8.6Convolution backprojection

Once we have an FBP algorithm, there should be no problem at all to obtain a convolution backprojection algorithm. All we have to do is to take the inverse Fourier transform of the weighted ramp filter to get a convolution kernel.

This is easier said than done. The problem is that we have a closed-form expression for the frequency domain transfer function for the windowed ramp filter; however, we do not have a closed-form expression for the spatial domain convolution kernel. Numerical evaluation of the convolution kernel costs time.

Fig. 8.6: FBP reconstruction results using dynamic MRI data. Top: Reconstructions without using the reference image Y. Bottom: Reconstructions with assistance from the reference image Y. [Thanks Drs. Adluru and DiBella of the University of Utah for the MRI scan]

If we do not have a closed-form convolution kernel, the second best thing is to obtain an explicit expression for an approximate convolution kernel. Here we use the example in Section 8.5.1 to illustrate our strategy. The noise-weighted ramp filter transfer function is given as

$H\left(\omega \right)=\frac{\left|\omega \right|}{1+{\gamma }_0\left|\omega \right|},$

where

${\gamma }_0=\frac{\beta }{w_m}.$

We do not have an explicit inverse Fourier transform expression for (8.30). We need to find an expansion of $H\left(\omega \right)=\frac{\left|\omega \right|}{1+{\gamma }_0\left|\omega \right|}$ and each term in the expansion should have a closed-form inverse Fourier transform expression.

Now the question is: What basis functions should we choose? The Fourier series expansion does not work, because H(ω) is not periodic. The Taylor expansion does not work well, because the polynomials do not look like the function $H\left(\omega \right)=\frac{\left|\omega \right|}{1+{\gamma }_0\left|\omega \right|}$ which is monotonically increasing from 0 to 1 in the positive axis of ω.

If we factor the ramp filter |ω| out, 1/(1 + γ0|ω|) is monotonically decreasing and looks like an exponential function e–γ0⋅ω for ω > 0. Based on this strategy, we propose the following expansion for ω >0:

$\frac{1}{1+{\gamma }_0\cdot \omega }\approx \frac{1}{3}\left(e^{-{\gamma }_0\omega }+e^{-{\gamma }_1\omega }+e^{-{\gamma }_2\omega }\right),$

where the parameters γ1 and γ2 have to be determined. The range of ω is [0, 1/2]. Approximation (8.32) is already exact at ω = 0. We further request that eq. (8.32) to be exact at ω =1/4 and ω = 1/2. Thus, we have two unknowns (1 and 2) and two equations:

$\begin{array}{l}\frac{1}{1+{\gamma }_0/2}=\frac{1}{3}\left(e^{-{\gamma }_0/2}+e^{-{\gamma }_1/2}+e^{-{\gamma }_2/2}\right)\\ \frac{1}{1+{\gamma }_0/4}=\frac{1}{3}\left(e^{-{\gamma }_0/4}+e^{-{\gamma }_1/4}+e^{-{\gamma }_2/4}\right).\end{array}$

Solving these two equations (8.33) and (8.34) yields

${\gamma }_1=-4.\text{In}\left(\frac{A+\sqrt{2B-A^2}}{2}\right)$

and

${\gamma }_2=-4.\text{In}\left(\frac{A+\sqrt{2B-A^2}}{2}\right),$

where

$A=\frac{3}{1+{\gamma }_0/4}-e^{-{\gamma }_0/4}$

and

$B=\frac{3}{1+{\gamma }_0/2}-e^{-{\gamma }_0/2}.$

Using the above results and an integral table, the closed-form filter kernel h(n) can be obtained by performing 1D Fourier transform and sampling at the integer points:

