# To calculate the orientation of different particles in an image of a packed bed can be solved by Markov chain Monte Carlo simulation

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AASHNA SUNEJA 2021년 9월 18일
편집: AASHNA SUNEJA 2021년 9월 18일
I want to calculate the orientation of particles in a packed bed from the images obtained from CT. This can be calculated using the Markov chain monte carlo simulation which requires the estimate of seven parameters and about which I am totally clueless. I am attaching a sample image and the detailed procedure to calculate the orientation.
MARKOV CHAIN MONTE CARLO SIMULATION
Markov chain Monte Carlo (MCMC) here to help extract pellet information. The position, p, of a pellet P in a packed structure can be described by a set of seven parameters p ) {x, y, z, ax, ay, az, a}, where {x, y, z} is the location and {ax, ay, az, a} represents the rotation axis and rotation angle relative to the local coordinate system of the pellet. In the digital approach, the digital image of pellet P, denoted by s, can be described by a set of gray-scale voxels s) {s1, s2,..., sn}, where n ) n1 × n2 × n3 and n1, n2, and n3 are the dimensions of the bounding box of P. For a proposed pellet Q to fit P, the digitization of Q forms another set of gray-scale pixel image t ) {t1, t2,..., tn}. If we assume Gaussian noise for image s, the likelihood of fitting Q to P is then L(s;{x,y,z,ax,ay,az,a}))( 1√2πσ)2Πi)1nexp(-di2σ2) where di ) |si - ti| and σ2 is the noise variance. If we assume that there is no preferential factor affecting the position of P, we can use uniform priors for any of the parameters of p, i.e., uniform distributions for locations, rotation axis, and rotation angle. The combination of the priors and the likelihood above gives the posterior density π({x, y, z, ax, ay, az, a}|s). One way to calculate the posterior means (the most likely values of the parameters) is by the technique of MCMC simulation. The simulation produces a Markov chain whose equilibrium distribution is the posterior (target) density π( · ). If Q is randomly placed in the packed column and if the generated Markov chain is long enough, the chain will eventually settle at its equilibrium state. The most likely values of p ) {x, y, z, ax, ay, az, a} can then be estimated from π( · ). Detailed descriptions of MCMC and the sampling strategy are beyond the scope of the work. We refer interested readers to additional material for a practical introduction to the technique.
I want to know how to calculate the rotation axis and angle.

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