Comparison of different iterative reconstruction algorithms for X-ray volumetric inspection More info about this article: http://www.ndt.net/?id=22973 Georgios Liaptsis 1,2, Alan L. Clarke 1 and Perumal Nithiarasu 2 1 TWI Technology Centre (Wales), Harbourside Business Park, Harbourside Road, Port Talbot, SA13 1SB, UK 2 Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Swansea, SA1 8EN, UK Abstract E-mail: georgios.liaptsis@affiliate.twi.co.uk The widely established technique of traditional industrial X-ray Computed Tomography (CT) uses transformed based algorithms to reconstruct volumetric data. The most widely used algorithm is considered to be the Feldkamp, Davis and Kress (FDK) algorithm. However, the FDK algorithm suffers from a number of limitations. It requires thousands of projections to have good quality reconstructed images. This increases the acquisition time and subsequently the cost. Also, they are completely unsuitable for advanced scanning geometries with limited projections, complex trajectories or robotised radiography. These difficulties can be overcome using iterative reconstruction algorithms. In the present work we demonstrate a comparison of all existing iterative reconstruction algorithms as well as a new multiplicative version. A parametric study was carried out in order to find the advantages of each algorithm. The algorithms were analysed for their performance in different inspection scenarios. 1. Theoretical background 1.1 Introduction For both CT and CL (Computed Laminography), various reconstruction algorithms have been developed which significantly increase the image quality or decrease reconstruction (computational) time. Transform based methods use the Radon transform (1) particularly the inversion of the transform known as Fourier Slice Theorem (FST). The most representative algorithm of the transform based methods is the Filtered Back Projection (FBP) algorithm. Initially, the FBP algorithm was developed in 2D using a fan beam model. The most widely used of 3D reconstruction algorithms is the Feldkamp, Davis and Kress (FDK) algorithm (2) which uses the 3D cone beam model. Iterative reconstruction techniques use an entirely different approach for tomographic imaging. They are considered to be much simpler than the transformed-based methods but are in general much slower. The slow reconstruction speed makes it almost inapplicable for CT but is considered highly applicable for limited angle techniques where fewer projections are acquired. A rule of thumb is that iterative techniques need half of projections than transformed based algorithms (3).
1.2 Iterative reconstruction techniques Iterative reconstruction algorithms assume that the object's cross-section consists of an array of unknown values for which algebraic equations are established. Conceptually, it is much simpler than transform based methods but it lacks the speed of implementation. Furthermore, it is considered inapplicable to traditional CT, because of the number of projections for a full 360º scan is very large and the algorithm cannot be processed in an acceptable time (4). However, when applied to methodologies requiring fewer projections this limitation no longer valid and it is fully applicable for laminography, limited projection and any other limited angle techniques (5). The most representative algorithm of the iterative reconstruction methods is the Algebraic Reconstruction Technique (ART) algorithm. ART was initially proposed by R. Gordon in 1970 to develop a technique for three dimensional electron microscopy and X-ray photography (6). The implementation equation for ART is: ( +1) = ( ) + 2 =1... (1) where qi is the calculated projections based on the previous iterations calculation of grid values: ( ) = =1... (2) Where λ is the relaxation factor, k is the iteration value, j is the cell index value, i is the ray index value and n is the cell index value for a specific ray. The term Win includes all the weight coefficients of a particular beam i from all the cells. The term Wij includes all the weight coefficients for a particular cell j from all beam projections. The equation is iterative in nature, in which the next iteration s cell value is equal to the current value plus the correction value. The ART equation is used as the basis for the development of different computational implementations of reconstruction algorithms. These implementations have been known by their acronyms SIRT (7), SART (8) and three kind of MART: MART1 (6), MART2 (9) and MART3 (10). 2. Proposed reconstruction technique 2.1 Iterative reconstruction equation breakdown Most of the existing iterative reconstruction methods have been implemented and tested. There are several differences among the techniques. SART, SIRT and ART apply the correction with addition while the three kinds of MART are Multiplicative. SART and SIRT are Simultaneously applied to all the voxels while the other techniques are applied to each individual voxel at a time. Figure 1 demonstrate the similarities and differences of the algorithms in Venn diagram. The components of the equations, even though they are mostly similar, are applied differently within each method. All these differences have been the subject of a thorough investigation. The strategy was to understand every 2
