Tomography by iterative convolution: empirical study and
application to interferometry
Charles M. Vest and Ivan Prikryl
An algorithm for computer tomography has been developed that is applicable to reconstruction from data
having incomplete projections because an opaque object blocks some of the probing radiation as it passes
through the object field. The algorithm is based on iteration between the object domain and the projection
(Radon transform) domain. Reconstructions are computed during each iteration by the well-known convo-
lution method. Although it is demonstrated that this algorithm does not converge, an empirically justified
criterion for terminating the iteration when the most accurate estimate has been computed is presented.
The algorithm has been studied by using it to reconstruct several different object fields with several different
opaque regions. It also has been used to reconstruct aerodynamic density fields from interferometric data
recorded in wind tunnel tests.
1. Introduction
Computer tomography is the reconstruction of objectdensity fields from measured values of projections, i.e.,line integrals, of the field. This problem arises in theinterpretation of experimental data in a wide variety ofapplications. In the present case our interest is pri-marily in the interpretation of holographic interfero-grams of aerodynamic flows. During the last decadecomputer tomography, which mathematically is basedon the Radon transform, has become a mature tech-nique with extensive literature. The theory, practice,and application of this technique are presented in arecent book by Deans, 1 which contains a large bibliog-raphy, and in other references such as the special issueof the Proceedings of the IEEE on computerized to-mography. 2
According to the theory of the Radon transform, across section of an object field can be reconstructed ifwe know the values of its integrals along all possiblelines through it. In practice, of course, we must dealwith discrete values of these integrals for a finite num-ber of projections. When interferometry is used tostudy the distribution of density in a compressible gasflowing around a test object in a wind tunnel, these data
The authors are with University of Michigan, Department of Me-
chanical Engineering & Applied Mechanics, Ann Arbor, Michigan
are limited in a troublesome way because the test objectblocks many of the optical rays which create the pro-jections. The consequent badly posed problem is saidto have incomplete projections. Tomographic recon-struction from incomplete projections has receivedsurprisingly little attention in the literature, perhapsbecause the problem does not have a unique solution.Nonetheless, development of techniques to produceaccurate estimates of density fields from such incom-plete projections is of practical importance.
To effect a reconstruction from incomplete projec-tions it is necessary to somehow extrapolate the pro-jection data into the region of missing data, e.g., theregion in which light is blocked by the test model. Zienet al.3 carried out such extrapolations using the criterionthat the integral (zero-order moment) of all projectionsof the object field must be equal. This criterion is oneof many conditions which must be satisfied by a func-tion if it is the Radon transform of an object. There alsoare series expansion methods such as that used by Chaand Vest,4 which in theory are probably the best ap-proach because the opaque region is filled in by afunction so that the projections satisfy all the necessarycriteria for a Radon transform. Lewitt and Bates5 havedeveloped computer algorithms, based on series ex-pansions and a modified backprojection method, whichcan effectively reconstruct images from incompleteprojections. In a sense, these are generalizations of themethod of Zien et al.3 All these methods can be cum-bersome in practice, and the series-expansion methodsinvolve fairly complicated programming and can be-come computationally intensive when the object fieldis complicated.
Fig. 2. Representation of the iterative convolution algorithm.
Fig. 1. Object field and its projections.
In this paper we discuss an iterative convolutionmethod in which the opaque region is filled numericallywith an artificial field which evolves during an iterativeprocess. During each iteration a tomographic recon-struction is required. These are performed by a con-volution method. Because convolution methods forreconstruction are simple and require relatively lowstorage and computation, the resulting code is efficientand can be run on a microcomputer. The iterativeconvolution method involves an iteration between theprojection (Radon transform) domain and the objectdomain. Such an approach is suggested by the iterativeFourier transform algorithm proposed by Sato et al.6
for problems of reconstruction from a limited numberof complete projections and by the work of Fienup andothers on the related problem of estimation of objectsfrom measurements of the modulus of their Fouriertransforms. A comparative review of algorithms for thislatter problem is in Ref. 7. The iterative convolutionmethod was first suggested by Braga and Vest8 for casesin which projections are available only over a limitedrange of directions. It was much less successful in thatapplication than in the case of incomplete projectionsdiscussed in this paper. Recently, Medoff et al.9 re-ported the use of iterative convolution algorithms im-plemented on a commercial CT scanner to producecross-sectional x-ray images in a variety of situations inwhich data were missing. They also, presented an op-erator framework within which to approach severaltypes of limited-data image reconstruction problem.
