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Practical aspects of implementing three-dimensionaltomography inversion for volumetric

flame imaging

Weiwei Cai,1 Xuesong Li,2 and Lin Ma3,*1Virginia Tech, Blacksburg, Virginia 24060, USA

2Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA3Laser Diagnostic Lab, Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg,

Virginia 24061, USA

*Corresponding author: [email protected]

Received 19 September 2013; accepted 20 October 2013;posted 23 October 2013 (Doc. ID 197784); published 18 November 2013

Instantaneous three-dimensional (3D) measurements have been long desired to resolve the spatial struc-tures of turbulent flows and flame. Previous efforts have demonstrated tomography as a promisingtechnique to enable such measurements. To facilitate the practical application, this work investigatedfour practical aspects for implementing 3D tomographic under the context of volumetric combustiondiagnostics. Both numerical simulations and controlled experiments were performed to study: (1) thetermination criteria of the inversion algorithm; (2) the effects of regularization and the determinationof the optimal regularization factor; (3) the effects of a number of views; and (4) the impact of theresolution of the projection measurements. The results obtained have illustrated the effects of thesepractical aspects on the accuracy and spatial resolution of volumetric tomography. Furthermore, all theseaspects are related to the complexity and implementing cost (both hardware cost and computationalcost). Therefore, the results obtained in this work are expected to be valuable for the design andimplementation of practical 3D diagnostics. © 2013 Optical Society of AmericaOCIS codes: (280.1740) Combustion diagnostics; (100.6890) Three-dimensional image processing;

(120.0280) Remote sensing and sensors.http://dx.doi.org/10.1364/AO.52.008106

1. Introduction

Three-dimensional (3D) measurements have beenlong desired for the study of many thermal-fluidtopics to resolve the 3D spatial structures inherentin turbulent flows [1,2]. Furthermore, instantaneousmeasurements with proper temporal resolution areof particular importance due to the dynamic andtransient nature of turbulent flows [1,2]. Possibleoptions that can potentially meet both the spatialand temporal requirements seem to be very limited

and most of the existing efforts can be broadlydivided into two categories [3,4]. The first categoryof techniques obtains 3D measurements by rapidlyscanning a planar imaging technique such as Miescattering [5] or laser induced incandescence [6].The second category of techniques obtains 3Dmeasurements volumetrically by performing a 3Dtomography [7–10]. A comprehensive survey andcomparison of these possible options is beyond thescope of this current paper, and an independentreview of this topic would be of significantimportance and value for optical diagnosticians. Thiscurrent work therefore directly focuses on the tomog-raphy approach. More specifically, the current work

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8106 APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013

http://dx.doi.org/10.1364/AO.52.008106

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Copyright by the Optical Society of America. Weiwei Cai, Xuesong Li, and Lin Ma, "Practical aspects of implementing three-dimensional tomography inversion for volumetric flame imaging," Appl. Opt. 52, 8106-8116 (2013); doi: 10.1364/ao.52.008106

examines several practical aspects of implementing3D tomographic inversions under the context ofcombustion measurements based on previous re-sults. Existing tomography inversion methods arepredominately developed by and for the medicalimaging community [11]. Consequently, directapplication of these methods to flow and combustionimaging often results in nonoptimal performance dueto both the fundamental and practical differencesbetween flow/flame and medical imaging. This workparticularly discusses the following four practicalaspects of 3D tomographic inversion under thecontext of volumetric combustion measurements.

First, this paper discusses the termination criteriaof inversion methods and their effects on the accu-racy and efficiency of tomographic reconstruction.An effective termination criterion is an importantaspect of any inversion method, because it directlyaffects the computational cost and the reconstructionaccuracy. An ideal criterion should be one thatguarantees convergence and terminates immedi-ately when further calculation provides negligibleimprovements. Theories are available to analyzethe termination criteria, but practical issues suchas measurement noise and inconsistent projectiondata restrict the usefulness of the theories and thetermination criteria often need to be determined em-pirically [9,11,12]. This work studied the terminationcriteria using both simulated and experimental data,and demonstrated the effective termination of aninversion method based on the simulated annealing(SA) algorithm.

