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Biomed. Phys. Eng. Express 4 (2018) 047005 https://doi.org/10.1088/2057-1976/aac9af

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Filtered backprojection implementation of the immediately-after-backprojection filtering

Gengsheng LZengDepartment of Engineering,Weber StateUniversity, Ogden, Utah 84408,United States of AmericaDepartment of Radiology and Imaging Sciences, University ofUtah, Salt LakeCity, Utah 84108,United States of America

E-mail: larryzeng@weber.edu

Keywords: analytic image reconstruction algorithm, image reconstruction, tomography, iterative image reconstruction algorithm

AbstractIn an iterative image reconstruction algorithm, it was demonstrated that the contrast-to-noise ratio inthefinal reconstruction could be improved if a low-pass filter was applied to the backprojction of theprojection-domain discrepancy, and then this backprojectionwas used to update the image from theprevious iteration. The goal of this paper is to extend thismethod to theweightedfilteredbackprojection (FBP) algorithm.

1. Introduction

In 2006, we developed a method to increase thelesion’s contrast-to-noise ratio in an iteratively recon-structed image (Zhang and Zeng 2006). A typicaliterative image reconstruction algorithm consists of atleast one pair of projector and backprojector. Ourmethod was to apply a low-pass filter to the back-projection, and the low-pass filtered backprojectionwas used to update the image from the previousiteration. That method was clearly different from thepost-filtering method, in which an image is firstreconstructed and then a filter is applied to the finalimage. The original idea of that 2006 method wasmotivated by the encouraging work of using blobs toreplace the non-overlapping pixels or voxels (Matejand Lewitt 1995, Hanson and Wecksung 1985, Lewitt1990, Lewitt 1992,Wang et al 2004).

By reading that 2006 paper (Zhang and Zeng2006), one may not be convinced that the immedi-ately-after-backprojection filtering method can beused to approximate the blob pixel or voxel model.However, one can clear see that themethod is effectivein improving the lesion’s contrast-to-noise ratio.

Recently the FBP algorithm can be associated withan ‘iteration number,’ k, in the sense that one canchoose the ‘iteration number’ k so that the FBP recon-struction is an approximate to the kth iteration of theiterative Landweber reconstruction if the relaxationparameter is carefully adjusted (Zeng 2012, Zeng andZamyatin 2013, Zeng 2014, Zeng 2016). We refer to

the FBP algorithm with an ‘iteration number’ as theweighted FBP or windowed FBP (wFBP) algorithm.The purpose of this current paper is to extend theimmediately-after-backprojection filtering method tothe wFBP algorithm. We understand that the wFBPalgorithm does not use a square pixel model, and theblob model does not make any sense to wFBP.We stillhope that the immediately-after-backprojection filter-ing strategy is able to improve the lesion’s contrast-to-noise ratio even for the wFBP algorithm. In this paper,we will use the iterativemaximum-likelihood expecta-tion-maximization (MLEM) algorithm as a bench-mark to perform some comparison studies. Ourproposed algorithmwill be described in section 2.

2.Methods

2.1. Iterative Landweber algorithmThe iterative Landweber algorithm is a well-knowngradient decent algorithm (Landweber 1951). It hasmany versions. We propose a new version of theLandweber algorithm with additional immediately-after-backprojection filtering V, and we call it a quasi-Landweber algorithmwhich is be expressed as

a= + -- -( ) ( )( ) ( ) ( )X X VA P AX , 1k k T k1 1

where X( k) is the image array of the kth iteration resultexpressed as a vector, P is the projection arrayexpressed as a vector, A is the imaging matrix (alsoknown as the projection matrix), V is a low-pass filtermatrix, and α>0 is the update step size (also known

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as the relaxation parameter). To guarantee conv-ergence, the relaxation parameter α must be smallenough. In (1), AT represents the backprojectionmatrix. The matrix V in (1) is the immediately-after-backprojecton filter. The recursive expression (1) canbe transformed to a non-recursive form as follows.

å

aa aa a

a a

a a

a

= + -= + -= + -

´ + -

= -

+ -

- -

-

-

=

-⎡⎣⎢

⎤⎦⎥

( )( )( )

[ ( ) ]

( )

( ) ( )

( ) ( ) ( )

( )

( )

( )

X X VA P AX

VA P I VA A X

VA P I VA A

VA P I VA A X

I VA A A P

I VA A X . 2

k k T k

T T k

T T

T T k

n

kT n T

T k

1 1

1

2

0

1

0

If the square matrix (I—M) is non-singular, wehave the identity:

å = - -=

--( ) ( ) ( )M I M I M , 3

n

kn k

0

11

which can be readily verified by pre-multiplying (I–M)on both sides. In this paper, we assume the initialcondition X(0) to be zero and the matrix VATA ispositive definite. Thus the non-recursive form (2) canbe furtherwritten as a closed formwithout theΣ sign:

a a aa

= - -= - -

-

-

( ) [ ( ) ]( ) [ ( ) ]

( )

( )X VA A I I VA A VA P

VA A I I VA A VA P.4

k T T k T

T T k T

1

1

2.2. Fourier-domain representationThe intension of this section 2.2 is not to give amathematical proof or to develop any new theory. Itserves as a summary of our methods from the last 7years. It compares two discrete algorithms: the itera-tive Landweber algorithm and the FBP algorithm. TheFBP algorithm must be derived on a continuous-to-continuous imaging model, though the implementa-tion is discrete. It relates the similarities of the twoalgorithms. We then use one algorithm to approx-imate another discrete algorithm. In a previous pub-lication (Zeng et al 2013), a relation between theiterative Landweber algorithm and the FBP algorithmis establish by using calculus of variations.

The derivation of a Fourier-domain representa-tion for the iterative Landweber algorithm was repor-ted before (Zeng 2012, Zeng and Zamyatin 2013,Zeng 2014). For the sake of making this paper self-

contained, the main steps of the derivation are lis-ted here.

In tomography, the matrix ATA is a projection-backprojection operator. When it operates upon animage X, it can be approximated as a convolution ofthe image X with a two-dimensional (2D) 1/r kernel,where r is the distance to the origin (Zeng 2010). In theFourier domain, this 1/r convolution kernel corre-sponds to the w∣∣ ∣∣/1 transfer function (Zeng 2010).One can translate (i.e., approximate) thematrix repre-sentation to the Fourier (i.e., frequency domain)representation as shown in table 1. We usew w w= ( ),x y to represent the frequency vector in the2D Fourier domain. In the 1D Fourier domain, we useω to represent the frequency.

This matrix domain and Fourier domain approx-imation is good only for the ideal situation, where theprojector A represents the exact Radon transform. Adiscrete projector A is an approximation of the Radontransform. A better approximation can be achieved byusing a smaller pixel size and a larger image array.

Realizing that ATP is the pure backprojection ofthe sinogram P without ramp filtering, (4) is simply‘backproject first, then filter’ algorithms (Zeng 2010).In other words, in the Fourier domain, algorithm (4)can be implemented as the application of the filterw a w w- - ∣∣ ∣∣[ ( (∣∣ ∣∣) ∣∣ ∣∣) ]/v1 1 k to the pure back-projection of the sinogramdataP.

Thanks to the central slice theorem in tomography(Zeng 2010), a ‘backproject first, then filter’ algorithmcan be readily transformed into an FBP algorithm. Inour case, this FBP algorithm is almost identical to theconventional FBP algorithm, except that the conven-tional ramp filter |ω| is modified by a window function(i.e., a weighting function):

w a w w= - -( ) [ ( (∣ ∣) ∣ ∣) ] ( )W v1 1 , 5k

where the low-pass filter v(|ω|) can be freely chosen bythe user. In our previously published papers, thefunction v(|ω|) was always set to constant one (i.e., anall-pass filter). In fact, one can readily verify that thewindow functionW(ω) itself is a low-pass filter, and itsbandwidth increases as the k increases.

Formula (5) is themain result of this paperwith anewlow-passfilter v(|ω|). The implementationof the newFBPalgorithm is rather straightforward.The interested readerscan refer to (Zeng andZamyatin 2013) or (Zeng 2017) formoredetailed instructions.

Table 1.Correspondingmatrix and Fourier representations.

Matrix representation Fourier representation (transfer function)

Projection/backprojection ATA w∣∣ ∣∣/1

Identity transform I 1

Low-pass filter V w( )v-( )VA AT 1 w

w

( )( )v

a- --( ) [ ( ) ]VA A I I VA A VT T k1w a- - w

w

⎡⎣⎢

⎤⎦⎥( )( )

1 1v k

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng

2.3. Computer simulationsA two-dimensional (2D) circular lesion phantom, asshown in figure 1, was generated for the computersimulations studies. The imaging geometry was paral-lel with 180 uniformly-spaced views over 180°. Theimage array size was 180×180 (pixels). The detectorhad 180 detection bins, and the bin size was the sameas the pixel size. The phantom consisted of a largecentered disk (value 1) of radius 54, three circular hotlesions (value 2) of radius 2.7, 4.5 and 7.2, respectively,and two circular cold lesions (value 0.2) of radius 2.7and 4.5, respectively. The large disk is also referred toas the background. In order to avoid the inverse-problem crime, the projections (i.e., line integrals)were generated using smaller virtual detector bins. Thesize of an actual detector bin was 10 times the size of avirtual bin. Poisson noise was incorporated into theprojections to simulate the cases of emission com-puted tomography such as PET (positron emissiontomography) and SPECT (single photon emissioncomputed tomography). Then 10 virtual detectionbins were combined to form the projection in 1 actualdetection bin. Two noise levels were used. Thephantom pixel values were reduced to 50% for thelower count case. The total number of acquired photoswas 1.7×106 for the higher-count data, and the totalnumber of acquired photos was 8.5×105 for thelower-count data. Attenuation and scattering effectswere not included in the simulations studies.