$\begin{array}{l}h\left(n\right)={\displaystyle \underset{-1/2}{\overset{1/2}{\int }}H\left(\omega \right)e^{2\pi in\omega }d\omega }\\ =\frac{2}{3}{\displaystyle \underset{0}{\overset{1/2}{\int }}\omega \left(e^{-{\gamma }_0\omega }+e^{-{\gamma }_1\omega }+e^{-{\gamma }_2\omega }\right)}\cos \left(2\pi n\omega \right)d\omega \\ =\frac{-2}{3}{\displaystyle \sum _{k=0}^2\frac{{\left(-1\right)}^n{e}^{-\frac{{\gamma }_k}{2}}\left({\gamma }_k^3+4{\gamma }_k{\pi }^2{n}^2+2{\gamma }_k^2-8{\pi }^2{n}^2\right)}{{\left({\gamma }_k^2+4{\pi }^2{n}^2\right)}^2}}.\end{array}$

Figure 8.7 illustrates that the approximation relationship (8.32) is fairly accurate. There are six curves in Figure 8.7: three cases of true curves and corresponding three cases of approximated curves. We can only see three curves in Figure 8.7. This is because the true and approximated curves match so well and on the top of each other.

Fig. 8.7: Verification of the approximation expression (8.32) with different values of γ0.

If there is no noise weighting, that is, wm ≡ 1 in eq. (8.32), expression (8.39) gives a convolution kernel for a Bayesian convolution backprojection algorithm, which can be implemented in the following simple two steps:

1.Convolve the projections p(n, θ) with the convolution kernel h(n) as defined in eq. (8.39), obtaining q(n, θ).

2.Backproject q(n, θ) to get the final reconstruction.

When ray-by-ray noise weighting is required, wm is a function of ray, and γ0 is no longer a constant. In this case, expression (8.39) still valid, but h(n) is shift variant and ray dependent. Thus, it is now not proper to refer h(n) to as a convolution kernel, because “convolution” assumes the shift invariant property. Other than not using the term “convolution,” the implementation steps are still the same as the simple two steps:

1.For each the projection p(n, θ), determine a kernel hn,((k) as defined in eq.(8.39), obtaining a value q(n, θ) as

$q\left(n,\theta \right)={\displaystyle \sum _{k}p\left(k,\theta \right)h_{n,\theta }\left(n-k\right).}$

2.Backproject q(n, θ) to get the final reconstruction.

Equation (8.40) still looks like the convolution formula. What is the difference? At a fixed view angle θ, the true convolution uses the same kernel h(n) to calculate q(n, θ) for every index n on the detector. On the other hand, the kernel hn,((k) changes in calculation of q(n, θ) for different n, as indicated in eq. (8.40). The computational complexity of eq. (8.40) is the same as that of a real convolution.

8.7Non-quadratic constraints

Many people choose to use the iterative image reconstruction algorithms because they are so versatile. For example, they work well when the objective function have non-quadratic constraints as long as they are convex. Most edge-preserving constraints are non-quadratic, but are convex. The FBP algorithms are linear algorithms; they can only solve linear equations. If the objective function is quadratic, the gradient of the objective function is linear. By setting the gradients to zero we obtain a set of linear equations. To our knowledge, the FBP algorithms are unable to solve nonlinear equations.

Of course, one can apply nonlinear filters as pre-processing or post-processing procedures. For example, one can apply a nonlinear median filter to de-noise the projections and apply a nonlinear edge-preserving filter to de-noise the FBP reconstructed image.

In many situations, the iterative reconstruction outperforms pre and post filtering. In this section, we propose a fast method to apply nonlinear filtering during reconstruction.

A unique feature in an iterative algorithm is that the result depends on the initial image. The conventional FBP algorithm does not accept the initial image. We want to change that!

Let us recall the beginning of Section 8.1, where the Landweber algorithm has two parts:

$X^{\left(k\right)}=F^{\left(k\right)}P+H^{\left(k\right)}{X}^{\left(0\right)}.$

The first part X(k) = F(k)P only depends on the projections P and can be implemented as an FBP algorithm. The second part only depends on the initial image X(0), which has been assumed to be zero in the discussion in the first part of this chapter. From now on the initial image X(0) does not have to be a zero.