11. Iterative Convolution AlgorithmThe convolution algorithm is one of the most conve-
nient and efficient techniques of computer tomography.Its use requires that each projection be complete; henceit cannot be applied directly when part of the projec-tions is blocked by an opaque object in the object field.Therefore, it is necessary to create artificial data to fillin the missing portions of the projections. The algo-rithm presented here does so through an iterative pro-cess. Because the process is iterative, it also permits
the use of other types of a priori information or con-straints such as positivity.
The iterative convolution algorithm consists of thesix steps below. It is assumed that we are recon-structing an object field f(r,4) which is known to be zeroeverywhere outside a circular region of radius r. Thedata are values of the projections AYO) as shown in Fig.1. The algorithm also is represented in Fig. 2.
(1) Initial estimation of the missing projectiondata.
(2) Reconstruction by the convolution method.(3) Revision in the object domain: The computed
object function from step (2) is set to zero for r > r. Allnegative values of the object field can be replaced byzero if the field is known to be positive everywhere.
(4) Computation of a new projection by line inte-gration through the computed and revised objectfield.
(5) Revision in the projection (Radon transform)domain: Computed projection data are replaced bymeasured data everywhere they are available.
(6) Iteration by returning to step (2) unless somesuitable termination criterion is satisfied.
The Appendix contains an analysis of the conver-gence properties of this procedure. In fact, it is shownthat in general the algorithm does not converge. Em-pirical studies indicate that the technique initiallygenerates sequentially better estimates of the field,reaches a best estimate, and then diverges. The anal-ysis in the Appendix indicates that the rate of initialconvergence depends to a large extent on the spacingbetween sampled values of the projection which are usedduring reconstruction.
Our algorithm differs from that presented by Medoffet al.9 in that we do not assign fixed values to the objectfunction in the region occupied by the test object duringobject-domain revision to define the object functionuniquely. In the applications of interest to us, we couldnot do so without introducing an undesired disconti-nuity of the object function.
In applying the iterative convolution algorithm wemust find a way to choose the number of samplingpoints in each projection, define a technique for initiallyfilling in the projections in the region of missing data,
specify the particular convolution method for recon-struction, and develop an appropriate criterion forterminating the iteration.
A. Selection of the Number of Projection SamplingPoints
We assume that we have N projections spaced atequal intervals over 180°. The data used for computingreconstructions of the object field are to be M equallyspaced samples of each projection (including the regionin which the measured projections are blocked by anopaque object). The spacing AY between data pointsin a given projection and AO, the angle between pro-jections, are
AY= 2r/(M - 1),
AO = w/N.
(1)
(2)
Using the Whitaker-Shannon sampling theorem,Klug and Crowther1 0 showed that if the object functionhas a bandlimit p0, the intervals AY and AO shouldsatisfy the following inequalities:
AY < 1/(2 po);
AO • 7r/(2lrporo - 3).
(3)
(4)
Substituting Eqs. (3) and (4) into Eqs. (1) and (2) weobtain
M - 1 Ž 4rO; (5)
N + 3 > 2rropo. (6)
Finally, combining Eqs. (5) and (6) as equalities, thefollowing relation results
(N + 3)/(M -1) = r/2. (7)
The significance of this relation for our present purposesis discussed below. It should be noted that because theobject function is space-limited, it cannot be bandlim-ited; hence reconstruction errors due to aliasing mayprove troublesome if the object function has substantialFourier components at frequencies beyond po. Ofcourse, this statement holds regardless of the particularreconstruction algorithm.
