Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography

Figure 1.  Illustration of the calculation of 2d radiological paths. The variable $\varphi$ is the angle of inclination and $(ia, ib)$ is the index of the pixel. The variable $L$ is the intersection length of a ray with two vertical gridlines and $w$ is the proportion between intersection length of current pixel and $L$. The current pixel intercepted by the ray is $(ia, ib)$. $w>1$ in Algorithm. 1 corresponds to the left figure where the next pixel intercepted by the ray is $(ia, ib+1)$ and $w <1$ corresponds to the right figure where the next pixel is $(ia, ib+1)$. The proportion $w$ is calculated alternatively in Algorithm. 1 including only a few addition or subtraction operations and $w$ is used to decide whether the next pixel intercepted by the ray corresponds to the case in the left figure or the right figure

Figure 2.  Illustration of the calculation of 3d radiological paths. $w_{x}$ and $w_{y}$ are the proportion of the projected ray in $zx-plane$ and $zy-plane$ respectively. $w_{x}$ and $w_{y}$ are calculated by algorithm. 1. $L'$ is the constant length of the ray between the plane $iz = ic$ and $iz = ic+1$. The current voxel intercepted by the ray is $(ia, ib, ic)$. $w_{x} <w_{y}$ in Algorithm. 2 corresponds to case that the next two voxels intercepted by the ray are $(ia+1, ib, ic)$ and $(ia+1, ib+1, ic)$. $w_{x}\geq w_{y}$ corresponds to case that the next two voxels are $(ia, ib+1, ic)$ and $(ia+1, ib+1, ic)$. $w_{x}$ and $w_{y}$ are used to decide which voxels are intercepted by the ray after the current voxel $(ia, ib, ic)$

Figure 3.  Reconstructions of 3D Shepp-Logan phantom using the alternating minimization scheme described in Algorithm 3 and artifact reduction strategy in Sec. 2.3. The size of Shepp-Logan phantom is $300\times 300\times 300$ and two slices are shown in (a)~(e). (a)(b) are slices of original phantom. (a) corresponds to the red line in (b) and (b) to the red line in (a). Euler angles are used to generate projections: rotation $\phi$ changes between $-60^{\circ}$ and $60^{\circ}$ at $1^{\circ}$ intervals for each measurement, azimuthal angle $\theta$ is fixed at $0^{\circ}$ and in-plane rotation $\psi$ at $165^{\circ}$. (c)+(d) and (e)+(f) show the reconstruction image and segmentations without the artifact reduction strategy and (g)+(h) and (i)+(j), with a cutoff $\kappa_{\sigma}$ in the operator $\mathcal{K}_{\sigma}$ defined in (14). We chose the parameter $\sigma = \Phi /4$. While streak artifacts near $\pm 60^{\circ}$ in (c)+(d) and streak artifacts in (e)+(f) can be observed, the implementation of the artifact reduction strategy smooths some artifacts in (g)+(h) and (i)+(j)

Figure 4.  Reconstructions and segmentations of the Shepp-Logan phantom $f^{\dagger}$ from noisy Radon data. Left column: reconstruction $f^{MS}$ from noisy data $\mathcal{R}f^{\dagger} + \delta N(0, 1)$ using the artifact reduced Mumford-Shah algorithm ($\alpha = 2, \beta = 0.0002, \varepsilon = 0.0001 $); middle column: The reconstructed edge indicator function $v$; right column: reconstructions from same noisy data using WBP method in TOM software toolbox. For stronger noise it is more difficult to reconstruct the low contrast regions for both methods. From the comparison between the left and right column, we can see that our reconstruction method with Mumford -Shah functional can depress the noise compared to WBP method while the edges are preserved. If an edge is detected in $v$, the reconstruction $f^{MS}$ has a sharp edge

Figure 5.  Reconstructions of cryo-specimen using the alternating minimization scheme described in Algorithm 3 and artifact reduction strategy in Sec. 2.3. Left column is the reconstructed images, the middle column is the edge sets with our algorithm, and the right column is the reconstructed images with WBP algorithm in TOM toolbox. The images in the red block are zoomed in and put in the top right corner. The edges in the middle column are enhanced in the left column compared to the WBP reconstructions in the right column. Compared to the results of WBP in right column, the results in the left show the effect of noise reduction while the edges are preserved

Figure 6.  Density maps and ResMap of the reconstructed images of WBP method and our method. The mean resolution of WBP is 17.17Å while the mean resolution of our reconstruction with alternative minimization algorithm is 15.07Å