According to eq. (8.4), H(k) is defined as

$H^{(k)}={(I-\alpha A^{T}WA)}^k,$

which can be implemented in the Fourier domain with a 2D transfer function in the polar coordinate system:

$H(\omega )={\left(1-\frac{\alpha w}{\left|\omega \right|}\right)}^k\text{when}\omega \ne 0,$

and we define H(0) = 0. As |ω| → ∞, H(ω) → 1. As k increases, the bandwidth of the high-pass filter gets narrower. A fast implementation of the second term in eq. (8.41) can be

1.Take 2D fast Fourier transform (FFT) of X(0).

2.Multiply the Fourier transformed image by H(ω).

3.sTake 2D inverse FFT.

Thus eq. (8.41) can be implemented in two parts: a windowed FBP algorithm for the projections P and a high-pass filter for the initial image X(0). Let us introduce our strategy with an example.

Say, we have a task of image reconstruction. This task can be achieved by using 2,000 iterations of the Landweber algorithm, which can also be done by using a windowed FBP algorithm with k = 2, 000. We are now experts for that.

Let us give algorithm (8.41) a name: the 2-input FBP, one inputbeing P and the other input being the initial image X(0). For the first input P, we apply an FBP; for the second input X(0), we apply a high-pass filter. Instead of using the windowed FBP with k = 2, 000, we can achieve the same goal by applying the 2-input FBP (with k = 500) 4 times. The result of one application of the 2-input FBP algorithm is the initial image of the next application.

It is nice to notice that the first part of the algorithm “F(500)P” only needs to be calculated once, even though the entire algorithm needs to be applied 4 times. However, the second part “H(500)X(0)” needs to be calculated 4 times, because each time the initial image X(0) is different.

You may think that it is silly to break down one application of the “k = 2, 000” algorithm into four applications of the “k = 500” algorithm. Yes, it is silly. It will not be silly if we apply an edge-preserving filter at the end of each application of the 2-input FBP algorithm. There are many edge-preserving de-noising filters such as bilateral filter and guided filter. You can use any of your favorite one. The flowchart of our suggested algorithm is shown in Figure 8.8 (still using our example).

Let us symbolically represent the chosen nonlinear filter as G. Thus, the algorithm illustrated in Figure 8.8 can be expressed as

$\begin{array}{l}{Y}^{(1)}=G[F^{(500)}P+0],\\ Y^{(2)}=G[F^{(500)}P+H^{(500)}{Y}^{(1)}],\\ Y^{(3)}=G[F^{(500)}P+H^{(500)}{Y}^{(2)}],\\ Y^{(4)}=G[F^{(500)}P+H^{(500)}{Y}^{(3)}].\end{array}$

Here Y(4) is the final reconstruction. We use notation Y(n) (instead of X(n)) is to avoid the confusion with the result of the algorithm (8.41) which does not use any nonlinear filters. In other words, notation X(k) is for the results of the iterative algorithm (8.41) without any nonlinearity involved; notation Y(n) is for the results of the algorithm (8.45) with a nonlinear filter G:

Fig. 8.8: A non-quadratic edge-preserving constraint is incorporated in the 2-input FBP algorithm, where the FBP part only performs once and the image filtering part performs multiple times.

$Y^{(n)}=G[F^{(k)}P+H^{(k)}{Y}^{(n-1)}].$

8.8A viewpoint from calculus of variations

The discussion in the previous sections gives an impression that the new FBP algorithms can only be derived from the iterative Landweber algorithm. In fact, these new FBP algorithms can be derived without using the iterative Landweber algorithm. In this section, we give an example of deriving a new FBP algorithm by using calculus of variations.

