Minimal-scan filtered backpropagation algorithms for diffraction tomography

$Page 1: Minimal-scan filtered backpropagation algorithms for diffraction tomography$
2896 J. Opt. Soc. Am. A/Vol. 16, No. 12 /December 1999 X. Pan and M. A. Anastasio

Minimal-scan filtered backpropagation algorithmsfor diffraction tomography

Xiaochuan Pan and Mark A. Anastasio

Graduate Program in Medical Physics, The Department of Radiology, The University of Chicago, 5841 SouthMaryland Avenue, Chicago, Illinois 60637

Received March 31, 1999; revised manuscript received August 12, 1999; accepted August 16, 1999

The filtered backpropagation (FBPP) algorithm, originally developed by Devaney [Ultrason. Imaging 4, 336(1982)], has been widely used for reconstructing images in diffraction tomography. It is generally known thatthe FBPP algorithm requires scattered data from a full angular range of 2p for exact reconstruction of a gen-erally complex-valued object function. However, we reveal that one needs scattered data only over the angu-lar range 0 < f < 3p/2 for exact reconstruction of a generally complex-valued object function. Using thisinsight, we develop and analyze a family of minimal-scan filtered backpropagation (MS-FBPP) algorithms,which, unlike the FBPP algorithm, use scattered data acquired from view angles over the range 0 < f< 3p/2. We show analytically that these MS-FBPP algorithms are mathematically identical to the FBPPalgorithm. We also perform computer simulation studies for validation, demonstration, and comparison ofthese MS-FBPP algorithms. The numerical results in these simulation studies corroborate our theoreticalassertions. © 1999 Optical Society of America [S0740-3232(99)01712-3]

OCIS codes: 100.3190, 100.3010, 100.6950.

1. INTRODUCTIONDiffraction tomography (DT) is an inversion scheme thatcan be used for obtaining the spatially variant refractive-index distribution of a scattering object. Applications ofDT can be found in various scientific fields such as medi-cal imaging,1,2 nondestructive evaluation of materials,3,4

and geophysics.5,6 In DT, under the weak scattering con-dition, one can invoke the Born or Rytov approximation7,8

to derive the Fourier diffraction projection (FDP) theo-rem, which relates the Fourier transform (FT) of the mea-sured scattered data to the FT of the object function. TheFDP theorem has provided the basis for development ofvarious linear algorithms for image reconstruction inDT.3,9,10

For two-dimensional (2D) transmission DT that em-ploys the classic scan configuration, based on the FDPtheorem, Devaney9 developed the well-known filteredbackpropagation (FBPP) algorithm for exact reconstruc-tion of an object function. Because evanescent waves areneglected in DT, the object function can be determinedonly up to a maximum spatial frequency of A2n0 .3,9–11

We will refer to the reconstruction of such a low-pass fil-tered object function as the reconstruction of the objectfunction. We shall also refer to the circular region of ra-dius of A2n0 centered at the origin of the Fourier space ofthe object function as the complete Fourier coverage ofthe object function. Although the FBPP algorithm canreconstruct exactly a real object function from scattereddata acquired from view angles in [0,p],12 it generally re-quires scattered data measured from view angles in[0,2p] for exact reconstruction of a complex-valued objectfunction.

Here, however, we reveal that the FT of scattered dataacquired from view angles in [0,3p/2] can provide com-plete coverage of the Fourier space of the object function.

0740-3232/99/122896-08$15.00 ©

Using this important observation, we develop and analyzea family of minimal-scan filtered backpropagation (MS-FBPP) algorithms for exact reconstruction of an objectfunction. Unlike the FBPP algorithm, which requiresscattered data acquired from view angles in [0,2p], theseMS-FBPP algorithms utilize only scattered data mea-sured at view angles in [0,3p/2].

This paper is arranged as follows: In Section 2 webriefly review the FDP theorem and the FBPP algorithm.In Section 3 we introduce the concept of the minimal com-plete data set. In Section 4, on the basis of the insightsgained in Section 3, we propose a family of MS-FBPP al-gorithms for exact reconstruction of an object functionfrom the minimum complete data set. In Section 5 weimplement and analyze these MS-FBPP algorithms. InSection 6 we provide a discussion of the study.

