Practical Hybrid Convolution Algorithm for Helical CT Reconstruction Alexander A. Zamyatin, Katsuyuki Taguchi, and Michael D. Silver
Abstract-- Great strides have been taken in the last few years in the development of both approximate and exact reconstruction algorithms for helical cone-beam CT. The proposed algorithm uses so-called hybrid convolution which is a sum of ramp filtering and Hilbert transform of projection data. We show that it satisfies the requirements of a modern practical CT reconstruction algorithm. In this work we evaluate the new method and compare it to several previously published algorithms. We evaluate spatial resolution and noise properties, and also the cone beam artifact. Evaluation demonstrated that the proposed algorithm outperforms the helical Feldkamp algorithm and another candidate in terms of image noise-to-resolution balance, as well as the cone beam artifact. We also discuss implementation issues and propose a simplified version for over-scan reconstruction. I.
INTRODUCTION
K
ATSEVICH [1], [2], introduced an exact cone beam algorithm of the filtered-backprojection (FBP) type. Instead of conventional ramp filtering it employs modified Hilbert transform (by modified we mean that kernel is h(sinγ) rather than h(γ)) of the partial derivative of cone beam data. Filtering has to be performed in the special family of filtering planes, [1]-[4], which requires additional rebinning steps before and after convolution. For practical purposes, however, manufacturers use approximate cone beam algorithms due to simplicity of implementation and flexibility of the ramp kernel. But it turns out that algorithms based on Hilbert transform have some new properties, lacking in the conventional Feldkamp-type algorithms [5] - [8], which lead to more efficient reconstruction. A fan beam reconstruction algorithm, based on Hilbert transform reconstruction formula was introduced in [9]. They pointed out that for exact reconstruction only projections corresponding to the region-of-interest (ROI) are required, which opens a possibility to less-than-a-short-scan reconstruction. Previously, for a short scan reconstruction, the set of projections over the whole field-of-view (FOV) was required (π + fan angle). Also the data sufficiency condition is Manuscript received October 20, 2004. This work was supported by Toshiba Medical Systems. A. A. Zamyatin and M. D. Silver are with Bio-Imaging Research, Inc, Lincolnshire, IL 60089 USA (telephone: 847-279-5100, e-mail:
[email protected],
[email protected]). K. Taguchi is with Toshiba America Medical Systems, Inc, Tustin, CA 92780 USA (telephone: 847-279-5100, e-mail:
[email protected]).
relaxed. In other words some part of image can be accurately reconstructed from a limited set of projections, in contrast to traditional “all or nothing” way in fan beam reconstruction. The other advantage of the above algorithm (Noo’s algorithm) is that weighting of redundant fan beam data is performed after convolution. It makes the algorithm more efficient, compared to any Feldkamp-type algorithm, since data does not have to be re-convolved for each slice. Yet another advantage of Noo’s algorithm was discovered during our evaluation on the noisy water cylinder phantom. It turns out that noise variance is more uniform throughout the image compared to Feldkamp algorithm. PSF is also less space variant for Noo’s algorithm. This can be explained by the fact that backprojection weight is the inverse distance, not the inverse square distance, and so-called magnification effect is reduced. However, as it was pointed out in [10], Hilbert transform based algorithms introduce stronger smoothing compared to ones with ramp filtering. This is due to the additional numerical differentiation step. It is usually implemented as 2 point or 3 point difference, which results in loss of resolution. In view of this observation, Kudo proposed an algorithm for fan beam data that consists of both ramp and Hilbert filtering, similar to the algorithm proposed in this paper. However, Kudo’s algorithm is more similar to Feldkamp’s, with its disadvantage of inverse square weight (and less noise and PSF uniformity) plus it involves derivative of the weight function, which make this algorithm less efficient for practical purposes. We propose an algorithm that has the following advantages: 1) Half-scan weighting after convolution. Therefore, data does not have to be re-convolved for each slice, which allows faster reconstruction. 2) Flexible reconstruction range. Algorithm has a possibility of less-than-a-short-scan [9], or super-short-scan [10] reconstruction or, more general, any reconstruction range satisfying the relaxed sufficiency condition [9]. 3) Shift-invariant filtering. Data is filtered on projectionby-projection basis independently of reconstructed image position. 4) Ramp filtering. The filtering step consists of hybrid convolution, which is a sum of Ramp- and Hilbert-convolved data. The major contribution comes from the ramp-filtered data, which results in improved image sharpness compared to methods with only Hilbert convolution.