Let the image to be reconstructed be f(x, y) and its Radon transform be [Rf ](s, θ), defined as

$[Rf](s,\theta )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x,y)\delta (x\cos \theta +y\sin \theta -s)dxdy,}}$

where δ is the Dirac delta function, θ is the detector rotation angle, and s is the line-integral location on the detector. The Radon transform [Rf ](s, θ) is the line integral of the object f(x, y). Image reconstruction is to solve for the object f(x, y) from its Radon transform [Rf ](s, θ). Let the noisy line-integral measurements be p(s, θ). The objective function v depends on the function f(x, y) as follows:

$v(f)={\Vert Rf](s,\theta )-p(s,\theta )\Vert }^2+\beta \Vert f-g^2\Vert ,$

where the first term enforces data fidelity, the second term imposes prior information about the image f, the parameter β > 0 controls the level of influence of the prior information to the image f, and g is a reference image Using integral expressions, v in eq. (8.47) can be written as

$\begin{array}{l}v(f)={\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\theta }{\int }}{[{\displaystyle \underset{x=-\propto }{\overset{\propto }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\propto }{\int }}f(x,y)\delta (\cos \theta +y\sin \theta -s)dydx-p(s,\theta )}}]}^{2}dsd\theta }}\\ +\beta {\displaystyle \underset{x=-\propto }{\overset{\propto }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\propto }{\int }}{[f(x,y)-g(x,y)]}^{2}dydx.}}\end{array}$

Now we are ready to use the calculus of variations to find the optimal function f(x, y) that minimizes the objective function v(f) in eq. (8.48). The initial step is to replace the function f(x, y) in eq. (8.48) by the sum of two functions: f(x, y)+ %'(x, y), where '(x, y) can be any arbitrary function. The next step is to evaluate dvd% %=0 and set it to zero. That is,

$\begin{array}{l}0=2{\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}\{\left[{\displaystyle \underset{x=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\infty }{\int }}f(x,y)\delta (x\cos \theta +y\sin \theta -s)dydx-p(s,\theta )}}\right]}}\\ \times \left[{\displaystyle \underset{x=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\infty }{\int }}\eta (x,y)\delta (x\cos \theta +y\sin \theta -s)dydx}}\right]\}dsd\theta \\ +2\beta \times {\displaystyle \underset{x=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\infty }{\int }}[f(x,y)-g(x,y)dydx.}}\end{array}$

In practice, the function f(x, y) is compact, bounded, and continuous almost everywhere. After changing the order of integrals, we have

$\begin{array}{l}0={\displaystyle \underset{x=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{y=-\infty }{\overset{\infty }{\int }}\eta (x,y)}}\times \{{\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}\left[{\displaystyle \underset{\widehat{x}=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{\widehat{y}=-\infty }{\overset{\infty }{\int }}f(\widehat{x},\widehat{y})\delta (\widehat{x}\cos \theta +\widehat{y}\sin \theta -s)d\widehat{y}d\widehat{x}-p(s,\theta )}}\right]}}\\ \delta (x\cos \theta +y\sin \theta -s)dsd\theta +\beta [f(x,y)-g(x,y)]\}dydx.\\ \end{array}$

Equation (8.50) is in the form of ${\displaystyle \iint \eta (x,y)c(x,y)dydx=0.\text{Since }\eta \text{(}x,y\text{)}}$ can be any arbitrary function, according to the calculus of variations, one must have c(x, y)=0, which is the Euler–Lagrange equation. The Euler–Lagrange equation in our case is

$\begin{array}{l}{\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}\left[{\displaystyle \underset{\widehat{x}=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{\widehat{y}=-\infty }{\overset{\infty }{\int }}f(\widehat{x},\widehat{y})}\delta (\widehat{x}\cos \theta +\widehat{y}\sin \theta -s)d\widehat{y}d\widehat{x}-p(s,\theta )}\right]}}\delta \left(x\cos \theta +y\sin \theta -s\right)dsd\theta \\ +\beta [f(x,y)-g(x,y)]=0.\end{array}$