B. Initialization of Blocked Projection Data
Each projection is incomplete; i.e., part of the data ismissing because optical rays have been blocked by theopaque object. To initiate the iterative process we mustdevelop a set of artificial projection values to fill in theseblank regions. The algorithm we used to generate theseinitial values was developed on the basis of empiricalstudies of the process. These initial values were chosento meet three requirements:
(1) The values represent a smooth continuation ofthe measured projection data into the blocked region.
(2) The maximum absolute value of this artificialdata is not larger than necessary to satisfy requirement(1).
(3) The zero-order moments (integrals) of all pro-jections are equal.
The latter is a requirement for a function to be aRadon transform.
C. Computation of the Object Function
Reconstructions of the object function in step (2) ofthe iterative algorithm were done by the convolutionmethod using the original Ramachandran-Lakshmi-narayanan filter.11 That is, the objection function iscomputed as
(8)f(r,O) = E g[A(r,0)AY,Ok]AO,k
where the convolution integral g is represented by thesummation
g(mAYOk) = {l/ 4 [(mAY O4)]
-- E rI(m + p)A\Y,uk/Pu1//Y7r p odd
and
#(AY) = r sin( - Ok)
For a given r and X,
/ = (f cosOk -a sinOk)/(AY)
and
Ok = k(AO)
for
k = 0,1,2,. N - 1 and 3 = r sin(¢)
a = r cos(o).
(9)
(10)
(11)
(12)
Because the convolution integral is computed at thepoints m(AY),Ok rather than at the points AuAY,Ok re-quired for evaluation of Eq. (8), the following linearinterpolation scheme was used:
g[ktAY,Ok] = (m + 1 - ,)g(mAY,Ok)
+ (-m)g[(m + 1)AY,0k].
D. Computation of Projections
(13)
In step (4) ot each iteration it is necessary to calculateprojections of the current estimate of the object func-tion. We did this by using a 15-point Gaussian quad-rature. This method is suitable because the samplepoints at which the object function is needed can benonuniformly spaced, and because it requires a rela-tively small number of sample points to attain a givenaccuracy. While computing the projections in thismanner, it is assumed that the object field is not dis-continuous. This procedure would have to be alteredif there are discontinuities such as strong shocks in thefield.
E. Termination Criterion
Both the analysis in the Appendix and empirical re-sults indicate that in general the iterative process ini-tially generates a sequence of increasingly accurate re-constructions, reaches a most accurate estimate, thendiverges in a sequence of increasingly inaccurate re-constructions. It is critically important, therefore, toestablish a criterion for terminating the iterations at ornear the most accurate estimate. A large part of the
Fig. 3. One of the object functions used in the empirical study of the iterative convolution algorithm. This function is given by Eq. (14):(a) the function; (b) the function with the segment r < r = 0.6 ro missing.
research which is summarized in this paper was devotedto empirically studying the convergence of the algorithmfor a variety of test object functions and opaque regions.A variety of termination criteria was suggested andtested. The one which proved most reliable is quitesimple:
In the jth iteration we compute the jth object func-tion and rmsoj, which is the rms error of the estimatedobject function on the boundary r = r. This can becomputed because the actual object function is zeroeverywhere on this boundary. In the jth iteration wealso compute the ( + 1)th projection function andrmspj 1, which is the rms error of this projection func-tion over the region where measured projections areavailable; rmsp = 0 because it is the rms error of theinitialized projection function. (We found that theminimum values of rmso and rmsp generally occurredin different iterations.)
Let rmsp be found to be minimum in the jth iteration;i.e., rmspj+ 1 is a minimum. If j 1, we accept theobject function computed in the jth iteration if rmspJ+2> rmspj or in the ( + 1)th iteration if rmspj+2 < rmspj.If the minimum rmsp occurs at the first iteration, j =1, we accept the object function computed in the firstiteration if rmso2 rmso1, or in the second iteration, j= 2, if rmso2 < rmso1 .