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5) Backprojection with the inverse distance weight. This results in better noise uniformity, compared to traditional methods with an inverse distance squared weight. 6) Simplicity of implementation. The algorithm is of traditional FBP form, where only the filtering step is a little more complicated than usual ramp filtering. II. DERIVATION OF THE FAN BEAM ALGORITHM Let g(s, γ) denote fan beam data, where s is the projection angle, γ is the fan angle, source trajectory is denoted by a(s), q(⋅) is the ramp kernel and h(⋅) is the Hilbert kernel, x is the reconstruction point, Λ is the projection range satisfying the relaxed sufficiency condition. Noo proposed the following formula (see [9]) w( s, γ ) ∂ ∂ 1 f ( x) = g ( s, γ ) * h(sin γ ) ds + Λ 2π | x − a ( s ) | ∂s ∂γ γ =γ ( s, x ) which can be split into two integrals as follows: w( s , γ ) ∂ 2π f (x) = ds g ( s, γ ) * h(sin γ ) Λ | x − a ( s ) | ∂s γ =γ ( s , x )
∫
∫
+
w( s , γ ) ∂
∫ | x − a( s) | ∂γ g (s, γ ) * h(sin γ ) Λ
ds .
(1)
γ =γ ( s , x )
Now let us make two remarks: Remark 1: It is a well know fact that combination of differentiation and Hilbert transform is equivalent to ramp ϕ ′(t ) ∗ h (t ) = −ϕ (t ) ∗ q(t ) for some convolution, i.e. differentiable function ϕ(t). Note that in the second integral differentiation and convolution are performed with respect to the same variable. However, we cannot apply this relation directly (since kernel argument is sin γ), and we need to modify it to use with modified kernels. In Section A we derive this relation, which allows us to avoid differentiation in the fan angle direction. Remark 2: Fan beam data is actually given as g(s, u), where u is the detector coordinate, u = R γ, where R is the source-todetector distance. Therefore, ∂ ∂g ( s, u ) ∂g ( s, u ) ∂ g ( s, u ) = + + R, (2) ∂ ∂s s ∂ ∂u γ i.e. the second differentiation term has a magnification factor of R. This means that amplitude and contribution of the second term is much greater than that of the first term. It follows that by replacing Hilbert transform of derivative with ramp filtering in the second term, we will be able to significantly increase resolution without changing the structure of the algorithm, i.e. keeping its benefits like super-short scan and inverse distance backprojection. A. Main result Let us first introduce known relations that will be used in this section. The Hilbert kernel is given by h(t) = – 1/πt, and its Fourier transform is given by H( ω) = i sign ω . The ramp
kernel q(t) is defined through its Fourier transform Q( ω) = |ω |, i.e. q(t) = F–1[Q(ω)]. The following relations are used: (3) Q(ω) = – iω H( ω), (4) H’(ω) = 2iδ (ω), (5) Q”(ω) = 2δ (ω), where δ(ω) is the delta-function given by equation F[δ] = 1. From relations (3)-(5), and sign’( ω) = 2δ ( ω), one can derive (6) iω H”( ω) = 2δ (ω) The above relations are written formally and are understood in the sense of distributions, i.e. as acting on compactly supported functions such as g(s, γ). To put the second integral into ramp-convolution form, we will use the following result. Lemma. With the above notations, one has: d/dγ g(s, γ)∗h(sin γ) ≅ – g(s, γ)∗q(sin γ) – G(0) /2π,
(7)
where G(⋅) is the Fourier transform of g(s, γ) with respect to the second argument. Proof: Let us start with Taylor series expansions: 1 γ = 1+ γ 2 + O γ 4 , (8) sin γ 6 and
( )
2
γ 1 = 1 + γ 2 + O γ 4 , sin 3 γ to approximate modified kernels 1 γ h(sin γ) = h(γ) ≅ h(γ) + γ 2 h(γ), sin γ 6 and
( )
2
(9)
(10)
γ 1 q(γ) ≅ q(γ) + γ 2 q(γ). q(sin γ) = (11) 3 sin γ Note that by the convolution of derivative property we have: d/dγ g(s, γ)∗h(sin γ) = g(s, γ)∗ d/dγ h(sin γ). Now let’s consider Fourier transforms: 1 1 F[q(sin γ)] = F[q(γ) + γ 2 q(γ)] = Q(ω) – Q”(ω) 3 3 2 (12) = Q(ω) – δ (ω). 3 1 d h(sin γ)] = iω F[h(γ) + γ 2 h(γ)] F[ dγ 6 1 = iω H( ω) – iω H”( ω) 6 1 = – Q(ω) – δ (ω) 3 = – F[q(sin γ)] – δ (ω). (13) Here we used the multiplication property of the Fourier transform: F[t n u(t)] = i n F (n)[u(t)]. Now, using the linearity of the Fourier transform and the fact that F–1[G( ω)δ( ω)] = G(0)/2π, assertion of the lemma easily follows.
