Counter examples for unmatched projector/backprojector in an iterative algorithm
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It is rather controversial whether it is justified to use an unmatched projector/backprojector pair in an iterative image reconstruction algorithm. One common concern of using an unmatched projector/backprojector pair is that the optimal solution cannot be reached. This concern is misleading and must be clarified. We define a figure-of-merit in the image domain as the distance between the reconstructed image and the true image, as the normalized mean-squared-error (NMSE). The NMSE is used to determine whether an unmatched matched projector/backprojector pair can provide a better image than a matched projector/backprojector pair. Hot and cold lesion’s contrast-to-noise ratio is also used as an alternative secondary figure-of-merit for algorithm comparison. Computer-generated counterexamples are used to test the performance for matched and unmatched projection/backprojection pairs for different reconstruction algorithms. The projectors are ray-driven, and the backprojectors are ray-driven and pixel-driven. For the attenuation-free data examples, the unmatched pixel-driven backprojector outperforms the matched ray-driven backprojector. For the attenuated data example, the matched ray-driven backprojector performs better. The ray-driven backprojector can be slightly improved by using an attenuation coefficient that is larger than the true one; in this case the backprojector becomes unmatched. Unmatched projector/backprojector pairs are fairly flexible. If the backprojector is properly chosen, good results can be obtained. However, we have not found a general rule to select a good backprojector.
KeywordsTomographic image reconstruction Iterative algorithms Projector/backprojector pair
Every iterative medical image reconstruction algorithm consists of a projector/backprojector pair. A projector can be represented as a matrix A. A matched backprojector is the transpose matrix AT. In other words, the matched backprojector is the adjoint operator of the projector. An example of matched projector/backprojector pair is the distance-driven pair [1, 2]. Almost all iterative algorithms are derived by assuming a matched projector/backprojector pair.
A projector should be chosen to mimic the physical data generation procedure as close as possible. In many applications, researchers may use a backprojector that is somewhat different from the matrix AT, referring to it as an unmatched backprojector. People have various reasons to use an unmatched backprojector [3, 4, 5, 6, 7, 8, 9, 10]. An unmatched backprojector may simplify computer/GPU implementation, may reduce computational cost, may reduce image artifacts, may increase the algorithm convergence rate, and so on.
Good results can be obtained by using valid unmatched backprojectors. For example, if the ramp filter is used in the backprojector, the backprojector is valid, the algorithm is fast, but the resultant images are quite noisy [11, 12].
An invalid backprojector is sometimes useful in the early iterations.
Choosing a valid backprojector may not be a very critical factor in a practical image reconstruction problem.
A “converged” solution usually is very noisy and not desirable.
In practice, we should pay attention to the initial “convergent” trend and choose a rapid projector/backprojector pair, which may be an invalid pair, and then use regularization methods to guide or stop the iteration process.
The main goal of our 2000 paper was to establish the conditions, under which an unmatched projector/backprojector pair was valid. A valid projector/backprojector pair does not change the convergence property of the reconstruction algorithm. A valid pair may make the algorithm converge faster or slower. A naïve conception is that a valid pair with an unmatched backprojector does not give as good reconstruction as the matched backprojector can provide. The main goal of the current paper uses counter examples to point out that this naïve conception is not true. This point was not discussed in our 2000 paper.
In the paper by Kamphuis et al. an unmatched projector/backprojector pair was used to speed up the iterative image reconstruction, obtaining almost the same image quality . In fact, the most common motivation of using an unmatched projector/backprojector pair in medical imaging is to significantly reduce the computation burden, even though the image quality is slightly compromised. Our paper is mainly concerned about the image quality.
Of course, if not carefully chosen, an unmatched projector/backprojector pair is most likely performs poorly and may introduce artifacts, as observed by Rahmim et al. in a motion tracking application .
Some researchers believe that the optimal solution can only be reached by matched projector/backprojector pairs [15, 16, 17, 18]. They also believe that the unmatched projector/backprojector pairs may give a faster algorithm, but the price to pay is the higher noise and image artifacts.