Ill. Empirical Study of the AlgorithmThe characteristics and procedures discussed in the
preceding section were based on an extensive numericalstudy of the algorithm. This study was carried out byusing the algorithm to reconstruct analytically definedobject functions from simulated data generated bycomputing line integrals through them. Three objectfunctions were used in the study. In all cases the objectfunction was zero everywhere outside a circle of radius
ro. In each numerical experiment an opaque object ofcircular or triangular cross section was placed in theobject field to block a significant region of the projec-tions. Reconstructions were then made while someparameters or procedures were varied, and the averageand maximum errors in the reconstructed fields werecomputed.
To present typical results of this study, consider theobject function shown in Fig. 3. Figure 3(a) is a repre-sentation of the object function
(14)Figures 4-6 indicate the results of reconstructions of thisobject function when a circular opaque object with ra-dius r = 0.6ro is present in the center of the object do-main [see Fig. 3(b)]. In these figures, reconstructionerror is defined as the difference between the computedand actual values of the object function at a point di-vided by the maximum value of the function that occursin the object domain. Average errors are defined as theaverage of these errors over a grid of 648 points withinthe circle of radius r. Note that the function to be re-constructed is relatively complicated, and that only 40%of each projection is available.
Figure 4 shows the dependence of the maximum(broken lines) and average (solid lines) percent errorsin the reconstruction of this object function on thenumber of iterations. The two sets of curves in thisfigure correspond to two different schemes for initial-izing the projections in the region where data are miss-ing. The scheme denoted A in this figure is that de-scribed in the preceding section. As in this case, this
Fig. 4. Dependence of reconstruction errorfor two different schemes for initializing
data.
70
60
50
40
30
20
10
8 10
on number of iterationsthe missing projection
8-,
M = 7
N--
15\
\ \I
\ \I
\ \ \ -
-3\\\ \ \\
X 7\ ,n_
Avg.
. . . . I . I I I I I I0 I 4 6l. 1 1 1 4 .0 2 4 6 a 10 1 2 14
NUMBER OF ITERATIONS
Fig. 6. Dependence of reconstruction error on amount of data. The
number of data points M is fixed for each curve, and the number ofprojections N is matched to M using Eq. (7).
40
30
20
10
I N = 221 1 30
14 1 1
I I| J Max.
I I
_ I I , 38
22
14 30
.46
l I I l
0 2 4 6 8 10
NUMBER OF ITERATIONS
Fig. 5. Dependence of reconstruction error on number of iterations
and number of projections. The number of data points, M = 27, is
fixed.
initialization scheme generally gave the most accuratereconstructions after the smallest number of iterations.It was, therefore, used in subsequent work. The otherset of curves in this figure corresponds to a differentinitialization scheme in which stationarity of the zero-order moment of the projections is not imposed. Thepoints T in each curve indicate the iteration which thetermination criteria would select as the final result.Note that in each case it does indeed select an iterationwhich is either the most accurate or very close to it.
In the experiments described by Fig. 4 the numberof projections N and the number of data points M ineach projection were held constant. Figure 5 shows theresults of a study in which M = 27, but the number ofprojections N was varied. N appears as a parameter foreach curve. For this case, Eq. (7) indicates that N = 38would be a reasonable amount of data to record. Theresults in Fig. 5 indeed indicate that increasing N ini-tially increases the accuracy of reconstruction, but in-creases beyond N = 38 do not improve the accuracy.
Figure 6 indicates the effect of the total amount ofdata on the accuracy of reconstruction. The numberof data M in each projection is constant for each of thecurves of error as a function of the number of iterations.In each case the number of projections is matched to Mby applying Eq. (7). The circular opaque object usedin the study summarized by Fig. 6 had radius r, = 0.7ro, which is slightly larger than that in the precedingfigures.