Second, this paper discusses the effects of regulari-zation on the reconstruction of various flames.Regularization, the incorporation of available apriori information in the inversion, is an effectivetechnique to improve the performance of tomogra-phy, applicable for both medical [13] and combustionimaging [14,15]. However, after studying variousregularization techniques on various flame patterns,the results obtained in this work show that theregularization techniques developed for medicalimaging do not apply effectively for flame measure-ments due to the structural differences betweenmedical targets and flames. For example, the formertypically features sharp edges while the latter (espe-cially a highly turbulent flame) features distributedand irregular patterns.

Third, this paper discusses the effects of number ofviews on reconstruction quality. In contrast to medi-cal imaging where projection data at many views(e.g., more than thousands) are available as inputfor the inversion, projection data in combustionapplications are often available at a very limitednumber of views, both because of the dynamic natureof flames and the practical difficulty of obtainingoptical access [16–18]. Past works have performedboth computational and theoretical studies on the ef-fects of the number of views on the spatial resolutionand fidelity of the reconstruction [9,19]. They alsoexplored possible ways of obtaining more views in

combustion systems using imaging fibers [20]. Thiswork examined the effects of a number of views fortomographic 3D combustion imaging combiningcomputational, theoretical, and experimentalresults.

Fourth, this paper discusses the effects of theresolution of the projections measurements on thespatial resolution and fidelity of the reconstruction.Projection data for 3D tomography are recorded (byCCD cameras for example) with finite spatial resolu-tion. The projection measurements are often binnedto enhance the signal-to-noise ratio in combustionapplications, and such binning degrades the spatialresolution of the projection data. This work there-fore, examined the effects of CCD resolution and bin-ning on 3D tomographic inversion.

The rest of the paper is organized as follows. Sec-tion 2 summarizes the mathematical formulationand experimental arrangement used in this work.Section 3 reports the results obtained in all fouraspects from both numerical simulation and experi-mental studies. Section 4 summarizes the paper anddiscusses our ongoing research.

2. Mathematical Formulation and ExperimentalArrangement

A. Problem Formulation and Inversion Algorithms

The mathematical formulation of 3D tomographyhas been previously detailed elsewhere [10,21],and a brief summary is provided here to facilitatethe discussion. Figure 1 schematically illustratesthe mathematical formulation. As shown, thevolumetric distribution of the target quantity[denoted as F�x; y; z�] is discretized into voxels. Forexample, in this work, the experimental work reliedon the chemiluminescence emission of CH� andF�x; y; z�, hence represents the concentration ofCH� in this case. But as to be seen, the mathematicalformulation is general and applicable to tomographybased on other signal generation processes as well,including chemiluminescence from other flameradicals (or emission spectroscopy in general), Miescattering, or laser-induced fluorescence (LIF). Thus

Fig. 1. Mathematical formulation of the TC problem and exampleflames used to validate the TC technique experimentally.

20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS 8107

we expect the results (at least the simulation results)to be applicable to tomographic imaging based onother signal generation processes. An image systemcollects the signal to be recorded by a CCD array, andthe two-dimensional image formed on the CCD arrayis called a projection (P). The projection formed onthe CCD is given by

P�r; θ;ϕ� �XNx

ix

XNy

iy

XNz

iz

F�xi; yi; zi�

· PSF�xi; yi; zi; r; θ;ϕ�; (1)

where r (distance to the origin), θ (azimuth angle),and ϕ (inclination angle) specify the orientationand location of the projection measurement as shownin Panel a; ix; iy; iz are the indices of the voxel cen-tered at �xi; yi; zi�; Nx, Ny, and Nz are the total num-ber of grids along the x; y; z direction, respectively;and PSF represents the point spread function de-fined as the projection formed by a point source lo-cated at �xi; yi; zi� with unity intensity. Equation (1)essentially calculates the projection as a weightedsummation of the PSF across all voxels with thesought function as the weight. The 3D tomographyproblem can now be formulated as:

Given a set of projections (Ps) measured at variousr; θ, and ϕ, find F�x; y; z�.