This paper argues that the common beliefs can be misleading, because the objective function measures the discrepancy between the forward projections and the measured data in the projection domain with some image domain constraints. The “optimal solution” should be defined in the image domain. The matched/unmatched projector/backprojector issues are still not well understood. We make no attempts to explain why a certain unmatched pair is preferable for a particular problem/algorithm choice. The investigation carried out in this paper is focused on single photon emission computed tomography (SPECT), even though the results can be applied to other modalities as well. In this paper, we use some counterexamples to disprove the statement that optimal solutions can only be achieved by matched projector/backprojector pairs.
Iterative algorithms are used to reconstruct an image by minimizing an objective function (or, equivalently, maximizing a likelihood function). The iterative algorithm minimizes the objective function step by step. When the objective function is minimized, the conventional optimal solution is reached. The objective function is set up in the projection domain. Due to noise, this conventional optimal solution is noisy.
For example, the (MLEM maximum likelihood expectation maximization) algorithm is proven to increase the likelihood function at each iteration [19, 20]. The conventional optimal solution of the MLEM algorithm is the maximum likelihood solution. In practice, all projection data contain noise, and the optimal maximum likelihood solution is too noisy to be useful.
It is observed from Fig. 1 that there is an initial converging trend, a minimum discrepancy point, and a later diverging trend. The minimum discrepancy solution is not the maximum likelihood solution due to noise. It is beneficial to stop the MLEM iteration at the minimum discrepancy point and capture a less noisy image.
We do not yet have a relationship between the maximum likelihood solution and the minimum discrepancy solution. We argue that a projection/backprojection pair that leads to the maximum likelihood solution may not lead to the optimal minimum discrepancy solution. By optimal minimum discrepancy solution we mean that any other projector/backprojector pair cannot reach a smaller discrepancy. In this paper, the algorithm is fixed; the projector is fixed; only the backprojector is allowed to vary. Since the matched projector/backprojector pair is able to reach the maximum likelihood solution, our argument can be rephrased as follows: a matched backprojector may not be able to provide the minimum discrepancy solution. An unmatched backprojector may outperform the matched backprojector in the sense that an unmatched backprojector may give a smaller discrepancy at its minimum discrepancy point.
A uniform two-dimensional (2D) circular phantom with a diameter of 120.32 pixels was used in the computer simulations. The phantom, based on SPECT imaging, contained two smaller cold discs and two smaller hot discs, all with a diameter of 25.6 pixels. Let the image intensity of the larger circular disc be 1 unit. The cold discs had an intensity value of 0.5, and the hot discs had an intensity value of 1.5. Two situations were considered. In the first situation, the projections were attenuation-free. In the second situation, a uniform circular attenuator with a diameter of 120.32 pixels and attenuation coefficient of 0.05 per pixel was used in data generation and in image reconstruction. The projections were generated analytically without using pixels, to avoid committing the inverse crime. The inverse crime occurs when the same (or almost the same) theoretical ingredients are employed to synthesize as well as to invert data in an inverse problem.
There were 180 projection views over 360°. The detection array had 128 bins. The images were reconstructed in a 128 × 128 array. Poisson noise was incorporated in the projection data and three levels of noise were generated. Noise level 1 had 1,025,000 projection total photon counts, level 2 had 2,050,000 counts, and level 3 had 4,100,000 counts.
For the attenuation-free situation, the projector was the ray-driven, line-length weighted. We tested two backprojectors: the matched ray-driven, line-length-weighted backprojector and the unmatched pixel-driven backprojector that is normally used in the FBP (filtered backprojection) algorithm. Both the EM algorithm and the Green’s MAP–EM–TV algorithm were used for image reconstruction, with 100 iterations.
For the situation where the projections were attenuated, the projector was the ray-driven, line-length and attenuation weighted. We tested three backprojectors: the matched ray-driven, line-length and attenuation-weighted backprojector, the unmatched ray-driven, line-length and attenuation-weighted backprojector with attenuation coefficient changed from 0.05 per pixel to 0.064 per pixel, and the unmatched pixel-driven backprojector without attenuation modeling. The EM algorithm was used for image reconstruction, with 100 iterations.
