image in the frequency domain with a two-dimensional ramp filter. The major drawback of BpjF is that the mean value of the nxonstruction is not correct.
CAN THE BACKPROJECTION FILTERING ALGORITHM BE AS ACCURATE AS THE FILTERED BACKPROJECTION ALGORITHM? Gengsheng L. Zeng and Grant T. Gullberg Department of Radiology, University of Utah Salt Lake City, Utah 84132, USA
Abstract
Fop many years people have realized that the image nxmnstructed by the filtered backprojection (FBP) algorithm is more accurate than the image reconstructed by the backprojectionfiltering(BpjF) algorithm. The FBP algorithm is implemented in two steps: (1) convolve the projections with a kernel function, then (2) backproject the modified projections. The BpjF algorithm is also implemented in two steps: (1) backproject the projections. then (2) filter the backprojected image in the frequency domain with a two-dimensional ramp filter. The major drawback of BpjF is that the mean value of the nxonstruction is not correct. This error is caused by the way the two-dimensionaldiscrete ramp-filter is formed by sampling the COntinuOUS ramp-filter, which results in zeroing the DC component. The Nyquist Theorem is used to show that this approach of farming a discrete ramp-filter is not accurate and a new method to form the discrete ramp-filter is developed. The newly developed filter is the same as the commonly used d i s c ~ t eramp-filter except for the DC value and preserves the correct mean value of the image.
I. INTRODUCTION There are basically two types analytical algorithms to reconstruct the tomographic images, that is, the filtered backprojection (FBP) algorithm and the backprojection (BpjF) filtering algorithm. The FBP algorithm is implemented in two steps: (1) convolve the projections with a kernel function, then (2) backproject the modified projections. The BpjF algorithm is also implemented in two steps: (1) backproject the projections, then (2) filter the backprojected image in the frequency domain with a two-dimensional ramp filter.
FBP algorithm with the filtering step perfarmed in the fkquency domain [3.4].The current remedy is to use the d i s c ~ t Fourier e transform of the truncatedumvolver to replace the discrete ramp-filter.However, the explicit expressionfor the two-dimensionalconvolver is not availablefor the band-limited ramp filter, people have no choice but using the sampled ramp filter in the frequency domain in the Bp3; algorithm. In this paper, we investigate a means to defme the two-dimensional discrete film function which preserves the correct DC value of the image.
II.THEORY AND METHOD The DC-shift problem is found not only in the BpjT algorithm but also in the FBP algorithm when the filtering step is implemented in the frequency domain via FFTs. We claim that the DC-shift problem is caused by the way the 1D and 2D discrete ramp-filters are formed - sampling the continuous counterparts. One m a y ask:“Why is it wrong to sample the continuous ramp-filter?’ By ramp-filter, we mean the windowed (apodized) ramp-filter in this paper. The ramp-filter and its 1D inverse Fourier transform are denoted by Q(w) and q(t).respectively, shown in Figure 1.
Since q(t) is band-limited, that is, Q(w) has a finite support: Q(w)= 0 as Iwl > W, we can sample q(t) at the Nyquist
t
when we wrote a paper on varying focal length fan-beam algorithm 111 which used the backprojection filtering (BpjF) technique, one of the reviewers suggested that the observation of the DC-shift by the BpjF algorithm was interesting and deserving a further investigation. The DC-shift phenomenon has caught people’s attention for years 121. however to our knowledge no effective solutions have been reported. People face the same problem when implementing the ?he research work presented in this manuscript was partially supported by NIH Grant R 0 1 HL 39792.
0-7803-2544-3195 $4.000 1995 IEEE
Figure 1. One dimensional (apodized) ramp-filter Q(w) and its inverse Fourier transformq(t).
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two steps: (1) apodizing the ramp-filter, and (2) sampling the apodized filter as shown in Figure 2.
sampling interval o f 1/ (2W) without introducing any errors. The sampled q(t) is used in the FBP algorithm as the onedimensional convolver. On the other hand. Q(w) is not bandlimited. that is q(t) does not have a finite support. Ideally speaking. samplii Q(w) always introduces errors. Taking advantage of the fact that q(t) -> 0 as 14 + W . we can assume
-
that q (t) 0 as It1 > T for some large number T. Therefore, one can sample Q= w
(3)
However, the equation Q ( w ) = w does not apply for w = 0, because in this case the ''Wdegenerates to a disk and its area is
(y)2n
(4)
Similar to equation (1). we have
2n
1
(5)
1 Q ( 0 ) = -Aw
(6)
8
If the frequency sampling is continuous, i.e. Aw -> 0, we have Q(0)-> 0. This result is consistent with the well-known ramp filter in the continuouscase. For discrete sampling, our new filter when
w+O
A BpjF algorithm was implemented for the parallel projection geometry in the following steps. Firstly, the projection data (Nelement vectors) were backprojectedinto the image array (a 2N x 2N matrix).Secondly, the image array was transformed into the frequency domain by a 2N x 2N twodimensional fast Fourier transform 0. Thirdly, the transformed image was multiply point-by-point with the filter e w in equation (8). Fourthly, the multiplied function d fi-equency domain image was transformed back to the image domain by a 2N x 2N two-dimensional inverse fast Fourier transform Finally, the central N x N image was displayed.