As mentioned above, this formulation is generallyapplicable to signals generated from various mecha-nisms including chemiluminescence [10], emission[18], scattering [22,23], and LIF [24]. In the casewhere the signal is angle dependent (like Miescattering), the PSF can be modified to include thescattering phase function and then Eq. (1) remainsvalid. However, note that Eq. (1) is limited to linearsignals, and is not applicable to problems withnonlinear effects such as multiple scattering.

Various algorithms have been developed to solvethe tomographic inversion problem as formulatedabove [11], and our work has implemented severalof these algorithms including the algebraicreconstruction technique (ART) as described in [9],the multiplicative algebraic reconstruction tech-nique (MART) as described in [25], and the orderedsubset expectation maximization (OSEM) algorithmas described in [26]. We have also developed a newalgorithm based on the SA technique [27] to solvethe 3D tomography problem, and our new algorithmtomographic inversion by SA, is code named TISA.The TISA algorithm formulates the tomographyproblem into an optimization problem as shownbelow:

minimize D �Xr;θ;ϕ

�Pm�r; θ;ϕ� − Pc�r; θ;ϕ��2

with respect to F�x; y; z�; (2)

where Pm represents the measured projections at�r; θ;ϕ�; Pc the projection calculated at �r; θ;ϕ� with

a given distribution according to Eq. (1), andD there-fore, the overall difference between the measuredand calculated projection. Equation (2) is then mini-mized by the SA technique. Before comparing thevarious algorithms on the four specific aspects afore-mentioned, we make some general comments basedon our previous results. Combustion imagingpresents a set of unique challenges and also opportu-nities for the development of inversion algorithms.We have performed numerical comparisons of thesealgorithms on a range of phantoms (i.e., targets Fsthat are artificially created) and different signal gen-erating mechanisms, and the results have suggestedreasonable performance from all algorithms and alsolimitations on phantoms featuring highly turbulentand irregular patterns [10,12,14,23,28,29]. The firstchallenge of combustion imaging involves the avail-ability of projection data at a very limited numberof views, ranging from 2 [15,16] to about 50[9,18,20,30,31]. In contrast, other applications (e.g.,medical imaging) have significantly more projections(thousands and more) available. Therefore, thedesired algorithm should converge under limitedprojection data (which are often also noisy). The sec-ond challenge involves the computational costneeded for processing and analyzing flow andcombustion measurements. Tomographic inversionitself is computationally intensive, comparable tocomputational fluid dynamics (CFD) in many ways[32]. The computational cost is further compoundedby the need to take measurements at high speed(multikilohertz) to resolve the temporal dynamicsof turbulent flows and flames. Therefore, the desiredalgorithm should be computationally efficient andalso amiable for large scale parallelization [32–34].Despite these challenges, flows and flames alsopresent some unique opportunities for tomographicimaging, and the ideal algorithm should be able toincorporate these opportunities. For example, unlikemedical targets, flows and flames follow a set of gov-erning equations, presenting an opportunity to incor-porate these governing equations in the inversionprocess to improve the reconstruction accuracy[32,35] and reduce the computational cost [36].

B. Experimental Arrangement

The experimental study in this work was performedusing laboratory flames and multiview arrangementto conduct 3D tomographic chemiluminescence (TC),similar to that discussed in [10]. Chemiluminescenceemitted from CH� radicals in the flame from variousview angles was recorded either sequentially by onecamera (if the flame is stable) or simultaneously bymultiple cameras (if the flame is unstable). Thenbased on the measured projections, a tomographicreconstruction was performed to obtain the 3D flamestructure.

Three of the example flames used in the experi-mental study are shown in Fig. 1: (A) a simplelaminar flame generated by a Bunsen burner,(B) a patterned flame generated by a customized


McKenna burner, and (C) a v-gutter stabilized flame.These flames were either designed to have stable andwell-defined spatial features (such as flames A andB) to quantitatively validate 3D measurements baseon TC, or designed to be turbulent and dynamics(such as flame C) to demonstrate TC’s temporalresolution. For example, flame A had a known coneshape and a flame front thickness of 0.30 mm. FlameB is a stable and disk-like flame with a diameter of∼61 mm and a thickness of ∼1 mm. A honeycombwas place on the burner to create controlled patternson the flames. The honeycomb’s cells are squareswith size of 1.25 × 1.25 mm and certain cells wereblocked to create the desired pattern. Variouspatterns have been created and studied in this work.Flame B shown in Fig. 1 has patterns of a rectangu-lar region with a size of 8.75 × 10 mm, a column ofcells to form a vertical line with 1.25 mm thickness,and two rows of cells to form a horizontal line withand 2.5 mm thickness. After the tomographicreconstruction, these spatial features (such as the lo-cation, thickness, and size of the flame front) wereextracted and compared to the known values toquantify the accuracy of the TC measurements, aswill be detailed in Section 3.

