (25), which can now be interpreted in terms of the spatial transfer function as pð~ o; oÞ ¼ pðox ; oy ; oz ; oÞ ¼

bs I o io Gð~ oÞfð~ oÞZðoÞ Cp

(30)

where the Fourier transform of the spatial Green’s function is the spatial transfer function: Gð~ oÞ ¼

ðo2k

1 . k2s Þ

(31)

Another way of thinking about Eq. (30) is to write it as pð~ o; oÞ ¼

bs I o ioGð~ oÞffð~ oÞZðoÞg ¼ Gtotal ð~ o; oÞffð~ oÞZðoÞg Cp

(32)

where G total ð~ o; oÞ ¼

bs I o ioGð~ oÞ Cp

(33)

is the ‘total’ system transfer function acting on the product of the input pulse and the heterogeneity function.

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5. Green’s and transfer function via spherical polar coordinates From Eq. (28), repeated here for convenience, the spatial Green’s function is given by Z 1 1 ei~o ~r Gð~ rÞ ¼ d~ o. (34) 3 ð2pÞ 1 o2k k2s In this integral, ~ r is a fixed quantity and the integration is taken over all ð~ oÞ ¼ ðox ; oy ; oz Þ. The spatial Fourier transform variables for this integral can be converted into their own set of polar coordinates: ox ¼ r sin co cos yo oy ¼ r sin co sin yo oz ¼ r cos co (35) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ o2x þ o2y þ o2z is the frequency radial coordinate and (co, yo) are the angular coordinates in the frequency space. The spherical polar coordinates are given by (r,co,yo) with 0pr, 0pcop, 0pyo2p. The subscript ‘o’ is used to denote that these are coordinates in the frequency domain but may be dropped when the context is clear. Here c is the colatitude measured from the z-axis and y is the azimuth or longitude. For compactness of notation, we also use the change of variable m ¼ cosðco Þ so that dm ¼ sinðco Þdðco Þ. Within the integral, ~ r is a fixed quantity, so that without loss of generality we can take ~ r to ~ ~ be in the positive z direction. It then follows that o r ¼ rr cosðco Þ ¼ mrr, where r ¼ j~ rj. Given this change of variables, the Green’s function becomes Gð~ rÞ ¼

1 ð2pÞ3

dy 0

r2 r2 k2s

eimrr dm dr.

(36)

The inner integral can be evaluated as Z

eimrr dm ¼

1 eimrr 1 irr ½e e irr . ¼ irr 1 irr

The Green’s function in Eq. (36) then becomes Z 1 Z 1 1 r 1 r irr irr Gð~ rÞ ¼ ½e e dr ¼ eirr dr 2 2 2 2 2 irð2pÞ 0 r ks irð2pÞ 1 r k2s Using residues, the last integral becomes Z 1 r pi ½eirr dr ¼ ðeiks r þ e iks r Þ 2 2 2 1 r k s

(37)

(38)

(39)

Hence the spatial Green’s function can be written as Gð~ rÞ ¼ GðrÞ ¼

1 iks r ðe þ e iks r Þ. 8pr

(40)

We note that in the limit as ks-0, the wave equation approaches the Poisson equation. It thus follows that an appropriate Green’s function for the wave equation should

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approach the Green’s function for the Poisson equation, which is given by G poisson ðrÞ ¼

1 . 4pr

(41)

From Eq. (40), we see that limks !0 GðrÞ ¼ ð1=8prÞð2Þ ¼ 1=4pr as required. 5.1. Causal Green’s function In general eiks r and e iks r are considered wavefronts in the incoming and outgoing directions and thus only one of these will obey causality. To aid in the selection of a causal solution, the Sommerfeld radiation condition is often employed. The Sommerfeld radiation condition states that the sources in the field must be sources not sinks of energy. Therefore, energy radiated from sources must scatter to infinity and cannot radiate from infinity into the field. Mathematically, a solution u(x), where x is the spatial variable, to the Helmholtz equation is considered to be radiating if it satisfies q ðn 1Þ=2 lim jxj þ ik uðxÞ ¼ 0 (42) xj!1 qjxj where n is the dimension of the space. This is the radiation equation based on an implied time variation of eiot, which is implicit in the chosen definition of the Fourier transform. For example, with the chosen definition of the Fourier transform, the transform of (t) is iof~ðoÞ, clearly implying the eiot dependence. Had a different definition of the Fourier transform been used, the implied time variation would have been e iot, in which case the sign of i in Eq. (42) would have to be reversed. Based on Eq. (42), with an eiot dependence, only the e iks r is considered to be a radiating solution. This amounts to disregarding the incoming wave and selecting the outgoing wave. Furthermore, to satisfy the Poisson equation limit, which is akin to a conservation of energy, the ‘‘strength’’ of the outgoing wave must be doubled. It is for these reasons that the spatial (Helmholtz) Green’s function is finally written as e ikr . (43) 4pr Since the spatial portion of Gtotal ð~ o; oÞ is completely contained in Gð~ oÞ, it then follows that GðrÞ ¼

G total ð~ r; oÞ ¼

bs I o b I o io e iks r ioGð~ rÞ ¼ s C p 4pr Cp

(44)

where Gð~ rÞ is the spatial inverse Fourier transform of Eq. (31), as deﬁned by Eq. (28) and found in Eq. (43). 6. Photo-acoustics Fourier theorem We return to the main statement of the solution, the equation for the spatial pressure distribution Z 1 ~ fð~ oÞ b I o ioZðoÞ 1 pð~ r; oÞ ¼ s ei~o ~r d~ o. (45) 3 2 Cp ð2pÞ 1 ðok k2s Þ