(m.
1 A0Aw
nus,
I
III. RESULTS
I
provides a correct DC-gain, so that there will be no DC-shift in the reconstructed images. The filter presented in equation (7) can be use in the FBP algarithm to replace the 1D ramp filter, and can also be used in the BpjF algorithm to replace the post
The reason touse a 2N x 2N array (instead of N xN) is to reduce the aliasing during the FFT and IFFT. Ideally speaking the backprojection array should be w x W. Any finite dimensional array will introduce some aliasing artifacts. This is one reason that BpjF algorithm does not perform as well as the FBP algorithm. However, if the backprojection array is large enough (or equivalently, the object is small enough), the aliasing will be too small to be noticed. In our computer simulations, two two-dimensional uuifcxm disc phantom (see Figure 4) was used, one with a diameter of 105 and the other with a diameter of 210. There were 256 projection angles over 360". There are 256 bins on the parallel-beam detector. The backprojectian array was 512 x 512, and the final image array was 256 x 256. The cut-off frequency of the ramp-filter was 0.5 of the Nyquist frequency. No noise was added to the projection data and no roll-off window (other than the sharp chop-off as shown in Figure 2) was applied to the ramp-filter.
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For each phantom, three reconstructions were obtained:
(1) the CBP (Convolution Backprojection) reconstruction. (2) accurate reconstructions. Due to the aliasing and DC value the BpjF reconstruction with Q(0) = 0, and (3) the BpjF problems, the BpjF reconstructions are inferior. However, the reconstruction with Q(0)= Aw/8, where Aw = 1/512. The new DC gain greatly improved the accuracy of the central profiles of the reconstructed images are shown in Figure r e c " C t i o n . The smaller Phantom resulted in m m reconstructions than the larger one. This implies that a larger 5. backprojection array will help to reduce the &sing artifact &d It is observed that the CBP algorithm provides the most improvethe imageaccuracy.
d = 210
d = 105
1.5
":'UTI # I
1.0. ideal phantom
0.5
0.0
0.0
1
-0.5
1
25 1.5 1.0. 0.5
CBP reconstructions
0.0
-0.5
1 256
1.00049 1
;I
0.999957
1
-0.5
BpjF reconstructions
w/ Q ( 0 ) = 0
1
0.97287 1 -0.5\ 1.5 BpjF reconstructions 1 W l Q ( 0 ) = EAw
1.0-
256
L
~~
1
0.913990 -0.511
256
1
0.5 1 AW = -
0.0
5 12
0.987352 -0.5;
1
256
Figure 5. Central profiles of the two ideal phantoms and reconstructions. The backprojected arrays are of5 12 x 5 12. The profiles are drawn through the disc center across the image. Only the pixel values from pixel number 129 to pixel number 384 are shown. 1235
IV. CONCLUSIONS The BpjF algorithm has two problems to affect its reconstruction accuracy. One problem is aliasing due to the use of a finite array to approximatethe infinite backprojection array. The aliasing artifacts can be greatly reduced by using a large enough finite backprojection array. Usually a 4N x 4N array is adequate if the object size is N. Another problem is due to the undesirablefilter function which zeros theDC component ofthe image. We suggest to use Awn as the DC gain in the “ramp” filter. Computer simulations show that these remedies improve the reconstruction accuracy significantly.
V. ACKNOWLEDGMENTS We thank Biodynamics Research Unit,Mayo Foundation for use of the Analyze software package.
VI. REFERENCES [l] G. L. Zeng and G. T. Gullberg,“A Backprojectionfiltering algorithm far a spatially varying focal length collimator.” IEEE Trans. Med. Imaging, vol. 13, no. 3. September 1994, pp. 549-556. [2] S. Suzuki and S. Yamaguchi, “Comparison between an image reconstructionmethod of filtering backprojectionand thefiltered backprojection method,”Applied Optics,vol. 27. . 110.14,July 1988. p ~2867-2870. [3] A. Rosenfeld and A. C. Kak,Digital Picture Processing (2nd Ed), Volume 1, Academic Press. New York, 1982. 141 S . W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, Implementation and Applications, Edited by G. T.Herman, Sprhger-Verlag.Berlin. 1979. pp. 9-79. [5] C. R. Crawford, “CT filtration aliasing artifacts,” IEEE Trans. Med. Imaging, vol. MI-10. 1991.p~.92-102.
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