In summary, this section introduces themathematical formulation and experimentalapproach used in this work. Based on these, the nextsection will examine the four specific aspects of 3Dflame imaging quantitatively.

3. Results and Discussion

Both numerical and experimental studies have beenperformed for the discussions in this section. Thenumerical study used various phantoms, withfour of them shown in Fig. 2, and the experimentalstudy was performed as discussed in Section 2.B.The phantoms were designed with the intention tosimulate the phantoms used in this work, so that

the performance of the tomography measurementscan be evaluated both experimentally and numeri-cally. For example, phantom 1 in Fig. 2 was createdto simulate the patterned McKenna burner flamegenerated experimentally, phantom 2 to simulatethe stable Bunsen flame, phantom 3 to simulate aturbulent jet flame, and phantom 4 to simulate aturbulent flame stabilized by a v-gutter. ThoughFig. 2 shows one layer of each phantom, all phantomswere created and used volumetrically in this work.Also note that the same color scale as used here inFig. 2 (red and blue, respectively, correspond to highand low CH� concentration) is applied throughoutthis work.

Before discussing any of the specific aspects, wefirst establish that the inversion algorithms areinsensitive to the initial starting condition. Variousinitial guesses were used to start the algorithmstested, including TISA, ART, MART, and OSEM.All these algorithms were observed to be insensitiveto the initial guess. Figure 3 shows a set of exampleresults where the TISA algorithm was applied tophantom 4 with three different initial guesses ofthe target F: a 3D Gaussian distribution, a 3Duniform distribution, and a 3D random distribution.As Fig. 3 shows, the function value (normalized bythe largest F observed) changed differently duringiterations with different initial guesses, but thereconstruction error (e) all converged to the samevalue. This work defines the following reconstructionerror to quantify the inversion accuracy:

e �P

ix

Piy

Piz jFrec

ix;iy;iz− Fix;iy;iz jP

ix

Piy

Piz jFix;iy;iz j

; (3)

where Frec represents the reconstructed distributionof the target radical. Similar behavior was observedwith all algorithms on both phantoms and experi-mental measurements.

A. Termination Criterion

An effective termination criterion is important forany inversion method because it directly affectsthe computational cost and also the reconstruction

Fig. 2. Phantoms used for numerical simulations. Fig. 3. Insensitivity to initial guess.


accuracy. An ideal termination criterion is one thatguarantees convergence while aborting immediatelywhen further calculation provides negligible im-provements. Unfortunately, there is no universal cri-terion, to our knowledge, that works effectively for allpractical applications where empirical criteria areoften used. As an example, the following terminationcriterion has been used for the ART algorithms [37]:

��XNx

ix

XNy

iy

XNz

iz

Fk�xi; yi; zi� −XNx

ix

XNy

iy

XNz

iz

Fk−1�xi; yi; zi��

≤ ε · β ·XNx

ix

XNy

iy

XNz

iz

Fk�xi; yi; zi�; (4)

where Fk and Fk−1 are the reconstructed F in the kthand �k − 1�th iteration, respectively; ε is a small pos-itive number and was empirically suggested to be inthe range of �10−6; 10−3�; and β is the relaxation factorin the ART algorithm. This criterion essentially ter-minates the ART algorithm when the overall changein F during two consecutive iterations is below asmall proportion of the overall magnitude of the re-constructed F. Figure 4 illustrates the limitation ofthis criterion by simultaneously tracking D and eduring ART iterations. Here D refers to the overalldifference between the measured and calculated pro-jections during the kth iteration, i.e.,

