IEEE TRANSACITONS ON NUCLEAR SCIENCE, VOL 37, NO. 2, APRIL 1990 773

ACCELERATED IMAGE RECONSTRUCTION FOR A CYLINDRICAL POSITRON TOMOGRAPH USING FOURIER DOMAIN METHODS

C. W .S tearns, D. A.Chesler, and G.L.Brownel1 Physics Research Laboratory

Massachusetts General Hospital, Boston ,M A*

Abstract I

We present a method to reconstruct projection images through assembly in the Fourier domain, rather than by backprojection. By bypassing the rate-limiting backpro- jection step of filtered backprojection, the new algorithm operates more quickly than previous methods. Fourier do- main operations are also used to form forward-projected views of the imaged object; portions of these views are re- quired to operate a full three-dimensional reconstruction using the “cross-plane” projection planes. These Fourier domain algorithms will form the basis for the reconstruc- tion algorithm in PCR-11, a volumetric positron imager currently under construction at MGH.

1 Introduction

PCR-I1 is a volumetric P E T instrument currently under construction at MGH [l]. PCR-I1 will achieve high sensi- tivity by accepting the so-called “cross-plane’’ coincidence events, those which do not lie in the transverse planes of the instrument. In all, PCR-I1 will collect projection mea- surements from over 27 million different coincidence pairs. The image reconstruction for PCR-I1 is not separable into a stack of two-dimensional image reconstructions because of the contribution of the cross-plane rays. A full three- dimensional reconstruction algorithm is necessary to pro- cess the PCR-I1 data set into a volumetric image.

Pelc and Chesler developed an algorithm for three- dimensional filtered backprojection [2]. Other investiga- tors have recently approached the three-dimensional image reconstruction problem through space-domain based algo- rithms [3],[4],[5]. We have chosen to develop a reconstruc- tion algorithm in the Fourier domain, in order to avoid the backprojection operation which limits the speed of t,hese algorithms.

‘Address for correspondence: Charles W. Stearns, GE Medical Systems, P.O. Box 414, W-641, Milwaukee, WI 53201.

2 Fourier Domain Reconstructioll We have previously reported preliminary results using the Fourier domain algorithm [6]. The new algorithm, like fil- tered backprojection, is based on the Projection-Slice The- orem, which demonstrates that the Fourier transform of a projection plane view is a slice sample of the Fourier trans- form of the original object. The principle of the new algo- rithm is to avoid the backprojection step of filtered back- projection by directly “placing” the Fourier transform of the filtered projection plane onto a rectilinear grid in the Fourier domain, in the location specified by the Projection- Slice Theorem. The image is assembled in the Fourier do- main, rather than the space domain, and a final three- dimensional inverse F F T is performed at the conclusion of the algorithm to produce the image.

An observation reported in [6] was a loss of recovery of image signal intensity for points far from the origin of im- age space. This was attributed to the smoothing effects of the Fourier domain placement step. The duality of the forward and inverse Fourier transforms provides an expla- nation for the recovery loss. Space domain smoothing is easily interpreted as a high-frequency attenuation of the signal; by duality, frequency domain smoothing represents a high-space attenuation of the signal. Also, in the same manner that a space domain signal can be “unsmoothed” by applying a high-pass filter to the signal, a frequency do- main smoothing can be compensated by applying a high- space-pass filter to the signal.

The shape of the recovery loss, and hence the multi- plicative correction factor, is based on the form of the smoothing operation which is performed in the placement step of the algorithm. When a linear weighting function is used in the placement operation (represented by a triangle function with a width of one Fourier domain sample unit), the expected correction factor in the space domain is the Fourier transform of a triangle function, or:

sin2 (z) A(w) = (a2

along each dimension which is involved in the interpola- tion. In this equation, w refers to the dimension of interest, and W is the grid size along that dimension.

0018-9499/90/0400-0773$01.00 Q 1990 IEEE

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Position along x=y

Figure 1: Application of the recovery correction factor does not produce equal recovery as the source is moved from the center of image space. Hollow points represent the uncorrected data; filled points are corrected as discussed in the text.

Figure 1 presents the results of four separate image re- constructions, with the imaged object placed at the origin, ( l2 ,12 ,0 ) , (24,24, 0), and (3G136, 0). The image intensity along the line t = y on the z = 0 plane is plotted; since there is blurring along both k, and k, , we apply the correc- tion factor twice, for a sin4(u)/(u4) form of the correction function. This theoretically-derived multiplicative correc- tion factor does not precisely undo the recovery loss, but overcompensates to a slight degree. Previously, this was attributed to the “incompleteness” of the blurring function in the Fourier domain (i.e,, some Fourier domain samples were aligned precisely with the grid, and therefore were not blurred), and a quantitative correction was fit to re- constructed images by increasing the value of W in the correction equation to about 5% more than the true grid dimension.

