Nonconvex image reconstruction via expectation propagation

The problem of efficiently reconstructing tomographic images can be mapped into a Bayesian inference problem over the space of pixels densities. Solutions to this problem are given by pixels assignments that are compatible with tomographic measurements and maximize a posterior probability density. This maximization can be performed with standard local optimization tools when the log-posterior is a convex function, but it is generally intractable when introducing realistic nonconcave priors that reflect typical images features such as smoothness or sharpness. We introduce a new method to reconstruct images obtained from Radon projections by using expectation propagation, which allows us to approximate the intractable posterior. We show, by means of extensive simulations, that, compared to state-of-the-art algorithms for this task, expectation propagation paired with very simple but non-log-concave priors is often able to reconstruct images up to a smaller error while using a lower amount of information per pixel. We provide estimates for the critical rate of information per pixel above which recovery is error-free by means of simulations on ensembles of phantom and real images.

Figure 1

Computed tomography scans (left panels) and their corresponding empirical distributions (right panels), going from full resolution images of $512\times 512$ (corresponding to the narrower, dark red empirical distribution) to lower resolutions $256\times 256, 128\times 128, 64\times 64$, and $32\times 32$ (the latter corresponding to the wider, light red empirical distribution). Panels (a) and (b) show two different viewpoints of the abdomen while (c) and (d) two different views of a knee.

Figure 2

(a) Fraction of wrong assigned pixels versus sampling rate $\alpha$. Dashed (thick) lines are used for $p=6$ ($p=3$) phantoms while different symbols denote different reconstruction techniques. (b) Estimate of the error versus noise-to-signal ratio $\sigma /L$ with $\sigma$ the standard deviation for the Gaussian noise for binary images. In both panels, the points correspond to sample averages over 50 randomly generated phantoms, while the error bars are one standard deviation from the average.

Figure 3

(a) $E_2$ error for continuous images as a function of the sampling rate $\alpha$ for the Shepp-Logan phantom. The dashed line corresponds to error-free reconstruction threshold. (b) Reconstruction error versus noise-to-signal ratio $\sigma /L$ using synthetic random phantoms for fixed $\alpha =0.745$ and $p=6$ (an example of phantom image is depicted in the inset).

Figure 4

$E_2$ error for real tomographic images as a function of the sampling rate $\alpha$ for (a) the mouse's head, (b) $\mathrm{CT}\_\mathrm{head}\_38$, (c) $\mathrm{CT}\_\mathrm{head}\_80$, and (d) $\mathrm{CT}\_\mathrm{head}\_100$.

Figure 5

Full resolution images for (a) the mouse's head, (b) $\mathrm{CT}\_\mathrm{head}\_38$, (c) $\mathrm{CT}\_\mathrm{head}\_80$, and (d) $\mathrm{CT}\_\mathrm{head}\_100$.

Figure 6

Table containing the reconstructions of $\mathrm{CT}\_\mathrm{head}\_100$ using ${\mathrm{EP}}_{\mathtt{diff}}$, TV, ${\mathrm{EP}}_{\mathtt{int}}$, and QP as a function of the sampling rate $\alpha$. Top right region, delimited by the red thick line, contains the perfectly reconstructed images.

Figure 7

Reconstruction error in the $(\alpha ,\beta )$ plane for all our reconstruction techniques differing in the priors: (a) ${\mathrm{EP}}_{\text{bin}}$, (b) ${\mathrm{EP}}_{\text{sparse}}$, and (c) ${\mathrm{EP}}_{\text{int}}$ panels for binary images and (d) ${\mathrm{EP}}_{\text{diff}}$, (e) ${\mathrm{EP}}_{\text{sparse}}$, and (f) ${\mathrm{EP}}_{\text{int}}$ for continuous images.