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Using the coordinate transformation from Cartesian to spherical polar coordinates as given in Eq. (35), the integral in Eq. (45) will be rewritten. The subscript ‘o’ is dropped for brevity. We start by assuming that the vector ~ r is in the positive z direction so ~ ~ that ¼ (0,0,r) ¼ r and o r ¼ rr cosðcÞ ¼ mrr, where r ¼ j~ rj. The more general case for ~ r will be subsequently considered. For the case where ~ r is in the positive z direction, the spatial inverse Fourier transform integral in Eq. (45) can be written as 1 ð2pÞ3

Z 2p Z 1 Z 1 ~ ~ oÞ fð~ fðr; m; yÞ imrr 2 1 i~ o ~ r e d~ o¼ e r dr dm dy. 2 3 2 r2 k2s ð2pÞ 0 1 ðok k s Þ 1 0 1

(46)

By the first mean-value theorem for integration, there must exist some value of (m ,y ) for 1pmp1, 0py p2p such that Z 1~ ~ m; yÞ fðr; fðr; mn ; yn Þ ½eirr e irr 2 1 imrr 2 r dr e r dr dm dy ¼ 2 2 irr r2 ks ð2pÞ 0 r2 k2s 0 1 0 Z 1 ~ fðr; mn ; yn Þ irr 1 ¼ e r dr (47) ð2pÞ2 ir 1 r2 k2s ~ ~ mn ; yn Þ ¼ fð r; mn ; yn Þ, which In the last line, we have made use of the fact that fðr; implies that the Fourier transform of the heterogeneity function is defined to have an even extension in the radial dimension. Using Cauchy’s integral formula, the last integral can be simplified to yield 1 ð2pÞ3

pð~ r; oÞ ¼

bs I o ioZðoÞ 1 ~ ~ s ; mn ; yn Þe iks r . ½fðks ; mn ; yn Þeiks r þ fðk Cp 8pr

(48)

The integral in Eq. (48) must converge since the inhomogeneity must have compact support, and must thus eventually become zero. This expression contains both eiks r and e iks r spherical waves. The incoming eiks r waves are disregarded based on causality and the strength of the remaining wave is doubled, as previously discussed. Therefore, it follows that iks r bs I o io b I o io ~ s ; mn ; yn Þ e ~ s ; mn ; yn ÞGðrÞ ¼ s ZðoÞfðk ZðoÞfðk Cp Cp 4pr ~ s ; mn ; yn Þ ¼ G total ðr; oÞZðoÞfðk

pð~ r; oÞ ¼

(49)

where Gtotal(r, o) is defined in Eq. (33) as ðbs I o io=C p ÞGðrÞ. Eq. (49) makes the important statement that the resulting pressure field at ~ r is proportional to the Fourier transform of the heterogeneity function, evaluated on the sphere r ¼ ks. This is a sphere in Fourier space centered at zero and of radius ks. It should also be noted that in Eq. (49), Gtotal(r, o) acts as a transfer function—that is it multiplies the input and heterogeneity functions to yield the resulting output pressure in physical space. This is interesting because Gtotal(r, o) is actually a Green’s function, implying the relationship between input and output should be one of convolution in physical space, not multiplication. So far, this result has only been demonstrated for the case where ~ r is in the positive z direction. To consider an arbitrary position vector in space, it is noted that any arbitrary ^ To see this, we first rotate ~ position can be reached by rotating the vector ~ r ¼ ð0; 0; rÞ ¼ rk. r

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804

by c about the x-axis clockwise. This can be represented with a rotation matrix 2 3 1 0 0 6 7 A ¼ 4 0 cos c sin c 5. 0 sin c cos c

(50)

The resulting vector is now rotated counterclockwise about the z-axis by y p/2. This can be represented by the rotation matrix 2 3 sin y cos y 0 6 7 (51) B ¼ 4 cos y sin y 0 5. 0 0 1 Thus the total rotation matrix is given by the product of the two matrices (in the correct order!) 2 3 sin y cos c cos y sin c cos y 6 cos c sin y sin c sin y 7 BA ¼ 4 cos y (52) 5. 0

sin c

cos c

The original vector ~ r ¼ ð0; 0; rÞ ¼ rk^ is now given by 3 2 32 3 2 r sin c cos y sin y cos c cos y sin c cos y 0 7 6 76 7 6 BA~ r ¼ 4 cos y cos c sin y sin c sin y 54 0 5 ¼ 4 r sin c sin y 5 0

sin c

cos c

(53)

r cos c

which is precisely the spherical coordinate transformation and thus any point in the plane can be located in this manner. This proves that an arbitrary point in space can be reached ^ which was the vector used to prove by appropriate rotation of the vector ~ r ¼ ð0; 0; rÞ ¼ rk, the result in Eq. (49). The reachability of an arbitrary point in space from ð0; 0; rÞ ¼ rk^ via rotation is important as the rotation invariance property of the Fourier transform can now be invoked. Speciﬁcally, this property says that if F ð~ oÞ is the multi-dimensional Fourier transform of f ð~ rÞ for any vector ~ r, then the Fourier transform of f(R) is given by F ðR~ oÞ, where R is a rotation matrix. Using this, Eq. (45) becomes Z 1 ~ fðR~ oÞ i~o ~r b I o ioZðoÞ 1 pðR~ r; oÞ ¼ s e d~ o. (54) 3 2 Cp ð2pÞ 1 ðok k2s Þ Note that ok2 is unaffected by rotation since it is the length of the frequency vector. When Eq. (54) is transformed into spherical polar coordinates, the R rotation matrix changes the angular coordinates c and y in the argument of f but leaves the r coordinate unchanged. Thus the result given by Eq. (49), namely that the pressure ﬁeld is proportional to the Fourier transform of the heterogeneity function on a sphere of radius ks, holds for an arbitrary vector ~ r. In this case, we can write ~ s ; mn ; yn Þ pðR~ r; oÞ ¼ G total ðr; oÞZðoÞfðk R R

(55)

where mR,yR are the spherical coordinate angles obtained after rotation by the matrix R and thus reﬂect the angular dependence of the pressure on the left-hand side of the equation.