part of the equations and the effect they have to the convergence speed, reconstructed image quality and efficiency. The ultimate goal was to suggest alterations to the existing methods and even create a new reconstruction technique. Figure 1 Similarities and differences among iterative reconstruction algorithms. In order to alter reconstruction equations, similarities among the techniques need be clarified. An iterative algorithm can be divided into two parts, the intensity value of the previous iteration and the correction. The first part is self-explained, we start with an initial guess for the first iteration since there are no previous iterations. In the next iteration, the initial guess is replaced with the intensity value of the previous iteration. The second part, the correction, is at the heart of the algorithm and most of the calculations occur here. The projection value pi is subtracted (or divided in MART) with the calculated projection value qi. The result of this calculation is multiplied by the Wij which is the summation of weights from all the beams that traverse a particular voxel j. For the multiplicative equations this is enough and the last step is to be subtracted from 1 (except MART3). For the additive techniques one extra normalisation step is required. This is especially true for SART and SIRT which needs the normalisation as the correction contains the summation of corrections for all the voxels. This breakdown makes it clear that in each iteration, only a small number of the equation factors are changed from one iteration to the other. The projection values pi is an input, and Win and Wij arrays once calculated, do not need computation again. The only factor that changes is qi, particularly the Vn component since Win remains the constant. Considering this, the algorithms can be implemented in a very efficient way. Another aspect is the normalisation factor that occurs in simultaneous algorithms. The main advantage of these algorithms is the convergence speed. In each iteration, the correction part contains information from all the voxels. This does not occur in the "per voxel" equations (basically ART and three MARTs), however they are considered more flexible and more accurate (11). It was decided to alter these per voxel equations in order to achieve the quick convergence. Therefore, the corrections need to be summed, such as in in SART and SIRT. In order to normalise the correction for each individual 3
voxel, the correction needs to be divided by the number of corrections found for the voxel j. We called this term Nj. 2.2 Modified reconstruction algorithms The new equations are demonstrated below. The modified versions will be called mart, mmart1, mmart2 and mmart3, which are demonstrated in equations 3, 4, 5 and 6, respectively. ( +1) = ( ) + =1 2 =1... (3) ( +1) = ( ) 1 1 =1... (4) ( +1) = ( ) 1 1 =1... (5) ( +1) = ( ) =1... (6) 3. Numerical simulations 3.1 Reconstruction algorithms comparison In order to validate the computational methods, numerical simulations were carried out. The Shepp-Logan phantom (12) of 256x256 pixels was used as the test object. The setup parameters were chosen to be as close as possible to the real CT system located at TWI. The input values were: source-to-detector distance=1257.22mm, source-to-object distance=628.61mm and detector pixel size=0.2mm. The scanning geometry was CT of 360 projections. Figure 2 shows the reconstructed images from the various algorithms as well as the original phantom. For sake of comparison with the existing technique, the SIRT algorithm was also implemented. The most obvious observation is that the mmart2 (see Figure 2e) suffer from severe artifacts which make the image completely unreadable. This is maybe attributed to the term Wij which is added and somehow interfere with the correction. No further discussion for mmart2 will follow. However, the other algorithms performed well with no obvious issues. The reconstructed images are looking very similar to the human eye. In order to have some quantitative data, the Root Mean Square Error was calculated across the iteration range and demonstrated at Figure 3. 4
Figure 2 Reconstructed images of Shepp-Logan phantom using different algorithms (a) Original, (b) SIRT, (c) mart, (d) mmart1, (e) mmart2 and (f) mmart3. Figure 3 Root Mean Square Error comparison. Starting with mart, it has the slowest conversion and also the higher error after 200 iterations. The second higher error appears on the mmart1 although it has the quickest conversion due to the absence of the term Wij in the equation. Also, there are some circular artifacts as well as artifacts at the point of rotation. The best results in terms of error were provided by mmart3 which performed better than standard SIRT. SIRT has a steep convergence over the initial 20 iterations but for further iterations, mmart3 has a stable lower error. A noteworthy observation is that the multiplicative algorithms have clear background around the phantom (value of 0) which is very important advantage compared to the additive algorithms. The clear background is due to the multiplicative nature of the algorithms, which is not sensitive to small variations. 5
