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A directional regularization method for the limited-angle Helsinki Tomography Challenge using the Core Imaging Library (CIL)

  • *Corresponding author: Jakob Sauer Jørgensen (jakj@dtu.dk)

    *Corresponding author: Jakob Sauer Jørgensen (jakj@dtu.dk
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  • This article presents the algorithms developed by the Core Imaging Library (CIL) developer team for the Helsinki Tomography Challenge 2022. The challenge focused on reconstructing 2D phantom shapes from limited-angle computed tomography (CT) data. The CIL team designed and implemented five reconstruction methods using CIL (https://ccpi.ac.uk/cil/), an open-source Python package for tomographic imaging. The CIL team adopted a model-based reconstruction strategy, unique to this challenge, with all other teams relying on deep-learning techniques. The iterative algorithms implemented with CIL showcased exceptional performance, with one algorithm securing third place in the competition. The best-performing algorithm employed careful CT data pre-processing and an optimization problem with single-sided directional total variation regularization combined with isotropic total variation and tailored lower and upper bounds. The reconstructions and segmentations achieved high quality for data with angular ranges down to 50 degrees, and in some cases acceptable performance even at 40 and 30 degrees. This study highlights the effectiveness of model-based approaches in limited-angle tomography and emphasizes the importance of proper algorithmic design leveraging on available prior knowledge to overcome data limitations. Finally, this study highlights the flexibility of CIL for prototyping and a comparison of different optimization methods.

    Mathematics Subject Classification: Primary: 65K10, 94A08; Secondary: 65R32.

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  • Figure 1.  The 5 training datasets provided. The top row shows the full 360° sinograms A-E. The bottom row shows the ground truth reconstructions A-E

    Figure 2.  Left: Histogram of the pixel transmission values before and after re-normalization. Right: Zoom around 1.0

    Figure 3.  The sinogram, image data of the FDK reconstruction and the system geometry for the original (top) and zero-padded (bottom) data

    Figure 4.  Top Left: Absorption vs. path length for training dataset E. This shows the polynomial fit to pre-processed data and the estimated monochromatic correction. Top Right: Line profiles through the FDK reconstructions of dataset E with and without beam-hardening correction. Bottom left: FDK reconstruction of dataset A after beam-hardening correction has been applied. Bottom right: Line profiles through the FDK reconstruction of dataset A with and without beam-hardening correction

    Figure 5.  Performance comparison of Otsu and multi-Otsu segmentation on $ 60^\circ $ data. Left: TV reconstruction. Center: Otsu segmented reconstruction. Right: multi-Otsu segmented reconstruction. Top: Training data A. Bottom: Training data B

    Figure 6.  Reconstruction with FDK of the training sample A with 50$ ^\circ $ data. Left: reconstruction with linear gray color map in the range [-0.097, 0.104]. Center: segmentation. Right: Difference between segmentation and ground truth segmentation provided by the organizers. Correctly classified pixels are white, true background pixels misclassified as acrylic are red, and true acrylic pixels misclassified as background are blue

    Figure 7.  Reconstruction with isotropic TV of sample A with 50$ ^\circ $ data. Left: reconstruction displayed with linear gray color map in the range [-0.009, 0.064]. Center: segmentation. Right: Difference between segmentation and ground truth segmentation. Correctly classified pixels are white, true background pixels misclassified as acrylic are red, and true acrylic pixels misclasssified as background are blue

    Figure 8.  Fitting of the circle with the iterative method described in section 4.4, for the training sample B. In this example 4 iterations are required. In each plot we display the following: the fitted circumference, in red; the edges used by the algorithm in the current step to calculate the circle center and radius, in blue; the reconstruction of the complete data set; and the circle, whose radius is 4 pixels less than the fitted one, which determines which of the edges will be removed at the next iteration

    Figure 9.  Reconstruction with isotropic TV and disk constraint of sample A with 50$ ^\circ $ data. Left: reconstruction displayed with linear gray color map in the range [0.0, 0.0409]. Center: segmentation. Right: Difference between segmentation and ground truth segmentation. Correctly classified pixels are white, true background pixels misclassified as acrylic are red, and true acrylic pixels misclassified as background are blue

    Figure 10.  From left to right: isotropic TV reconstruction with a green arrow showing the direction of the center X-ray beam in the fan in the middle of the angular range for the limited angle dataset (i.e., the central direction), rotated isotropic TV reconstruction such that the y axis is parallel to the central direction (in green), isotropic plus anisotropic TV reconstruction as submitted to the challenge, isotropic plus anisotropic TV reconstruction at convergence (see section 4.8)

    Figure 11.  Reconstruction with isotropic and anisotropic TV with disk constraint of sample A with 50$ ^\circ $ data. The first row presents the results of the anisotropic TV algorithm as submitted to the challenge. The second row presents the results of the same algorithm at full convergence of the PDHG algorithm, resulting in an improvement of the score of about 4% with respect to the submitted algorithm (see section 4.8). The reconstructions are displayed with linear gray color map in the range [0.0, 0.0409]

    Figure 12.  Results of Algorithm 24B, compared to ground truth, for evaluation datasets

    Figure 13.  Quantitative comparison of the five algorithms 24A–24E submitted by the CIL team (solid lines, with the best-performing algorithm 24B in black) and the best performing algorithm from each of the competing teams, 15A, 16B, 9, 8B, 17A, 14, 13 and 7 (dashed lines). Left: Average scores over the three phantoms at each of the seven levels from $ 90^\circ $ to $ 30^\circ $ data. Right: Scores transformed by $ -\log(1-\texttt{score}) $ to highlight differences between initial scores close to 1

    Figure 14.  Each column shows one of the evaluation samples, with ground truth on the top and output of algorithm 24B on the bottom. Superimposed as a green arrow is the central direction of the X-ray cone. Left: level 4 ($60^\circ)$, sample C. Center: level 4 ($60^\circ)$, sample B. Right: level 6 ($40^\circ$), sample C. If most edges are roughly not perpendicular to the central direction of the X-ray fan, these are captured by data, and score is high (Left: 0.96374). If most edges are roughly perpendicular, they are harder to reconstruct, and score is lower (center: 0.92751, right: 0.74646). The right column shows that only the structures parallel to the central direction of the X-ray can be recovered

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