D �Xr;θ;ϕ

jPkm�r; θ;ϕ� − Pk

c �r; θ;ϕ�j2; (5)

where Pkm and Pk

cPm stand for measured and calcu-lated projection obtained by the end of the kthART iteration. As Fig. 4 shows, D decreases mono-tonically with k, which increases monotonically withdecreasing ε because more iterations are needed tofind F that can better match the calculated projec-tions to the measured ones. However, e does notdecrease monotonically with increasing k (and conse-quently not decreasing ε either). In the results shownin Fig. 4, setting ε � 10−3 terminated the inversion

too early, before the minimal e was reached. Incontrast, setting ε � 10−4 terminated the inversiontoo late, after it passed the minimal e and resultedin a less accurate reconstruction after almost4× more computational cost compared to the cri-terion with ε � 10−3. The results obtained in Fig. 4were phantom 1, eight randomly positioned viewsand 5% artificial Gaussian noise in projections.The ART, MART, and OSEM algorithms were testedon various phantoms with different termination cri-teria. The results confirmed the difficulty of design-ing an effective termination criterion. All thealgorithms showed success on some cases, and alsoencountered issues like those illustrated in Fig. 4on other cases.

In comparison, the new TISA algorithms that wedeveloped can be terminated consistently with asimple criterion in all the cases we tested. TheTISA algorithm solves Eq. (2) by the SA algorithm[38]. The SA algorithm is a proven technique for min-imizing complicated functions. The SA algorithmseeks the minimum of a function by simulatingthe way liquids anneal and crystallize, a naturalprocess during which a large number of moleculesfind the state of minimal total energy [27]. The SAalgorithm minimizes D, defined in Eq. (2) also byiterations, and the following criterion was found tobe effective:

jDk−Dk−1j ≤ ε ·Dk; (6)

where Dk and Dk−1 are the difference between the si-mulated and measured projections as defined inEq. (2) during the kth and k − 1th iteration, respec-tively. Similar to Fig. 4, Fig. 5 tracks D (normalizedby D1) and e simultaneously under the same condi-tions as those used in Fig. 4. The results in Fig. 5show that, with Eq. (6), bothD and e decreasedmono-tonically with decreasing ε (i.e., increasing k andcomputation cost) in the TISA algorithm. Resultsobtained in other cases with the TISA algorithmshowed the same trend as seen in Fig. 5.

Fig. 4. Evolution of e and normalized residual illustrating issueswith termination criterion in the ART algorithm.

Fig. 5. Evolution of e and normalized F illustrating the mono-tonic decrease of e in the RHybrid algorithm.


B. Regularization

It has been well recognized that a priori information,if available, can be incorporated in the tomographicinversion via regularization to improve the inversion[12,13,27]. Here we studied the regularization of theART and TISA algorithms (the regularized algo-rithms were code named RART and RTISA, respec-tively). Two types of regularizations were studied:smoothness and total-variation. The smoothnessregularization considers the degree of smoothnessof the sought F in the inversion, as detailed in[12,14]. The total variation (TV) regularization ofthe target function F is defined as [13]

RTV �F� �Xix;iy;iz

��Fix;iy;iz − Fix−1;iy;iz�2 � �Fix;iy;iz − Fix;iy−1;iz�2 � �Fix;iy;iz − Fix;iy;iz−1�2

q: (7)

According to Eq. (5), the TV of F represents the sum-mation of the gradient magnitude of F over all voxels.Inclusion of RTV in the reconstruction has beenshown to preserve the smoothness or the edges ofthe sought F [10,13]. In the RART algorithm, theRTV term was minimized at the end of each ART iter-ation with respect to F, and the updated F was thenused as the input for the next ART iteration. In theRTISA algorithm, the RTV term was simply added tothe difference defined in Eq. (2) to form a newmasterfunction to be minimized, i.e.,

min f � D� γ · RTV; (8)

where f is the new master function and γ the regu-larization parameter to adjust the relative impor-tance between the D and RTV terms. The optimalselection of γ is critical for the application of anyregularization technique, which controls the relativeweights of the a priori information (e.g., smoothnessor TV) and the a posteriori knowledge (i.e., themeasurements) in the tomographic inversion process[27]. This work found the so-called L-curve method,developed for solving ill-posed linear equations[39], to be effective in determining the optimal γfor relatively simple flames.