Equation (8.51) can be further rewritten as

$\begin{array}{l}{\displaystyle \underset{\widehat{x}=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{\widehat{y}=-\infty }{\overset{\infty }{\int }}f(\widehat{x},\widehat{y})}\left[{\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}\delta (\widehat{x}\cos \theta +\widehat{y}\sin \theta -s)\delta }(x\cos \theta +y\sin \theta -s)dsd\theta }\right]}\\ ={\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}p(s,\theta )\delta (x\cos \theta +y\sin \theta -s)dsd\theta +\beta g(x,y).}}\end{array}$

Notice that

${\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}p(s,\theta )\delta (x\cos \theta +y\sin \theta -s)dsd\theta ={\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}p(s=\theta )}}}|{}_{s=x\cos \theta +y\sin \theta }d\theta =b(x,y)$

is the backprojection of the projection data p(s, θ), and the backprojection is denoted as b(x, y). It must be pointed out that b(x, y) is not the same as f(x, y) even when p(s, θ) is noiseless, because no ramp filter has been applied to p(s, θ).

Also notice that

$\begin{array}{l}{\displaystyle \underset{\theta -0}{\overset{\pi }{\int }}{\displaystyle \underset{s=-\infty }{\overset{\infty }{\int }}\delta (\widehat{x}\cos \theta +\widehat{y}\sin \theta -s)\delta (\cos \theta +y\sin \theta -s)dsd\theta }}\\ ={\displaystyle \underset{\theta =0}{\overset{\pi }{\int }}\delta ((x-\widehat{x})\cos \theta +(y-\widehat{y})\sin \theta )d\theta =\frac{1}{\sqrt{{(x-\widehat{x})}^2+{(y-\widehat{y})}^2}}}\end{array}$

is the point spread function of the projection/backprojection operator at point (x, y) if the point source is at (

Using eqs. (8.53) and (8.54), eq. (8.52) becomes

${\displaystyle \underset{\widehat{x}=-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{\widehat{y}=-\infty }{\overset{\infty }{\int }}f(\widehat{x},\widehat{y})}}\left[\frac{1}{\sqrt{{(x-\widehat{x})}^2+{(y+\widehat{y})}^2}}+\beta \cdot \delta (x-\widehat{x},y-\widehat{y})\right]d\widehat{y}d\widehat{x}=b(x,y)+\beta g(x,y).$

The left-hand side of eq. (8.55) is a 2D convolution. Taking the 2D Fourier transform of eq. (8.55) yields

$F({\omega }_x,{\omega }_y)\times \left[\frac{1}{\sqrt{{\omega }_x^2+{\omega }_y^2}}+\beta \right]=B({\omega }_x,{\omega }_y)+\beta G({\omega }_x,{\omega }_y),$

or

$F({\omega }_x,{\omega }_y)=[B({\omega }_x,{\omega }_y)+\beta G({\omega }_x,{\omega }_y)]/\left[\frac{1}{\sqrt{{\omega }_x^2+{\omega }_y^2}}+\beta \right].$

Here the uppercase letters are used to represent the Fourier transform of their spatial domain counterparts, which are represented in lowercase letters; ωx and ωy are the frequency variables for x and y, respectively.

Equation (8.57) is in the form of “backprojection first, then 2D filtering” reconstruction approach and the Fourier domain 2D filter is $\sqrt{{\omega }_x^2+{\omega }_y^2}/\left(1+\beta \sqrt{{\omega }_x^2+{\omega }_y^2}\right).$ By using the central slice theorem, an FBP algorithm, which performs 1D filtering first and then backprojects, can be obtained. The Fourier domain 1D filter for this new FBP algorithm is the central slice of the 2D filter $\sqrt{{\omega }_x^2+{\omega }_y^2}/\left(1+\beta \cdot \sqrt{{\omega }_x^2+{\omega }_y^2}\right),$ which gives

$\text{RAMP(}\omega \text{)=|}\omega \text{|/[1+}\beta \cdot \left|\omega \right|\text{]}\text{.}$

When β = 0, eq. (8.58) reduces to the ramp filter ω of the conventional FBP algorithm.