Lesion contrast-to-noise ratio (CNR) curves
According to the MLEM algorithm derivation, the backprojector matrix is the transpose of the projection matrix. Thus the maximum likelihood solution is achieved by a matched projector/backprojector pair. If an unmatched projector/backprojector pair is used, the solution of the MLEM algorithm may not result in the maximum likelihood solution.
When we compare with the true image, the maximum likelihood solution is noisy and is far away from the true image. The solution closest to the true image (we refer to it as the optimal solution or minimum discrepancy solution) is achieved with a low iteration number. This optimal solution may or may not be obtained by a matched projector/backprojector pair.
This paper uses some counterexamples to disprove the hypothesis that an optimal solution can only be achieved by matched projector/backproject pairs. The projectors in this paper are ray-driven line-length weighted. For the attenuation-less projection data, the unmatched pixel-driven backprojector gives better results than the matched ray-driven backprojector. The contrast-to-noise ratio (CNR) curves also show the better performance given by the pixel-driven backprojector. The results are consistent for all three noise levels.
However, we cannot conclude that the unmatched pixel-driven backprojector is always better than the matched ray-driven backprojector. This point is illustrated by the examples with attenuated projection data. The matched ray-driven backprojector models the attenuation effects, while the unmatched pixel-driven backprojector does not model the attenuation effects. As a result, the matched backprojector gives better results. This example does not imply that the matched backprojector is the best.
When projections are attenuated, we observe that the ray-driven backprojector outperforms the pixel-driven projector that does not model the attenuation effects. However, if we use an unmatched ray-driven backprojector by slightly increasing the value of attenuation coefficient (from 0.05 to 0.064), the performance is slightly improved.
The role of the projector is much more important than the backprojector. The projector ought to emulate the measurement physics accurately. The backprojector ought to be similar to the adjoint transform of the projector (i.e., the matched backprojector). Thus the projector is more relevant for image quantity. Slightly deviation of the backprojector from the matched backprojector may slightly alter the algorithm’s performance. We do not know yet how to find a better backprojector in general, because a good backprojector may be application dependent, for example, depending on the object to be imaged. This paper does not suggest that a pixel-driven backprojector may be better, or using a larger attenuation coefficient in the backprojector is better. From the attenuated data simulation results, we see some inconsistencies between the NMSE criterion and the CNR criterion. In clinical studies, there are no true images to compare with. One can use lesion contrast-to-noise or detectability to evaluate the reconstruction. In situations where the true images are given (e.g., in computer simulations), mutual information can be used to compare the similarities between the reconstructed images with the true images. If the true image and the reconstructed image are perfectly registered, the most popular way to measure their distance is to use the Euclidean norm, which is essentially the NMSE. The NMSE criterion is our primary criterion.
Many theoretical results/properties are available for iterative algorithms at their convergence. For example, the noise sensitivity of the solution can be studied by the singular value decomposition and its associated condition number . If the iterative algorithm stops at its minimum discrepancy point, we know almost nothing about the properties of the minimum discrepancy solution. In this paper, we use counterexamples to disprove the hypothesis that an optimal solution can only be achieved by matched projector/backprojector pairs. Unmatched projector/backprojector pairs are fairly flexible. If the backprojector is properly chosen, good results can be obtained. However, it is an open problem to find this minimum discrepancy point when the true solution is not known beforehand.
Once the reconstruction algorithm and the projector are chosen, our computer simulations suggest that the matched backprojector may not give the optimal minimum discrepancy solution and an unmatched backprojector may achieve the optimal minimum discrepancy solution. It is almost impossible to judge the performance when comparing two algorithms using real data, because the optimal stopping point for each algorithm is unknown. The value of this paper is the theoretical insight. At least we know that better results exist than the result that is provided by the matched projector/backprojector pair. How to find those better results is still an open problem.
Research reported in this publication was supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under Award Number R15EB024283. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.
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Conflict of interest
Dr. Zeng reports a grant from the National Institute of Biomedical Imaging and Bioengineering (NIBIB) of the National Institutes of Health. This research was supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB) of the National Institutes of Health under award number R15EB024283.
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