Before detailing the choice of γ, Fig. 6 first shows aset of results to illustrate the usefulness of regulari-zation (and also its limitations). We applied the ARTalgorithms to various phantoms with and withoutthe TV regularization, and two sets of exampleresults are shown in Fig. 6. These results wereobtained with phantoms 2 and 4 as shown in Fig. 2.Eight simulated projections were used in thesimulations, with 5% Gaussian noise artificiallyadded to the projections to simulate measurementuncertainty. The upper panel of Fig. 6 shows the

Fig. 6. Application of regularization in the TC technique. Projections from eight random views were used with 5% Gaussian noise added(these same conditions were used in the results in Figs. 7 and 8).


reconstruction of phantom 2, which as mentionedearlier, was created to simulate a cone-shaped stablelaminar flame generated by a Bunsen burner. Notethat even though the phantom is axially symmetric,the tomographic reconstruction did not assume sucha priori knowledge. From left to right is the phantomitself, the ART reconstruction, and the RARTreconstruction, respectively. As can been seen, theART reconstruction had artifacts (e.g., the disconti-nuities and cavities in the reconstruction) and theoverall error was e � 5.6%. The application of theTV regularization significantly reduced the overallerror to e � 2.8% and eliminated much of theartifacts seen in the ART reconstruction.

The lower panel of Fig. 6 shows the reconstructionof phantom 4, which was created to simulate aturbulent flame stabilized on a v-gutter. Again fromleft to right is the phantom itself, the ARTreconstruction, and the RART reconstruction,respectively. As can be seen, the ART and RARTreconstructions were almost identical to each other.The overall error of the ART reconstruction wase � 5.6%, and that of the RART reconstructionis e � 5.4%.

Several observations can be made from the resultsshown in Fig. 6, and these observations were validwhen we applied regularization to other algorithms(e.g., the TISA algorithm) and other phantoms. First,with a proper choice of γ (to be discussed immediatelybelow), the application of regularization reduced e forall algorithms on all phantoms tested. Second,however, the reduction was more pronounced onsmooth (or “laminar”) phantoms than on irregular(or “turbulent”) phantoms, because the TV regulari-zation preserves the smoothness and sharp edges. Inthe examples shown in Fig. 6, phantom 2 is smoothand has clear edges and therefore the TV regulariza-tion is effective, but phantom 4 does not feature anyclear edge or smooth distribution, causing theineffectiveness of the TV regularization as reflectedin Fig. 6. It is an important research topic to find aregularization that can work effectively on turbulent

targets, and we are exploring the incorporation ofgoverning equations as regularization in our ongoingwork.

Figures 7 and 8 provide more insights into resultsshown in Fig. 6, and also illustrate the selection of γusing the L-curvemethod. In practice, the soughtF isunknown, and therefore e is not available to guidethe selection of γ. The L-curve method recognizes thisissue and therefore relies on quantities that can bepractically obtained to determine γ: the differencebetween the measured and calculated projections(D) and the regularization term itself (e.g., RTV). Inthe L-curve method, the inversion problem is solvedmultiple times, each time with a different γ, and Dand the regularization term are recorded each time.Panel (a) of Fig. 7 shows a set of results of D and RTVrecorded when the TV regularization was applied tophantom 2 using the RTISAmethod. The plot had anapproximate L shape, and the L-curve method usedthe γ corresponding to the corner of the L curve as theoptimal values. This work calculated the maximumcurvature to determine the corner, and the pointsnear the corner were shown as solid square symbolsin Panel (a) of Fig. 7. Panel (b) of Fig. 7 shows the eobtained under the γs used (because a known phan-tom was used here so e can be calculated). The solidtriangle symbols correspond to the γs near the cornerof the L curve shown in Panel (a), illustrating thatthe minimal e indeed occurred at the γs determinedby the L-curve method. The L curve exemplified inPanel (a) of Fig. 7 is essentially a trade-off curve.When γ is negligibly small (i.e., 1.5 × 10−5), the inver-sion was performed only to minimize D withoutconsidering the regularization, resulting in smallD and large RTV. When γ is exceedingly large (i.e.,15), the inversion was performed only based on theregularization (i.e., to minimize the TV) without con-sidering the measurements, resulting in large D andsmall RTV. The success of the L-curve method in thecase shown in Fig. 7 lies in the existence of a distinctcorner during the transition from small to large γ, asFig. 7. L curve for phantom 2 (a smooth flame).

