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Joint System and Algorithm Design forComputationally Efficient Fan Beam CodedAperture X-ray Coherent Scatter Imaging

Ikenna Odinaka, Joseph A. O’Sullivan, David G. Politte, Kenneth P. MacCabe, Yan Kaganovsky,Joel A. Greenberg, Manu Lakshmanan, Kalyani Krishnamurthy, Anuj Kapadia, Lawrence Carin,

and David J. Brady

Abstract—In x-ray coherent scatter tomography, tomographicmeasurements of the forward scatter distribution are used toinfer scatter densities within a volume. A radiopaque 2D patternplaced between the object and the detector array enables thedisambiguation between different scatter events. The use of a fanbeam source illumination to speed up data acquisition relativeto a pencil beam presents computational challenges. To facilitatethe use of iterative algorithms based on a penalized Poissonlog-likelihood function, efficient computational implementation of theforward and backward models are needed. Our proposed imple-mentation exploits physical symmetries and structural propertiesof the system and suggests a joint system-algorithm design,where the system design choices are influenced by computationalconsiderations, and in turn lead to reduced reconstructiontime.Computational-time speedups of approximately 146 and 32 areachieved in the computation of the forward and backwardmodels, respectively. Results validating the forward model andreconstruction algorithm are presented on simulated analytic andMonte Carlo data.

I. I NTRODUCTION

X-ray coherent scatter imaging involves the reconstructionof an object’s volumetric scatter density from tomographicmeasurements of the scattered x-ray data. The forward scatterdistribution from an object is modeled as a superposition ofthe intensities of the scatter from all object points. The scatterfrom each object point occurs due to photons illuminating theobject from all incident angles at different energy-dependent

This paper has been submitted to IEEE Transactions on ComputationalImaging for consideration.

Ikenna Odinaka (ikenna.odinaka@duke.edu), Yan Kaganovsky,Joel A. Greenberg, Lawrence Carin, and David J. Brady are with theDepartment of Electrical and Computer Engineering, Duke University,Durham, NC 27708.

Joseph A. O’Sullivan is with the Department of Electrical and SystemsEngineering, Washington University in St. Louis, Saint Louis, MO 63130

David G. Politte is with the Mallinckrodt Institute of Radiology, WashingtonUniversity in St. Louis, Saint Louis, MO 63110

Kenneth P. MacCabe is with Physical Numerics, Raleigh, NC 27675Manu Lakshmanan is with the Department of Biomedical Engineering,

Duke University, Durham, NC 27708Kalyani Krishnamurthy is with Pendar Medical LLC, Cambridge, MA

02138Anuj Kapadia is with the Department of Radiology, Duke University

Medical Center, Durham, NC 27710K.P. MacCabe and K. Krishnamurthy were with the Department of Electri-

cal and Computer Engineering, Duke University, Durham, NC 27708, whenthis work was done.

This work has been supported by the U.S. Department of HomelandSecurity Science and Technology Directorate Contract Number: HSHQDC-11-C-00083

intensities, as permitted by the source configuration. Thedetailed forward model is given below, but it is immediatelyapparent that the central challenge in x-ray coherent scatterimaging is the separation of the scatter back into a volumetricscatter density.

One approach to separate the scatter from the multiplexedmeasurements is to obtain different views of the object byusing a rotating gantry [1]. An alternative and novel approachis the use of a coded aperture between the object and thedetector [2], [3], which prevents the need for a rotatinggantry. Two important design elements in coded aperture x-ray coherent scatter imaging are a primary mask betweenthe source and the object, and a coded aperture (secondarymask) between the object and the detector array. Whereas theprimary mask serves to shape the incident beam, the secondarymask is responsible for blocking the transmitted x-rays anddisambiguating scatter angles.

Previous work in coded aperture x-ray coherent scatterimaging has shown the success of a pencil beam system(primary mask with a single small hole) [4]. We scale up toa fan beam system (primary mask with a slit) to acceleratemeasurement acquisition (see Fig. 1(a)). For this, new codesand algorithms are needed for efficient data collection andinference. Bradyet al. [2] give a detailed description of thedesign of new codes for a fan beam system. Here, we focus onefficient data inversion. MacCabeet al. [5] analyzed the fanbeam system with a 2D energy-integrating detector array anda 2D secondary mask (see Fig. 1(b)) from the standpoint ofrecovering the object’s spatial and angular distributions, underthe assumption that the object may be factorized in space andangle. Moreover, their system matrix was small enough to bestored in main memory on a standard computer. In this paper,we focus on general spatial and spectral (momentum transfer)distributions and we consider a much larger system, where thestorage of the system matrix may not be feasible. Unlike theangular distribution, the spectral distribution can be used as aunique signature for identifying materials [6].

Coded aperture design is based upon a combination ofobjectives. As many photons as possible should be allowed tohit the detector to increase the signal-to-noise ratio. However, asmaller number of holes in an aperture may allow for a moreaccurate determination of object location and scatter angle.For example, if the aperture has a single hole, the illuminateddetector points and the aperture hole define scattered rays

2

Beam-Stop

Object

Scatter

Point

Off-Center

Ray

Central

Ray

Secondary

Mask

Detector

Array

Fan

Beam

Primary

Mask

X-r

Sou

(0, 0, 0)

(a) Fan beam geometry schematic

(b) Secondary mask image

x

X-ray Source

y

Object Pt 1

Object Pt 2

Secondary

Mask

Detector

(c) Disambiguating scatter

(d) Translation symmetry

Fig. 1: In (a), the focal spot of the source is located at the origin of the coordinate system, locations in the object volume anddetector array are given asr andr′, respectively. The scatter angle and scatter vector are denoted byθ ands, respectively.ris a unit vector along the direction from the source to an object point, no is a unit normal vector to the face of the rectangularobject slice illuminated by the fan beam, andnd is a unit normal vector to the front of the detector array. Therectangularobject slice is exaggerated to illustrate the unit normal vector. The secondary mask in the fan beam geometry schematic isisolated in (b). (c) shows an example of the secondary mask differentiating scatter from different object locations to the samedetector location. The scattered photons from object point1 are blocked by the secondary mask, while those from object point2 are transmitted. A single slice through the secondary maskand the detector array, parallel to the y-axis, is depicted in (c).(d) shows the effect of translation symmetry on object pixelwidth selection.v is a vector corresponding to the translation ofan object pixel by one pixel width along they direction.ρy is the translation symmetry step size.

3

that can be traced back to unique object locations. Designsof coded apertures range from intuitive to analytical. On theanalytical side, there is a desire for a uniform sensitivitytoa wide range of scatter locations and scatter angles. Suchanalytical designs are described elsewhere [2]. There is also acomputational imaging aspect of the secondary mask design.The secondary mask and the inference algorithm may bedesigned jointly, with a goal of optimizing a measure of thereconstruction performance. In this view, the ad hoc intuitivearguments for secondary mask selection play a subsidiary roleto quantitative measures of performance. While this is ourultimate goal, this paper focuses on inference algorithm de-velopment using a penalized maximum likelihood estimationapproach.

The inference algorithm relies critically on a computationalrepresentation of the forward model for the data. Our forwardmodel has been derived analytically and tested using analyticand Monte Carlo simulations. We propose an efficient repre-sentation of the forward model based upon physical symme-tries and structural properties (e.g. smoothness considerations)of the system, described in detail below. The physical symme-tries assumptions are readily verifiable in analytical and MonteCarlo models, but less so in experiments. The ultimate benefitof the model and approach described here will be determinedin part by the calibration process for verifying alignment andsymmetry. Even if the symmetry assumptions break downin practice, other aspects of the efficient computation of theforward model such as smoothness consideration may bevalid. The computational implementation also involves a trade-off between on-line and off-line computations which can beoptimized for a target computer.

There is a connection between the computational time ofthe iterative algorithm and judicious choices for the samplingin the image domain. If the voxel size is chosen to be aninteger multiple of the spacing of pixels on the detector,many geometry computations can be reused. A commontechnique for accelerating the convergence of an iterativealgorithm involves partitioning the detector measurements andusing each partition (subset) in turn to update the estimatedimage parameters. This technique is called ordered subsets(OS). The symmetries available in the ideal case motivate aparticular choice of subsets in an ordered subset expectation-maximization OSEM-type algorithm.

Reconstruction results for simulated data based on the ana-lytic forward model and Monte Carlo simulation demonstratethe algorithmic performance for a particular choice of thesecondary mask. Computational-time speedups of about 146and 32 are obtained for the forward and backward models,respectively. In addition, the spatial distribution and the mo-mentum transfer profiles of the simulated object are recoveredquickly and fairly accurately, with only a few ordered subsetEM-type iterations required.