The characteristics of contrast recovery and noisepropagation in a reconstruction algorithm can beshown by the contrast recovery coefficient (CRC)(Liow and Strother 1991) of each lesion versus the nor-malized standard deviation (σ/Mback) of the back-ground. TheCRC is defined as

=-

-

( )( ) ( )CRC

1

16

M

M rec

M

M phan

les

back

les

back

whereMles andMback represent the mean of the lesionand the mean of the background, and the subscripts‘rec’ and ‘phan’ denote the reconstruction and the truephantom, respectively. In computer simulations, wecalculated the CRC for each of the five lesions for everyreconstruction using the noiseless projections. Thelesion value Mles was the value at the center of thelesion. The background value Mback was the value atthe center of the large disc. For the hot lesions

- = - =( ) ( )/ /M M 1 2 1 1 1.les back phan For the coldlesions - = - = -( ) ( )/ /M M 1 0.2 1 1 0.8.les back phan

The image noise was evaluated using the normal-ized standard deviation of the noise in the central51×51 square region of the large disc reconstructedwith the noisy data as

ås=

--

=

( ) ( )M M N

x x1 1

17

back back i

N

i i1

2

where σ is the standard deviation, N is the number ofimage elements used in the calculation, xi is the valueof the ith pixel of the image reconstructed from noisydata, and xi is the expectedmean value of the ith pixel.The purpose of the normalization is to eliminate theinfluence on the noise measurement of non-uniformvalues of the image within the regions that aresupposed to be uniform.

2.4. SPECT experimentAHoffman brain phantomwas used in a SPECT study.The phantom was injected with Tc-99m and scannedfor 20 min. Three low-energy high-resolution collima-tors were used in a three-detector IRIX scanner. Thedetector pixel size was 2.3 mm and the detectedphotons were stored in a 256×256 array on eachdetector. Data acquisition was the step-and-shootmode with 120 views over 360°. One slice of thephantom data was used for image reconstruction, andthe total photon counts for that slice were 1.4×106.The reconstructed image was stored in an 88×88array.

3. Results

In the computer simulations, the proposed weightedFBP algorithm (denoted as wFBP-V) is compared withthe weighted FBP (wFBP) algorithm (Zeng andZamyatin 2013) as well as the iterative maximumlikelihood expectation maximization (MLEM) algo-rithm (SheppandVardi 1982,Lange andCarson1984).The wFBP algorithm is a special case of w =(∣ ∣)v 1 in(5). The conventional FBP algorithm (Shepp andLogan 1974) is a special case of the wFBP by setting ¥k in (5). Poisson noise weighting is incorporated

in all three algorithms.The low-pass filter w(∣ ∣)v chosen in this paper is

defined as

ww p

wp

=+

- <⎡⎣⎢

⎤⎦⎥( ) ( )

( )

v1 cos

2with

2 2.

8

10

Figures 2 and 3 show contrast versus noise plotsfor each lesion and for two different noise levels,respectively. Figure 2 contains plots for the highercount (i.e., less noisy) projections, and figure 3 con-tains plots for the lower count (i.e., noisier) projec-tions. The lesion contrast is characterized by the CRC(contrast recovery coefficient) values as defined in (6).

Figure 1.The computer generated lesion phantom.

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng

The noise is characterized by the normalized standarddeviation at the center of the large background disc asdefined in (7).

Each curve represents a series of reconstructionsabout one lesion. Each curve consists of 40 points, andeach point corresponds to a reconstructed image. For

Figure 2.Higher-count data reconstruction contrast versus noise comparison of three algorithms.

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng

each iteration number or k value, the algorithm recon-structs an image, the contrast of the lesion of interest iscalculated, and the noise variance at a uniform region

is also calculated from this reconstruction. The noisevariation is the horizontal coordinate of a point on thecurve, and the contrast is the vertical coordinate of the

Figure 3. Lower-count data reconstruction contrast versus noise comparison of three algorithms.