A brief study was made to compare the accuracy ofreconstruction obtained using the iterative convolutionmethod with that obtained using a series expansionmethod. In particular, reconstructions from the samedata were carried out by the iterative convolutionmethod and by a technique in which the object field isexpanded in a Fourier series in the azimuthal directionand in a series of Legendre polynomials in the radialdirection. This series expansion method is describedin detail in Ref. 4. Table I contains the results of thereconstruction of the object function defined by Eq. (14)with different opaque objects. Average and maximumerrors are presented for N' = M' = 6 and N' = M' = 12,
Table 1. Comparison of Errors Among Reconstruction Obtained byIterative Convolution and Series Expansion Methods. a
Series expansion IterativeOpaque object N' = M' = 6 N' = M' = 12 convolution
Circle (r = 0.3ro) 47.5 25.1 19.510.7 6.1 1.8
Circle (r = 0.6ro) 60.0 100.0 27.511.0 16.9 3.7
Small triangle 43.2 52.4 16.110.8 8.8 2.9
Large triangle 50.1 54.9 22.610.5 8.4 3.8
a The object function is given by Eq. (14). In each entry the uppernumber is the maximum error in the field, and the lower number isthe average error.
where N' and M' are the numbers of Fourier harmonicsand Legendre polynomials in the series representationof the object field, respectively. In this example, theiterative convolution method was considerably moreaccurate; however, when the object function was of asimpler form, the series expansion method was ofcomparable, or sometimes higher, accuracy.
Additional numerical results are in Ref. 12.
IV. Application to Interferometric Wind Tunnel DataBoth the iterative convolution method and the series
expansion method described in the previous sectionwere used to reconstruct aerodynamic density distri-butions from data obtained by holographic interfer-ometry. The data were recorded by George Lee ofNASA Ames Research Center in preliminary tests of aholographically instrumented wind tunnel. The testobject was a circular cylinder of 25.4-mm radius with ahemispherical nose of the same diameter. We presenthere two typical results.
Figure 7 shows the radial distribution of densitychange in a plane 12.7 mm behind the stagnation point(tip of the nose) when the mainstream flow has a Machnumber of 0.6. In this case the cylinder is at zero angleof attack, so the flow is radially symmetric. Therefore,the data can be inverted by Abel transformation. Thesolid curve in Fig. 7 is the reconstruction by the iterativeconvolution method, and the dashed curve was obtainedby inverse Abel transformation of the data. The twotechniques give nearly identical results.
Figure 8 contains reconstructions of density in theasymmetric flow field near the cylinder when it is at a5.5° angle of attack in a flow of Mach number 0.6. Theupper plots in this figure are the density distribution ina plane perpendicular to the flow and tangent to thenose of the test object. The lower curves are the densitydistribution in planes perpendicular to the flow 1.27 and2.54 nm behind the stagnation point. The solid curveswere computed using the iterative convolution method,and the dashed curves were computed using the seriesexpansion method described above. Data leading tothis figure were recorded by dual-plate holographicinterferometry, with interferograms being recorded over180° in intervals of 100. It should be noted that therewas considerable experimental error in these data. This
-O . 08 _
-0. 12
- Iterative Convoltion
- nverse Abel Transfos-
-0.16 _
-0.202 3 4 N 6
RADIAL POSITION (c)
Fig. 7. Reconstructed distribution of axisymmetric density changein the plane 12.7 mm behind the tip of a cylindrical projectile in a
Mach 0.6 airflow at zero angle of attack.
-o -8 - 4 -2 0 2 4 6 8 10
RADIAL POSITION {c)
Fig. 8. Reconstructed asymmetric distribution of density differencein a Mach 0.6 airflow about a projectile at 5.5° angle of attack.
figure discloses a typical phenomenon: the series ex-pansion method tends to give rise to considerable os-cillation in the radial direction relative to reconstruc-tions by the iterative convolution method. On the otherhand, the iterative convolution method leads to some-what greater oscillations in the azimuthal direction thandoes the series expansion method. In some cases theseries expansion method failed to provide a recon-struction, whereas the iterative convolution methoddid.
In summary, the iterative convolution method ap-peared to provide reasonable reconstructions thatgenerally appeared to be superior to series-expansionreconstructions made from identical interferometricdata. A small number of these reconstructions arepresented here as illustrative examples, but it shouldbe noted that the data were very noisy, and considerableuncertainty existed regarding the absolute interferencefringe order in some views.