The remainder of this paper is organized as follows: InSection II, we describe the forward model. Sections IIIand IV describe efficient computations of the forward andbackward models, respectively. The measurement model andimage recovery problem are described in Section V, whileSection VI develops the regularized ordered subset EM-type

reconstruction algorithm. The results of applying the orderedsubset EM-type algorithm to analytically and Monte Carlosimulated data are presented in Section VII, while concludingremarks are given in Section VIII. Appendices A and Bdescribe, in detail, the derivation of the forward model andthe image recovery algorithm, respectively.

II. FORWARD MODEL

In the fan beam coded aperture coherent scatter model, thex-ray source transmits photons in a fan within a plane. Asshown in Fig. 1(a), each photon illuminates and interacts withthe object. The photons are either absorbed, transmitted, orscattered by the object. The transmitted and scattered photonspropagate from the object to the detector array and are eitherblocked or transmitted by the intervening secondary mask.

As shown in Fig. 1(a), the focal spot of the source is locatedat the origin of the coordinate system and the fan beam residesin the z = 0 plane. Object scatter locations are indexed by(x, y) coordinates and corresponding object pointr = [x, y, 0],with positive x-coordinate pointing from the source throughthe object to the secondary mask and the detector. Since thesource is located at the origin,r also serves as the source toobject point vector, with magnituder and unit vector givenas r. The detector plane is perpendicular to the central rayof the x-ray fan which goes along the x-axis. A point on thedetector isr′ = [Xd, y

′, z′]. The unit normal vector to thedetector plane isnd as shown in Fig. 1(a). The object pointand the detector point determine the scatter vectors = r′ − r,with magnitudes and unit vectors. The scatter vector andthe source to object point vector determine the scatter angleθ, shown in Fig. 1(a).

The flux measured by the energy-integrating detector arrayat locationr′ is given as

g(r′) =∫ ∫

H (r′, r, q) f(r, q)dr dq, (1)

where

H (r′, r, q) = CGso(r)God(s)T (r, r′)∆θS(θ, q) (2)

Gso(r) =x

(x2 + y2)1.5

God(s) =|nd · s|s2

.

S(θ, q) =q(

1 + cos2 θ)

cos(

θ2

)

sin2(

θ2

) Φ

(

hcq

sin(

θ2

)

)

.

H (r′, r, q) is the forward operator, which includes geometricfactors such as the source-to-object geometry factorGso(r),the object-to-detector geometry factorGod(s), the coded aper-ture mask modulation (transmission) factorT (r, r′), and thescatter angle spread∆θ. The forward operator also includesa spectral factorS(θ, q) due to photon-matter interaction anda normalization constantC. Φ(E) is the spectra of the x-raysource,h is Planck’s constant, andc is the speed of light.The object scatter densityf(r, q), which is to be recovered,is a function of the object spatial locationr and momentumtransfer q, where q is given in A−1. See Appendix A for

4

details on the derivation of the multiplicative factors that com-prise H (r′, r, q). Although the forward model is expressedusing continuous variables and integrals, it is implementeddiscretely.

The linear forward model given in Eq. 1 ignores theself-attenuation of the incident source-to-object photons andscattered object-to-detector photons. Moreover, the model doesnot account for Compton scattering, which dominates coherentscattering at larger scatter angles. Ignoring self-attenuation andCompton scattering produces a linear model that is adequatefor weakly attenuating objects and scatter measurements ac-quired at low scatter angles [7], [8].

III. C OMPUTATION OF THEFORWARD MODEL

In anticipation of an iterative algorithm for estimatingthe object scatter density, presented in Section V, efficientcomputational implementations of the forward and backwardmodels are required. The factors in Eq. 1 may be reordered toavoid repeated computations as follows

g(r′) =∫

CGso(r)God(s)T (r, r′)∆θW (r, θ)dr, (3)

whereW (r, θ) =

∫

S(θ, q)f(r, q) dq (4)

is the effective spectral factor. The integral in Eq. 4 takesscatter density, at a given object point, as a function ofmomentum transfer and produces scatter density as a functionof scatter angle. A straightforward implementation of theforward model in Eq. 3 yields a computational structure givenin Algorithm 1.

Depending on the programming language (scripting versuscompiled) and the amount of main memory available, anysubset of ther, r′, and q loops can be vectorized. For largeproblems, where storing the system matrixH is not practical,all the loops cannot be vectorized. For such problems, most ofthe computations need to be performed online and efficiently,with as many computations as possible reused in the code.In this paper, we have chosen to vectorize only ther′ andq loops. As a consequence of the vectorization of theqloop, the innermost momentum transfer loop is implementedas a matrix-vector product. Note that the same vectorizationapproach applies to the non-optimized and accelerated versionsof the forward model.

Three ways to improve the efficiency of the code are(1) scatter angle interpolation, (2) exploiting geometricsym-metries in computing the system factors, and (3) balancingonline and offline computations. The result of applying theseelements of efficiency to the non-optimized forward modelis given in Algorithm 2. The algorithm shows the final formof the forward model computational structure, incorporatingoffline computation of the spectral, source-to-object geometry,and mask modulation factors, interpolation of scatter angles,translation symmetry, and left-right and up-down mirror sym-metries. We found that the most significant improvementin computational time is due to scatter angle interpolation.Further details are described below.

Algorithm 1 Computational structure for forward model withno optimization

Given object scatter densityf(r, q)for each object pointr do

for each detector pointr′ doComputeGso(r)ComputeGod(s)ComputeT (r, r′)Compute∆θCompute scatter angleθ(r, r′)for each momentum transferq do

ComputeS(θ, q)W (r, θ) += S(θ, q)f(r, q)

end forg(r′) += Gso(r) ×God(s)× T (r, r′)

×∆θ ×W (r, θ)end for

end forg(r′) ×= CReturn detector imageg(r′)

A. Geometric Symmetries

The object-to-detector geometry factorGod(s) in Eq. 2relies on the computation of the scatter vectors = r′ − r,which has lengths = |s| and the cosine factor at the detectorsurface|nd · s|, where s = s

s. For the geometry illustrated

in Fig. 1(a), the cosine factor equals the magnitude of thex component ofs. We now consider three different ways toexploit the symmetries present in the system geometry. Weassume that the incident fan beam lies in a plane that isperpendicular to the detector plane and that the central rayof the fan beam intersects the center of the detector array, asshown in Fig. 1(a). To illustrate the symmetries, we replacethe loop overr in Algorithm 1 with a loop over thex andydirections in the object domain, as shown in lines 5 and 6 ofAlgorithm 2.

1) Translation Symmetry:For the system envisioned in thispaper, there is a large array of small detector pixels. A typicaldetector may have an array of approximately 1500 by 2000pixels, with pixel widths of 0.19 mm. One choice in thereconstruction is the selection of pixel widths in the objectdomain. These pixel widths should be motivated from a firstprinciples analysis of achievable object resolution. However,some flexibility in the exact pixel width remains. Supposethat the displacement between object pixel centers in theydirection (across the fan beam) is an integer multiple of thedetector pixel width. For our study, we use a factor of 16 toget 16*0.19 = 3.04 mm width in they direction. Letv be avector corresponding to the translation of a pixel in the objectby one pixel width along they direction. Then, the equality(see Fig. 1(d))

(r′ + v) − (r+ v) = (r′ − r) = s

implies that for each scatter vector from object locationr todetector locationr′, there is an identical scatter vector fromobject locationr+ v to detector locationr′ + v. Thus, given

5

Algorithm 2 Computational structure for forward model incorporating the proposed optimizations.f andg are the set of valuesin the object domain and detector corresponding tof(r, q) and g(r′), respectively.x • y denotes the inner product betweenvectorsx andy, ⊙ represents element-wise multiplication.fliplr(·), flipud(·), andvertcat(·, ·) are mathematical operators forreversing a matrix column-wise from left to right, row-wisefrom top to bottom, and for vertically concatenating two matrices,respectively.U andD are labels for the up and down halves of the detector array, whereasL andR are labels for the left andright sides of the object domain. For example,TUL(r) is the mask transmission values for scatter vectors from a spatial locationr on the left side of the object domain to the pixels in the upperhalf of the detector array.M andN are the number of pixelsin the detector array along thez andy directions, respectively.God(1 : M/2, 1 : N) represents a matrix of object-to-detectorgeometry factors from a given object voxel to half of the detector array alongz, with M/2 rows andN columns.1 : N isa list of numbers from1 to N in increments of1. ρy is the translation symmetry step size corresponding to the number ofdetector pixels along they direction equivalent to the width of an object pixel alongy. S(θ, :) is a vector of spectral factorsover all q for a fixed scatter angleθ while f(r, :) is a vector off(r, q) values over allq for a fixedr. C is the normalizationconstant.

