By Jillian Kunze
Figure 1. A sample of the tree Populus tremula inside a micro-CT scanner. Figure courtesy of Tatiana Bubba.
The sophistication of computed tomography (CT) and micro-CT scanners—which operate on a smaller scale than CT scanners but have a much higher resolution—has significantly increased over the course of the last several decades. This technology has enabled the high-resolution imaging of plant structures in two and three dimensions, even down to the cellular level (see Figure 1). "This is very important from a physics and biological perspective, because it allows us to have in vivo information about the water and carbon status inside a plant,” Tatiana Bubba of the University of Bath said. “Ultimately, it is crucial to have information on these to establish climate models."
However, plant structures do not absorb X-rays very well, which can limit these kinds of imaging studies. One way that scientists deal with this shortcoming is by injecting plants with iodine to serve as contrast agent, thereby increasing the absorption of X-rays and producing images with better contrast. The injected iodine also moves throughout the plant’s phloem, turning this into a dynamic computed tomography problem. This makes the process of reconstructing an image from multiple projections—i.e., multiple sets of data that are gathered by rotating the X-ray source about the target at many different angles—much more complex.
There are many known approaches to reconstruct an image of a scanned object from measurements when the object is static, but dynamic tomography is complicated by its dependency on time. Figure 2 illustrates this concept with an object that changes from a square to a circle as the X-ray source rotates around it; there would most likely not be such a dramatic change to the object in a real application, but the idea remains that the target undergoes some kind of change as time passes.
During a minisymposium presentation at the 2022 SIAM Conference on Imaging Science, Bubba explained mathematical techniques for interpreting imaging data of phloem transport through plant stems. She described joint work done as a part of the AtMath (Atmospheric Mathematics) collaboration with Tommi Heikkilä, Samuli Siltanen, Hanna Help, Simo Huotari, and Yann Salmon (all from the University of Helsinki).
Biologists and physicists are currently very interested in non-destructive imaging of plants, for which micro-CT benchtop scanners are very attractive. Contrary to positron emission tomography (PET) and magnetic resonance imaging (MRI), CT scans expose plants to less harmful radiation. Micro-CT also does not require radioactive tracers—unlike PET and MRI—but can still obtain high-resolution images. The technology is commonly available in plant ecophysiology libraries. However, a drawback is that full scans of the micro-CT are too slow to capture the iodine tracer’s movement. “One way that one could reduce both radiation dose and duration of imaging is to lower the number of scanning angles,” Bubba said. “Putting all this together, we not only have a dynamic tomographic problem, but we also have an under-sampled one because we want to reduce the radiation; this leads to a severely ill-posed problem.”
Figure 2. A simplified example of dynamic tomography. As the X-ray source rotates around the object and takes multiple projection images, the object itself also changes from a square to a circular shape. Figure courtesy of Tatiana Bubba.
Bubba and her collaborators wanted a simple model that could address their problem, did not require particular information on the motion that was occurring, and could detect the iodine’s onset and subsequent spread throughout the plant. So, they elected to use a sparsity enforcing, motion-aware model based on shearlets, which is a multiresolution representation system that was first developed about 15 years ago. “Shearlets are a fine system compared to wavelets or the Fourier transform that basically allow us to decompose a signal or an image into its building blocks at a different resolution or different scaling, and also a different orientation that is encoded by the shearing matrix,” Bubba said.
A shearlet-based approach can preserve the edges in an image, denoise an image even when there are not many projections to work with, and pick up on parts of an object that remain static over time. That is useful for this particular application, as the plant structure should not change significantly while the iodine flows throughout the plant (though the plant could shrink if it receives too much radiation). The approach that Bubba described in her talk involved using the three-dimensional (3D) shearlet-based regularization technique to reconstruct two-dimensional (2D) cross-sections of the plant over time, though the ultimate goal is to reconstruct the entire plant’s 3D volume over time.
The researchers first benchmarked their image reconstruction approach using a digital plant stem phantom, then tried imaging two different real objects. The first was a gel phantom — a non-living object made out of gel that they could infuse with iodine, which would spread throughout the gel in a more controllable way than in a living plant. They could also take more time to measure the object without damaging it, which allowed them to obtain images at 17 time frames. Finally, they measured an actual living tree at 11 time frames (see Figure 1).
Figure 3. Experimental results of iodine (shown in yellow) spreading through physical objects imaged with an X-ray micro-CT scanner and processed with various techniques. 3a. Images of the cross section of a gel phantom compiled from 30 projections taken at time steps 2, 7, and 12 and processed using three-dimensional (3D) and two-dimensional shearlet techniques. 3b. Images of the cross section of a living tree compiled from 30 projections taken at time steps 1, 7, and 11 and processed using 3D shearlet and Haar wavelet techniques. Figure courtesy of Tatiana Bubba.
The researchers compared several different methods for reconstructing these images: their 3D shearlet method (which accounts for changes across time) as well as a 2D shearlet approach and Haar wavelet technique (which are both applied independently at each time frame, and so do not consider the temporal connection across time frames). Figure 3 provides results that were reconstructed from only 30 projections, with the yellow portions in the images representing the onset of iodine. Figure 3a depicts the gel phantom imaged at time steps 2, 7, and 12 and reconstructed using the 3D shearlet and 2D shearlet techniques. The 3D shearlet technique detects the time at which the iodine begins to spread, though the image also depicts the iodine spreading outside where it would be expected. The 2D shearlet reconstruction, however, was hampered by the movement in the object and created an image with the iodine smeared out even more. The Haar wavelet approach produced a relatively good outcome in this case.
The differences between the reconstruction techniques were especially apparent when imaging the living tree. Figure 3b depicts a cross-section of the tree imaged at time steps 1, 7, and 11 and reconstructed using the 3D shearlet and Haar wavelet techniques. The 3D shearlet method clearly shows the onset of iodine; it does not reconstruct a particularly sharp image, but this is at least partially attributable to the relatively small number of projections. The 2D shearlet approach once more smeared out the areas with iodine, while the Haar wavelet technique had difficultly suppressing noise and did not mark the area of iodine as well.
Overall, the researchers were able to combine regularization in both spatial and temporal domains to improve the quality of reconstruction, although doing so did incur a higher computational cost. “The results are, in general, promising towards the development of micro-CT as a tool to study phloem transport using an iodine tracer,” Bubba said. In the future, the researchers might try regularizing using other approaches that are more computationally efficient, which could enable four-dimensional reconstructions — scans of a plant that include three spatial dimensions in addition to time.
Further Reading
Bubba, T.A., Heikkilä, T., Help, H., Huotari, S., Salmon, Y., & Siltanen S. (2020). Sparse dynamic tomography: A shearlet-based approach for iodine perfusion in plant stems. Inverse Probl., 36(9), 094002.
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Jillian Kunze is the associate editor of SIAM News. |