2. FOURIER DIFFRACTION PROJECTIONTHEOREM AND FBPP ALGORITHMA. Fourier Diffraction Projection TheoremIn 2D transmission DT imaging that employs the classicalscan configuration,9,13 as shown in Fig. 1, a scattering ob-ject is illuminated by monochromatic plane-wave radia-tion of frequency n0 , and the transmission scattered dataare measured over a straight line, which is perpendicularto the direction of propagation of the incident planewaves. From the scattered data obtained at variousangles f, one seeks to reconstruct the object functiona(r), which is generally a complex-valued function.

We use us(j, f) to denote the scattered data measuredalong the line h 5 l, as shown in Fig. 1, and Us(nm , f) todenote the one-dimensional (1D) FT of us(j, f) with re-spect to j. One can define a modified 1D FT of the scat-tered data as

1999 Optical Society of America

X. Pan and M. A. Anastasio Vol. 16, No. 12 /December 1999 /J. Opt. Soc. Am. A 2897

M~nm , f! 5 Us~nm , f!jn8

2p2n02U0

exp~2j2pn8l !, (1)

where n8 5 An02 2 nm

2. Because the quantities on theright-hand side of Eq. (1) are known or can be measured,M(nm , f) can thus be treated as the measured data.Under the Born approximation, one can obtain the well-known FDP theorem,3 which can be expressed math-ematically as

M~nm , f!

5 5 E2`

` E2`

`

a~r!exp$2j2p@nmj 1 ~n8 2 n0!h#%dr

if unmu < n0

0 if unmu . n0

, (2)

where the polar coordinates (r, u) and the rotated coordi-nates (j, h) are related through j 5 r cos( f 2 u) and h5 2r sin( f 2 u). The FDP theorem in Eq. (2) statesthat M(nm , f ) provides the values of the 2D FT of a(r)along the semicircular arc AOB of radius n0 , as shown inFig. 2. As the view angle f varies from 0 to 2p, the twosegments OA and OB of the semicircle AOB generate twodistinct complete coverages,3,11 as shown in Figs. 3(a) and3(b), of the 2D Fourier space of a(r).

B. FBPP AlgorithmUsing the FDP theorem, Devaney developed9 the FBPPalgorithm that reconstructs a(r) directly by combiningboth of the complete coverages in Figs. 3(a) and 3(b) andthat can be expressed mathematically as

a~r, u! 51

2E

f50

2p Enm52n0

n0 n0

n8unmuM~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf, (3)

where na 5 sgn(nm)Anm2 2 nm

2, nm 5 j(n8 2 n0), and

a 5 sgn~nm!arcsinS 1

2n0

Anm2 2 nm

2D . (4)

Fig. 1. Classical scan configuration of 2D (transmission) DT.The incident plane wave propagates along the h axis, and thescattered data are measured along the line h 5 l.

It should be noted that a is an odd function of nm . Asshown in Eq. (3), the integration over f is carried outfrom 0 to 2p. Therefore, for exact reconstruction of agenerally complex-valued object function a(r), the FBPPalgorithm requires that the complete data set [i.e., fullknowledge of M(nm , f)] be taken at view angles f in[0, 2p].

3. MINIMAL COMPLETE DATA INDIFFRACTION TOMOGRAPHYWe now examine more closely the coverages of the 2DFourier space of a(r) generated by the segments OA andOB in Fig. 2. The coverages produced by the segmentsOA and OB as f varies from 0 to 3p/2 are shown in Figs.4(a) and 4(b), respectively. It can be observed that eachof these two coverages is an incomplete coverage of the 2DFourier space of a(r). However, one can superimposethese two incomplete coverages in Figs. 4(a) and 4(b) toobtain complete coverage of the 2D Fourier space of a(r),as shown in Fig. 4(c). It should be noted that such a su-perposition of these two incomplete coverages in Figs. 3(a)and 3(b) results in a complete coverage that is doublysampled in certain regions and singly sampled in others.The ability to construct the complete coverage shown in

Fig. 2. The FDP theorem states that the 1D FT of the scattereddata along the line h 5 l is equal to the 2D FT of the object func-tion along a semicircle AOB with a radius of n0 in its Fourierspace.