1: PrecomputeGso(r) for object spatial locationsr2: PrecomputeS(θ, q) for predefinedq andθ samples3: Precompute four symmetry-based mask factor images:TUL(r), TDL(r), TUR(r), TDR(r)4: procedure g = FORWARD PROJECT(f )5: for each object location alongx do6: for each object location alongy in the left halfdo7: WL(θ) = S(θ, :) • f([x, y, 0], :)8: WR(θ) = S(θ, :) • f([x,−y, 0], :)9: if first location along they direction then

10: Compute God(1 :M/2, 1 : N)11: else12: Update God(1 :M/2, ρy + 1 : N) = God(1 :M/2, 1 : N − ρy)13: RecomputeGod(1 :M/2, 1 : ρy)14: end if15: Compute∆θ16: Compute scatter anglesθ(r, r′)17: Interpolate WL(θ) andWR(θ) and store inW interp

L andW interpR , respectively

18: FetchGso([x, y, 0])19: Fetch TUL([x, y, 0]), TDL([x, y, 0]), TUR([x,−y, 0]), andTDR([x,−y, 0])20: gUL += Gso([x, y, 0])⊙God ⊙ TUL([x, y, 0])⊙∆θ ⊙W interp

L

21: gDL += Gso([x, y, 0])⊙God ⊙ TDL([x, y, 0])⊙∆θ ⊙WinterpL

22: gUR += Gso([x, y, 0])⊙God ⊙ TUR([x,−y, 0])⊙∆θ ⊙W interpR

23: gDR += Gso([x, y, 0])⊙God ⊙ TDR([x,−y, 0])⊙∆θ ⊙W interpR

24: end for25: end for26: gU = gUL + fliplr(gUR)27: gD = gDL + fliplr(gDR)28: g = C ⊙ vertcat(gU , flipud(gD))29: end procedure

scatter vectors computed for one value ofy in the object,the scatter vectors for adjacent object locations are nearlyall determined, with only scatter vectors corresponding todetector pixels near the edge of the detector array needingrecomputation. This gives an efficient update for the scattervector computation and consequently, the object-to-detectorgeometry factorGod computation.

Let us assume there areM/2 rows in each half of thedetector array along thez-direction, N detector columns(y direction), andρy represents the number ofy-directionaldetector pixels that correspond to 1y-directional object pixel.Lines 9 to 14 in Algorithm 2 show the modification of thenon-optimized forward model computation due to translation

symmetry. For the first location in the object along they direc-tion, the object-to-detector geometry factorGod is computedfor all detector pixels under consideration. For the next objectpixel alongy, the current values ofGod starting from columnρy+1 toN are identical to the previous values ofGod startingfrom the first column to the last butρy column. The currentvalues ofGod for the firstρy columns need to be recomputed,since they have no precomputed correspondence. From thealgorithm, we see that previous object-to-detector geometryfactor computations are re-used, and only a small fractionof the factor needs to be recomputed. For a 1500 by 2000detector array and a translation symmetry step size of 16, thereduction in object-to-detector geometry factor computation

6

is approximately 98.9%. In practice, the actual time savingsobtained depends on the difference between the time it takesto compute the object-to-detector geometry factor and the timeit takes to read it from main memory.

2) Left-Right Mirror Symmetry:Another form of symmetrythat we can exploit in the computation of the forward modelis left-right mirror symmetry. Since the central ray of thefan beam intersects the center of the detector array, they-coordinates of the focal point of the source and the centerof the detector array are equal. We can consider an objectreconstruction region whose center’sy-coordinate is alignedwith that of the source and detector array. For such an objectregion, we can select an even number of pixels along theydirection. With this choice, the scatter vectors from the lefthalf of the object region are a mirror reflection of those onthe right half of the region. The scatter angles and magnitudeof the scatter vectors are equal for both halves of the objectregion.

As a result of the left-right mirror symmetry, we only needto compute the scatter anglesθ, geometric factorsGso, God,and ∆θ, and the spectral factorS(θ, q) for one half of theobject region, giving a factor of two speedup in computation,barring the costs of computing the mask transmission geo-metric factorT . Line 6 in Algorithm 2 signals the beginningof exploiting left-right mirror symmetry. Lines 26 and 27are needed to combine the left-right symmetric results of theforward model.

3) Up-Down Mirror Symmetry: In addition to left-rightmirror symmetry, further improvements in the computationalefficiency of the forward model are possible. Since the centralray of the fan beam intersects the center of the detector array,the z-coordinate of the focal point of the source equals thatof the detector array center. To simplify the discussion, weassume the detector array has an even number of pixels alongz (see Fig. 3), so that thez-coordinate of the center lies atthe border between adjacent pixels. Consequently, the scattervectors from the object to the top portion of the detector arrayare reflections of those to the bottom half.

Since only half of the detector arrays are used in thecomputation of the scatter angle, geometric factorsGso, God,and ∆θ, and the spectral factorS, a further factor of twospeedup in computational may be achieved, ignoring the costof computing the mask factorT . The use of up-down mirrorsymmetry in Algorithm 2 begins with line 10. Line 28 isneeded to combine the top and bottom halves of the results ofthe forward model.

One of the difficulties with left-right and up-down mirrorsymmetries is that they are easily violated in practice. Thefanbeam may not lie in a plane perpendicular to the detector planeand the central ray of the fan beam may not intersect the centerof the detector array. If these deviations are minor, rotation andinterpolation may be utilized to reinstate the perpendicularitybetween the fan beam and detector planes. Moreover, thedetector plane measurements may be transformed by shiftingthe measurement window alongy or z so as to align thesource and detector array center, with potential data loss dueto cropping. In addition, if the object is not located aroundthecentral ray of the fan beam, an excess amount of pixels may be

needed to cover the illuminated slice of the object. Given thepotential significant speedup in the computation, care shouldbe taken in the design and calibration of the coherent scatterimaging system to ensure that the assumptions for left-rightand up-down mirror symmetries hold.

Speedups due to left-right and up-down mirror symmetriesmay be obtained in the computation of the mask modulationgeometry factorT . This computational savings hinges onthe fact that the secondary mask is symmetric about thecentral ray of the fan beam. However, this may interferewith the principles involved in the design of the secondarymask. In essence, the computational choices affect the designof the coherent scatter imaging system. If we anticipate acomputationally efficient forward model which includes onlinecomputation of the mask modulation geometry factor, thenthe secondary mask should be designed to allow for someor all of the mirror symmetries. On the other hand, givena fixed secondary mask, the algorithm should be designedto accommodate the potential absence of mirror-symmetricsecondary masks, as we will see later. This algorithmic choiceis directly influenced by the system design.

B. Scatter Angle Interpolation

To accelerate the computation of the forward model inEq. 3, the spectral factorS(θ, q) is precomputed by usinga set of predefined uniformly sampled scatter angles. Wethen approximate the spectral factor at other scatter angles byinterpolating. Given that the set of momentum transfer valuesis also predefined, the spectral factor matrixS(θ, q) can beprecomputed. To get the approximate values ofS(θ, q) usedin the q-loop, nearest neighbor interpolation overθ may beutilized.

An example of using scatter angle interpolation to approxi-mate the true values ofS(θ, q) is shown in Fig. 2. The curvescorrespond to the true and interpolated spectral factors for afiltered 125 kVp source atq = 0.2A−1. 250 uniformly sampledscatter angles from 0.2 toπ/6 radians are used in generatingthe interpolated spectral factors. From the figure, we can seethat the spectral factor is smooth and slowly varying withθ,except in areas surrounding the characteristic peaks of thex-ray source. This suggests that interpolation will lead tofairly accurate values of the spectral factor, for a standardpolychromatic x-ray source. Since the effective spectral factorW is a linear combination of spectral factors, it can also beadequately approximated by interpolation.

Note that such practical polychromatic sources introduceblurring in the resulting scatter angles, for each momentumtransfer value. That is, according to Bragg’s equation (seeEq. 10), a monochromatic source would yield a single scatterangle for each momentum transfer. However, for a polychro-matic source, each momentum transfer has a range of scatterangles that result, of known intensity. For a monochromaticsource, the spectral factor is no longer slowly varying, so thatinterpolation is no longer a valid strategy. However, for sucha source, each value ofq will have a corresponding uniquevalue ofθ associated with it, making acceleration techniquessuch as interpolation, unnecessary.

7

0.1 0.2 0.3 0.4 0.50

20

40

Scatter Angle [rad]

S(θ

,q)[A

.U.]

TrueInterpolated

Fig. 2: True and angle-interpolated spectral factorsS(θ, q) thattransforms object scatter density as a function of momentumtransfer to object scatter density as a function of scatter angle.The spectral factors were obtained using a filtered 125 kVpsource atq = 0.2A−1.

Lines 2 and 17 of Algorithm 2 show the changes to thenon-optimized forward model computation due to scatter angleinterpolation. From the algorithm, we can see that the spectralfactorS(θ, q) is precomputed and the effective spectral factorW is interpolated. The precomputation of the spectral factormatrix and the interpolation of scatter angles avoids the re-computation of the spectral factor for each(q, θ) pair.

