Reconstruction and segmentation from sparse sequential X-ray measurements of wood logs

Sebastian Springer; Aldo Glielmo; Angelina Senchukova; Tomi Kauppi; Jarkko Suuronen; Lassi Roininen; Heikki Haario; Andreas Hauptmann

doi:10.3934/ammc.2023002

Article Contents

2023, Volume 1, Issue 1: 1-20. Doi: 10.3934/ammc.2023002

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Reconstruction and segmentation from sparse sequential X-ray measurements of wood logs

1.
International School for Advanced Studies (SISSA), Trieste, Italy
2.
Computational Engineering Department, Lappeenranta-Lahti University of Technology, Lappeenranta, Finland
3.
Finnos Oy, Lappeenranta, Finland
4.
Research unit of Mathematical Sciences, University of Oulu, Oulu, Finland
5.
Department of Computer Science, University College London, London, UK

^*Corresponding author: Sebastian Springer
^*Corresponding author: Sebastian Springer

Received: February 2023

Revised: August 2023

Early access: August 25, 2023

Published online: August 25, 2023

This work was supported by the Centre of Excellence of Inverse Modelling and Imaging (CoE), Academy of Finland, decision numbers 336787 and 336796. AH was was supported by the Academy of Finland project 338408. SS was supported by the Academy of Finland, project number 334 817.

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Abstract

In industrial applications, it is common to scan objects on a moving conveyor belt. If slice-wise 2D computed tomography (CT) measurements of the moving object are obtained we call it a sequential scanning geometry. In this case, each slice on its own does not carry sufficient information to reconstruct a useful tomographic image. Thus, here we propose the use of a Dimension reduced Kalman Filter to accumulate information between slices and allow for sufficiently accurate reconstructions for further assessment of the object. Additionally, we propose to use an unsupervised clustering approach known as Density Peak Advanced, to perform a segmentation and spot density anomalies in the internal structure of the reconstructed objects. We evaluate the method in a proof of concept study for the application of wood log scanning for the industrial sawing process, where the goal is to spot anomalies within the wood log to allow for optimal sawing patterns. Reconstruction and segmentation quality are evaluated from experimental measurement data for various scenarios of severely undersampled X-measurements. Results show clearly that an improvement in reconstruction quality can be obtained by employing the Dimension reduced Kalman Filter allowing to robustly obtain the segmented logs.

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Mathematics Subject Classification: Primary: 94A08, 60G35; 62H30; 65R32.

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Figure 1. A schematic drawing of the fan-beam acquisition geometry. Here $ S $ represents one of the X-ray sources while $ D = 1154.2 $ is the plate on which detectors were installed. The parameter $ r_{S} = 859.46 $ is the distance between the X-ray source and centre of rotation, $ r_{D} = 705.37 $ is the distance between the centre of rotation and the detector, $ h_{S} = 232.86 $ is the source shift, $ h_{D} = -24.65 $ is the detector shift, $ \alpha_{D} = 0.16 $ is the detector tilt, and $ n_{D} = 768 $ is the number of detector elements

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Figure 2. Wooden phantoms used to perform the system matrix calibration

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Figure 3. (a) and (c) are the reference figures obtained using FBP and dense sampling with 360 measurement angles while (b) and (d) are the respective reconstructions obtained using DrKF and a fixed system of 3 X-ray source-receiver pairs

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Figure 4. Two reference slices used to compare the reconstruction/segmentation quality

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Figure 5. DrKF reconstructions of the Reference A with acquisition geometry rotating with random speed

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Figure 6. DrKF reconstructions of the Reference A with acquisition geometry rotating by 1 degree between consecutive slices

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Figure 7. DrKF reconstructions of the Reference A with acquisition geometry rotating by $ \Delta/4 $ degree between consecutive slices, where $ \Delta $ is the angular difference between consecutive sources in the acquisition geometry

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Figure 8. DrKF reconstructions of the Reference B with acquisition geometry rotating by $ \Delta/4 $ degree between consecutive slices, where $ \Delta $ is the angular difference between consecutive sources in the acquisition geometry

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Figure 9. Segmentations of the Reference A and the reconstructions presented in fig. 7

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Figure 10. Segmentations of the Reference A by using DPA and a standard multi OTSU

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Figure 11. Comparison of quantitative values for for different acquisition schemes. Values are obtained by averaging in the block of 11 slices centered at the reference in fig. 4a. (A) Dice squared coefficient, where the reference segmentation was obtained from the full angular FBP reconstruction. With $ Z $ set to 3.4 for number of angles smaller than 7 and 2.4 for the other cases. (B) Average PSNR

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Figure 12. Values are obtained by averaging in the block of 11 slices centred at the reference in fig. 4a. (A) Comparison of the segmentation method used with a standard multi OTSU. (B) Impact of the reduced dimension $ r $ (amount of singular values) on the computational (blue) time and the reconstruction quality (orange).

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