3.2 Analytical image reconstruction
4.1.2 Iterative image reconstruction
In order to be able to consider a more accurate path approximation (using the MLP), event-by-event reconstruction using the data in list-mode format was performed. The ART algorithm with an additional total variation (TV) minimization constraint was used, as detailed in Section3.3.1. The main challenge in the reconstruction using this algorithm is the adjustment of the relaxation parameter (λin Equation 3.32), the TV weighting (w1 in Equation 3.35) and the number of iterations of TV descent. The main difficulty is that the optimal parameters depend on the noise in the data and on the considered object. This particular problem is a general problem of algebraic reconstruction techniques. In the case of proton imaging, for example, different algebraic image reconstruction algorithms have been tested by Penfold [2010] and each time the parameters need to be adjusted. The same problem can be found in limited angle X-ray tomography using ASD-POCS (described in Section 3.3.3.2) [Barquero and Brasse, 2012].
In order to determine the optimal image, two criteria were used. The first is the relative error between the reconstructed image and the phantom, defined as:
RE = P
j|xphj −xj| P
j|xphj | (4.1)
4.1. PREAMBLE: IMAGE RECONSTRUCTION OF THE RELATIVE STOPPING POWER
Figure 4.5: Image reconstruction of the Forbild phantom after 8 iterations of ART with the following parameters: λ= 0.001,w1 = 0.1 and 2 iteration of TV descent. The image reconstructed using FBP with the binning case (d) was taken as initial estimate. Colour range of [0:1.5].
wherexphj is the value in thejth voxel of the phantom andxj the value of the same voxel in the reconstructed image. Note that this figure of merit is useful to compare different reconstructions of the same phantom but is not adapted to compare reconstructions of different objects. The second considered criterion was visual assessment of noise and detail in the images.
Parameters were tested and adjusted manually in an empirical fashion to find the
“optimized” ones. For simplicity, the relaxation factor was kept constant throughout cycles and iterations. The data was always organized as 1 proton/mm2 of source per projection for each cycle. A treatment of each proton history once represents an iteration of the algorithm. It was shown that starting the iterative reconstruction procedure with an initial estimate already close to the object, such as an analytically reconstructed image, accelerates the convergence of the algorithm [Penfold,2009]. The starting point of the reconstruction was taken as the FBP image reconstructed using the data binned according to the position upon interaction on the upstream tracking plane (case (d) in the previous section, illustrated on Figure4.3(d)).
Satisfactory results were found for a relaxation factor λ = 0.001, a TV weighting factor w1 = 0.1 and two iterations of TV descent after each cycle. The reconstructed image after 8 iterations is shown on Figure 4.5. Figure4.6 shows the relative error as a function of the iterations for this set of parameters. The point at 0 iteration represents the relative error of the FBP image used as initial estimate of the object.
It can be seen that the ART iterations over the FBP image allow to reduce the error and that the algorithm seems to converge. Figure4.7represents the reconstructed RSP profile through the biggest left ear resolution pattern, as drawn previously on the phantom in Figure 4.3.
While the relative error between the reconstructed image and the phantom is reduced by using the iterative algorithm, it can be noticed that the spatial resolution is not. This
3 4 5 6 7 8 9
0 2 4 6 8 10
Relative error (%)
Iterations
Figure 4.6: Relative error (RE) between the reconstructed image and the phantom as a function of the iterations. Reconstruction parameters were set to λ= 0.001, w1 = 0.1 and 2 iteration of TV descent. The point at 0 iteration represents the relative error of the FBP image used as initial estimate of the object.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
60 80 100 120 140
Reconstructed RSP
Position (mm)
Phantom FBPART
Figure 4.7: Profile through the left ear pattern of the phantom, FBP (binning case (d)) and ART reconstructions.
4.1. PREAMBLE: IMAGE RECONSTRUCTION OF THE RELATIVE STOPPING POWER
result is surprising because a more accurate path estimation would intuitively lead to a better spatial resolution. However, it can be explained by the fact that the weighting factor of the TV may not be optimal. As the TV minimisation results in a reduction of the gradients, inappropriate weighting may result in a blur.
Discussion
Because it will not impact the main conceptual arguments of this thesis work, the optimization of the parameters for the ART algorithm was not performed.
However, should such an optimization be of interest, two main things need to be considered. The first is the image used as a starting point, here the FBP image. It can be put forward that this image shows some streak artefacts which might affect the ART algorithm. The second thing to consider is the optimization of the parameters for the ART algorithm. The search for the optimized parameters could be performed automatically. This would require a set of figures of merit to test the different aspects of image quality. Indeed, the RE considered here represents a general criterion on the whole image, favouring the low frequencies, and is not indicative of the spatial resolution or noise. A comprehensive quantification of image quality would in addition need to consider noise, spatial resolution and potential artefacts. A huge amount of computation time will also be necessary to systematically test the possibilities. The reconstruction algorithm set-up here, in particular the ray-tracing and MLP computation, could be implemented for computation on graphical processing units (GPU). The gain in computation time would make it possible to consider such a systematic search. In order not to have to test all possibilities, a search following an experimental design could be performed, as can be done in experimental sciences. Furthermore, it needs to be considered that the optimized parameters determined on a simulation with perfect detectors and with such a semi-anthropomorphic phantom may be different from the ones required to optimize image quality in clinical conditions.
Nevertheless, the path approximation is of key importance in order to achieve the best spatial resolution possible with proton imaging. In this context, the MLP is the most accurate path approximation that has been proposed to this day. The sequel of this chapter aims at further investigating the MLP, from two different points of view. The computation of the MLP is performed assuming all materials have the scattering and energy loss properties of water. This is often the case for first approximations. However, in regions with important proportions of bone or air, such as the thorax or nasal cavity, this approximation may not be the most appropriate. Therefore, a MLP estimation in non-homogeneous medium consisting of slabs was derived and studied in Section 4.2.
Section 4.3aims at investigating the impact of the tracking system parameters (spatial resolution, material budget, positioning of the tracking planes) on the path estimation.