C. Online-Offline Computations

The choice of which factors of the forward model tocompute online or offline depends on the amount of mainmemory available, the cost of computing the factor online,and the speed of loading the factor from main memory. If thefactor is too large to fit in main memory, then part or all ofthe factor should be computed online. Moreover, if the costof computing the factor is equivalent to the speed of loadingthe factor from main memory, then the computations shouldbe performed online.

Part of the choice in online computations involves themask modulation geometry factor. If this factor has desirablesymmetry properties as outlined above, then many aspects ofits use may be computed online. When the mask is completelydetermined through experimental measurements, then a lot ofthe potential symmetries break down. In addition, the typesofmasks that we have considered for the fan beam Monte Carlostudy do not have these desirable symmetries. Taking theseobservations into account, we have chosen to precompute thebinarized mask factors offline and load them into our code atrun-time. To accommodate larger reconstructed object regions,larger detector arrays, and/or smaller main memory sizes, themask factors may need to be computed online.

The source-to-object geometry factorGso(r), can be pre-computed and stored. StoringGso(r) requires a very small

amount of memory, since it is indexed only by the numberof object spatial pixels. On the other hand, precomputing theobject-to-detector geometry factorGod(s) will require a muchlarger amount of memory for the fan beam system underconsideration, since it is indexed by both the object spatial anddetector pixels. An alternative to precomputing all ofGod(s)is interpolation between a subset of precomputedGod(s) orprecomputing only a fraction of the factors and computing therest online, but we do not pursue these avenues in this paper.Note that storing the mask modulation geometry factorTrequires far less memory than the object-to-detector geometryfactor since it is binary. Lines 1, 3, 18, 19, in Algorithm 2show changes to the computational structure of the forwardmodel due to offline computations. The mask modulationgeometry factor is computed in four parts corresponding to thetransmission of scattered photons from either half of the objectdomain (left or right) to either half of the detector array (up ordown), through the secondary mask. For example,TUL(r) isthe mask transmission values for scatter vectors from a spatiallocationr on the left side of the object domain to the pixelsin the upper half of the detector array.

IV. COMPUTATION OF THEBACKWARD MODEL

A non-optimized computational structure for the backwardmodel is given in Algorithm 3.

Algorithm 3 Computational structure for backward modelwith no optimization

Given detector imageg(r′)for each object pointr do

for each momentum transferq dofor each detector pointr′ do

ComputeGso(r)ComputeGod(s)ComputeT (r, r′)Compute∆θCompute scatter angleθ(r, r′)ComputeS(θ, q)f(r, q) += Gso(r) ×God(s)× T (r, r′)

×∆θ × S(θ, q)× g(r′)end for

end forend forf(r, q) ×= CReturn object imagef(r, q)

An efficient implementation of the backward model isparamount for an overall efficient iterative algorithm for objectscatter density estimation. As was the case for the forwardmodel, the geometry and spectral factors offer several opportu-nities for efficient computation. The scatter angle interpolation,symmetry classes, and online-offline trade-off identified forthe efficient computation of the forward model can be easilyincorporated into an efficient computation of the backwardmodel. Algorithm 4 gives such an implementation. Note thatunlike in Algorithm 2 where the scatter angle interpolationwasperformed on the effective spectral factorW after integrating

8

Algorithm 4 Computational structure for backward model incorporatingthe proposed optimizations. See caption of Algorithm 2for notation.

PrecomputeGso(r) for object spatial locationsrPrecomputeS(θ, q) for predefinedq andθ samplesPrecompute four symmetry-based mask factor images:TUL(r), TDL(r), TUR(r), TDR(r)procedure f = BACK PROJECT(g)

Extract four symmetry-based detector images fromg: gUL, gDL, gUR, gDRfor each object location alongx do

for each object location alongy in the left halfdoif first location along they direction then

Compute God(1 :M/2, 1 : N)else

UpdateGod(1 :M/2, ρy + 1 : N) = God(1 : M/2, 1 : N − ρy)RecomputeGod(1 :M/2, 1 : ρy)

end ifCompute∆θFetchGso([x, y, 0])Fetch TUL([x, y, 0]), TDL([x, y, 0]), TUR([x,−y, 0]), andTDR([x,−y, 0])gL = Gso([x, y, 0])⊙God ⊙∆θ ⊙ (TUL([x, y, 0])⊙ gUL + TDL([x, y, 0])⊙ gDL)gR = Gso([x, y, 0])⊙God ⊙∆θ ⊙ (TUR([x,−y, 0])⊙ gUR + TDR([x,−y, 0])⊙ gDR)Compute scatter anglesθ(r, r′)for each momentum transferq do

Interpolate S(θ, q) and store inS interp

f([x, y, 0], q) = C ×(

gL • S interp)

f([x,−y, 0], q) = C ×(

gR • S interp)

end forend for

end forend procedure

out the q dimension, the scatter angle interpolation of thespectral factorS in Algorithm 4 occurs independently for eachq. This results in the backward model being slower than theforward model.

In the next section, we setup the optimization problem inwhich the forward and backward models play a crucial rolein the recovery of the underlying image.

V. I MAGE RECOVERY PROBLEM DESCRIPTION

The measurementsyi from an x-ray coherent scatter imag-ing system are modeled as independent Poisson distributedrandom variables

yi ∼ Poisson(J∑

j=1

Hi,jfj + ri), i = 1, . . . , I (5)

whereI is the number of measurements,J is the number ofimage voxels.H ∈ R

I×J+ is the system matrix (discretized

forward model) withHi,j denoting theij th entry. The columnvector f ∈ R

J+ is the lexicographic ordering of the hyper-

spectral image to be recovered, withfj denoting thej th entry.r ∈ R

I+ are the known background measurements, with theith

entry denoted byri. Let J = B ×Q, whereB is the numberof spatial bins in the image andQ is the number of spectral(momentum transfer) channels.

We consider a penalized Poisson negative log-likelihoodfunction of the form

J(f) = L(f) + βR(f), (6)

whereR(f) is the regularizer,β > 0 is the regularizationcoefficient, and

L(f) =

I∑

i=1

di(li)

is the negative log-likelihood function, with

di(l) = (l + ri)− yi ln (l+ ri) + ln (yi!)

and li =∑Jj=1Hi,jfj, i = 1, . . . , I.

A standard edge-preserving regularizer, with independentspectral bins, is chosen forR(f) and is given by

R(f) =

J∑

j=1

∑

k∈Nj

wj,kψδ (fj − fk) , (7)

whereψδ (·) is the edge-preserving potential function, withscale parameterδ, which is symmetric, convex, and possessesdesirable smoothness properties [9], [10], [11],Nj is the set ofneighbors of thej th image voxel, andwj,k is a neighborhoodweight to compensate for different physical units of the spec-tral and spatial dimensions, and different voxel sizes in eachdimension. We assume a spatially piecewise smooth object. Ingeneral, the momentum transfer profile (MTP) of a material

9

may not be piecewise smooth, so that only spatial neighborsare permitted inNj for each image voxel.

The constrained convex optimization problem of interest isthen

minimizef

J(f)

subject to f ≥ 0.(8)

VI. RECONSTRUCTIONALGORITHM

To solve optimization problem 8, we consider a sequence ofsimpler optimization problems obtained by lifting the objectivefunction around the previous image estimate. In particular, weobtain a surrogate objective function which is fully separablewith respect to the image parameters. This choice allows usto utilize the optimized code for online forward and backwardmodels computation. Algorithm 5 shows the steps involved insolving the optimization problem in (8) using the sequenceof convex optimization problems given in (25) as detailed inAppendix B.

Algorithm 5 EM-type image reconstruction algorithm.FORWARD PROJECT(·) and BACK PROJECT(·) are obtainedfrom Algorithms 2 and 4, respectively.1 is a vector of ones.⊘ denotes element-wise division.

Given the measured datay and backgroundrInitialize the neighborhood structureN = Nj, w =wj,kInitialize the imagef0

Precomputeb(1) = BACK PROJECT(1)for t = 0 to T − 1 do

Computez = FORWARD PROJECT(f t) + r

Computeb(2) = BACK PROJECT(y ⊘ z)Computef t+1 using Eq. 27

end for

To accelerate the convergence rate of the EM-type algo-rithm, we employ ordered subsets, a range decomposition(measurement space partition) method. Algorithm 6 showsthe structure of the ordered subset EM-type algorithm. Thereare several choices for the measurement space partition. Oneparticular choice preserves the symmetry classes identified inSection III. To preserve left-right mirror symmetry, when adetector pixel belongs to a subset, the pixel correspondingto its mirror reflection about the liney = 0 should belongto the same subset. Moreover, to preserve up-down mirrorsymmetry, when a detector pixel belongs to a subset, the pixelcorresponding to its mirror reflection about the linez = 0should belong to the subset. In order to satisfy both left-rightand up-down mirror symmetries, if one detector pixel is in asubset, then its three mirror reflections must also be includedin that subset. In addition, to utilize translation symmetry toreduce computation, pixels that areρy pixels away (along they direction) from each of the four symmetric pixels (originalpixel + 3 reflection pixels) must also be included in the subset.