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6.1. Statement of the generalized photo-acoustics Fourier theorem The implication of this last statement is that we can go from writing in the spatial Fourier domain ~ oÞZðoÞ pð~ o; oÞ ¼ G total ð~ o; oÞfð~

(56)

to writing in the physical spatial domain ~ s ; mn ; yn Þ. pð~ r; oÞ ¼ G total ðr; oÞZðoÞfðk

(57)

o; oÞ is a transfer function in the all frequency (time and space) domain. The function G total ð~ Eq. (57) implies that Gtotal(r, o) also behaves as a transfer function in the spatial domain. The statement as given in Eq. (57) is particularly important for several reasons. First, it is immediately obvious how a change in Green’s function or its description can be incorporated. Furthermore, Eq. (57) states that the pressure function at any location is directly proportional ~ s ; mn ; yn Þ where fðk ~ s ; mn ; yn Þ is the 3D spatial Fourier transform of the heterogeneity to fðk function evaluated on the sphere r ¼ ks ¼ o=cs and at some angular values (m ,y ) such that 1pm p1, 0pyp2p. Lastly, the Fourier rotation theorem can be used to gather more information about the spatial Fourier transform of the heterogeneity function on that sphere. We recall that this property says that if F ð~ oÞ is the multi-dimensional Fourier transform of f ð~ rÞ for any vector ~ r, then the Fourier transform of f(R) is given by F(R), where R is a rotation matrix. From symmetry, the Green’s functions G(r) and Gtotal(r,o) are unaffected by rotation, as is clearly the time-dependent Z(o). Therefore, the effect of rotation is to locate a different set of angles R(m ,y ) ¼ (mR,yR), obtained via rotation, on the same r ¼ ks ¼ o=cs sphere in spatial Fourier space such that ~ s ; mn ; yn Þ. pðR~ r; oÞ ¼ G total ðr; oÞZðoÞfðk R R

(58)

Similarly, making a measurement at the same angular position but at a different value of r will only produce a change in the proportionality constant given by Gtotal(r, o) and yield the ~ s ; mn ; yn Þ. same spatial location on the heterogeneity Fourier transform sphere fðk 7. Spherically symmetric object functions We consider a special case of Eq. (45) where the Fourier transform fð~ oÞ is spherically symmetric so that it is a function of r only and does not depend on the angles m or y. Under this assumption, the angular integration can be performed. In the special case of spherical symmetry of the heterogeneity function, we can go from writing in the spatial Fourier domain pð~ o; oÞ ¼ G total ð~ o; oÞfð~ oÞZðoÞ

(59)

to writing in the physical spatial domain ~ sÞ pðr; oÞ ¼ G total ðr; oÞZðoÞfðk

(60) ~ where it must be recalled that fðks Þ is the 3D spatial Fourier transform of the heterogeneity function evaluated at r ¼ ks ¼ o=cs . Furthermore, from Eq. (33) that the net effect of Gtotal in the time domain is a time delay from the e iks r term and a derivative from the io term. Hence, from Eq. (60), it can be seen that the temporal pressure response will be a time-delayed version of the derivative of the

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temporal convolution of the input pulse with the heterogeneity function. One of the huge beneﬁts of choosing the input pulse to be a Dirac–delta function is that convolution with the Dirac–delta function returns the function itself. Therefore, if the input pulse is chosen to be a Dirac–delta function, the pressure response observation will almost directly be an observation of the heterogeneity function itself. Since in the special case of spherical symmetry there is no angular dependence of pressure, then the pressure need only be measured at a single point. For detection at a radius of r ¼ rd o pðrd ; oÞ ¼ Gtotal ðrd ; oÞZðoÞf~ . (61) cs By measuring the pressure at a single point, information about the Fourier transform of the heterogeneity function has been obtained at the point r ¼ ks ¼ o=cs . Therefore, measurements at several different frequencies, o, will need to be made in order to obtained sufﬁcient information about the heterogeneity function in order to permit inverse Fourier inversion. Thus, the frequency content of the input pulse Z(o) determines the Fourier plane coverage of the heterogeneity function and the resolution possible. If the input pulse is band limited, then the reconstruction of the heterogeneity function will also be a bandlimited version. 7.1. Some special cases Some special cases will now be individually considered. 7.1.1. Special case: point source at the origin Consider the special case where there is a point source at the origin. In that case 1 dðrÞ (62) 4pr2 is the Dirac–delta function at the origin, written in spherical polar coordinates. More importantly, the Fourier transform of f is 1 and Eq. (61) becomes fðrÞ ¼ d3 ð~ rÞ ¼

pðrd ; oÞ ¼ Gtotal ðrd ; oÞZðoÞ. Inverting Eq. (63) gives bs I o 0 rd pðrd ; tÞ ¼ Z t . 8prd C p cs

(63)

(64)

In this case, it is a time-shifted version of the derivative of the input pulse that is detected at an arbitrary point. 7.1.2. Special case: point source at the origin and input Dirac– delta pulse Consider the special case where there is a point source at the origin and the input pulse is also a Dirac–delta function occurring at t0 so that ZðtÞ ¼ dðt t0 Þ ! ZðoÞ ¼ e iot0 . In this case, the measured pressure is simply the Green’s function itself pðrd ; oÞ ¼ Gtotal ðrd ÞZðoÞ ¼

bs I o io e iks rd iot0 e . C p 4prd

(65)

