Numerical studies on Feldkamp-type and Katsevich-type algorithms for cone-beam scanning along nonstandard spirals

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1 Numeical studies on Feldkamp-type and Katsevich-type algoithms fo cone-beam scanning along nonstandad spials Jiehua Zhu a,b, Shiying Zhao a, Hengyong Yu c, Yangbo Ye a, b, Seung Wook Lee d, *Ge Wang a, b a CT/Mico-CT Lab, Depatment of Radiolog b Depatment of Mathematics, Univesity of Iowa, Iowa Cit IA54, USA; c College of Communication Engineeing, Hangzhou Dianzi Univesit Zhejiang , China; d Koea Atomic Enegy Reseach Institute, Daejeon , South Koea ABSTRACT In this pape, we pefom numeical studies on Feldkamp-type and Katsevich-type algoithms fo cone-beam econstuction with a nonstandad spial locus to develop an electon-beam mico-ct scanne. Numeical esults ae obtained using both the appoximate and exact algoithms in tems of image quality. It is obseved that the two algoithms poduce simila quality if the cone angle is not lage and/o thee is no shap density change along the z - diection. The Katsevich-type algoithm is geneally pefeed due to its natue of exactness. Keywods: Computed tomogaphy (CT), cone-beam, nonstandad spial scanning, Feldkamp-type algoithm, Katsevisch-type algoithm. 1. INTRODUCTION Electon beam computed tomogaphy (EBCT), also known as ulta-fast computed tomogaph is a CT scanne specifically developed fo imaging of the heat. In the EBCT design, an electon beam is focused magnetically diected at the tungsten taget. Then, the taget yields a focus spot geneating X-ay photons. These photons penetate a patient, and ae detected by an aay of detectos fo use in tomogaphic econstuction. EBCT can vey accuately identify and quantify the calcium deposits in o aound coonay ateies, which is an indication of coonay heat diseases. Howeve, thee ae two majo weaknesses with the cuent EBCT techniques. Fist, it is not in cone-beam geomety and does not suppot spial/helical scanning, while it has been ecognized that spial/helical cone-beam scanning is advantageous fo the next geneation of biomedical CT. Second, the X-ay spot is not intensive sufficiently to poduce the image quality the mechanical otation based scannes can achieve. Recently Wang and Ye 1 poposed to upgade the EBCT design with a spial cone-beam scanning capability and adapt the achitectue fo small animal imaging, especially fo cadiac studies of the mouse. This electon-beam mico-ct (EBMCT) pototype is intended to impove the cuent tempoal esolution of mico-ct by an ode of magnitude, allowing moe than 10 tomogaphic econstuctions pe cadiac cycle of the mouse with adequate image quality. Non-standad spial cone-beam econstuction algoithms, of both appoximate and exact types, ae of ou inteest fo EBMCT. In the standad spial cone-beam image econstuction, both exact and appoximate algoithms have thei elative meits in tems of image quality. Ideall the exact algoithm econstucts an image accuately. Pacticall the appoximate algoithm often poduces satisfactoy esults, and is computationally efficient. Theefoe, both types of the algoithms have been extensively studied ove past two decades. In 1984, Feldkamp et al. 3 poposed an appoximate cone-beam algoithm fo cicula cone-beam scanning. The Feldkamp algoithm has poved to be quite excellent, especially fo a small cone angle. In 1991, Wang et al. 4 genealized the Feldkamp algoithm fo cone-beam scanning along vaious scanning loci, which woks well fo spheical, od-like o plate-shaped objects with vaious scanning loci including nonstandad spials. In 001, Katsevich 5 deived the fist theoetically exact econstuction fomula fo the spial cone-beam geomety in the filteed backpojection fomat. The wok by Katsevich is based on the PI-line and Tam-Danielsson detection geomety. Fo ou pupose of EBMCT, we studied PI-line geomety and genealized the Tam-Danielsson window in the nonstandad spial case of constant adius but vaiable pitch 6 and constant pitch but vaiable adii 7. Recentl we poposed a Katsevich-type algoithm fo vaiable adius spial scanning Developments in X-Ray Tomogaphy IV, edited by Ulich Bonse, Poc. of SPIE Vol (SPIE, Bellingham, WA, 004) X/04/$15 doi: /