If we consider the horizontal and vertical indexes of thedetector pixels, the vertical indexes can be used to satisfyup-down symmetry constraints while the horizontal indexes

can be used to address left-right and translation symmetryconcerns. Due to up-down mirror symmetry, we can considerpartitioning half (up or down) of the vertical detector pixelindexes. The partitioning of the other half follows directlyfrom up-down mirror reflections. For a detector array withMrows, we can partition either the top or bottomM/2 rows usingany strategy. We have chosen to select members of a subset byusing a fixed step sizeρz. To balance the vertical subsets,ρzshould be a factor ofM/2. The detector pixels with verticalindexesm : ρz :M/2 andM−m+1 : −ρz :M/2+1 belongto the same subset, where1 ≤ m ≤ ρz is the smallest verticalindex in the subset. Note that we have used MATLAB’s listingnotationa : b : c to mean numbers froma to c in steps ofb,a inclusive.

In order to satisfy left-right symmetry constraints, the samestrategy used for the vertical indexes can be used for the hor-izontal indexes. However, to also satisfy translation symmetryconstraints, extra precaution must be taken. The differencebetween the horizontal indexes of detector pixels that belong tothe same subset must be a multiple ofρy. For a detector arraywith N columns, satisfying translation symmetry implies thatthe detector pixels with horizontal indexesn : ρy : N belongto the same subset, where1 ≤ n ≤ ρy/2 is the smallesthorizontal index in the subset. In addition, to satisfy left-rightmirror symmetry, the detector pixels with horizontal indexesN − n + 1 : −ρy : 1 also belong to that subset. To balancethe horizontal subsets,ρy should be a factor ofN and even.

Algorithm 6 Ordered subset expectation-maximization typeimage reconstruction algorithm. FORWARD PROJECTp(·) andBACK PROJECTp(·) are obtained from Algorithms 2 and 4,respectively, by utilizing the appropriate partition of the mea-surement (detector) space.1 is a vector of ones.⊘ denoteselement-wise division.

Given the measured datay and backgroundrPartitiony andr into P disjoint subsetsyp andrpInitialize the neighborhood structureN = Nj, w =wj,kInitialize the imagef0

Precomputeb(1)p = BACK PROJECTp(1), ∀p = 1, . . . , P

for t = 0 to T − 1 dof t,0 = f t

for p = 1 to P doComputezp = FORWARD PROJECTp(f

t,p−1) + rp

Computeb(2)p = BACK PROJECTp(yp ⊘ zp)

Computef t,p using Eq. 27end forf t+1 = f t,P

end for

For illustration, we consider the partitioning of a scaled-down version of the detector array with 32 rows and 64columns. The partitions shown in Fig. 3 satisfy the require-ments for preserving symmetries, when the translation symme-try step size isρy = 16 and the fixed step size in the verticaldirection is 8 pixels. In the figure,mh and mv representthe horizontal and vertical midpoints of the detector array,

10

Fig. 3: Layout of the symmetry-preserving choice of 64 subsets based on a mini detector array with 32 rows and 64 columns.The translation symmetry step size isρy = 16. mh andmv represent the horizontal and vertical midpoints of the detectorarray and correspond to the linesz = 0 andy = 0, respectively. Pixels with the same combination of color and pattern belongto the same subset.

corresponding to the linesz = 0 and y = 0, respectively.Pixels with the same combination of color and pattern belongto the same subset. Using a fixed step size of 8 pixels gives 8different subsets along the vertical direction. The choiceof 16as the translation symmetry step size gives a total of 8 (= 16/2)horizontal subsets. This gives a total of 64 subsets as shownin Fig. 3. For example, in each row, the pixels with horizontalindexes (1, 17, 33, 49) and (64, 48, 32, 16) belong to the samesubset. Also, in each column, the pixels with vertical indexes(1, 9) and (32, 24) belong to the same subset. The step sizesthat were used for partitioning the mini detector array wereapplied to the full detector array used in the simulations thatfollow.

VII. R ESULTS

The forward and backward models, integral parts of theOSEM-type iterative algorithm for estimating object scatterdensity (see Algorithm 6), were validated on simulated ana-lytic and Monte Carlo data. The subsets were chosen in thesame way as the illustrative example in the previous section.For both simulations, the source was located at the origin andthe center of the flat-panel energy-integrating detector arraywas 1546.5 mm away along the positive x-axis. The sourcewas operated at 125 kVp, and the spectrum was filtered by0.5 mm of aluminum, before being shaped into a fan by the

primary aperture (slit). The secondary mask was placed 100mm in front of, and parallel to, the plane of the detector array.The detector array had 1536 rows and 2048 columns, witha pixel pitch of 0.19 mm in bothz and y directions. Thecenter of the reconstructed object was located 1035 mm fromthe source, along the positive x-axis. A region of 70 mm by85.12 mm, in thexy-plane, was reconstructed, with a pixelpitch of ∆x = 2.5 mm and∆y = 3.04 = (16 × 0.19)mm. 79 evenly spaced momentum transfer bins from 0.01to 0.4 A−1 were used. There wereNθ scatter angle sampleschosen uniformly from 0 toπ/6 radians, excluding 0. Thesecondary mask shown in Fig. 1(b) was used in both theanalytic forward model and Monte Carlo simulated data. Sincethe mask is not amenable to the mirror symmetry classes, thefour symmetric mask modulation factors are precomputed, asstated in Algorithm 2.

A. Analytic Simulation

The analytic forward model was used to model an objectcontaining two vials of strong scatterers in close proximity,illustrated in Fig. 4(c). Rectangular vials of sodium chloride(NaCl) and aluminum (Al) crystalline powder were placedalong the direction of the fan beam’s central ray. Each vialoccupies a rectangular region 10 mm by 12.16 mm in size.The vials are separated by 20 mm along the positive x-axis.

11

The object scatter density was simulated by inserting eachmaterial’s momentum transfer profile at the appropriate spatiallocation. The momentum transfer profile of each material wasobtained experimentally by using an x-ray diffractometer [12]and then interpolated to match the sampling of the momen-tum transfer space. The resulting momentum transfer profiles(MTPs) of NaCl and Al are shown as the red reference curvesin Figs 4(e) and 4(f), respectively. The spatial distributionof the simulated object (f(x, y) =

∫

f(x, y, q) dq), obtainedby summing over all the momentum transfer channels ofthe object scatter density, is shown in Fig. 4(c). The non-optimized forward model shown in Algorithm 1 was appliedto the object scatter density in creating the noise-less scatterdata, to prevent the simulated data from being corrupted byscatter angle interpolation, which is utilized by the optimizedforward model shown in Algorithm 2. Poisson noise was laterintroduced with a maximum photon count of 50, across alldetector pixels. The simulated noisy data is given in Fig. 4(a).

Next, we characterize the computational-time speedup of theforward and backward models introduced by each element ofthe proposed optimization described in Section III, relative tothe non-optimized models in Algorithms 1 and 3. The analyticsimulated data and object was used for this characterization.

To characterize the speedup in computational time dueto each element of the proposed optimization discussed inSection III, the variants of the forward model were appliedto the simulated vial object, whereas those of the backwardmodel were applied to the noise-less simulated data. Bothfull-data and ordered-subsets implementations of the forwardand backward models were tested. For the ordered-subsetimplementations, the models were iterated over all the sub-sets. The forward and backward models were implementedin MATLAB R© R2015a, on a dual processor, 6 cores perprocessor, 128 GB Windows machine. Table I shows the timetaken to apply each model once on the appropriate data and thespeedup in computational time relative to the time taken by thenon-optimized model. The speedup is computed as the ratio ofthe computational time of an algorithm to the computationaltime of the non-optimized version. The table also shows theerror associated with each element of optimization relative tothe non-optimized version. The results for the ordered-subsetsimplementations are given to the right of the results for thefull-data implementations. For the models that utilize scatterangle interpolation, the number of scatter angle bins used ininterpolation isNθ = 250.