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In practical applications angle γ is bounded and usually does not exceed 0.52 (30°). Thus approximation in (10) and (11) does not introduce any significant artifacts. Let us introduce the following convolution operators: - Modified ramp filter: Qm[g(s, γ)] = g(s, γ)∗q(sin γ). - Modified Hilbert filter: Hm[g(s, γ)] = g(s, γ)∗h(sin γ). - Modified ramp filter with DC offset: Q0[g(s, γ)] = – ( g(s, γ)∗q(sin γ) + G(0) /2π ). Then the proposed algorithm can be written as: 1 w( s , γ ) ∂ f ( x) = ds Q 0 [ g ( s, γ )] + H m [ g ( s, γ )] 2π Λ | x − a ( s ) | ∂s γ =γ ( s , x ) or simply w( s, γ ) 1 f ( x) = R[ g ( s, γ )]γ =γ ( s , x ) ds , (14) 2π Λ | x − a ( s ) | where ∂ R [g ( s, γ )] = Q 0 [g ( s, γ )] + H m g ( s, γ ) (15) ∂s is called the “hybrid” convolution operator. The function w(s, γ) is the redundancy weight function. In our evaluations we use the weight function proposed in [7], [11], [12] for the proposed algorithm and HFDK. For the Hilbert algorithm evaluation the weight function described in [9] (Noo weighting) was used.
∫
∫
III. EXTENSION TO CONE-BEAM GEOMETRY For relatively small cone angle the algorithm can be extended to cone beam data using the usual techniques, such as backprojection from the 2D data array, 1D convolutions along detector rows, or rebinned tangential lines (to reduce cone beam artifacts). See [5], [6], [8], [13] for more details. The formula takes form: w( s , γ ) 1 f ( x) = R [ g ( s, γ , α )]γ =γ ( s , x ) ds , (16a) 2π Λ | x − a ( s ) | or w( s , γ ) 1 f ( x) = R [ g ( s, γ , α ) cos α ]γ =γ ( s , x ) ds , (16b) 2π Λ | x − a ( s ) |
∫
∫
shift, one needs to calculate the projection sum and add it to the convolved data with a factor of 1/2π. However, this filter can be implemented more efficiently if one uses the fast Fourier transform. Then the DC offset can be implemented in the Fourier domain by adding G(0) to the DC component (0th sample) of the complex product Z( ω) = G(ω)Qm( ω), where Qm( ω) is the modified ramp kernel (please do not confuse kernels with operators; kernels are functions and denoted with italic font style). Note that Z(0) = 0, since Qm(0) = 0 for ramp kernel. B. Hybrid convolution It is convenient to implement the whole hybrid convolution process, including differentiation, as one module to save time on reading and writing data between different sub-steps. It is also important to implement convolution on projection-byprojection basis. In projection-by-projection implementation it is allowed to keep two or three projections in memory. Note that Kudo’s algorithm does not allow for projection-byprojection implementation, because convolved projections need to be weighted before they finally added together. In other words, hybrid convolution presented in this paper is a shift-invariant filtering. C. Reconstruction flow The proposed algorithm has very efficient FBP structure. It can be implemented in the following steps: I. Apply hybrid filter for the projection data. II. For each image slice: 1. Redundant data weighting 2. Backprojection with inverse distance weight. Hybrid convolution can be represented by the following chart: DATA IN
Projection subtraction Modified Hilbert filtering
where α is the cone angle and a(s) is the helical trajectory, or some other trajectory satisfying 3D sufficiency conditions.
