Spectral Image Corruption

To train a denoiser or deblurrer that should later clean EIT reconstructions, one needs pairs (clean, degraded). Real EIT reconstructions are expensive to produce, so a cheap synthetic degradation is used that imitates their typical artefacts: loss of fine detail, spatially varying blur, and structured noise.

Discrete cosine transform (DCT). For an $n\times m$ image $x$, the 2D DCT-II $C=\mathrm{DCT}(x)$ expresses $x$ in cosine modes $\cos \left(\frac{\pi k(2i+1)}{2n}\right)\cos \left(\frac{\pi \ell (2j+1)}{2m}\right)$. These are the eigenvectors of the discrete Laplacian with reflecting (Neumann) boundaries (see Discrete Cosine Transform). As the squared magnitude of frequency $(k,\ell )$ take the corresponding eigenvalue, scaled to pixel frequencies,

$$|\omega {|}^2=(\frac{2n}{\pi }{)}^2{\sin }^2\frac{\pi k}{2n}+(\frac{2m}{\pi }{)}^2{\sin }^2\frac{\pi \ell }{2m}\approx k^2+{\ell }^2\text{for small }k,\ell .$$

Iterated corruption. Repeat $T$ times:

1. add spatial white noise: $x\leftarrow x+{\sigma }_{\text{sp}}\xi$;
2. transform: $C\leftarrow \mathrm{DCT}(x)$;
3. add frequency-dependent noise: $C_{k\ell }\leftarrow C_{k\ell }+{\sigma }_{\text{fr}}(1+|\omega {|}^2){\xi }_{k\ell }$;
4. damp high frequencies: $C_{k\ell }\leftarrow e^{-\lambda |\omega {|}^{2s}}{C}_{k\ell }$;
5. transform back: $x\leftarrow {\mathrm{DCT}}^{-1}(C)$.

Step 4 alone (for $s=1$) is the solution operator of the discrete heat equation $\dot{x}={\Delta }_{h}x$, a linear Gaussian-type blur. It preserves the mean and satisfies the discrete maximum principle, so it creates no new extrema. With $|\omega {|}^2=k^2+{\ell }^2$ instead, the truncated cosine series of the Gaussian multiplier rings near edges. Combined with repeated noise injection, the result is a random, nonstationary degradation whose strength is controlled by $T$, $\lambda$, $s$ and the noise levels.

Caveat. This is only a proxy. EIT artefacts depend on depth (the resolution loss grows towards the centre; see Decay of Boundary Measurements) and on the specific algorithm. Training on actual reconstructions, or learning inside the reconstruction loop, captures the operator-specific degradation better (see Learned Regularization).

In ModularEIT.jl: corrupt_image.

References

1. N. Ahmed, T. Natarajan, K. R. Rao (1974). Discrete Cosine Transform. IEEE Trans. Comput. C-23(1), 90–93. doi:10.1109/T-C.1974.223784
2. G. Strang (1999). The Discrete Cosine Transform. SIAM Review 41(1), 135–147. doi:10.1137/S0036144598336745