From the table, we see that the greatest improvement incomputational time is due to scatter angle interpolation; itgives an improvement of a factor of about 33 (39 for OS) and9 (7 for OS) in the application of the forward and backwardmodel, respectively, relative to the non-optimized models. Thenext largest improvement in computational time is due toeither left-right or up-down mirror symmetry, which give animprovement of a factor of about 2 in both models. Theimprovement due to translation symmetry and online-offlinetrade-off are marginal at best. The forward and backwardmodels are accelerated by a factor of approximately 172 (146for OS) and 37 (32 for OS), respectively, by utilizing all theelements of our optimized algorithms. Note the non-linearity

in speedup of the cumulative effect of the elements of ourproposed optimization. The scatter angle interpolation intro-duces errors of approximately 6% and 1% in the computationof the forward and backward models, respectively, relativetothe non-optimized algorithms. These relative errors can bereduced by increasing the number of scatter angle samplesused for the interpolation or using a higher order interpolation.However, the error reduction comes at the cost of increasedcomputational times. WhenNθ = 2000, the times to computethe full forward and backward models are 65.28 s (78.40 s forOS) and 289.62 s (291.74 s for OS) respectively.

The ordered-subsets implementations of the fully-optimized(AO) and non-optimized (NO) forward and backward modelswere used in recovering the simulated vial object based onthe OSEM-type algorithm in Algorithm 6. Fig. 4 shows theresults of estimating the object scattering density using 20iterations of the OSEM-type algorithm. Each mean momentumtransfer profile was obtained by averaging the MTPs acrossthe known location occupied by a material. From the figures,we can see that the spatial distribution given in Fig. 4(d) andthe momentum transfer profiles are recovered fairly accuratelyin a few iterations. There is a slight shift in the peak of theMTP recovered using the fully-optimized models with 250scatter angle bins (SABs) relative to those obtained using thenon-optimized models, especially at lower momentum transfervalues, corresponding to the approximation error incurreddue to scatter angle interpolation. To diminish the shift, afiner sampling of the scatter angles may be used. From thefigure, we see that the reconstruction results based on thefully-optimized models with 2000 SABs are a better matchto those of the non-optimized models. The estimated meanmeasurements were obtained by applying the fully-optimizedforward operator, with 250 SABs, on the estimated objectscatter density. The estimated mean measurements are inagreement with the simulated noisy Poisson measurements.

The estimated spatial distributions and momentum transferprofiles closely match those of the reference configuration andmaterials. However, from the recovered momentum transferprofile of NaCl and Al shown in Figs 4(e) and 4(f), respec-tively, we can see that a few iterations has only accuratelyrecovered the largest peak.

Ordered subset promises an acceleration of the convergencerate of a regular EM-type algorithm that is comparable to thenumber of subsets [13]. Fig. 5 shows the value of the objectiveas a function of the iteration number. We can see that anacceleration factor of about 76 is obtained by using the 64subsets described in Section VI.

12

TABLE I: Computational time, speedup, and relative error for different levels of optimization of the forward and backwardmodels. Speedup is defined as the ratio of the computational time of an algorithm to the computational time of the non-optimizedversion. MATLAB R©’s tic-toc functions were used in timing the variants of the forward and backward models. The numberof scatter angle bins used in interpolation isNθ = 250. The NRMSE is computed relative to the results (in the measurementor object space) of the non-optimized model. To obtain the NRMSE, the RMSE is normalized by the square root of themean of the square of the entries of the result obtained usingthe non-optimized model. The results for the ordered-subsetsimplementations are given to the right of the results for thefull-data implementations. Note that the cumulative effect of allthe optimizations on the speedup is not linear.

OptimizationModel

Forward BackwardTime (s) Speedup NRMSE (%) Time (s) Speedup NRMSE (%)

FDI OSI FDI OSI FDI OSI FDI OSI FDI OSI FDI OSI

NO 10516.88 9466.80 1.00 1.00 0.00 0.00 10512.32 9365.06 1.00 1.00 0.00 0.00

SAI 319.45 240.24 32.92 39.41 6.20 6.20 1191.05 1330.34 8.83 7.04 0.67 0.67TS 10385.48 9434.27 1.01 1.00 0.00 0.00 10438.93 9472.91 1.01 0.99 0.00 0.00LRMS 5271.56 4822.61 2.00 1.96 0.00 0.00 5317.47 4763.28 1.98 1.97 0.00 0.00UDMS 4993.05 4860.33 2.11 1.95 0.00 0.00 4975.95 4791.84 2.11 1.95 0.00 0.00OOT 10353.90 9525.87 1.02 0.99 0.00 0.00 10315.77 9441.22 1.02 0.99 0.00 0.00AO 61.04 64.70 172.30 146.32 6.20 6.20 287.40 288.98 36.58 32.41 0.67 0.66

NO = No Optimization; SAI = Scatter Angle Interpolation; TS =Translation Symmetry; LRMS = Left-Right Mirror Symmetry; UDMS = Up-Down MirrorSymmetry; OOT = Online-Offline Trade-off; AO = All Optimizations; NRMSE = Normalized RMSE; RMSE = Root Mean Square Error;FDI = Full-DataImplementation; OSI = Ordered-Subsets Implementation

0 2 4 6 80

1

2

3

4x 10

8

Iteration Number

Obj

ectiv

e F

unct

ion

Val

ue

OSEM−typeEM−type

(a) Initial iterations

100

101

102

103

7

8

x 106

Iteration Number

Obj

ectiv

e F

unct

ion

Val

ue

2 iters 152 iters

OSEM−typeEM−type

(b) Subsequent iterations

Fig. 5: Objective function value versus iteration number forregular and ordered subset EM-type algorithms. The optimiza-tion curves were split into two regimes to illustrate the meritsof ordered subsets. The fully-optimized operators, with 250SABs, were utilized.

B. Monte Carlo Simulation

The object utilized in the Monte Carlo simulation was a 5mm by 50 mm by 50 mm rectangular slab of graphite powderwhose momentum transfer profile is shown as the referencein Fig. 6(d). The Monte Carlo detector measurements includessingle coherent scattering events from the slab of graphitecrys-talline powder, multiple scattering events, Compton scatteringevents, and scatter from the secondary aperture. Our forwardmodel only accounts for single coherent scattering from theobject, and the other scattering events constitute unmodelednoise.

Since the image recovery algorithm tries to explain thedetector measurements based on only coherent scattering,artifacts are introduced in the reconstruction. To reduce theseartifacts in the momentum transfer region of interest (ROI),from 0.01 to 0.4A−1, we performed the reconstruction usinga larger region extending to 0.6A−1 and later cropped to theROI. Fig. 6 shows the results of estimating the mean detectorphoton counts and the object scatter density, using 20 iterationsof the OSEM-type algorithm and the non-optimized (NO) andfully-optimized (AO) forward and backward models. Again,we see that the spatial distribution and the momentum transferprofile are recovered quickly and fairly accurately. Noticeableartifacts are observed in the spatial distribution at the cornersof the reconstruction region, due to the mismatch between themodel used for the Monte Carlo simulation and the analyticalmodel. For the same reason, the reconstruction using the fully-optimized (AO) models, with 250 SABs, incidentally appearsto fit the reference object better than the fully-optimized,with 2000 SABs, and the non-optimized models, as shownin Fig. 6(d).

These results demonstrate that without prior knowledge ofthe location of scatterers in an object, the fan beam system,in conjunction with the OSEM-type algorithm and efficient

13

(a) Simulated noisy measurements (b) Estimated mean measurements

y [mm]

x[m

m]

−40 −30 −20 −10 0 10 20 30 401000

1010

1020

1030

1040

1050

1060

1070

(c) Simulated spatial distribution

y [mm]

x[m

m]

−40 −30 −20 −10 0 10 20 30 401000

1010

1020

1030

1040

1050

1060

1070

(d) Estimated spatial distribution

0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

Momentum Transfer [A−1]

ScatteringIn

tensity

[A.U

.]

ReferenceNOAO 2000 SABsAO 250 SABs

(e) Normalized MTP for NaCl

0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

Momentum Transfer [A−1]

ScatteringIn

tensity

[A.U

.]

ReferenceNOAO 2000 SABsAO 250 SABs

(f) Normalized MTP for Al

Fig. 4: Estimation of object scatter density from vials of NaCl and Al crystalline powders using ordered-subsets implementationsin Algorithm 6. Each momentum transfer profile (MTP) was obtained by averaging the MTPs across each known locationoccupied by a material. The estimated mean measurements were obtained by applying the fully-optimized forward operator,with 250 scatter angle bins (SABs), on the estimated object scatter density. NO and AO stand for no optimization and alloptimizations, respectively. Note that both NO and AO implementations use ordered subsets.

implementations of the forward and backward models, can beused to efficiently estimate their scattering densitiesf(x, y, q).

14

(a) Measurements at detectors (b) Estimated mean measurements

y [mm]

x[m

m]

−40 −30 −20 −10 0 10 20 30 401000

1010

1020

1030

1040

1050

1060

1070

(c) Estimated spatial distribution

0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

Momentum Transfer [A−1]

ScatteringIn

tensity

[A.U

.]