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This can be easily inverse Fourier transformed to the time-domain to yield b Io 0 rd pðrd ; tÞ ¼ G total ðrd ÞZðoÞ ¼ s d t t0 4prd C p cs b Io 1 rd d t t0 ¼ s 4prd C p ðt t0 ðrd =cs ÞÞ cs

807

(66)

where the result d0 ðxÞ ¼ dðxÞ=x has been used. Eq. (66) is a scaled, time-shifted delta–Dirac pulse, with the magnitude of the time-delay given by t0 þ ðrd =cs Þ. 7.1.3. Special case: Gaussian heterogeneity function It is assumed that the heterogeneity function is a (spherically symmetric) Gaussian, so that 2 1 fðrÞ ¼ pffiffiffi e ðr=aÞ a p

for some positive a. The 3D Fourier transform of this can be found to be o 2 2 2 2 2 ~ fðrÞ ¼ e ða r =4Þ ) f~ ¼ e ða r =4cs Þ . cs It then follows that Eq. (61) becomes o b I o io e ird ðo=cs Þ ða2 o2 =4c2s Þ ~ pðrd ; oÞ ¼ G total ðrd ; oÞZðoÞf e ZðoÞ. ¼ s cs Cp 4prd

(67)

(68)

(69)

Clearly, the temporal pressure response will be a time-shifted version of the derivative of the convolution of the input temporal pulse with the Gaussian heterogeneity function. If it is also assumed that the input pulse is a Dirac–delta function so that Z(o) ¼ 1, then Eq. (69) can be inverse transformed back into the time domain to give pðrd ; tÞ ¼

bs I o c2s ðcs t rd Þ ðt ðrd =cs ÞÞ2 ðc2s =a2 Þ e . p3=2 rd C p a3

(70)

Clearly this is a time-shifted version of the first derivative of the input Gaussian. For an assumed heterogeneity with a width related to a, the time-response has a width related to a/cs. This is not an unreasonable result and it is emphasized that this assumes a Dirac–delta temporal input. 7.1.4. Special case: Gaussian heterogeneity function and Gaussian input pulse Now assume that the input pulse is not a true delta function but is also a Gaussian pulse so that 2 2 2 2 1 ZðtÞ ¼ pffiffiffi e b t ) ZðoÞ ¼ e ðb o =4Þ . b p

(71)

Substituting into Eq. (69) and inverse Fourier transforming back into the time domain yields ! bs I o ðcs t rd Þc2s ðcs t rd Þ2 pðrd ; tÞ ¼ exp . (72) C p p3=2 rd ðc2s b2 þ a2 Þ3=2 c2s b2 þ a2

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The functional form in Eq. (72) is very similar to that in Eq. (70), which is reasonable since a delta pulse can be approximated with a narrow enough Gaussian pulse. However, the potential problem with Eq. (72) is that if the width of the input temporal pulse is not known very well, then it will be difficult or impossible to extract the exact width of the heterogeneity Gaussian, which is typically the quantity being sought in medical imaging applications. From Eq. (72), it can be seen that the response to an input pulse of width proportional to b and a heterogeneity of width proportional to a is a pulse qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of width b2 þ ða2 =c2s Þ. In comparison to the case where the input pulse is a Dirac–delta function, the pressure response pulse has been widened by the Gaussian nature of the input pulse, in direct proportion to the width of the input pulse. However, any uncertainty in the precise width of the input pulse would imply uncertainty into the size of the heterogeneity being detected. If ba/cs so that b is negligible compared with a/cs, the effect of the width of the input pulse will have very little effect on the pressure response and can essentially be ignored. It is noted that this condition is the same as that for the previously mentioned requirement of stress confinement. 7.2. Solving for the heterogeneity function The Fourier transform of the heterogeneity function can be isolated in Eq. (61) o pðrd ; oÞ C p 4prd þiks rd ~ f e . ¼ cs ZðoÞ bs I o io Or

1 ~ o pðrd ; oÞ C p 4prd ird o=cs f e . ¼ c3s cs ZðoÞ c3s bs I o io

(73)

(74)

The left-hand side of Eq. (74) is written to make it clear that the scale properties of the (3D) Fourier transform will be used. Taking the inverse Fourier transform of both sides of Eq. (74) will yield 1 ~ o 1 pðrd ; oÞ C p 4prd ird o=cs F 1 f e ¼ F . (75) 3D 3D c3s cs ZðoÞ c3s bs I o io Using the scale properties of the 3D Fourier transform, this gives C p 4prd 1 pðrd ; oÞ ird o=cs C p 4prd 1 1 iopðr; oÞ ird o=cs e F1D e fðcs rÞ ¼ 3 F3D ¼ 3 ZðoÞio ZðoÞio cs bs I o cs bs I o 2pr 2C p rd 1 pðrd ; oÞ ird o=cs e F ;o ! r . ¼ 3 cs bs I o r 1D ZðoÞ This can be written as 2C p rd r þ rd Q fðrÞ ¼ 2 cs b s I o r cs where as before, we deﬁne pðrd ; oÞ ;o ! r . QðrÞ ¼ F11D ZðoÞ

(76)

(77)

(78)

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Note that if the input pulse is a delta function so that Z(t) ¼ d(t)-Z(o) ¼ 1, then QðrÞ ¼ F 1 1D fpðrd ; oÞ; o ! rg ¼ pðrÞ.

