Simulation of Parallel Beam CT Reconstruction Algorithm Based on Grating Phase Contrast Imaging

Transcription

1 American Scientific Reearch Journal for Engineering, Technolog, and Science (ASRJETS ISSN (Print , ISSN (Online Global Societ of Scientific Reearch and Reearcher Simulation of Parallel Beam CT Recontruction Algorithm Baed on Grating Phae Contrat Imaging Rong Feng a*, Yang Yadong b, Ma Xuehao c a,b,c School of Electronic and Information Engineering, Tianjin Poltechnic Univerit, Tianjin, , China a huheng677@163.com b @qq.com c @qq.com Abtract X-ra phae contrat imaging ha higher reolution in the imaging of light element material than aborption imaging, and i increaingl ued in man field uch a medicine, biolog and material cience.grating phae contrat imaging tand out among man phae contrat imaging method due to it unique advantage.thi paper briefl introduce the filtering back projection recontruction algorithm of the traditional parallel beam CT, and then introduce the Hilbert filter back projection algorithm uitable for the parallel beam CT of the grating phae contrat. Finall, the imulation of the parallel beam CT recontruction algorithm baed on the grating phae contrat imaging i carried out.through the detailed introduction of the realization principle of traditional CT imaging and phae contrat CT imaging, it can be een that there i no eential difference between the two implementation principle.the recontruction algorithm of traditional CT imaging i mature, and then onl need to be lightl improved to be uitable for CT recontruction baed on grating phae contrat imaging. Keword: Parallel beam CT; Grating phae contrat imaging; Filtered back projection; Hilbert filter back projection. 1. Introduction Conventional X-ra imaging i baed on the aborption principle of X-ra b ubtance. Since ome ubtance compoed of light element aborb X-ra weakl, high contrat image cannot be obtained b conventional X-ra imaging technique.when X-ra pa through a ubtance * Correponding author. 51

2 not onl the intenit i attenuated but the phae alo change.the interaction of X-ra with matter can be expreed b a complex refractive index, ie: 1 δ + iβ [1], Where δ i the phae factor, β i the aborption factor.in hard X-ra, the phae factor of a ubtance compoed of light element i uuall 1000 to 100,000 time that of the aborption factor[2], o that a higher reolution phae contrat pattern can be obtained b obtaining the phae change amount of the X-ra after paing through the light element material. At preent, the main phae contrat imaging method include interference imaging[3~5], coaxial imaging[6~10], diffraction enhanced imaging[11~13], and rater imaging[14~18].the firt three imaging method mut ue a coherent light ource, and the imaging field of view i mall, thu limiting it promotion and development.in 2006, F.Pfeiffer realized the application of common X-ra ource in grating imaging b uing Talbot-Lau effect[19], which reduced the requirement of grating imaging for light ource coherence, improved imaging field of view, and made it practical.due to it unique advantage, the grating imaging method ha become a hot topic in recent ear. At the ame time, the phae extraction method baed on the grating imaging method and the CT imaging recontruction algorithm have alo made man breakthrough.huang and hi colleague ued Hilbert Filtered Back-Projection Algorithm (HFBP to directl recontruct the phae information ditribution of the object through the obtained phae firt-order differential information[20].the HFBP algorithm wa firt ued for CT recontruction of diffraction-enhanced imaging. Since the diffraction-enhanced imaging method and the grating phae contrat imaging method are both the firt-order differential information of the projection, the CT recontruction algorithm baed on diffraction-enhanced imaging i alo uitable for the grating imaging method. The HFBP algorithm draw on the filtered back-projection (FBP of traditional CT. At the ame time, baed on the HFBP algorithm, a erie of CT recontruction algorithm uitable for grating phae contrat imaging are developed b uing the idea and method of traditional CT.Therefore, the FBP recontruction algorithm in traditional CT and the HFBP algorithm in phae contrat CT are particularl important.thi article will briefl introduce the related content of parallel beam aborption CT and parallel beam phae contrat CT baed on grating imaging.finall, the imulation experiment of HFBP algorithm baed on grating imaging method i carried out. 2. parallel beam aborption CT The Autrian mathematician Radon propoed recontructing image through projection and became the theoretical bai of CT technolog. Therefore, acquiring projection data i an important part of CT technolog.the projection of the aborption CT i the line integral of the attenuation coefficient µ of the ubtance paing through the object path in the projection direction, which can be expreed a: p = g( x, dl (1 Where l i the ra propagation path, g ( x, i the attenuation coefficient of the object to the ra, and p i the reulting projection.the Fourier tranform of the projection i equal to the central ection of the Fourier tranform of the object, which i the famou central lice theorem.baed on the projection back projection and 52