In addition, there were low-level artifacts in the pictures, as depicted in Figure 2. As the imaged object is moved from the center of image space, the artifact grows in size. In the worst case depicted in the figure, the maximum amplitude of the artifact is 6% of the maximum voxel in- tensity. Since this artifact propagates throughout image space, it will influence quantitative measurements taken from images reconstructed using the Fourier domain algo- rithm.

3 Axial Shearing on the Projec- tion Planes

In the work reported in [6], the recovery roll-off occurred in all three dimensions. This is because the Fourier domain

Figure 2: Artifacts present near zero signal amplitude as the source is moved from the center of image space. Top row. Source at ( O , O , O ) and (12, 12,O). Bottom row. Source at (24,24,0) and (36,36,0).

transformation ( k , , k,,) --t ( k , , 12, , k , ) , generally produces non-integer values for k, , k , and k , , and the Fourier do- main “smoothing” must be active in all three dimensions. However, the projection plane coordinate system may be modified to eliminate the need for IC,-axis blurring. If the axial coordinate along the projection plane is redefined:

The Fourier domain samples will be spaced according to the equation:

IC,,+ = k,, sin0. (3)

The k , component of the projection data location, derived from the Projection-Slice Theorem, is then defined by:

k , = k,, sin 0 = k U + . (4)

Since the values of kv* are integers on the projection plane, kz always lies on the image grid, and no interpolative place- ment step is necessary along the t , direction.

If the placement operation no longer effects smoothing along the k, direction, then there should be no recovery attenuation along the z-axis in the reconstructed image. To demonstrate this effect, a set of reconstructions was performed with the imaged object placed at the origin, ( O , O , G ) , ( O , O , 12), and ( O , O , 18). Data profiles along the z-axis are presented in Figure 3. There is no recovery loss as the object is moved from the origin.

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12 ,

&I 2 -2 - - 1 6 - 8 0 8 1 6 2 4 3 2

Position along Z axis Figure 3: Reconstruction using axial shearing in the off-plane projection views produces equal recovery along the z-axis without application of a correction factor.

;1 -2 - 2 0 0 2 0 4 0 6 0

Position along x=y Figure 4: With 2X oversampling, the recovery correction factor effectively equalizes the four reconstructed objects.

4 Oversampling in the Fourier Domain

The low-level artifacts, as well as the inaccuracy of the theoretical recovery correction factor, are caused by the inaccuracy of the placement step in the Fourier domain. The projection planes used in the reconstruction algorithm are the same size as the reconstructed image volume ( i . e . , a 128x128~64 image volume is viewed by 128x64 projec- tion planes); the Fourier transform of the projection plane is therefore sampled at the same grid size as the Fourier transform of the reconstruction. This provides too few Fourier domain samples for the placement step to oper- ate effectively. The placement step is more effective if the transformed projection plane is sampled (in the Fourier domain) more finely than the reconstructed image grid on which it is to be placed.

Increasing the Fourier domain sampling density is ac- complished by zero-padding the projection direction, along the directions which the blurring effects of the placement step will be operating. Because the axial shearing of the projection planes removes the need for blurring along the tz direction, zero-padding is only needed along the trans- verse direction. The transverse direction on the projection plane is zero-padded to increase its dimension from W to bW. The Discrete Fourier Transform of the projection plane then has KW samples in the transverse direction, providing oversampling by a factor of t for the placement operation.

The effects of oversampling on the recovery correction are shown in Figure 4. This figure represents the same experiment as that depicted in Figure 1, except that zero- padding to achieve a factor of two oversampling was per-

formed on each projection plane. The four objects are now, after application of the theoretically derived correction fac- tor, equal in amplitude.

In addition, oversampling reduces the low-level artifact present in the images. Figure 5(a) shows the low-level artifact, a t the same scale as in Figure 2, in the case of twofold oversampling. The maximum extent of the arti- fact is about 1.5% in the image of the object a t (36 ,36 ,0 ) . The artifact can be reduced further by increasing the over- sampling (Figure 5( b)) ; fourfold oversampling reduces the artifact size to below 1%.

Oversampling increases the running time of the recon- struction algorithm; t-fold oversampling increases the run- ning time of the algorithm by about a factor of k. The relative running speed measured for the reconstruction of a 128 x 128 x 64 image from 896 projection planes is pre- sented in the following table:

Back p ro j ec t ion No oversampling 2X oversampling

The exact running time factors will vary from computer to computer, depending on the exact configuration and degree of code optimization. However, the Fourier domain algorithm, even with oversampling, is considerably faster than filtered backprojection.