Fig. 8. Application of regularization to phantom 4 (a turbulentflame).


shown in Panel (a). Such a distinct corner representsan optimal balance between D and R. Any furtherincrease in γ leads to a sharp rise in D, and anydecrease in γ results in negligible change inD. There-fore, the γs near the corner represent a state wherethe maximum “amount” of regularization can beadded in the inversion without affecting the role ofthe measurements.

In contrast, the data shown in Fig. 8 obtained on aturbulent phantom do not exhibit such a distinctcorner. As shown in Panel (a), the transition fromsmall to large γ was gradual in this case, resultingin the failure to identify the optimal γ and explainingthe marginal usefulness of regularization observedin Fig. 6.

Application of the smoothness regularization tothe phantoms showed a similar trend as seen inFigs. 7 and 8. Both the smoothness and TV regulari-zation were effective in improving the inversion onphantoms that are smooth and/or have clear edges,but not effective on turbulent phantoms. The designof an effective regularization technique for turbulentobjects is an important research need.

Lastly, the studies describe in Sections A and Babove were also performed using experimental data,and the same observations were made as those madewith numerical phantoms. The experimental flameswere not as accurately known as the numerical phan-toms. Therefore, the studies involving experimentaldata relied on some characteristic features extractedfrom the flames, and such extraction is bestexplained in Sections C and D below.

C. Number of Views and Resolution of ProjectionMeasurements

The number of views and resolution of the projectionmeasurements are also two important aspects for the

practical implementation of 3D diagnostics. They di-rectly impact the requirement of optical access andcost (both hardware cost and computational cost).Therefore, this section investigates their effects onthe quality of the 3D measurements.

Figure 9 shows the reconstruction using experi-mental data of a flame generated using the setup dis-cussed in Section 2.B. To examine the effects ofnumber of views (N) and resolution of the projectionsmeasurements, the reconstruction was performedunder four different cases by varying the numberof views (N � 4 and 8) and applying binning to theprojections (2 × 2 binning and no binning). Withoutbinning, projections measured by the CCD camera(1376 × 1040 pixels with 6.45 × 6.45 μm pixel size)were directly used in the inversion. The measure-ment domain was discretized into 64 × 64 × 16 voxels(a total of 65,536 voxels). Under these conditions, thePSD described in Eq. (1) at one view angle was about8 GB in size when stored in double precision. The sizeof the PSD is approximately proportional to the num-ber of pixels in the projection measurements. There-fore, with 2 × 2 binning, the size of the PSD reducedto 2 GB per view angle. Such memory requirementand computational cost underline the importanceto carefully design the number of views and resolu-tion of the projections in practice. Figure 9 showsthat (1) eight views resulted in an overall more accu-rate reconstruction than four views and (2) with afixed number of views, reducing the resolution ofthe projections via binning does not significantlydeteriorate the overall quality of the reconstruction.One explanation for the second observation is thatwhile binning reduced the resolution and thusly re-duced the number of measurements available as in-puts for the inversion, it also reduced the uncertaintyin the measurements at the same time.

Figure 10 shows a quantitative analysis of theresults in Fig. 9 by focusing on the vertical columnblocked in the flame. As mentioned in Section 2.B,the blocked column had a width of 1.25 mm, repre-senting the smallest spatial feature created in the

Fig. 9. Layer 1 of the reconstructions from experimentally mea-sured projections.

Fig. 10. Reconstruction of experimental data with and withoutbinning the measured projections.


experiments. Based on the reconstructions shown inFig. 9, we extracted the width of the column bycalculating the gradient of the reconstruction andlocating the sharpest CH� concentration change.Figure 10 shows the reconstructed width of thecolumn under the conditions used in Fig. 9. Note thatthe reconstructed width may vary along the column(i.e., at difference y locations). Therefore, multiplevalues were obtained from each layer and Fig. 10shows the median (the solid symbols) and standarddeviation (the error bars) of these values.