This research was sponsored by NASA under grantNAG 2-118. The authors would like to thank George
Lee and Washington Braga for their advice and assis-tance.
Appendix: Convergence Analysis
Let En be the mean square difference between thetrue object function and the computed object functionat the end of the nth iteration. By the true objectfunction f(x,y) we mean one of the functions definedover the entire domain, including the region occupiedby an opaque object, that have at most discontinuitiesof the first kind and whose Radon transforms exist andequal the projection data everywhere these data areavailable.
Let F(Q,ij) and F(p,O) be the Fourier transforms of thetrue object function expressed in Cartesian and polarcoordinates, respectively. Let h(x,y) be the inverseRadon transform of h (Y,6), which is the revised formof the projection function ( Y,6) computed in step (5)of the algorithm. Furthermore let H(Q,,q) or equiva-lently H(p,O) be the Fourier transforms of h(x,y). Thesubscript n refers to values obtained during the nth it-eration.
Using the notation described above we carried out thefollowing analysis of the change of the mean squareerror of reconstruction from one iteration to the next.In what follows we use the well-known Parseval relationbetween 2-D functions and their Fourier transforms anda Parseval relation that can be derived between the 1-DFourier transform of a function and the 1-D projectionof the same function. 3
The mean square error is
En = f sf jf(X,Y) -f(X,Y) 2dxdy
= f f IF(,?) -FnQ(,,7)I2dxdy
- f0 S. IF(p,O) Fn (po)12
1pIdpdO
= a 3 S IF(pO) -Fn(p,O)I2 dpdO
= an 50 f I (Yo)- in (YO) I 2d Yd 0
2 an 5 X IY ) - fin(YO) 2dYdO
= an fJ f I F(pO) -Hn (po) I2dpdO
an J 5 IF(pO) - Hn(p,o) 2IpIdpdO
an fJ X IF( a) - HnQ( ) 2 dtdn
=n f- fJ I f(x,y) - h(XY)I12dxdy
2 J J f If(xy) - fn+i(x,y)I 2dxdy
an En+1 , (Al)
where an and bn are formally defined as
Ex X I F(p a- Fn(p~o 0)21p IdpdO
: S: I F(p,) -F(p,0) 2 dpdO
S - F(pO) - 121p|dpd
bN J IF(pO)-Hn(pO)12 dpdOs:.iIF(p,O) -
(A2)
(A3)
The iterative convolution procedure converges to atrue object function if the mean square error decreasesas the iterations proceed. According to Eq. (Al) themethod will converge if an 2 bn in all iterations. Ac-cording to Eqs. (A2) and (A3), this will occur if theFourier transform of f(x,y) is closer to the Fouriertransform of hn (x ,y) than to the Fourier transform offn(xy), particularly at higher frequencies. Because ofthe nature of the projection domain revision, hn (YO)has a discontinuity of the first kind at the boundaryseparating the regions of measured and computedprojections. This causes increasing importance of thehigher frequency components as the iterations proceed;however, in practice we discard transform componentsbeyond some frequency po. This may be one reason forthe ultimate divergence of the reconstructions whichwere observed in numerical studies if the inequalitiesare not strong enough. If the spacing in the grid ofsample points in the projection domain is increased, theapproximation of the measured part of the projectionfunction in any iteration can be improved. But this willexacerbate the problem with high frequency compo-nents by increasing the slope of hn(Y,O) between theregions of measured and computed data. The iterationat which divergence begins presumably is determinedby the balance between these two effects.
Essentially the same conclusion regarding the con-vergence behavior can be reached by an analysis basedon a Parseval relation between the object function andits Radon transform derived by Ludwig13:
3 I f(x,y) 2 dxdy= So X I1-1[IpI"2 (p,0)]I2 dpd0.
(A4)
Here ApA) represents the 1-D Fourier transform ofA(Y,), and i-l denotes an inverse l-D Fourier trans-
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