(79)

In other words, for a delta function input pulse, the pressure pulse as measured as a function of time, is directly the shape of the inhomogeneity function after appropriate scaling by 1/r. It should be noted that with the exception of the nondimensional rd/r term, this is essentially the same result as found in Eqs. (19) and (22), as would be expected. 8. Disadvantage of the Dirac–delta pulse It was noted in the discussion on the example with the Gaussian heterogeneity function that if the input pulse is also a Gaussian, it may be difficult to discern the size of the heterogeneity function if there is any uncertainty in the width of the input pulse. It has also been observed that if the input-pulse is a Dirac–delta pulse that observations made are almost directly those of the heterogeneity function itself. So, in a certain sense, the choice of input pulse as a Dirac–delta pulse is optimal. However, perfect delta functions are difficult to produce in real-world conditions and it is more likely that the input pulse begins to resemble a Gaussian function as the input pulse acquires a finite width. If the heterogeneity function is also roughly assumed to be spherical with a Gaussian-like shape, which is reasonable, then it becomes apparent from the previous discussion that any lack of information regarding the length of the input pulse will become uncertainty in the true size of the heterogeneity function. Hence, we note that there is an inherent conflict between the systems ability to resolve inhomogeneities and the systems ability to detect weak signals from deep inhomogeneities. The reason is shorter pulses enable an almost direct observation of the heterogeneity function with little uncertainty and also the ability to discriminate between closely spaced heterogeneities. However, longer pulses will allow the system to absorb energy over a longer period of time and thereby distinguish between the photo-acoustic signal from the heterogeneity and other interfering energy sources. That is, longer pulses tend to increase the signal-to-noise ratio. 9. Green’s function approach in cylindrical polar coordinates It can be shown for radially symmetric functions that the in-plane 2D Fourier transform becomes Z 1 F 2D ðrÞ ¼ F2D ðf ðrÞjr ! rÞ ¼ 2p f ðrÞJ 0 ðrrÞr dr (80) 0 2

ox +oy2

where r ¼ is the spatial radial frequency variable. Thus, if a function is most easily described in terms of cylindrical polar coordinates and has no angular dependence, then the full 3D Fourier transform can best be thought of as performing the in-plane 2D Fourier transform operation followed by the 1D Fourier transform for the z variable: Z 1Z 1 F 3D ðr; oz Þ ¼ F3D ðf ðr; zÞjr ! r; z ! oz Þ ¼ 2p f ðr; zÞJ 0 ðrrÞe izoz r dr dz 1

¼ F1D ðF2D ðf ðr; zÞjr ! rÞ; z ! oz Þ.

(81)

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We now consider the transfer function from Eq. (34), repeated here for convenience Z 1 1 ei~o ~r Gð~ rÞ ¼ d~ o. (82) ð2pÞ3 1 o2k k2s The spatial frequency variable can be converted to cylindrical polar coordinates so that ox ¼ r cos yo ; oy ¼ r sin yo ; oz ¼ oz . Note that the ozvariable remains unchanged and that the radial frequency variable is r, which is completely different from the r used in spherical polar coordinates. Similarly, the spatial variables can be converted to cylindrical polar coordinates so that x ¼ r cos y and y ¼ r sin y with the z variable remaining unchanged. Using this change of variable, Eq. (82) becomes Z 1 Z 1 Z 2p irr cosðyo yÞ 1 e Gð~ rÞ ¼ eioz z r dy doz dr. (83) 3 ð2pÞ 0 r2 þ o2z k2s 1 0 From the integral deﬁnition of the zero-order Bessel function Z Z 1 p ix cosðc yÞ 1 p ix cos y J 0 ðxÞ ¼ e dy ¼ e dy. 2p p 2p p

(84)

Eq. (83) becomes Z Z 1 Z 1 1 1 1 1 e gs jzj ioz z J 0 ðrrÞ e doz r dr ¼ J 0 ðrrÞ r dr Gð~ rÞ ¼ 2 2 2 2 2p 0 2gs ð2pÞ 0 1 r þ oz k s (85) where g2s ¼ r2 k2s , in keeping with the previously used notation. Making use of Eq. (10), the preceding equation becomes Z Z 1 ks e igsi jzj 1 1 e gsr jzj Gð~ rÞ ¼ Gðr; zÞ ¼ J 0 ðrrÞ r dr þ J 0 ðrrÞ r dr. (86) 2p 0 2p ks 2igsi 2gsr Clearly the second integral in Eq. (86) implies the production of evanescent waves that will die and the first integral are the propagating waves that are of greater interest since these are the waves that in greater likelihood will be detected. Making the change of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi variable u ¼ k2s r2 , the travelling wave portion of the Green’s function becomes Gðr; zÞjtravelling

1 ¼ 2p

Z 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iujzj e du. J 0 r k2s u2 2i

(87)

For z40 this becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 Gðr; zÞjtrav z40 ¼ B1 ðuÞJ 0 r k2s u2 e iuz du 4pi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 F B1 ðuÞJ 0 r ks u ju ! z z40 ¼ 4pi

(88)

where F denotes the 1D Fourier transform and the variables being transformed are clearly indicated and additionally, the ﬁlter function is deﬁned as 1; 0pupks B1 ðuÞ ¼ . (89) 0 otherwise

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Similarly, for zo0, Eq. (87) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 Gðr; zÞjtrav zo0 ¼ B2 ðuÞJ 0 r k2s u2 e iuz du 4pi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 F B2 ðuÞJ 0 r ks u ju ! z z40 ¼ 4pi

811

(90)

where B2 ðuÞ ¼

1; 0

ks pup0 . otherwise

(91)