Fig. 3. As the measurement angle f varies from 0 to 2p, the twosegments OA and OB of the semicircle AOB generate two distinctcoverages, (a) and (b), respectively, of the 2D Fourier space of theobject function.


Fig. 4(c) is significant because it reveals that one can ex-actly reconstruct the object function a(r) from measureddata that correspond to view angles f in [0,3p/2]. There-fore the data that correspond to view angles over therange 3p/2 , f < 2p are, in principle, not required forexact reconstruction of a generally complex-valued objectfunction.

The FBPP algorithm in Eq. (3) generally requires fullknowledge of M(nm , f) in the complete data space W5 @ unmu < n0 , 0 < f < 2p#, as shown in Fig. 5(a), for ex-act reconstruction of a generally complex-valued objectfunction. We will refer to such full knowledge ofM(nm , f) as a complete data set. However, the aboveobservation (see Fig. 4) about the complete coverage of the2D Fourier space of a(r) suggests that it is possible to useknowledge of M(nm , f) only in the subspace M 5 @ unmu< n0 , 0 < f < 3p/2#, as shown in Fig. 5(b), for exact re-construction of the object function.

As shown in Fig. 5(c), one can divide the complete dataspace W into four subspaces, A, B, C, and D, where A5 @ unmu < n0 , 0 < f , 2a 1 p/2#, B 5 @ unmu < n0 , p/21 2a < f , p 1 2a#, C 5 @ unmu < n0 , p 1 2a < f, 3p/2#, and D 5 @ unmu < n0 , 3p/2 < f , 2p#. Be-cause a is a function of nm , the subspaces A, B, and Cspecify nonrectangular regions of the data space, as dis-played in Fig. 5(c). According to the FDP theorem in Eq.(2), one can show11,14 that

M~nm , f! 5 M~2nm , f 1 p 2 2a!. (5)

Using Eq. (5), one can also show that, for any value ofM(nm , f) in subspace A, one can find an identical valueof M(nm , f) in subspace C, suggesting that knowledge ofM(nm , f) in subspace A is redundant to that of M(nm , f)

Fig. 4. As f varies from 0 to 3p/2, the two segments OA and OByield two incomplete coverages (a) and (b), respectively, of the 2DFourier space of the object function. Superimposing the two in-complete coverages in (a) and (b), one obtains (c) a complete cov-erage of the 2D Fourier space of the object function.

in subspace C. Similarly, one can show that knowledge ofM(nm , f) in B is redundant to that of M(nm , f) in sub-space D. In principle, it is true that values of M(nm , f)in regions A and B completely determine the object func-tion. However, with the classical scan geometry, regionsB and C cannot in practice be determined independently ofeach other. We therefore refer to knowledge of M(nm , f)in the rectangular region defined by the union M5 A ø B ø C 5 @ unmu < n0 , 0 < f < 3p/2#, as shown inFig. 5(b), as the minimal complete data set. The aboveobservation about the complete coverage of the 2D Fou-rier space of a(r) [see Fig. 4(c)] suggests that this mini-mal complete data set can be used to reconstruct exactly acomplex-valued object function.

4. MS-FBPP ALGORITHMSUsing a minimum complete data set, one may, in prin-ciple, generate a complete data set by simply copying therelevant values of M(nm , f) in subspace B to subspace D

Fig. 5. (a) Complete data space W that contains data from theview angles in [0, 2p]. (b) The subspace M that contains mini-mal complete data from the view angles in [0, 3p/2]. (c) Thesubspaces A, B, C, and D in the complete data space. Theboundary between subspaces A and B is specified by the equationf 5 p/2 1 2a. The boundary between subspaces B and C isspecified by the equation f 5 p 1 2a. The boundary betweensubspaces C and D is specified by the equation f 5 3p/2.


because, as discussed above, the values of M(nm , f) insubspace B are identical to those of M(nm , f) in subspaceD. Subsequently, one can reconstruct a(r) from this cop-ied complete data set by use of the FBPP algorithm.When the measured data are discrete, however, this ap-proach generally requires the use of a 2D interpolation formapping the measurement samples in subspace B to sub-space D. Such a 2D interpolation will generally compro-mise the accuracy of the reconstruction, especially whenthe measured data are sparsely sampled. To avoid sucha 2D interpolation it is desirable to reconstruct the objectfunction directly from the minimal complete data set.However, because the minimal complete data set containsno samples in subspace D, a direct application of theFBPP algorithm to the minimal complete data set wouldyield a distorted image, as will be shown below.