2 v In this pape, we pefom a numeical study on Feldkamp-type and Katsevich-type algoithms fo cone-beam scanning along a nonstandad spial of constant pitch and vaiable adius. The pape is oganized as follows. A genealized Feldkamp algoithm and a genealized Katsevich algoithm ae biefly summaized in Sections and 3, espectively. Numeical simulation esults obtained using the two algoithms ae pesented in Section 4. The elated issues ae discussed in the last section.. GENERALIZED FELDKAMP ALGORITHM Let v \ ( = ( cos s, sin s, h( ), a s b, (1) be a nonstandad spial, whee s epesents the otation angle, h '( > 0 fo any s in [ a, b] so that h ( is always inceasing. In a 3-D coodinate system ( x, z), it is assumed that an object is centeed at the coodinate system oigin and its chaacteistic function f ( x, z) is confined by a cylinde U of adius R, that is, f ( x, z) = 0 if x + y > R. The poblem is then to econstuct the function f ( x, z ) fom the X-ay pojection images. The genealized Feldkamp econstuction 4 is pefomed based on the following fomula: 1 π u g( x, z) = Rs ( p, ξ ) f p 0 ( v) v + p whee ( u, v) denotes the otated coodinate system descibed by u = xcos s + y sin s, (3) v = xsin s + y cos s R s ( p, ξ ) the two-dimensional equi-spatial cone-beam pojection data, f ( ) a econstuction filte, ~ z ( ξ =, (4) s and the local coodinate system [ x, ~ z ( ] associated with the angle s is defined by ~ z ( = z h(. (5) dpds, () + ξ The genealized algoithm () allows vaious scanning loci. Hee we focus on the nonstandad spial scanning with vaiable adius but constant pitch. Basicall the following thee steps ae followed to econstuct a z -slice: collecting data fom a spial tun centeed on the z -slice, filteing the data, and backpojecting the filteed data onto the z -slice. 3. GENERALIZED KATSEVICH ALGORITHM Assume that the scanning locus satisfies the conditions fo the uniqueness of the PI line in the case of vaiable adius spial scanning 7, the Katsevich algoithm can be genealized fo vaiable adius cone-beam scanning 8. The genealized Katsevich algoithm consists of the following steps: Step 1: Compute the fist deivative of cone-beam data with espect to vaiable s fo a fixed diection pointing to x U fom a souce y( on the scanning locus (1). Step : Detemine the PI-segment, whose endpoints ae denoted by s b (x) and s t (x), x U, by fist consideing a family of the PI lines intesecting the vetical line V which contains x, and then numeically locating the PI line that intesects the line V and y s ) with s ( s, s + π ). ( t Step 3: Pefom slant filteing of the fist deivative data. Step 4: Backpoject the filteed data accoding to weights invesely popotional to the distance fom x to the souce y( to econstuct f (x). t b b Poc. of SPIE Vol

3 4.1 Imaging geometies and phantoms 4. COMPUTER SIMULATION We selected spials with monotonically inceasing R ( and constant pitch: h v ps \ ( s ) = ( cos s, sin s, h( ), R ( = as + b, h( =. (6) π Note that the Achimedean spial R ( = as + b satisfies the PI-line uniqueness conditions 7. The simulation paametes wee designed accoding to the peliminay specifications of ou EBMCT system as listed in Table 1. The scanning 5 locus was then specified as = s +, h = 5 mm. 6π 85 p Table 1. Peliminay specifications of the electon-beam mico-ct pototype fo small animal cadiac imaging. Field of view 30-50mm Numbe of views pe otation Souce to axis distance 30-60mm Spatial & contast esolution ~100µm, ~5% Souce to detecto distance mm Scanning speed ~ 50 HZ As shown in Fig. 1, the adius of mouse was taken to be R = 0 mm, the length of its thoax was L = 50 mm, the detecto plate was D = 95 mm away fom the cente axis of the mouse. Because the genealized Feldkamp algoithm equies that all the points in the field of view could be seen fom the coesponding spial tun, we can easily detemine the minimum detecto size. Suppose that the cente of the thoax is placed at the oigin, as shown in Fig., to make the bottom slice zbot = 5 mm visible fom the stating point s1 = 3π on the lowe half tun, the width and height of the detecto plate shall be espectively D + s1 )) h( D + s1 )) W = = 4 mm, H = = 313 mm. (7) R ( s1 ) R s1) R To let this slice visible fom the ending point of the uppe half tun, s = π, we need W = 165 mm and H = 185 mm espectively. Then, we can conside the top slice z top = 5 similaly. It is staightfowad to obtain the uppe-half-tun values W = 117 mm and H = 107 mm, and the lowe-half-tun values W = 134 mm and H = 13 mm in the top-slice case. Theefoe, the minimum detecto plate should be 4mm by 313mm as equied by the genealized Feldkamp algoithm. Othe half-scan o PI-line based Feldkamp-type algoithms wee not included in this peliminay wok. Souce Scanning e-beam Mouse Spial locus Detecto plate Fig. 1.Electon-beam mico-ct using vaiable adius spial cone-beam scanning. 560 Poc. of SPIE Vol. 5535