Now, the Fourier transform of the Green’s Function is known to be 1=ðo2k k2s Þ, however, it is instructive to be able to derive an alternate form for it by taking the Fourier transform of G(r, z) as given in Eqs. (88) and (90). From Eq. (81), recall that for functions with cylindrical symmetry, the 3D Fourier transform can be thought of as F1D ðF2D ðf ðr; zÞjr ! rÞ; z ! oz Þ. Hence for z40, we obtain Gðr; oz Þjtrav z40 ¼ F3D Gðr; zÞjtrav z40

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 F B1 ðuÞJ 0 r ks u ju ! z ¼ F2D F1D 4pi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ F2D B2 ðoz ÞJ 0 r k2s o2z . (92) 2i where we have used the result FðFðf ðtÞÞÞ ¼ 2pf ð oÞ and B1( u) ¼ B2(u). Similarly, for zo0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B1 ðoz ÞJ 0 r k2s o2z . (93) Gðr; oz Þjtrav zo0 ¼ F3D Gðr; zÞjtrav zo0 ¼ F2D 2i The functions for which the 2D Fourier transform is sought in Eqs. (92) and (93) are radially symmetric (a function of r in cylindrical polar coordinates), with ks and oz as parameters. Thus, the 2D Fourier transform is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 B1;2 ðoz ÞJ 0 r k2s o2z J 0 ðrrÞr dr Gðr; oz Þjtrav ¼ F3D ðGðr; zÞjtrav Þ ¼ 2p 2i 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d r k2s o2z p (94) ¼ B1;2 ðoz Þ r i where the orthogonality of the Bessel function has been used in the last line. Eq. (94) is an alternate form of the 3D Fourier transform of the Green’s function. The benefit of writing it in this form are (i) the effects of the Green’s function become immediately apparent and (ii) this form can be directly applied to heterogeneity functions with cylindrical symmetry. In the first instance, it is immediately obvious from the Bj(oz), j ¼ 1,2 filter functions that only spatial frequencies where |oz|ks will be transmitted. It is also immediately obvious qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the Dirac–delta function that only the spatial frequencies such that r k2s o2z ¼ 0 result in propagating waves, by virtue of the sifting property of the Dirac–delta function. Note that this corresponds to a sphere in spatial frequency space, centered at the origin and of radius ks. This confirms the previously derived Fourier theorem.

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9.1. Example: heterogeneity function with cylindrical symmetry Recall from Eq. (30) that the pressure response is given in the Fourier domain as pð~ o; oÞ ¼

bs I o io ~ oÞZðoÞ. Gð~ oÞfð~ Cp

(95)

For a heterogeneity function with cylindrical symmetry, the use of cylindrical polar coordinates in the spatial and spatial frequency domains is most convenient so that fð~ oÞ ¼ fðr; oz Þ. Using the form for G as derived in Eq. (94), the pressure response becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d r k2s o2z bs I o io ~ p . (96) fðr; oz ÞZðoÞ B1;2 ðoz Þ pð~ o; oÞ ¼ r Cp i Eq. (96) can be inverse Fourier transformed in the radial direction such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 d r k2s o2z b I o po ~ oz Þ pðr; oz ; oÞ ¼ s J 0 ðrrÞr dr fðr; ZðoÞB1;2 ðoz Þ r Cp 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b I o po ~ ¼ s ZðoÞB1;2 ðoz Þf k2s o2z ; oz J 0 r k2s o2z Cp

(97)

From the expression in Eq. (97), it can be seen that the best detection geometry for this type of heterogeneity is a linear detection scheme with the linear array being arranged in the z direction, at a fixed radius. Furthermore, the preceding equation can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bs I o io p 2 2 ~ pðr; oz ; oÞ ¼ J 0 r ks oz B1;2 ðoz Þ ZðoÞf k2s o2z ; oz Cp i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b I o io ~ Gðr; oz ÞZðoÞf k2s o2z ; oz ¼ s Cp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ k2s o2z ; oz ¼ G total ðr; oz ; oÞZðoÞf (98) where b I o io Gðr; oz Þ G total ðr; oz ; oÞ ¼ s Cp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Gðr; oz Þ ¼ J 0 r k2s o2z B1;2 ðoz Þ. i

(99)

10. Summary of main mathematical results The main mathematical results are now summarized. It has been shown that in all coordinates systems, the frequency-domain expression for pressure is given by pð~ o; oÞ ¼ G total ð~ o; oÞffð~ oÞZðoÞg where G total ðn; oÞ ¼

bs I o ioGðnÞ. Cp

(100)

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G( ) is the spatial portion of the system transfer function, with the ( ) denoting that it can be written in various coordinate systems and the same holds true in the frequency or spatial domains or where the variables are mixed between frequency and regular space. 10.1. Photo-acoustics Fourier theorem The main result of this paper is the concise statement of the photo-acoustic Fourier theorem in various coordinate systems. In this theorem the Green’s function behaves as a pseudo-transfer function, being the appropriate proportionality constant between the measured pressure response, the spherical slice of the heterogeneity 3D Fourier transform and the input pulse. The results are presented here for the various coordinate systems. The basic idea of the result is that measured pressure is proportional to the product of a spherical slice of the 3D spatial Fourier transform of the heterogeneity function and the Fourier transform of the input function. The proportionality constant is the appropriate Green’s function behaving as a pseudo-transfer function, meaning that it is being multiplied in the spatial domain instead of convolved. 10.1.1. Cartesian coordinate system This result assumes detection on a z-oriented plane. Appropriate modiﬁcations can be made for planar detection schemes in the x or y directions. The spatial Fourier transform of the pressure response is then taken in the x, y directions only. The main result is pðox ; oy ; z; oÞ ¼ G total ðox ; oy ; z; oÞZðoÞF3D ffð~ rÞgjoz ¼ gsi with Gðox ; oy ; z; oÞ ¼

e gs jzj ; 2gs

g2s ¼ o2x þ o2y k2s

(101)

