Advanced Algorithms and Quantitative 3D Analysis

Reconstruction Algorithms

The Fourier sampling at low spatial frequencies is much higher compared to the sampling at low spatial frequencies. Therefore, a stright backprojection would result in an overestimate of the low frequencies while the high frequencies are underestimated, leading to a very blurry reconstruction.

One approach commonly used to overcome this problem is weighted backprojection (WBPJ), where a weighting filter is introduced reducing the contribution of the low spatial frequencies (Fig. 11).

Fig. 11: Comparison of straight backprojection (BPJ) and weighted backprojection (WBPJ).

Alternatively, iterative approaches such as the Simultaneous Iterative Reconstruction Technique (SIRT) are nowadays commonly used in materials sciences. In addition to reducing the oversampling, the SIRT approach also reduces the noise in the reconstruction compared to WBPJ.

However, with both reconstruction approaches, it is difficult to accurately reconstruct the object intensities in 3D. The exact shape of the weighting filter (a Fourier filter) introduces strong artifacts for the intensities. In the example of a zirconia filled polymer shown in Fig. 12, this effect can be seen as the image intensity of the vacuum around the sample is almost identical to the intensity of the polymer matrix. In the SIRT reconstruction of the same data set, the relative intensities are better preserved.

Fig. 12: Electron tomographic reconstruction of zironia nanoparticles in a polymer matrix: comparison of WBPJ and SIRT and missing wedge effect (N. Kawase, M. Kato, H. Nishioka, H. Jinnai, Ultramicroscopy, 2007, 107, 8-15). 

However, also the SIRT reconstruction does not produce easily quantifiable intensities that can be directly segmented by thresholding. Because of convergence limitations, there is a dependence of the reconstructed intensities of the averge density overlapping in the different projections. As one example, when reconstructing nanoparticles with different diameter, this results in decreasing intensities with decreasing particle diameter, which makes an accurate segmentation very difficult.

Fig. 13: 3D reconstruction of metal nanoparticles on a catalyst support. The graph shows the experimental correlation between particle size and reconstructed average intensity of the particles (from C. Kübel, D. Niemeyer, R. Cieslinski, S. Rozeveld, J. Mat. Sci. Forum, 2010, 638-642, 2517-2522).