Although the minimal complete data set contains allthe information necessary for exact reconstruction of theobject function, subspaces A and C contain redundant in-formation that needs to be properly normalized in the re-construction process. We now develop a family of MS-FBPP algorithms that can reconstruct exactly the objectfunction from the appropriately weighted minimal com-plete data set M8(nm , f). This weighted data setM8(nm , f) is defined as

M8~nm , f! 5 w~nm , f!M~nm , f!, (6)

where w(nm , f) can be a function of nm and f, which sat-isfies

w~nm , f! 1 w~2nm , f 1 p 2 2a! 5 1 (7a)

in complete data space W,

w~nm , f! 5 1 (7b)

in subspace B, and

w~nm , f! 5 0 (7c)

in subspace D. It should be noted that, althoughw(nm , f) in subspaces B and D are completely specifiedby Eqs. (7b) and (7c), respectively, the explicit forms ofw(nm , f) in subspaces A and C are unspecified for themoment. In principle, one can choose different w(nm , f)in subspaces A and C as long as these w(nm , f) satisfyEq. (7a). As one can see below, reconstructions obtainedby using different choices of w(nm , f) will respond differ-ently to the effect of discrete sampling. Therefore it isimportant to devise optimal choices of w(nm , f) for mini-mizing the amplification of the effect of discrete sampling.

Mathematically, the MS-FBPP algorithms are ex-pressed as

a ~w !~r, u! 5 Ef50

3p/2Enm52n0

n0 n0

n8unmuM8~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf.

(8)

Because one can have different (an infinite number of)choices of w(nm , f) in subspaces A and C, one can, in ef-fect, devise a family of MS-FBPP algorithms that arespecified by these different choices of w(nm , f).

Because w(nm , f) 5 0 in subspace D, Eq. (8) can be re-written as

a ~w !~r, u! 5 Ef50

2p Enm52n0

n0 n0

n8unmuM8~nm , f!


(9)

Comparing Eqs. (3) and (9), we can observe that the MS-FBPP algorithms have mathematical forms that are iden-tical to that of the FBPP algorithm. Furthermore, in Ap-pendix A we show that a (w)(r) is mathematically identicalto a(r) reconstructed with the FBPP algorithm in Eq. (3);i.e.,

a ~w !~r, u! 5 a~r, u!

51

2E

f50

2p Enm52n0

n0 n0

n8unmuM~nm , f!


(10)

The significance of Eq. (9) is that one can reconstruct theobject function a(r) exactly by simply applying the FBPPalgorithm to the weighted minimal complete data set inEq. (6).

5. RESULTSWe used computer simulation to validate the MS-FBPPalgorithms and the FDP theorem to calculate the discretescattered data from a uniform and complex-valued ellip-tical disk.11 The discrete complete data set comprised128 equally spaced view angles in [0,2p], and the discreteminimal complete data set comprised 96 equal spacedview angles in @0, 3p/2#. The size of the reconstructedimages was 128 3 128 pixels.

Using the FBPP algorithm, we reconstructed imagesfrom simulated complete and minimal complete data sets,which are shown in Figs. 6 and 7, respectively. The im-age reconstructed from the complete data set is observedto be free of any noticeable artifacts or distortions,whereas the image reconstructed from the minimal com-plete data set clearly contains distortion artifacts. Theseresults are consistent with the well-known fact that theFBPP algorithm generally requires a complete data setfor exact reconstruction of a complex-valued object func-tion.