3 the center lice theorem, a Filtered Back-Projection (FBP or Convolution Back-Projection (CBP algorithm for traditional aborption CT i developed. The FBP algorithm can be expreed a: π 0 j2πw( x coθ + inθ g( x, = P ( w e w dwdθ (2 θ Where g( x, i the attenuation coefficient ditribution function of the object to be recontructed, and (w i the Fourier tranform of the projection obtained b the X-ra of the direction angle θ paing through the object.the FBP algorithm firt obtain the projection data of the ra ource rotated 180 around the object, then elect the appropriate filter to filter the projection data, and finall backproject the filtered projection back to complete the image recontruction. P θ 3. parallel beam phae contrat CT The phae hift after the X-ra pae through the object can be expreed a[21]: 2π Φ = d ( x,, z dl l (3 Where l i the ra propagation path.it can be known from equation (1 and (3 that the phae hift after X-ra paing through the object ha the ame phical meaning a the intenit attenuation.therefore, a long a the phae hift information i obtained, the FBP or CBP algorithm of the traditional CT can be directl ued to directl recontruct the ditribution of the phae hift factor of the object.however, the grating phae contrat imaging method obtain the firt-order differential information of the phae hift, which can be expreed a[22]: 1 θ k Φ (4 Where k i the wave number of the X-ra.Since the obtained phae hift information doe not atif the rotation invariance, the CT recontruction cannot be directl performed uing the FBP or CBP algorithm. Figure 1: Schematic diagram of parallel beam canning 53

4 To implif the calculation, it i aumed that the phae hift factor of the object to be recontructed i onl related to the x- coordinate axi, regardle of the z axi.figure 1 i a chematic diagram of parallel beam canning.where x- repreent the coordinate tem of the object, -t repreent the rotation coordinate, l i the projection direction, a i the angle between the l and the axi, and the angle between the axi and the x axi of the rotation coordinate i alo a.since the grating phae contrat obtain the firt-order differential information of the phae hift and i perpendicular to the projection direction.when the X-ra projection direction i a hown in Figure 1, the combination of equation (3 and (4 can be obtained: d( x, θ = dl (5 Since x change with the projection direction and doe not atif the rotation invariance, the partial hift of the phae hift factor to the x and axe atifie the rotation invariance, o that: δ ( x, δ ( x, = coα + inα x (6 Where α i the rotation factor and i alo the angle between the X-ra projection direction and the -axi.since and x doe not change with the projection direction, i determined when the projection direction α i fixed.in the HFBP algorithm, the two-dimenional Fourier tranform of, δ ( x,, and x are: v, v, v, v.among them: x v = x j2πuθ( v (7 v = j2π vθ( v (8 Θ (9 ( v = Θ x ( vcoα + v inα From equation (7(8(9: v = j2π ( u coα + v inα Θ( v (10 Convert the coordinate tem ( u, v to the polar coordinate tem ( ρ, α, and the polar axi coincide with the u-axi, then: v = j2πρθ( v (11 54

