Porous media include a variety of materials, such as bone, cartilage, concrete, soil, and wood. All such materials allow the flow of water, or other liquids, and the understanding and modeling of this flow can be essential in areas of human health, construction, and groundwater studies. The image processing challenge in porous media is the reconstruction of the 3D architecture of void spaces given some sort of data, a particularly challenging task since many porous media contain pore structures on a wide range of scales.
Our approach to this class of problems has been to treat the problem as an inverse or estimation problem. Because the fine-scale porous field is discrete (pore / not pore), discrete-state solvers such as Simulated Annealing are quite effective, such as the following example from the work of Mohebi:
|Original Image||Observed or Measured Image||Reconstructed Result|
The above work left us with two challenges:
- How to address very large fields with structure on a wide variety of scales.
- How to address nonstationary fields, those with multiple distinct behaviours.
The work of Liu considered hierarchical models (challenge 1) having hidden fields containing a label describing some attribute of behaviour (challenge 2):
This work led to promising results on fairly complex fields:
|Original Porous Field|
|Low Resolution, Measured Field||Estimated Hidden Label 1||Estimated Hidden Label 2|
|Estimated Field - Liu||Estimated Field - Wavelet||Estimated Field - Wavelet|
The performance of the method proposed by Liu is quite strikingly better than that produced by other wavelet resolution-enhancement methods.
Related research areas
Mohebi, A., P. Fieguth, and M. A. Ioannidis, "Statistical fusion of two-scale images of porous media", Advances in Water Resources, vol. 32, no. 11, pp. 1567 - 1579, 2009. Details
Mohebi, A., and P. Fieguth, "Modeling and reconstruction of two-scale porous media using MRI measurement", Fourth Biot Conference on Poromechanics, New York, 2009. Details
Liu, Y., A. Mohebi, and P. Fieguth, "Modeling of multiscale porous media using multiple Markov random fields", Fourth Biot Conference on Poromechanics, New York, 2009. Details
Alexander, S. K., P. Fieguth, and E. Vrscay, "Discrete-state modeling of porous media over multiple scales", SIAM Conference on Mathematical and Computational Issues in the Geosciences, Avignon, 2005. Details
Alexander, S. K., P. Fieguth, and E. Vrscay, "Hierarchical annealing for random image synthesis", Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2003), Portugal, 2003. Details
Liu, Y., "Hidden Hierarchical Markov Fields for Image Modeling", Department of Systems Design Engineering, Waterloo, Ontario, Canada, University of Waterloo, 2011. Details