XRCT image processing for sand fabric reconstruction

Abstract

We explore computationally efficient techniques to improve the XRCT image processing of low resolution and very noisy images for use in reconstruction of the fabric of densely packed, natural sand deposits. To this end we evaluate an image preprocessing workflow that incorporates image denoising, single image super resolution, image segmentation and level-set (LS) reconstruction. We show that, although computationally intensive, the Non-Local Mean (NLM) filter improves the quality of XRCT images of granular material by increasing the signal-to-noise ratio without impairing visible structures in the images, and outperforms more traditional local filters. We then explore an image super-resolution technique based on sparse signal representation and show that it performs well with noisy data and improves the subsequent stage of binarization. The image binarization is performed using a Hidden Markov Random Fields (HMRF) with Weighted Expectation Maximization (WEM) algorithm which takes the spatial information into account and performs well on high resolution images, however it still struggles with low quality images. We then use the level set method to define the grain geometry and show that the Distance Regularized LS Evolution (DRLSE) is an efficient approach for data sets with large numbers of grains. Finally, we introduce a penalty term into the evolution of the LS function, to address the issue of adhesion of much finer particles, such as clay, on the surface of the reconstructed avatars, while maintaining the main morphological details of the grains.

Appendices

Appendix 1: Mathematical details of solving Lagrange dual problem

• 1. Objective function:

  $$\begin{aligned} D(\lambda )&= {} \min _D \mathcal {L}(D,\lambda ) \nonumber \\&= {} {\textbf {trace}}( X^T - XZ^T (ZZ^T + \Lambda )^{-1} (XZ^T) ^T - \Lambda ) \end{aligned}$$

  (A1)

• 2. Gradient:

  $$\begin{aligned} \frac{\partial D(\lambda )}{\partial \lambda _i} = \Vert XZ^T (ZZ^T + \Lambda )^{-1} e_i\Vert ^2 - 1 \end{aligned}$$

  (A2)

  Where $e_i\in \mathcal {R}^n$ is the i-th unit vector.

• 3. Hessian:

  $$\begin{aligned}{} & {} \frac{\partial ^2 D(\lambda )}{\partial \lambda _i \partial \lambda _j} = -2((ZZ^T + \Lambda )^{-1} \nonumber \\{} & {} \quad (XZ^T)^T XZ^T (ZZ^T+\Lambda )^{-1})_{ij} ((ZZ^T+\lambda )^{-1})_{ij} \end{aligned}$$

  (A3)

• 4. Since then, the Lagrange dual problem can be optimized using conventional solvers, i.e., Newton’s method or Conjugate Gradient. The optimum dictionary D is obtained via:

  $$\begin{aligned} D^T = (ZZ^T + \lambda )^{-1} (XZ^T)^T \end{aligned}$$

  (A4)

Appendix 2: Expectation-maximization algorithm

• 1. Start: Initialize parameter set $\Theta ^0$ (randomly sampled from normal distribution).

• 2. The E-step: Compute posterior probability $P(x_i=l \mid y_i, \Theta ^t)$ for each observed pixel $y_i$ with respect to parameters of l-th cluster, which is the first and second moment of inertial $\mu _l$ and $\Sigma _l$ in the MGM model. Mathematically, E-step seeks a distribution q such that:

  $$\begin{aligned} q^{t+1}=\arg \max _q F(q^t,\Theta ^t)=P(x\mid y,\Theta ^t) \end{aligned}$$

  (B5)

  Where $F(q^t,\Theta ^t)$ is the free energy at t-th iteration, it is a function of the expected complete log likelihood and the entropy of distribution q. $F(q^t,\Theta ^t)$ is the lower bound of log likelihood $\log {P(y_i)}$ for each pixel, and the bound is obtainable when q is the posterior of label $x_i$ with respect to the intensity level $y_i$. Sketch of proof:

  $$\begin{aligned} \begin{aligned} \log {P(y_i)}&= \log { \sum _{l=1}^L P(y_i, x_i=l \mid \Theta ) } \\&= \log { \sum _{l=1}^L \frac{q(x_i\mid y_i, \Theta )P(y_i, x_i \mid \Theta )}{q(x_i\mid y_i, \Theta )} } \\&= \log { \sum _q \frac{P(y_i, x_i \mid \Theta )}{q(x_i \mid y_i, \Theta )} } \ge \sum _q \log { \frac{P(y_i, x_i \mid \Theta )}{q(x_i \mid y_i, \Theta )} } \\&= \sum _{l=1}^L q(x_i\mid y_i, \Theta ) \log {P(y_i, x_i \mid \Theta )}\\&- \sum _{l=1}^L q(x_i\mid y_i, \Theta ) \log {q(x_i\mid y_i,\Theta )} \\&= F(q, \Theta ) \end{aligned} \end{aligned}$$

  (B6)

• 3. The M-step: Maximize free energy $F(q^{(t+1)},\Theta ^t)$:

  $$\begin{aligned} \begin{aligned} \Theta ^{t+1}&= \arg \max _{\Theta } F(q^{t+1}, \Theta ^t) \\&= \arg \max _{\Theta } \sum _{l=1}^L q(x_i \mid y_i, \Theta ) \log {P (y_i, x_i \mid \Theta )} \end{aligned} \end{aligned}$$

  (B7)

• 4. Repeat the E-step and the M-step until convergence (up to 30 iterations in this study).

Appendix 3: EM algorithm for the weighted mixture Gaussian model

• 1. The E-step computes the posterior probability of pixel $y_i$ from cluster l with parameter $\{\mu _l, \Sigma _l\}$ and weight $w_i$.

  $$\begin{aligned} q_{il}^{t+1} = \frac{\pi _l^t \hat{P}(y_i; \mu _l, \Sigma _l, w_i) }{\tilde{P} (y_i; \Theta , w_i)} \end{aligned}$$

  (C8)

  where $\hat{P}$ and $\tilde{P}$ are defined before

• 2. the M-step updates parameters space by maximizing free energy:

  $$\begin{aligned} \begin{aligned} \Theta ^{t+1}&= \arg \max _{\Theta }\sum _{i=1}^N \sum _{l=1}^L q_{il}^{t+1} \log {\pi _l \mathcal {N}(y_i; \mu _l, \frac{\Sigma _l}{w_i}) } \\&= \arg \max _{\Theta } \sum _{i=1}^N \sum _{l=1}^L q_{il}^{t+1} (\log {\pi _l} - \log {\mid \Sigma _l\mid ^{\frac{1}{2}}}\\&\quad- \frac{w_i}{2} (y_i-\mu _i)^T \Sigma _l^{-1} (y_i-\mu _i)) \end{aligned} \end{aligned}$$

  (C9)

  By canceling out the derivatives with respect to the model parameters, we obtained the following update formula for the mixture proportions, means and covariance matrices:

  $$\begin{aligned} \pi _l^{t+1}&= {} \frac{1}{N}\sum _{i=1}^N q_{il}^{t+1} \end{aligned}$$

  (C10)
  $$\begin{aligned} \mu _l^{t+1}&= {} \frac{\sum _{i=1}^N w_i q_{il}^{t+1} y_i}{\sum _{i=1}^N w_i q_{il}^{t+1}} \end{aligned}$$

  (C11)
  $$\begin{aligned} \Sigma _{l}^{t+1}&= {} \frac{\sum _{i=1}^N w_i q_{il}^{t+1}(y_i - u_i^{t+1})(y_i-u_i^{t+1})^T }{\sum _{i=1}^N q_{il}^{t+1}} \end{aligned}$$

  (C12)