5 From equation (1(2(3, it can be een that the phae hift factor δ ( x, can be recontructed into the FBP algorithm of the aborption CT, which can be expreed b polar ( ρ, α coordinate: δ ( x, = π j 2πρ ( x coα + inα ( v ρ e δρ 0 Θ (12 Subtituting equation (11 into (12 give: δ ( x, π = 0 v[ j gn( π] e j2πρ ( x coα + inα δρ (13 among them: 1, ρ > 0 gn( ρ = 1, ρ < 0 (14 It can be known from equation (13 that the firt-order differential data of the phae hift i multiplied b j gn( p in frequenc, which i equivalent to the Hilbert tranform.therefore, the HFBP algorithm i to obtain the firt-order differential information of the phae hift obtained b rotating the ra around the object b 180, then perform Hilbert filter tranformation on the obtained data, and finall backproject the filtered data back. 4. Simulation experiment Due to experimental condition, the projection data of the actual grating phae contrat imaging cannot be obtained.the projection data of the grating phae contrat i obtained b imulation, and the imulation tet of HFBP algorithm i carried out.the imulation model elect the Shepp-Logan head model a hown in Figure 2. The image reolution i 256x256, and the image area value repreent the phae factor of the object to be recontructed.the experimental tep are a follow: Obtain a gradient in the horizontal direction of the image and a gradient in the vertical direction. A hown in Figure 1, when the angle between the projection angle and the -axi i α, it can be known from equation (6 that the projection data in the grating phae contrat CT i the um of the product of the horizontal gradient and the vertical gradient of the image and the rotation factor, repectivel. The projection angle i from 0 to 179, and the interval i 1. Step (2 i repeated to obtain the firtorder differential information of the projection. The projection data i filtered b Hilbert, and the filtered projection data i back-projected back, and the CT recontruction map hown in Figure 3 i obtained. 55

6 Figure 2: Head model original image Figure 3: CT recontruction map It can be een from Figure 3 that the firt-order differential information of the projection of the phae contrat CT i obtained b imulation, and then the recontruction of the image i completed ver well b uing the HFBP algorithm. 5. Concluion B comparing the FBP algorithm in the aborption CT with the HFBP algorithm in the phae contrat CT, it can be een that there i no eential difference between the conventional CT and the phae contrat CT.The traditional CT theor i relativel mature. B referring to the traditional CT recontruction theor method and the idea of recontructing the phae factor ditribution directl uing the phae firt-order information in the HFBP algorithm, the traditional CT theor method can be applied to the grating phae contrat CT. Reference [1] Al-Nielen J, McMorrow D. Element of modern X-ra phic[m]. John Wile&Son Ltd, [2] Wilkin S W, Gureev T E, Gao D, et al. Phae-contrat imaging uing polchromatic hard X-ra[J]. Nature, 1996, 384(6607: 335. [3] Atuhi Momoe,Tohoru Takeda,Yuji Itai.Phae-contrat X-ra computed tomograph for overving biological pecimen andorganic material[j].rev.sci.intrum.,1995,66(2:1434~1436. [4] Atuhi Momoe,Jun Fukuda.Phae-contrat radiograph of nontained rat cerebellar pecimen[j]. Med.Ph.,1995,22(4:375~379. [5] Zheng Defeng,Wang Xiangzhao.Optimum deign of parallelplate interferometer for angular diplacement meaurement[j].chin.j.laer,2007,34(3:422~426. [6] A.Snigirev,I.Snigireva,V.Kohnet al..on the poibilitie of X-ra phae contrat microimaging b coherent high-energnchrotron radiation[j].rev.sci.intrum.,1995,66(12:5486~5492. [7] S.W.Wilkin,T.E.Gureev,D.Gaoet al..phae-contrat imaging uing polchromatic hard X-ra[J]. Nature,1996,384(28:335~338. [8] K.A.Nugent,T.E.Gureev,D.F.Cookonet al..quantitative phae imaging uing hard X-ra[J]. Ph.Rev.Lett.,1996,77(14:2961~2964. [9] P.Cloeten,W.Ludwig,J.Baruchelet al..holotomograph:quantitative phae tomograph with micrometer reolution uing hard nchrotron radiation X-ra[J].Appl.Ph.Lett.,1999,75(19:2912~ 56

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