5 Missing Data Reconstruction methods based on Fourier domain princi- ples, including both filtered backprojection and the new Fourier domain algorithm discussed in this paper, require

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Figure 5: Artifacts reduced by oversampling. In each case, the source is at (12,12,0) and (36,36,0) . Top row. 2X Oversampling. Bottom row. 4X Oversampling.

that each projection view is “complete”, seeing the en- tire field of view. This requirement is not met in the case of a cylindrical P E T detector such as PCR-11. The “in- plane” projection views (projection directions normal to the symmetry axis of the detector) are complete, but the “cross-plane” views are truncated at the ends of the in- strument, due to the finite extent of the PCR-I1 detector system. In order to apply either filtered backprojection or the Fourier domain algorithm for reconstruction of PCR-I1 images, a method to “fill in” for this missing data must be implemented.

The completeness of the “in-plane” projection views pro- vides the means to supply the missing data. Because these views are complete, a reconstruction can be performed us- ing only the in-plane projections. Note that this recon- struction is equivalent to the “stacked plane” reconstruc- tion methods used in multiplanar P E T instruments. This image then forms the basis for the missing projections. The required projection data may be produced by performing forward projections from this image (as in [5]), or the data may be obtained from the Fourier domain.

We have implemented a Fourier domain-based algorithm to select a projection plane from the Fourier transform of the image formed by reconstructing the in-plane pro- jections. The algorithm is the opposite of the placement step of the Fourier domain reconstruction algorithm; the Fourier transform of the projection plane is estimat.ed by interpolation from the image transform. As is the case in the placement step, the blurring of frequency domain val- ues leads to attenuation of the image signal away from the

Figure 6: Demonstration of Fourier domain method of pro- jection plane estimation. Above. T h e projection data. Below. Same projection plane, as estimated by Fourier domain method.

origin of image space. This is compensated for prior to the slice selection step, by applying the same correction fac- tor (see Equation 1, above) as is used the reconstruction algorithm.

Figure F demonstrates the efficacy of the Fourier domain method to estimate a projection plane. The upper portion of the figure is a “true” (computed) projection plane of the phantom (five small spheres on the y = 0 plane). The lower portion of the figure is the corresponding projection view, estimated from a reconstructed image of the phantom.

6 Conclusion

We have developed Fourier domain methods to perform the rate-limiting backprojection step of filtered backpro- jection, producing an algorithm which reconstructs a com- plete projection data set in less time than the equivalent filtered backprojection algorithm. Inaccuracies in image

recovery and artifacts in the image are reduced by over- sampling in the Fourier domain; this is easily accomplished by zero-padding the projection dat,a prior to entering the Fourier domain. In addition, we have applied these meth- ods to the problem of filling in for the “missing data” in the projection data set from a PET instrument of finite ex- tent. These techniques form the basis of the reconstruction algorithm which will be implemented for PCR-11.

Acknowledgements This work was supported by NIH grant 5 R01 CA32873 09, and by DOE grant DE FG02 87ER60519.

References [l] C.A. Burnham, D.E. Kaufman, D.A. Chesler, C.W.

Steams, D.R. Wolfson, and G.L. Brownell. Cylindri- cal PET detector design. I E E E Trans Nucl Sca, NS- 35(1):675-679, 1988.

[a] N.J. Pelc and D.A. Chesler. Utilization of cross-plane rays for three-dimensional reconstruction by filtered back-projection. J Comput Assast Toinogr, 3(3):385- 395, 1979.

[3] P.E. Kinahan, J .G. Rogers, R. Harrop, and R.R. John- son. Three-dimensional image reconstruction in object space. I E E E Trans Nucl Sci, NS-35( 1):635-638, 1988.

[4] D.W. Townsend, T. Spinks, M.S. Gilardi, A. Geissbuh- ler, J . Heather, and T Jones. Three-dimensional recoil- struction of PET data from a ring tomograph. IEEE Trans Nucl Scz, NS-36( 1):1056-1065, 1989.

[5] P.E. Kinahan and J.G. Rogers. Analytic three- dimensional image reconstruction using all detected events. I E E E Trans Nucl Sci, NS-36( 1):964-968, 1989.

[GI C.W. Steams, D.A. Chesler, and G.L. Brownell. Three dimensional image reconstruction in the fourier do- main. I E E E Trans Nucl Sci, NS-34( 1):374-378, 1987.