Several observations can be made based on theresults shown in Fig. 10. First, the spatial resolutionof the reconstruction, quantified by the width of theblocked column, improved with increasing number ofviews used. Figure 10 also shows the fit of the resultsto the Fourier slice theorem [40,41], which predictsthe spatial resolution of the reconstruction to beπλ∕N, where λ is a characteristic spatial scale. Asseen, the data were accurately captured by thetheorem. Second, different λ was determined whenbinning was applied to the measured projections,leading to a different resolving power of the tomo-graphic inversion. Therefore, even though the resultsshown in Fig. 9 suggest that binning did not causesignificant degradation to the overall reconstructionquality, the quantitative analysis shown in Fig. 10show that binning does affect the resolving powerof the tomographic inversion. Third, when projectiondata from eight views were used without binning, thetomographic inversion was able to resolve the mini-mum feature in the flame (1.25 mm), demonstratingthe spatial resolution of the 3D diagnostic technique.

Figures 11 and 12 show numerical resultsperformed using phantoms to simulate the experi-ments described above. As can be seen, the same

observations can be made from these numericalresults as those made from the experimental results.Finally, we make two notes before leaving thesediscussions. First, in the numerical simulations, boththe width of the blocked column and e can be calcu-lated due to the precisely known phantoms. Thereconstructed width and its fit showed the sametrend as those seen in Fig. 10 from the experimentaldata. Therefore, here Fig. 12 shows e from thenumerical simulations rather than the width ofthe blocked column. The results in Fig. 12 show thate can be fitted accurately by the Fourier slice theoremtoo, which could be useful for quantifying the inver-sion accuracy of targets with no distinct spatialfeatures. Second, the Fourier slice theorem wasdeveloped for the ART algorithm, and resultsobtained in this work suggested that it also appliesto the TISA algorithm.

Lastly, in practice, the number of views is animportant parameter in the design of tomographiccombustion diagnostics. These observations madehere illustrate the practical factors that should beconsidered in determining the optical number ofviews. These factors include fundamental considera-tions such as the desired spatial resolution and prag-matic factors such as the computational cost andoptical access. The results shown here should providevaluable guidance to the holistic consideration ofthese factors.

4. Summary and Discussion

This paper investigated four practical aspects for im-plementing 3D tomographic inversion under the con-text of volumetric flame imaging. These aspectsinclude: (1) the termination criteria of the inversionalgorithm; (2) the effects of regularization and the de-termination of the optimal regularization factor;(3) the effects of number of views, and (4) the impactof the resolution of the projection measurements.Both numerical simulations and controlled experi-ments were performed to study them. The results ob-tained have shown the difficulties of designing aneffective termination criterion, and suggested that

Fig. 11. Layer 1 of the reconstructions from simulated projec-tions.

Fig. 12. Reconstruction of experimental data with and withoutbinning the simulated projections.


the new TISA algorithm can be terminated effec-tively on all the cases tested. Regularization hasbeen demonstrated to significantly enhance theaccuracy of inverting smooth flames and/or flameswith clear edges. An L-curve method was found tobe able to determine the optimal regularizationparameter. Increasing the number of views andthe resolution of the projections has been shown toimprove both the accuracy and the resolving power,which agreed with theoretical predictions. Theresults obtained have illustrated the effects ofthese practical aspects on the accuracy and spatialresolution of 3D diagnostics based on tomographyinversion. Furthermore, these aspects, for instancethe number of views and the resolution of the projec-tion measurements, are all related to the complexityand implementing cost (both hardware cost and com-putational cost). Therefore, we expect the resultsobtained in this work to facilitate the practicalimplementation of 3D combustion diagnostics. Forexample, in our ongoing work, we are designingtemporally-resolved 3D diagnostics for practicalcombustors, which pose several challenges includingrestricted optical access, large measurement volume,and large data volume. The results discussed in thiswork provide key information to many aspects of thedesign, which aims at obtaining an optimal balanceof spatial resolution, temporal resolution, hardwarecost, and computational cost.

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