8.9Summary

–A linear iterative algorithm can be implemented as an FBP algorithm or a backprojection-then-filtering algorithm. The trick is the equivalence of the matrix multiplication with ATA and filtering with the transfer function 1/ |ω|.

–When you actually implement the iterative Landweber algorithm and the windowed FBP algorithm, you will notice that the step size α in the iterative Landweber algorithm and the α in the FBP algorithm are different. This is because ATA and 1/ |ω| are not exactly the same. The α for ATA is determined by the maximum singular value of the matrix A, while the α for 1/ |ω| is determined by the sampling interval of the frequency ω.

–One should use a large iteration number for a Bayesian algorithm. This is a diffi-cult and time-consuming task for the iterative algorithm. On the other hand, it is easy for the FBP algorithm, which we simply let k → ∞.

–There are many ways to derive the windowed FBP algorithm. We presented two approaches in this chapter: the iterative Landweber approach and the calculus of variations approach.

–Implementation of the new FBP algorithms requires the discrete forms of the windowed ramp filter. We only have the continuous forms of these filters. Remember that if you directly sample the continuous forms, the resultant discrete forms may cause DC bias in the reconstruction. This situation is similar to Example 4 of Chapter 2.

Problems

Problem 8.1	Implement the iterative Landweber algorithm. Run an example for a pair of (α, k). Then run your code again with (α/2, 2k). Then run your code again with (α/3, 3k). Do you get almost the same results?
Problem 8.2	Implement the windowed FBP algorithm. Run an example for a pair of (α, k). Then run your code again with (α/3, 2k). Then run your code again with (α/3, 3k). Do you get almost the same results?
Problem 8.3	Implement the iterative Landweber algorithm that has view-based noise weighting {wm}. Run an example for ({wm}, k). Then run your code again with ({wm/2}, 2k). Then run your code again with ({wm/3}, 3k). Do you get almost the same results?
Problem 8.4	Implement the FBP algorithm that has view-based noise weighting {wm }. Run an example for ({wm}, k). Then run your code again with ({wm/2},2k). Then run your code again with ({wm/3},3 k). Do you get almost the same results?

Bibliography

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[2]Zeng GL (2012b) A filtered backprojection MAP algorithm with nonuniform sampling and noise modeling. Med Phys 39:2170–2178.

[3]Zeng GL (2012c) Filtered backprojection algorithm can outperform maximum likelihood EM algorithm Int J Imaging Syst Technol 22:114–120.

[4]Zeng GL (2013) Comparison of a noise-weighted filtered backprojection algorithm with MLEM algorithm for Poisson noise. J.Nucl. Med. Tech., 41:283–288.

[5]Zeng GL, Zamyatin A (2013) A filtered backprojection algorithm with ray-by-ray noise weighting. Med Phys 40:031113; http://dx.doi.org/10.1118/1.

[6]Zeng GL, Li Y, DiBella ERV (2013a) Non-iterative reconstruction with a prior for undersampled radial MRI data. Int J Imaging Syst Technol 23:53–58.

[7]Zeng GL, Li, Y Zamyatin A (2013b) Iterative total-variation reconstruction vs. weighted filtered-backprojection reconstruction with edge-preserving filtering. Phys Med Biol 58:3413–3431.

[8]Zeng GL (2014a) Model-based filtered backprojection algorithm: A tutorial. Biomed Eng Lett 4:3–18.

[9]Zeng GL (2014b) Noise-weighted spatial domain FBP algorithm. Med Phys 41:051906, http://scitation.aip.org/content/aapm/journal/medphys/41/5/10.1118/1.4870989.

[10]Zeng GL (2015a) Comparison of FBP and iterative algorithms with non-uniform angular sampling. IEEE Trans Nucl Sci 62:120–130.

[11]Zeng GL (2015b) On few-view tomography and staircase artifacts. IEEE Trans Nucl Sci 62:851–858.

[12]Zeng GL, Divkovic Z (2016) An extended Bayesian-FBP algorithm. IEEE Trans Nucl Sci 63:151–156.