4 y z Half H s = π R s1 = 3π D W x s = π s ) D x s1 = 3π R Half H s 1 ) Fig.. Pojective views of the imaging geomety fo the bottom slice z = 5 mm. Top view fo the computation of the detecto width, side view fo the computation of the detecto height. In the genealized Katsevich algoithm, the minimum detection window is the egion in the detecto plate bounded by the cone-beam pojections of the uppe and lowe tuns of the nonstandad 3D spial locus stating fom a given souce position. We deduced the following bounday equations of the minimum detection window fo the souce position y( s 0 ) 7, ( R + s0))sin( s s0) ( R + s0))( s s0) h d1 =, d =. (8) s0) cos( s s0) s0) cos( s s0) Because the smallest scanning adius leads to the lagest cone angle and the geatest detection window, we can simply compute the size of the detecto plate fo the souce with the smallest scanning adius. In this wa we have the width 4mm and the height 138mm. In ou simulation, the paametes wee nomalized to unit elative to the object adius in the cases of the Shepp-Logan phantom and the disk phantom. The imaging geometies fo both the algoithms ae given in Table. Table. Nonstandad cone-beam scanning paametes used in the numeical simulation. Scanning adius GENERALIZED FELDKAMP = s + GENERALIZED KATSEVICH = s + 6π 6π Helical pitch 5mm 5mm Object adius 0mm 0mm Oigin to detecto distance 95mm 95mm Numbe of views pe otation Detecto size 4mm by 313mm 4mm by 138mm Numbe of detecto ows Hoizontal detecto sampling inteval Numbe of detecto columns Vetical detecto sampling inteval Reconstuction matix Poc. of SPIE Vol

5 The 3D Shepp-Logan phantom 9 consists of 10 ellipsoids. The effective X-ay absoption coefficient at a point is the sum of the elative paametes of the ellipsoids containing that point. Fig. 3 shows the slice z = 0. 5 of the Shepp-Logan phantom with the display inteval [ 0.95, 1.05]. The disk phantom consists of seven ellipsoids of half axes 0.7, 0.7 and 0.06, placed 0.0 apat along the z -axis. All ellipsoids in the disk phantom ae of density 1.0. The slice x = 0 of the disk phantom is shown in Fig. 3 with the display inteval [ 0, 1]. 4. Numeical esults Fig. 3. Repesentative slices of the Shepp-Logan phantom and the disk phantom. Slice z = 0. 5 of the Shepp-Logan phantom, and slice x = 0 of the disk phantom. Fig. 4 shows the econstuction images at z = 0. 5 of the Shepp-Logan phantom using the genealized Feldkamp algoithm and the genealized Katsevich algoithm, espectively. The mean-squaed-eo between the oiginal and econstucted slices and the econstuction time ae listed in Table 3. Fig. 4. Reconstucted slices of the Sheep-Logan phantom coesponding to Fig. 3. Genealized Feldkamp econstuction, and genealized Katsevich econstuction. Table 3. Expeiment data fom genealized Feldkamp and genealized Katsevich econstuctions. Genealized Feldkamp algoithm Genealized Katsevich algoithm Fig. 4 Fig. 5 Fig. 4 Fig. 5 Mean-squaed-eo CPU time (min.) Poc. of SPIE Vol. 5535