Modified Ramp filtering with DC shift
Projection addition DATA OUT
IV. IMPLEMENTATION A. Ramp filter There are a few ways to implement the modified ramp filtering with DC offset. The first can be used if one performs convolution in the spatial domain, without taking FFT. Note N
that G(0) = ∑ g ( s k , u n ) . Sum is used here instead of integral n =1
since projection data is discrete. Hence, to compensate the DC
Fig. 1. Hybrid convolution chart. Note that it is implemented as shiftinvariant, projection-by projection filtering.
V. NUMERICAL EVALUATION To evaluate the proposed algorithm we generate computersimulated cone beam projection data. Each projection has 896 detector elements per row with the number of rows as well as helical pitch and segment width depending on experiment. Projections are acquired at the rate 900 per revolution. The fan
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A. Spatial uniformity of the noise phantom Noise standard deviation is measured in 32 ROI’s of radius 20 pixels located at 210 mm from isocenter. Noise images are reconstructed from the same data set, representing projections of z-uniform water cylinder. Helical pitch ratio is 0.625 for the full-scan reconstruction and 1.25 for the half-scan reconstruction. It was noticed that in the half-scan mode the noise variation was greater in some directions and less in others. The direction of the most difference is related to the angle at which the X-ray source crosses the plane of reconstruction (see Figure 2). Plane of reconstruction A B
Std. dev., HU
reconstruction point. Hence we conclude that distance factor is important for noise uniformity. Plots below compare the three considered methods: 14 12 10 8 6 4 2 0 0
180 270 ROI angle, degrees Proposed
360
Hilbert
Fig. 3. Noise angular dependence: full scan 25 20 15 10 5 0 0
90
180 270 ROI angle, degrees
HFDK
Proposed
360
Hilbert
Fig. 4. Noise angular dependence: half scan
B. Point spread functions For this evaluation we chose a phantom with five thin long cylinders. Diameter of each cylinder is 0.1 mm. To calculate the point spread function (PSF) we used zoomed reconstructed images of size 256x256 with an FOV of 5.12 mm centered at the location of the cylinder. We report two full width at half maximum (FWHM) for the PSF’s at each of five radial positions: in the radial direction (labeled RAD) and tangential or angular direction (labeled ANG). HFDK-RAD HFDK-ANG Proposed-RAD Proposed-ANG Hilbert-RAD Hilbert-ANG
3.5
FWHM, mm
3
S
90
HFDK
Std. dev., HU
angle that covers scan field-of-view of 500 mm is 49.2°, and source to isocenter distance is 600 mm. Comparison is performed for multi-slice helical data. We compare three approximate cone beam algorithms: HFDK (helical Feldkamp algorithm) [5]-[8], which uses ramp filter only, Hilbert (proposed by Noo, et al in [9], and extended to cone beam in [13]) and the Proposed. 1D filtering in all algorithms is performed along detector rows, and no rebinning is used. We compare algorithms in three categories: spatial uniformity of the noise pattern, spatial resolution by means of PSF functions, and the cone beam artifact. In the noise experiment we evaluate the three methods in the full scan and half scan modes. The full-scan reconstruction avoids the dependence on the short-scan weighting function. In the resolution experiment we use full scan reconstruction for the same reason. In both experiments we choose z-uniform objects and a small helical pitch. Also we choose helical over circular cone beam reconstruction to avoid dependence on image slice z-position. In these numerical experiments we use 16 detector rows data with collimation size of 0.5 mm for PSF and 1 mm for noise study. To illustrate the cone beam artifact in the third numerical experiment we use simulated two data sets with 64 and 256 detector rows of 0.5 mm detector row pitch.