3.2 Noise investigation A study incorporating data noise was also performed to simulate more realistic situations. A quick investigation in the literature reveals that mainly two kinds of noise are used to simulate radiographic simulations. The first is Poisson noise which is specified by a mean value and the fluctuations are small (13, 14). The second is Gaussian noise specified by a standard deviation and the noise pattern follows the Gaussian distribution (15, 16). Gaussian noise can take both positive and negative values which can simulate the salt and pepper noise which appear in the radiographic projections. This is the main reason for using Gaussian noise in these simulations. Gaussian noise with a standard deviation of 600 has been added to each projection. In order to avoid negative pixel projections (which is not realistic) the absolute function has been applied. The average noise had value of 480 which compared to the average projection value of 8007, generates approximately 6% noise. The Normalised Mean Absolute (NMA) error has been calculated and demonstrated in Figure 4. The NMA error was chosen because it is sensitive to small errors in a large number of pixels. The error curve of SIRT follows a very peculiar tendency as up to iteration 40 the error decreases but increases for the rest iterations. Eventually SIRT has the highest error after 200 iterations. The conclusion drawn is that SIRT is not very tolerant to noise. The second highest error is found in mart which, as before, has the smallest convergence. The mmart3 has the smallest overall error value, slightly smaller than mmart1. Figure 4 Normalised Mean Absolute Error comparison from noisy projections. In order to test the performance of the algorithms in different material properties, different Region of Interest (ROI) sections were examined. ROI 1 was selected from the background area before the high density area, which forms ROI 2. ROI 3 is the internal area directly adjacent to ROI 2. Finally ROI 4 is selected from the tumour area. The averaged single pixel difference value after 200 iterations was calculated and presented at Figure 5. The mart gave by far the highest response to all ROIs. On the other hand, mmart1 gave the lowest in all ROIs. SIRT did not perform well at ROIs 1 and 2 but performed very well at ROIs 3 and 4. SIRT is very efficient for the internal, medium density, low contrasting regions and mmart3 performs well in the high contrasting areas around the boundaries of the phantom. This can be highly pertinent in applications where extraction of a CAD model tomographic data is required. 6
Figure 5 Averaged single pixel difference for four Region-of-interest areas along with the ROI positions. 4. Conclusions We have implemented modified versions of existing algebraic reconstruction techniques and compared them in terms of reconstruction image quality for both ideal and noisy scenarios. With the proposed alterations it is possible to apply the correction parameter simultaneously to all voxels in standard per-voxel algorithms. The new algorithms performed differently over the various regions of the phantom model. The mmart3 performed really well for the background (like all multiplicative algorithms) and at the boundaries of the object. This is ideal for applications in which a CAD model is required from the radiographic scan. The mmart1 performed well in the whole body of the phantom however it suffers from some artifacts in the central regions. The algorithms were also tested in noisy conditions in order to simulate more realistic scenarios. The multiplicative algorithms proved more resilient to noise than mart and SIRT. Another observation was that in SIRT, the error convergence curve started to increase after a specific iteration. This is definitely affected by the noise as in the noisefree data, this phenomenon was not happened. The important outcome of this research is that traditional per-voxel reconstruction algorithms can be transformed to simultaneous algorithms. This is very important in terms of convergence speed as well as implementation efficiency. Acknowledgements The authors thank the Sêr Cymru National Research Network in Advanced Engineering and Materials for granting a PhD studentship (NRN091) with funds from the Welsh Government and the Higher Education Funding Council for Wales (HEFCW). References 1. J. Radon, 'On determination of functions by their integral values along certain multiplicities', Ber. der Sachische Akademie der Wissenschaften Leipzig,(Germany), vol. 69, pp. 262-277, 1917. 2. L. A. Feldkamp, L. C. Davis, and J. W. Kress, 'PRACTICAL CONE-BEAM ALGORITHM', Journal of the Optical Society of America a-optics Image Science and Vision, vol. 1, pp. 612-619, 1984. 7
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