ReferenceNOAO 2000 SABsAO 250 SABs

(d) Normalized MTP for graphite

Fig. 6: Estimation of object scatter density from Monte Carlo measurements of a slice of graphite rectangular prism usingordered-subsets implementations in Algorithm 6. The momentum transfer profile (MTP) was obtained by averaging the MTPsacross the known location occupied by graphite. The estimated mean data was obtained by applying the fully-optimized forwardoperator, with 250 SABs, on the estimated object scatter density. See Fig. 4 for notation.

VIII. C ONCLUSIONS

The image recovery process relies heavily on a compu-tational representation of the forward model for the data.We have identified and described three ways of significantlyreducing the computational burden of the forward and back-ward models and the overall reconstruction algorithm, namely:scatter angle interpolation, symmetries in the system geometry,and balancing online and offline computations.

The forward and backward models and the correspondingalgorithms were validated using analytic and Monte Carlosimulations. Speedups of about 146 and 32 were obtainedin computing the forward and backward models using theordered-subsets implementations, respectively. The spatial dis-tribution and the momentum transfer profiles of the simulatedobjects were recovered fairly accurately using a few iterationsof the ordered subset EM-type reconstruction algorithm.

The design of the coherent scatter imaging system, includingthe choice of the detector array, its pixel pitch, and place-ment relative to the source, and the secondary mask and its

placement, influences the choices made in implementing theforward and backward models utilized in the reconstructionalgorithm. For example, the use of different symmetry classesis dictated by their presence in the designed physical system.Moreover, if the designed secondary mask and its placementfail to satisfy certain desirable properties that make the onlinecomputation of the mask modulation factor easy, it may bebetter to compute it offline, since its online computation maybecome the bottleneck in the computation of the forwardmodel.

On the other hand, the need for efficient algorithms alsoaffects several elements of the system design. For example,placing the center of the detector array so that the central rayfrom the source strikes it and adequate efforts ensuring properalignment of the imaging components, permit efficient onlineimplementations of the forward model and reconstructionalgorithms. Also, having a detector with a smaller pixel pitchpermits the potential use of larger translation symmetry stepsize, without significant deterioration of the recoverableobjectspatial resolution in they direction.

15

This paper describes the use of tomographic x-ray coherentscatter measurements in estimating volumetric scatter density,using a fan beam source distribution and a secondary mask.However, a subset of the methods we described for accelerat-ing the computation of the forward and backward models andrecovering the object scatter density can be easily carriedoverto other source distributions and more accurate forward modelsincorporating Compton scatter. For example, the scatter angleinterpolation technique will apply to any x-ray coherent scatterimaging system with a polychromatic source.

APPENDIX AFORWARD MODEL DERIVATION

In this section of the Appendix, we describe the multi-plicative factors that comprise the forward operatorH (r′, r, q)given in Eq. 1.

The x-ray source is assumed to be polychromatic. Whateversource filtering and energy-dependent factors such as detectorefficiency, present in the physical system are included in theeffective source model whose energy-dependent flux isΦ(E),where the x-ray energyE is given in keV. A photon, withenergyE, incident on the object atr is scattered through anangleθ, passes through or gets blocked by the coded aperture,and is measured by a detector pixel located atr′. For anenergy-sensitive detector array, the number of photons detectedis given by

g(E, r′) =∫ ∫

Φ(E)wo∆ΩsodσcohdΩ

∆ΩodAso(E, r)

Aod(E, r, r′)T (r, r′)δ

(

E − hcq

sin(

θ2

)

)

dr dq,

(9)

where

dσcohdΩ

=dσThompson

dΩf(r, q),

dσThompsondΩ

=r2e2

(

1 + cos2 θ)

.

This is a modified form of the forward model in [8] whichseparates the attenuation factor and solid angle∆Ω into twoparts; from source to object and from object to detector.∆Ωso is the solid angle subtended by the face of the object

voxel illuminated by the source photon while∆Ωod is the solidangle subtended by the face of the detector pixel illuminatedby the scattered photon. Although a fan beam is effectivelyplanar, we assume that each pixel in the fan beam within theobject space is effectively a voxel, with out-of-plane heightgiven by the thickness of the fan beam. We assume that thethickness of the fan beam is fixed across the entire object.wois the width of each object voxel in the direction parallel tothe normal vector to the face of the object voxel illuminatedby photons.Aso(E, r) is the attenuation of the photon fromthe source to the object,Aod(E, r, r′) is the attenuation of thescattered photon from the object to the detector, andT (r, r′)is the transmission of the scattered photon through the codedaperture from the object to the detector.dσThompson/dΩis the Thompson form of the differential cross section of a

free electron [7],dσcoh/dΩ is the differential cross sectiondescribing the scatter of an x-ray with energyE into a givensolid angle,re is the classical electron radius, andf(r, q) isthe square of the coherent scatter form factor [7], [8]. We callf(r, q) the object scatter density.

The momentum transfer parameterq is related to the x-rayenergyE through the equation [7]

q =E sin

(

θ2

)

hc, (10)

whereh is Planck’s constant,c is the speed of light, andhc =12.3984193 keV A. Equation 10 is often referred to as Bragg’slaw [8].

We assume that the detector array is energy-integrating,so that photons of all energies are accumulated at detectorlocationr′. The energy-integrated flux is then given as

g(r′) =∫ ∫ ∫

Φ(E)wo∆ΩsodσcohdΩ

∆Ωod

T (r, r′)δ

(

E − hcq

sin(

θ2

)

)

dr dq, dE, (11)

=

∫ ∫

(

∫

Φ(E)δ

(

E − hcq

sin(

θ2

)

)

dE

)

wo∆Ωso

r2e2

(

1 + cos2 θ)

f(r, q)∆ΩodT (r, r′) dr dq,

where we assume that the object is weakly attenuating so thatAso andAod can be ignored.

Bragg’s law acts to pick out(θ, q) pairs over a rangeof energies corresponding to the intersection of the incidentsource energies and the energies the detector is sensitive to.These energies are represented by the domain ofΦ(E). Usinga Dirac delta formulation, for a point detector, the integral overE may be simplified as

∫

Φ(E)δ

(

E − hcq

sin(

θ2

)

)

dE = Φ

(

hcq

sin(

θ2

)

)

. (12)

Due to a finite detector pixel size, there is a spread of scatterangles∆θ corresponding to each pair of object and detectorpoints. This corresponds to a spread in Bragg’s energy∆E,for each momentum transfer value. To account for the spreadin Bragg’s energy, we modify the integral in Eq. 12 by scalingthe Dirac delta as in

∫

Φ(E)δ

E − hcq

sin( θ2 )

∆E

dE = Φ

(

hcq

sin(

θ2

)

)

∆E. (13)

Using Bragg’s law, Eq. 10, the spread in Bragg’s energy is

∆E =hcq cos

(

θ2

)

2 sin2(

θ2

) ∆θ.

An estimate of∆θ is the angle between the scatter vectorsfrom the object point to the midpoint of the two edges of thefinite detector pixel along thez direction. This choice is intune with the symmetry cases identified in Section III.

16

The energy-integrated flux at detector pointr′ is then givenas

g(r′) =∫ ∫

Φ

(

hcq

sin(

θ2

)

)

hcq cos(

θ2

)

2 sin2(

θ2

) ∆θwo∆Ωso

r2e2

(

1 + cos2 θ)

f(r, q)∆ΩodT (r, r′) dr dq.

(14)

The source-to-object differential solid angle is given as

∆Ωso = Gso(r)Ao, (15)

where

Gso(r) =|no · r|r2

. (16)

Gso(r) is called the source-to-object geometry factor.nowhich is illustrated in Fig. 1(a), is a unit normal vector tothe voxel containing the object point,r is the unit vector inthe direction from the source to the object point,r = |r| isthe magnitude of the vector from the source to the objectpoint, andAo is the area of the illuminated face of the objectvoxel. Note thatAo andwo multiply to give the volume of theobject voxelVo. We assume that the object space is uniformlysampled, so that the voxels have the same volume. The frontface of the object voxel illuminated by a ray is oriented suchthat no is parallel to the direction of the fan beam’s centralray (the x-axis). As such,Gso can be simplified to

Gso(r) =x

(x2 + y2)1.5. (17)

The object-to-detector differential solid angle is given as

∆Ωod = God(s)Ad, (18)

where

God(s) =|nd · s|s2

. (19)

We callGod(s) the object-to-detector geometry factor.nd isa unit normal vector to the detector array,s is the unit vectorin the direction of the scatter vectors, from the object pointto the detector point.s = |s| is the magnitude of the scattervector andAd is the area of the detector pixel. We also assumethat the detector pixels have the same area.