The system transfer function is written in this coordinate system as Gð~ oÞ ¼

ðo2x

o2y

1 . þ o2z k2s Þ

(102)

10.1.2. Spherical polar coordinate system This is the most general version of the result, making no assumption about the detection or heterogeneity geometry. Spatial Fourier transforms of the pressure function are not taken. The main result is iks r

~ s ; mn ; yn Þ with GðrÞ ¼ e pð~ r; oÞ ¼ G total ðr; oÞZðoÞfðk ; 4pr The system transfer function in this coordinate system is Gð~ oÞ ¼

ðr2

1 . k2s Þ

m ¼ cosðcÞ

(103)

(104)

10.1.3. Cylindrical polar coordinate system Note that this result assumes radial symmetry in the heterogeneity function and is best used with a linear detection scheme where the line of detectors is in the cylindrical (z)

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direction. The spatial Fourier transform of the pressure response is then taken in this direction only. The main result is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ pðr; oz ; oÞ ¼ G total ðr; oz ; oÞZðoÞf k2s o2z ; oz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p with Gðr; oz Þ ¼ J 0 r k2s o2z B1;2 ðoz Þ. (105) i The system transfer function can be written in this coordinate system as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d r k2s o2z 1 p or as Gðr; oz Þjtrav ¼ B1;2 ðoz Þ Gð~ oÞ ¼ 2 2 2 r i ðr þ oz ks Þ

(106)

where the ‘trav’ subscript denotes the fact that this refers to travelling waves only and r refers to the in-plane radial variable (not spherical). The 1, 2 subscripts on the ﬁlter function are chosen depending on whether the measurements are made in the upper (B1) or lower (B2) half-plane.

11. Instrumental limitations The foregoing analysis has effectively assumed a perfect ability to measure. In real-world experiments, detectors and detector planes have ﬁnite sizes and all instruments have their own instrumental transfer functions. The advantage of a multi-dimensional transfer function approach to analysis is that multi-dimensional instrumental effects can be easily incorporated into the analysis by multiplying the system transfer functions by instrumental transfer functions. In this way, the effects of the measuring instruments themselves can be removed or mitigated. However, it must be stressed that proper knowledge of the instrumental transfer functions is required for a thorough analysis and it is thus important to determine these functions as part of the experimental design process.

12. Summary and conclusions In this paper, a consistent and uniﬁed approach to the photo-acoustic problem has been presented using a Green’s function/transfer function approach. Cartesian, cylindrical and spherical polar coordinate systems were considered and the solutions for special cases of each were presented. Most importantly, a generalized version of the Fourier slice theorem was presented and proved. This theorem is important in that it forms the basis for most reconstruction algorithms. The mathematical results are summarized in the penultimate section of this paper. These results can be used for forward simulations or for the design of reconstruction algorithms. The results indicate that the geometry of the detection hardware will greatly inﬂuence the optimal form of Green’s function chosen for use in analysis. Furthermore, the results also suggest that a non-uniform FFT or multidimensional Fourier transforms in curvilinear coordinates might be considered within the scope of the design of the inversion algorithm.

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Appendix A A.1. 2D Fourier transform of a radially symmetric function A simplifying class of 2D functions is known as radially symmetric. They can be expressed in terms of polar coordinates. Consider converting a 2D function of Cartesian coordinates f(x, y)into polar coordinates f(r, y). This can be done with the transformation equation: x ¼ r cos y y ¼ r sin y

(107)

where r is the radius and y is the angle in the un-transformed spatial domain. To consider the Fourier transform of a radially symmetric function, the Fourier transform variables are also converted into their own polar coordinates: ox ¼ r cos c oy ¼ r sin c

(108)

In the preceding equation, r and c are the radial and angular coordinates in the spatial Fourier domain. Using the transformations given by Eqs. (107) and (108), the 2D Fourier transform can be written as Z 1Z p F ðr; cÞ ¼ f ðr; yÞe irrðcos c cos yþsin c sin yÞ r dr dy (109) 0

If it is assumed that f is radially symmetric, then it can be written as function of r only and can thus be taken out of the integration over the angular coordinate and the previous equation becomes Z p Z 1 rf ðrÞdr e irr cosðc yÞ dy (110) F ðr; cÞ ¼ 0

Using the integral deﬁnition of the zero-order Bessel function Z Z 1 p ix cosðc yÞ 1 p ix cos y e dy ¼ e dy J 0 ðxÞ ¼ 2p p 2p p then Eq. (110) can be written as Z 1 f ðrÞJ 0 ðrrÞr dr F 2D ðrÞ ¼ F2D ðf ðrÞÞ ¼ 2p

(111)

(112)

which can be recognized as the Hankel transform so that F2D ðf ðrÞÞ ¼ Hðf ðrÞÞ, where H denotes the Hankel transform. Thus the special case of the 2D Fourier transform of a radially symmetric function yields the Hankel transform of that function. The functions f(r) and F(r) are a Fourier transform pair and can be written as f ðrÞ3F ðrÞ.