6 It can be obseved that the slice econstucted by the genealized Katsevich algoithm contains little shading atifacts, while the genealized Feldkamp algoithm uns faste but induces a dak egion in the top half pat. As illustated with the specified spial locus, the genealized Feldkamp algoithm also equies a lage detecto plate than the genealized Katsevich algoithm. This is because the full-scan based Feldkamp-type algoithm equies that cone-beam pojections fom opposite souce positions within one spial tun should ovelap fo full-scan based fan-beam econstuction afte the cone-beam data ae coected fo thei obliqueness 4. Expeiments wee also pefomed with the disk phantom to study the situation when the object contains a shap gadient along the z -diection. Longitudinal coss-sections of the econstucted disk phantom ae shown in Fig. 5, coesponding to Fig. 3. It can be obseved that with the disk phantom the genealized Feldkamp algoithm poduces substantial atifacts between the ellipsoids, while the genealized Katsevich algoithm consistently gives excellent image quality. Fig. 5. Reconstucted slices of the disk phantom coesponding to Fig. 3. Genealized Feldkamp econstuction, and genealized Katsevich econstuction. To evaluate the noise chaacteistics of these two algoithms, we added Gaussian noise with standad deviation 1.0 to synthetic pojection data. The noise level was evaluated by the mean-squae diffeence between econstuction with and without added noise. The noise measues with the genealized Feldkamp algoithm and the genealized Katsevich algoithm ae and , espectively. This esult indicates that the genealized Feldkamp algoithm is less sensitive to data noise, because it only equies high-pass filteing once while the genealized Katsevich algoithm equies high-pass filteing twice. Fig. 6. Reconstucted slices of a dynamic Shepp-Logan phantom. Genealized Feldkamp econstuction, and genealized Katsevich econstuction. Poc. of SPIE Vol

7 Futhemoe, we studied dynamic cone-beam econstuction using these two algoithms. Fo that pupose, the lagest bight ellipsoid in the Shepp-Logan phantom was moving unifomly along the hoizontal diection. Fig. 6 shows genealized Feldkamp and genealized Katsevich econstuctions, espectively. Thee ae much moe motion atifacts in the genealized Feldkamp econstuction than the genealized Katsevich countepat, because we used the full-scan based Feldkamp-type algoithm. Cleal the tempoal esolution with the Feldkamp-type econstuction can be easily impoved using half-scan o PI-line based Feldkamp-type algoithms, but the noise pefomance may be accodingly compomised. 5. DISCUSSIONS AND CONCLUSION The phantoms we used ae still simplistic. It is highly desiable that the phantom be as ealistic as possible in the CT simulation. Supequadics 10 ae a family of 3D objects, which can be used to model a vaiety of objects. They have eceived a majo attention in the fields of compute vision, compute gaphics, and obotics, because of thei compact fom and epesentation capabilities. 11 Recently we poposed an algoithm fo the computation of the X-ay tansfom of a quite geneal type of supequadics and demonstated its utility with a supequadic-based human thoax phantom. 1 Fig. 7 shows the slices of ou human thoax phantom at z = 0 and y = 0. It can be obseved that ou phantom is moe ealistic than the popula phantom used in the liteatue. Cuentl we ae developing a dynamic mouse model based on supequadics to test both appoximate and exact cone-beam algoithms. Fig. 7. Repesentative slices of a supequadics-based human Thoax phantom. Slice z = 0, and slice y = 0. In conclusion, we have evaluated the genealized Feldkamp algoithm and the genealized Katsevich algoithm fo EBMCT. Biefly speaking, the Feldkamp-type and Katsevich-type algoithms give compaable spatial esolution when the cone angle is not lage (the spial pitch is not lage), and/o the longitudinal vaiation of the object is not damatic. Geneally speaking, the Feldkamp-type algoithm uns faste, is less sensitive to noise, moe sensitive to motion, and poduces moe atifacts than the Katsevich-type algoithm. These esults ae consideed a necessay and impotant step towads dynamic volumetic mico-ct of the small animals. Futhe compaative studies ae undeway to eveal thei elative stengths and weaknesses systematically. ACKNOWLEDGEMENTS This wok was patially suppoted by the NIH/NIBIB gants EB00667 and EB REFERENCES 1. G. Wang and Y. Ye, Nonstandad spial cone-beam scanning methods, appaatus, and applications (Patent disclosue), Poc. of SPIE Vol. 5535

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