As discussed in Section 4, one can choose differentweighting functions w(nm , f) in subspaces A and C,which thus specify different MS-FBPP algorithms. Westudied two different MS-FBPP algorithms, which arespecified by two different weighting functions w(nm , f)in subspaces A and C. The first MS-FBPP algorithmwas specified by a simple weighting function, which, forunmu < n0 , is given by


w~nm , f! 5 51/2 0 < f < p/2 1 2a

1 p/2 1 2a < f < p 1 2a

1/2 p 1 2a < f < 3p/2

0 3p/2 < f < 2p.

.

(11)

Obviously, the weighting function defined in Eq. (11) sat-isfies Eq. (7). However, it possesses discontinuities atthe boundaries between different subspaces.

Using the MS-FBPP algorithm specified by Eq. (11), wereconstructed an image from the minimum complete dataset, which is displayed in Fig. 8. This image appearssimilar to that in Fig. 6, which was reconstructed by useof the FBPP algorithm from the complete data set. Onclose inspection, however, one can observe that this imagestill contains mild streak artifacts, implying that theweighted (discrete) minimal complete data set possessedinconsistencies that were not entirely canceled out in thereconstruction process. Because the FBPP and the MS-FBPP algorithms are exact only for continuously sampleddata, such inconsistencies can be attributed to the combi-nation of the discontinuities at the subspace boundariesof the weighting function and the discreteness of thesimulated minimum complete set of scattered data.

To diminish such distortion due to the discontinuity ofthe weighting function, one can devise a weighting func-tion that has no such discontinuities between differentsubspaces. From this perspective, we proposed anotherMS-FBPP algorithm, which is specified by the weightingfunction15

Fig. 6. (a) Real and (b) imaginary components of the imagereconstructed from the complete data set by use of the FBPPalgorithm.

w~nm , f!

5 5sin2Fp

4f

~p/4! 2 aG , 0 < f < p/2 1 2a

1, p/2 1 2a < f < p 1 2a

sin2Fp

4~3p/2! 2 f

~p/4! 1 a G , p 1 2a < f < 3p/2

0, 3p/2 < f < 2p.

.

(12)

This weighting function, although it satisfies Eqs. (7),possesses no discontinuity across the boundaries betweendifferent subspaces.

Using this MS-FBPP algorithm, we reconstructed animage from the minimum (discrete) complete data set,which is displayed in Fig. 9. This image contains virtu-ally no artifacts or distortions and is comparable in qual-ity with the image reconstructed by use of the FBPP al-gorithm from the complete data set (see Fig. 6). Thisresult corroborates our assertion that MS-FBPP algo-rithms can reconstruct accurate images from the minimalcomplete data set.

6. DISCUSSIONWe have shown that, in 2D transmission DT that employsthe classical configuration, the minimal complete data setacquired at view angles only in @0, 3p/2# contains the in-formation necessary for exact reconstruction of a gener-ally complex-valued object function. We subsequentlyproposed a family of MS-FBPP algorithms that, in prin-

Fig. 7. (a) Real and (b) imaginary components of the image re-constructed from the minimal complete data set by use of theFBPP algorithm.


ciple, can reconstruct exactly the object function directlyfrom the minimal complete data set.

The mathematical form of the developed MS-FBPP al-gorithms is identical to that of the FBPP algorithm [seeEq. (9)], except for their use of a weighting function tonormalize the redundant information inherent in theminimal complete data set. Consequently, when imple-menting the MS-FBPP algorithms one can utilize com-puter software that was originally developed for theFBPP algorithm.

In this study, although we have developed MS-FBPPalgorithms under the Born approximation, one canreadily develop MS-FBPP algorithms under the Rytovapproximation16 by appropriately redefining the modifieddata14 in Eq. (1). Although the MS-FBPP algorithms aremathematically equivalent to each other as well as to theFBPP algorithm, as demonstrated numerically in Figs. 8and 9, they respond to the effect of discrete sampling dif-ferently. In addition, when the scattered data containstatistical noise, the MS-FBPP algorithms specified bydifferent weighting functions are expected to propagatesuch data noise differently. Identification of the optimalweighting function that can minimize discrete samplingartifacts and statistical variation in the reconstructed im-age remains a topic for future investigation. Further-more, the algorithms that we have presented can be gen-eralized to DT with nonclassical scanning configurationsand to nonlinear DT that employs higher-order Born (orRytov) approximations.