(113)

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A.2. 3D Fourier transform of a spherically symmetric function In Eq. (61), the 3D Fourier transform for the assumed spherically symmetric object function is used. Let us now consider how to calculate this. Any spherically symmetric function, f ð~ rÞ can be written as f ð~ rÞ ¼ f ðrÞ where j~ rj ¼ r. The exponent in the Fourier ~ so that o ~ ~ kernel can be simpliﬁed by taking the z-axis to be parallel to o x ¼ rr cosðyÞ. We then have Z F ð~ oÞ ¼ F3D ðf ðrÞÞ ¼ f ð~ xÞe j~o ~x d~ x Z

0 1

¼ 0

¼ 2p 0

f ðrÞe jrr cosðcr Þ sinðcr Þr2 dcr dyr dr Z p 2 f ðrÞr dr e jrr cosðcr Þ sinðcr Þ dcr . 0

(114)

Using the change of variable m ¼ cosðcr Þ, the second integral becomes Z

e 0

jrr cosðcr Þ

sinðcr Þ dcr ¼

e 1

jrrm

1 e jrrm dm ¼ jrr 1

e jrr ejrr 2 sinðrrÞ ¼ 2 sincðrrÞ. þ ¼ rr jrr jrr

(115)

Recall the sinc function here is not normalized, so that sinc(x) ¼ sin(x)/x. Hence Eq. (114) becomes Z 1 F3D ðf ðrÞÞ ¼ F ð~ oÞ ¼ F ðrÞ ¼ 4p f ðrÞ sincðrrÞ r2 dr (116) 0

which is essentially 1D transformation from the spatial radial variable r to the spatial frequency variable r. A.3. Interpretation in terms of a 1D Fourier transform The 3D Fourier transform of a spherically symmetric function, repeated here for convenience, can also be interpreted in terms of a regular 1D Fourier transform Z 1 F3D ðf ðrÞÞ ¼ F ð~ oÞ ¼ F 3 ðrÞ ¼ 4p f ðrÞ sincðrrÞr2 dr. (117) 0

Writing the sine function in terms of the complex exponentials, Eq. (117) becomes

Z 1 f ðrÞ eirr e irr r dr F3D ðf ðrÞÞ ¼ F 3 ðrÞ ¼ 2pi r 0 Z Z 2pi 1 2pi 1 ¼ f ðrÞeirr r dr þ f ðrÞe irr r dr. (118) r 0 r 0 Now suppose we deﬁne an extension of f(r) to the entire real line so that f(r) ¼ f( r). Furthermore, a change of variable for the ﬁrst integral in Eq. (118) is implemented so that r- r (or properly, we let u ¼ r and then replace the dummy integration variable u with

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817

r again). This gives 2pi F3D Ă°f Ă°rĂ&#x17E;Ă&#x17E; Âź F 3 Ă°rĂ&#x17E; Âź r

f Ă° rĂ&#x17E;e 1

irr

2pi r dr Ăž r

f Ă°rĂ&#x17E;e irr r dr.

(119)

Using the property f(r) Âź f( r), can be clearly combined into a single Fourier integral so that the 3D transform of a spherically symmetric function f(r) is proportional to the 1D transform of rf(r): Z 2pi 1 f Ă°rĂ&#x17E;e irr r dr. (120) F3D Ă°f Ă°rĂ&#x17E;Ă&#x17E; Âź F 3 Ă°rĂ&#x17E; Âź r 1 Hence, using properties of 1D Fourier transforms, we can write F3D Ă°f Ă°rĂ&#x17E;Ă&#x17E; Âź F 3 Ă°rĂ&#x17E; Âź

2pi 2p q F1D Ă°rf Ă°rĂ&#x17E;jr ! rĂ&#x17E; Âź fF1D Ă°f Ă°rĂ&#x17E;jr ! rĂ&#x17E;g. r r qr

(121)

The inverse 3D transform for a spherically symmetric function can be shown to be Z 1 1 F 3 Ă°rĂ&#x17E; sinĂ°rrĂ&#x17E;r dr. (122) f Ă°rĂ&#x17E; Âź 2 2p r 0 Now, if f(r) is real and even then the Fourier transform of rf(r) will be imaginary and odd by known properties of 1D Fourier transforms. From Eq. (121), it can be seen that to obtain the 3D transform, the Fourier transform of rf(r) is multiplied by i/r so that the resulting 3D transform will be real and even. Hence F3D Ă°f Ă°rĂ&#x17E;Ă&#x17E; Âź F 3 Ă°rĂ&#x17E; is real and even. Using the known evenness of F3(r), the same trick can be used for the inverse 3D transform as for the forward transform. Namely, the sine function is written in terms of complex exponentials and then a change of variable from r- r is used in one of the two integrals. This implies that Eq. (122) can be written as Z 1 i f Ă°rĂ&#x17E; Âź 2 F 3 Ă°rĂ&#x17E; eirr r dr. (123) 4p r 1 Using known properties of Fourier transforms, this can be interpreted as f Ă°rĂ&#x17E; Âź

i 1 1 q 1 F1D Ă°rF 3 Ă°rĂ&#x17E;jr ! rĂ&#x17E; Âź fF Ă°F Ă°rĂ&#x17E;jr ! rĂ&#x17E;g. 2pr 2pr qr 1D

(124)

In summary, the 3D Fourier transform of a spherically symmetric function can be easily interpreted or computed in terms of the 1D Fourier transform. A.4. Scale property The scaling property for the 3D spherically symmetric Fourier transform is given by 1 r F3D Ă°f Ă°arĂ&#x17E;Ă&#x17E; Âź 3 F 3 . (125) a a Proof. We write from the deďŹ nition in Eq. (116) Z 1 F3D Ă°f Ă°arĂ&#x17E;Ă&#x17E; Âź 4p f Ă°arĂ&#x17E; sincĂ°rrĂ&#x17E;r2 dr. 0

(126)

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Now use a change of variable u Âź ar so that the preceding equation becomes Z 1 ru u2 du 1 r Âź 3 F3 f Ă°uĂ&#x17E; sinc F3D Ă°f Ă°arĂ&#x17E;Ă&#x17E; Âź 4p : & 2 a a a a a 0

(127)

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