2.5 2 1.5 1 0.5
Fig. 2. Spatial variance of noise standard deviation. A is the direction of the most spatial variance, B is the direction of the least spatial variance, S is the point where X-ray trajectory intersects the reconstruction plane.
Note that direction A is the direction of the most rapid change in the distance between X-ray source and a
0 0
50
100 150 Radius, mm
Fig. 5. Full width at half maximum comparison.
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200
250
C. Cone Beam Artifact For this study we used the modified clock phantom. Balls are grouped into two chains. One group is 75 mm from the center, and balls have radius 20 mm; the other is 150 mm from the center with radius 30 mm. The reconstructed field-of-view is 400 mm. Helical pitch ratio is 1.0. Window width is 10% of the full contrast.
Note that the cone beam artifact is reduced without any rebinning to tilted lines in the filtering step. VIII. ACKNOWLEDGMENT This work was inspired by discussions with Dr. Frederic Noo (University of Utah) and Dr. Alexander Katsevich (University of Central Florida). We would like to thank Dr. Frederic Noo for suggesting exchanging the order of weighting and filtering by Hilbert transform based approach. IX. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
Fig. 5. Cone beam artifact comparison. The pictures on the left are reconstructed using HFDK, and the pictures on the right are reconstructed using the proposed algorithm. TOP: 64 detector rows. BOTTOM: 256 detector rows. The horizontal filtering direction was used in both algorithms. One can see a significant reduction of cone beam artifact in the proposed method.
[10] [11] [12]
VI. DISCUSSION One can argue that we could not get rid of the differentiating step entirely, and differentiating remains with respect to the helical variable s. However, our empirical analysis shows that Hilbert term adds only low-frequency information to the image. All sharp structure is contributed by Ramp term. Hence this differentiation is not critical to image quality. Images obtained by the proposed algorithm are as sharp as obtained with Feldkamp reconstruction. Also, our algorithm allows using different modulated ramp kernels for specific applications, such head or lung imaging, which is a considerable practical advantage.
[13]
A. Katsevich. An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, V. 324, pp. 625-825, May 2004. A. Katsevich, Analysis of an exact inversion algorithm for spiral conebeam CT, Phys. Med. Biol, 2002, vol. 47, pp. 2583-2598. A. Katsevich, A. Zamyatin, Analysis of a family of exact inversion formulas for cone beam CT, submitted for publication. A. Zamyatin, Analysis of cone beam reconstruction in computer tomography, PhD Dissertation, University of Central Florida, December 2003. L.A. Feldkamp, L.C. Davis, and J.W. Kress. Practical cone-beam algorithm, J. Opt. Soc. Am, vol. 1, pp. 612--619, 1984. H. Kudo and T. Saito, Helical-scan computed tomography using conebeam projections, IEEE NSS-MIC 1992. M. D. Silver, K. Taguchi, and K. S. Han, Field-of-view dependent helical pitch in multi-slice CT, Proc. SPIE Med. Imag. Conf. 4320, 839-850, 2001. G. Wang, T. H. Lin, P. Cheng and D. M. Shinozaki, A general conebeam reconstruction algorithm, IEEE Trans Med. Imaging, MI-12, 48696, 1993. F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Phys. Med. Biol. 47(2002) 2525-2546. H. Kudo, F. Noo, M. Defrise and R. Clackdoyle, New super-short-scan reconstruction algorithms for fan-beam and cone-beam tomography, IEEE NSS-MIC 2002, M5-3. M. D. Silver, A method for including redundant data in computed tomography, Med. Phys., 27, 773-774, 2000. K. Taguchi, B. S. Chiang and M. D. Silver, A new weighting scheme for cone-beam helical CT to reduce the image noise, Phys. Med. Biol. 49, 2351-2364, 2004. H. Kudo, F. Noo, M. Defrise, and T. Rodet, New approximate filtered backprojection algorithm for helical cone-beam CT with redundant data, IEEE NSS-MIC 2003, M14-330.
VII. CONCLUSIONS The proposed algorithm preserves all the features, listed in introduction. It outperforms HFDK in both image quality and computational efficiency. The proposed algorithm has: - Less noise level - Less noise angular variation - Less cone beam shading
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