In certain practical situations it may be difficult and in-convenient to acquire measurements over a 2p angularrange about the scattering object. When this is the case,

Fig. 8. (a) Real and (b) imaginary components of the image re-constructed from the minimal complete data set by use of theMS-FBPP algorithm that is specified by Eq. (11).

the relaxed experimental requirements of the MS-FBPPalgorithms will be particularly desirable. Another im-portant and useful feature of the MS-FBPP algorithms istheir ability to decrease the data acquisition time by 25%compared with that for conventional algorithms (assum-ing a constant angular sampling density). This propertywill become significant if biological applications of DT be-come viable, because the artifacts that are due to move-ment in or temporal fluxuations of the scattering objectcan be reduced. As such, the algorithms and insight thatwe have presented may find application in various fieldsincluding tomographic microscopy and medical tomogra-phy.

APPENDIX AAccording to Eq. (8), the MS-FBPP algorithm is given by

a ~w !~r, u! 5 Ef50

3p/2Enm52n0

n0 n0

n8unmuM8~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf,

(A1)

where na 5 sgn(nm)Anm2 2 nm

2, n8 5 An02 2 nm

2, and ais given by Eq. (4). In terms of the contributions from thefour subspaces A, B, C, and D at positive and negative nm ,one can rewrite Eq. (A1) as

a ~w !~r, u! 5 T1 1 T2 , (A2)

where

Fig. 9. (a) Real and (b) imaginary components of the image re-constructed from the minimal complete data set by use of theMS-FBPP algorithm that is specified by Eq. (12).


T1 5 Enm50

n0 S Ef50

p/212a

1 Ef5p/212a

p12a

1 Ef5p12a

3p/2 D3

n0

n8unmuM8~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf, (A3)

T2 5 Enm52n0

n0 S Ef50

p/212a

1 Ef5p/212a

p12a

1 Ef5p12a

3p/2 D3

n0

n8unmuM8~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf. (A4)

Changing nm to 2nm in Eq. (A4) and taking note of Eq.(4), one obtains

T2 5 Enm50

n0 S Ef50

p/222a

1 Ef5p/222a

p22a

1 Ef5p22a

3p/2 D3

n0

n8unmuM8~2nm , f!

3 exp@2j2pnar cos~ f 1 a 2 u!#dnmdf. (A5)

Replacing f with f 1 p 2 2a in Eq. (A5) yields

T2 5 Enm50

n0 F Ef5p12a

3/2p

1 Ef53/2p

2p

1 Ef50

p/212aG3

n0

n8unmuM8~2nm , f 1 p 2 2a!


Substituting Eqs. (A3) and (A6) into Eq. (A2) and tak-ing note of Eq. (6) yield

a ~w !~r, u!

5 Enm50

n0 Ef50

p/212a

@w~nm , f! 1 w~2nm , f 1 p 2 2a!#

3n0

n8unmuM~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf

1 Enm50

n0 Ef5p/212a

p12a

@w~nm , f!#n0

n8unmuM~nm , f!


1 Enm50

n0 Ef5p12a

3/2p

@w~nm , f!

1 w~2nm , f 1 p 2 2a!#n0

n8unmuM~nm , f!

3 exp@ j2pvar cos~ f 2 a 2 u!#dnmdf

1 Enm50

n0 Ef53/2p

2p

@w~2nm , f 1 p 2 2a!#

3n0

n8unmuM~nm , f!


According to Eq. (7a), @w(nm , f) 1 w(2nm , f 1 p2 2a)# 5 1 in the first and third terms of Eq. (A7); ac-cording to Eq. (7b), w(nm , f) 5 1 in the second term ofEq. (A7) because f is in subspace B; and, according toEqs. (7a) and (7c), w(2nm , f 1 p 2 2a) 5 1 in thefourth term of Eq. (A7) because f is in subspace D. Sub-stitution of these final values of w(nm , f) into Eqn. (A7)yields

a ~w !~r, u! 5 Ef50

2p Enm50

n0 n0

n8unmuM~nm , f!