The secondary mask is situated between the object and thedetector array as illustrated in Fig. 1(a). It is assumed to lie ona plane parallel to the detector array, centered at the fan beam’scentral ray. There is a beam-stop on the secondary mask andin front of the fan beam [5] to attenuate the transmitted rays.This secondary mask spatially modulates the scattered x-rayflux. Given the object point and detector point, the intersectionof the scatter vector and the aperture is defined. The geometricfactor T (r, r′) represents the resulting spatial modulation ofthe scattered x-ray flux. For planar masks, in practice, weestimate the values of the binary secondary mask image usinga calibration scan with a flood illumination. The resultingmeasured image is rescaled from the detector plane back tothe secondary mask plane and then binarized. The intersectionof each scattered ray with the secondary mask plane providesa point of index into the binary in-plane mask image. Theresulting binary values giveT (r, r′).

Using the geometric factors defined above and the fact thatVo = Aowo andAd are constants, the energy-integrated fluxat detector pointr′ given in Eq. 14 can be recast as

g(r′) =∫ ∫

VoAdGso(r)God(s)T (r, r′)∆θ

Φ

(

hcq

sin(

θ2

)

)

hcq cos(

θ2

)

2 sin2(

θ2

)

r2e2

(

1 + cos2 θ)

f(r, q) dr dq,

=

∫ ∫

CGso(r)God(s)T (r, r′)∆θ

S(θ, q)f(r, q) dr dq, (20)

where

S(θ, q) =q(

1 + cos2 θ)

cos(

θ2

)

sin2(

θ2

) Φ

(

hcq

sin(

θ2

)

)

, (21)

is a spectral factor andC = 0.25hcVoAdr2e is a normalization

constant that can be obtained by a calibration scan and includesscalar factors such as the area of a detector pixel and thevolume of an object voxel.

APPENDIX BRECONSTRUCTIONALGORITHM DETAILS

To solve optimization problem 8, we consider a sequence ofsimpler optimization problems obtained by lifting the objectivefunction around an expansion point (e.g. the previous imageestimate). In particular, we lift the objective function toobtaina surrogate objective function which is fully separable withrespect to the image parameters. This choice allows us toutilize the optimized code for on-the-fly forward and backwardprojection.

We adopt the EM surrogate function [14] for the data-fittermL(f) and utilize De Pierro’s convexity trick [15] for theregularization term. The EM surrogate function for the data-fitterm is given as

L(f) =

I∑

i=1

J∑

j=1

[

−yiHi,j fj

li + riln

(

fj

fj

(

li + ri

)

)

+Hi,jfj

]

,

(22)where li =

∑J

j′=1Hi,j′fj′ and f is the previous imageestimate.

A surrogate function for the regularization term, using DePierro’s convexity trick [15] is given as

R(f) =

J∑

j=1

∑

k∈Nj

0.5wj,k

[

ψδ

(

2fj − fj − fk

)

+ψδ

(

2fk − fj − fk

)]

.

(23)

Using a quadratic surrogate for the edge-preserving poten-tial function of the form [9], [13]

ψδ(x) = ψδ(x) + ψδ(x)(x− x) +1

2ωψδ

(x)(x− x)2,

17

in Eq. 23, we obtain the following separable quadratic surro-gate function to the regularization term

R(f) =

J∑

j=1

∑

k∈Nj

0.5wj,k

[

ψδ

(

2fj − fj − fk

)

+ψδ

(

2fk − fj − fk

)]

.

(24)

wherex is the expansion point,ψδ(·) is the first derivative ofψδ(·) andωψδ

(x) , ψδ(x)/x is the curvature of the quadratic.The modified sequence of constrained convex optimization

problems of interest is then

f t+1 = argminf≥0

J(f), (25)

with J(f) = L(f) + βR(f) and f , f t.Taking the derivative ofJ(f) with respect tofj , we obtain

the gradient (∀j)

∂J(f)

∂fj= − fj

fjb(2)j + b

(1)j

+β

∑

k∈Nj

wj,k˙ψδ

(

2fj − fj − fk

)

+∑

m∈Nbj

wm,j˙ψδ

(

2fj − fj − fm

)

= − fjfjb(2)j + b

(1)j + β

[

fj

(

b(3)j + b

(4)j

)

(26)

−fj(

b(3)j + b

(4)j

)

+ b(5)j + b

(6)j

]

where

b(1)j =

I∑

i=1

Hi,j ,

b(2)j =

I∑

i=1

Hi,j

(

yi

li + ri

)

,

b(3)j = 2

∑

k∈Nj

wj,kωψδ

(

fj − fk

)

,

b(4)j = 2

∑

m∈Nbj

wm,jωψδ

(

fj − fm

)

,

b(5)j =

∑

k∈Nj

wj,kψδ

(

fj − fk

)

,

b(6)j =

∑

m∈Nbj

wm,jψδ

(

fj − fm

)

,

˙ψδ(x) , dψδ(x)/dx andN b

j is the set of image voxels thathave voxelj as a neighbor. For a symmetric neighborhoodstructure,N b

j = Nj andwj,k = wk,j . Note thatli is obtainedby applying the forward operator on the previous imageestimate. Moreover,b(1)j and b(2)j are obtained by applyingthe backward operator on a detector image of all ones and theratio of the actual measurement to the predicted measurement,respectively.

Using Lemma C.1 (see Appendix C), the gradient equation∂J(f)∂fj

= 0 admits the closed-form solution

fj =

−χ(2)j +

√

(

χ(2)j

)2+4χ

(1)j χ

(3)j

2χ(1)j

if χ(1)j > 0

χ(3)j

χ(2)j

if χ(1)j = 0

, (27)

where

χ(1)j = β

(

b(3)j + b

(4)j

)

,

χ(2)j = b

(1)j + β

[

−fj(

b(3)j + b

(4)j

)

+ b(5)j + b

(6)j

]

,

χ(3)j = fjb

(2)j .

APPENDIX CLEMMA STATEMENT AND PROOF

Lemma C.1. The extended function

f(x) =1

2ax2 + bx− c ln(x) + d, x ∈ [0,+∞], (28)

achieves a unique minimum at

x∗ =

−b+√b2+4ac2a if a > 0

cb

if a = 0, b > 0+∞ if a = 0, b ≤ 0

, (29)

for a ≥ 0, c ≥ 0, and b, d ∈ R, where we assume0 ln(0) =0 and ln(0) = −∞.

Proof: The first and second derivatives off(x) are (forx > 0)

df(x)

dx= ax+ b− c

x,

andd2f(x)

dx2= a+

c

x2,

respectively.Since d2f(x)

dx2 ≥ 0, f(x) is a convex function. Consider thecases ofa > 0, a = 0, b > 0, anda = 0, b ≤ 0

Case 1.a > 0Since d2f(x)

dx2 > 0 and limx→∞d2f(x)dx2 = a > 0, f(x) is

strongly convex. Thus,f(x) admits a unique minimizer.The critical point of f(x) occurs at the solution of the

gradient equationdf(x)dx

= 0 that lies within the domain off(x). The only endpoint of the domain off(x) is at x = 0.The unique minimizer occurs either atx = 0 or the criticalpoint.

Consider the sub-cases ofc = 0 and c > 0. Whenc = 0,the gradient equation becomesax+ b = 0, so thatx = − b

ais

a critical point and the unique minimizer, ifb ≤ 0. If b > 0,the endpointx = 0 is the unique minimizer.

Thus, whena > 0and c = 0,

x∗ =

− ba

if b ≤ 00 if b > 0

.

Whenc > 0, f(0) = +∞, so that the endpointx = 0 isnot an absolute minimum.

18

Consider the gradient equation (withx > 0)

ax+ b− c

x= 0. (30)

Multiplying both sides of Eq. 30 byx, we get the quadraticequation

ax2 + bx− c = 0. (31)

The discriminant of the quadratic expression isD = b2 +4ac > 0. In addition, since4ac > 0, |b| <

√b2 + 4ac, so that

Eq. 31 has exactly two solutions with opposite signs. Sincexmust be non-negative, the only valid solution to Eq. 30 and31 and the unique minimizer is

x∗ =−b+

√b2 + 4ac

2a. (32)

Equation 32 also handles the sub-case ofc = 0. For thatcase,x∗ = −b+

√b2

2a = −b+|b|2a . Whenb ≤ 0, |b| = −b, so that

x∗ = − ba. Whenb > 0, |b| = b, andx∗ = 0.

Thus,x∗, as given by Eq. 32, is the unique minimizer off(x), whena > 0.

Case 2.a = 0, b > 0In this case, the gradient equation becomesb − c/x = 0,

with solutionx = cb≥ 0. Since at the endpoint of the domain

of f , f(0) = +∞, x∗ = cb

is the unique minimizer off(x).

Case 3.a = 0, b ≤ 0In this case, the gradient function is alsob− c/x. However,

the function is strictly decreasing and unbounded below, sothat the minimum of−∞ is achieved atx∗ = +∞.

Thus,x∗ in Eq. 29 is the unique minimizer of the functionf(x) given in Eq. 28.

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