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng

point on the curve. In other words, the curves are scat-ter plots. All curves have the same number of points.Some curves look shorter than others because thepoints are denser on those shorter curves. The ‘goodpoints’ in the figures are close to the upper left corner,where the contrast is high and the noise is low. The‘bad points’ in the figures are close to the lower rightcorner, where the contrast in low and the noise is high.

For all cases infigures 2 and 3, the proposedwFBP-V algorithm is superiors to the other two algorithms.

The iterative ML–EM algorithm outperforms thewFBP algorithm for the hot lesions, while the iterativeML–EM algorithm and the wFBP algorithm havealmost the same performance for the cold lesions.

Some reconstructed images are displayed infigures 4 and 5 for the higher count and lower countprojections, respectively. The cold lesions in generalare slower to converge than the hot lesions. The itera-tion numbers or the k values were chosen as soon asthe CRC value formid-size cold lesion reached 0.95, in

Figure 4.Reconstructed images using higher-count projections with LEFT:wFBP-V,MIDDLE:MLEM, andRIGHT: wFBP. All threeimages have a CRCvalue of 0.95 for themid-size cold lesion. All images are displayedwith the same gray scale.

Figure 5.Reconstructed images using lower-count projectionswith LEFT:wFBP-V,MIDDLE:MLEM, andRIGHT: wFBP. All threeimages have a CRCvalue of 0.95 for themid-size cold lesion. All images are displayedwith the same gray scale.

Figure 6.Reconstructed images using SPECT experimental Hoffman brain phantomdatawith LEFT: wFBP-V,MIDDLE:MLEM,andRIGHT: wFBP.

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng

which we had kwFBP-V=20 000, kMLEM=90, andkwFBP=20 000, respectively.

For the real experimental Hoffman brain phantomstudy, three images are shown. They are reconstructedby the three algorithms which are the same as those forthe computer generated data. Only visual comparisonis provided infigure 6.

4.Discussion and conclusions

This paper is inspired by our older paper (Zhang andZeng 2006) where a low-pass filter was applied immedi-ately after the backprojection. This immediately-after-backprojectionfilterwas effective in improving the lesioncontract. Themain goal of this current paper is to extendthis immediately-after-backprojection filter strategy tothe weighted FBP (wFBP) algorithm. The wFBP algo-rithm is almost the same as the conventional FBPalgorithm, except that a window function (5) is used tomodify the ramp filter. The filter function v(ω) is newlyintroduced in this paper. This new function v(ω)approximates the effects of the immediately-after-back-projectionfilter in an iterative algorithm.

For the iterative algorithm (1), the matrix VATA isill-conditioned in tomography, and it may be singular.In our derivation, the matrix VATA is assumed to benon-singular and positive definite. Even if the matrixVATA is singular, it is still useful inmedical imaging aslong as it provides an early convergent trend, becausean early stopping is always used. In the algorithmdevelopment, we empirically establish a relationshipbetween an iterative algorithm and the FBP algorithm.Our earlier publications showed that we could use thisrelationship to use FBP to produce approximate ima-ges as an iterative algorithm could. We refer the read-ers to our earlier publications to see the similarreconstructions by these two algorithms (Zeng 2012).

Computer simulations presented in the paper showthat the new strategy is able to significantly improve thelesion contrast-to-noise ratio. We do not claim that theadded filter function v(ω) given in (5) is optimal in anysense. The function expression in (5)was obtained by thetrial-and-error method. However, any sort of low-passfilter v(ω) used in (5) is able to improve lesion’s contrast-to-noise ratio over the wFBP algorithm. It is stillunknown to us how to optimally select this function v(ω). The theory of enhancing lesion contrast-to-noiseratio is still yet to be developed, because the usual low-pass denoisingfilter reduces the contrast.

From a different perspective, we can also look atthe algorithm (1) as the conventional iterative Land-weber algorithm that has unmatched projector andbackprojector (Zeng andGullberg 2000); the projectoris A while the backprojector is the combination of Vand AT. The advantages and disadvantages of using anunmatched projector/backprojector pair have notbeen fully explored.

In our 2000 paper (Zeng andGullberg 2000), we pro-posed a criterion for the convergence: the eigenvalues ofVATA are all positive. In practice, this criterion is very dif-ficult to verify. In medical imaging, convergence is notourmain concern, because a converged algorithm gives avery noisy solution,which is too noisy to be useful in clin-ics. Stopping the algorithm early is a general acceptablerule. An algorithm is useful if it has an early convergenttrend. Our algorithm (1) has an early convergent trendand produces satisfactory images. This is the reason thatwe choose to continue todevelop it and extend it toFBP.

ORCID iDs

Gengsheng LZeng https://orcid.org/0000-0003-0790-6043

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Biomed. Phys. Eng. Express 4 (2018) 047005 GLZeng