(A8)

On the other hand, one can also rewrite the FBPP al-gorithm in Eq. (3) as

2a~r, u! 5 Ef50

2p Enm50

n0 n0

n8unmuM~nm , f!


1 Ef50

2p Enm52n0

0 n0

n8unmuM~nm , f!

3 exp@ j2pnar cos~ f 2 a 2 u!#dnmdf.(A9)

Replacing 2nm with nm and f with f 1 p 2 2a in thesecond term in Eq. (A9) and taking note of Eq. (6), one ob-tains

a~r, u! 5 Ef50

2p Enm50

n0 n0

n8unmuM~nm , f!


(A10)

Comparison of Eqs. (A8) and (A10) indicates that

a ~w !~r, u! 5 a~r, u!. (A11)

Therefore, under the conditions of continuous samplingand absence of data noise, the MS-FBPP algorithms givenin Eq. (8) are mathematically identical to the FBPP algo-rithm proposed previously by Devaney.9

ACKNOWLEDGMENTSThe authors thank Chien-Min Kao for interesting discus-sions. This work was supported in part by National In-stitutes of Health grant R29 CA70449 and the Paul C.Hodges Research Award.

Corresponding author Xiaochuan Pan can be reachedat the address on the title page or by phone, 773-702-1293; fax, 773-702-5986; or e-mail, [email protected].

REFERENCES1. M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K.

Barrett, B. A. Apivey, and D. A. Palmer, ‘‘A new consider-ation of diffraction computed tomography for breast imag-ing,’’ Acoust. Imaging 21, 379–390 (1995).

2. L. Gelius, I. Johansen, N. Sponheim, and J. Stamnes, ‘‘Dif-fraction tomography applications in medicine and seis-


mics,’’ in NATO Advanced Workshop on Inverse Problems inScattering and Imaging (Hilger, London, 1991).

3. R. Mueller, M. Kaveh, and G. Wade, ‘‘Reconstructive tomog-raphy and applications to ultrasonics,’’ Proc. IEEE 67, 567–587 (1979).

4. G. Kino, ‘‘Acoustic imaging for nondestructive evaluation,’’Proc. IEEE 67, 510–525 (1979).

5. A. Devaney, ‘‘Geophysical diffraction tomography,’’ IEEETrans. Geosci. Remote Sens. IGR-22, 3–13 (1984).

6. E. Robinson, ‘‘Image reconstruction in exploration geophys-ics,’’ IEEE Trans. Sonics Ultrason. SU-31, 259–270 (1984).

7. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978).

8. W. C. Chew, Waves and Fields in Inhomogeneous Media(IEEE Press, New York, 1994).

9. A. Devaney, ‘‘A filtered backpropagation algorithm for dif-fraction tomography,’’ Ultrason. Imaging 4, 336–350(1982).

10. M. Kaveh, M. Soumekh, and J. Greenleaf, ‘‘Signal process-ing for diffraction tomography,’’ IEEE Trans. Sonics Ultra-son. SU-31, 230–239 (1984).

11. S. Pan and A. Kak, ‘‘A computational study of reconstruc-tion algorithms for diffraction tomography: interpolationversus filtered backpropagation,’’ IEEE Trans. Acoust.Speech Signal Process. ASSP-31, 1262–1275 (1983).

12. A. Devaney, ‘‘The limited-view problem in diffraction to-mography,’’ Inverse Probl. 5, 501–521 (1989).

13. A. Devaney, ‘‘Reconstructive tomography with diffractingwavefields,’’ Inverse Probl. 2, 161–183 (1986).

14. X. Pan, ‘‘A unified reconstruction theory for diffraction to-mography with considerations of noise control,’’ J. Opt. Soc.Am. A 15, 2312–2326 (1998).

15. D. Parker, ‘‘Optimal short scan convolution reconstructionfor fanbeam CT,’’ Med. Phys. 9, 254–257 (1982).

16. G. Heidbreder, ‘‘Multiple scattering and the method of Ry-tov,’’ J. Opt. Soc. Am. 57, 1477–1479 (1967).

Documents

Minimal-scan filtered backpropagation algorithms for diffraction tomography