# Quad-Trees, Multiresolution image analysis, and boundaries

At MSc level / first year of PhD, it is proposed that the student assimilate basic material concerning CFTP and also the quad-tree analysis of Kendall and Wilson (2003), and then construct and implement a CFTP algorithm to segment ("remove noise from") a typical quad-tree-like image. The work programme would be:

- Read up CFTP from Kendall (2015) also referring to Kendall (2005) and Huber (2015).
- Use the template provided by Kendall (2015) to construct one's own implement of CFTP for segmentation of an image using Ising model. Kendall (2015) uses R: the implementation should use a language such as Python, for which it is relatively easy to obtain speed-ups using (for example) numerical packages.
- Read up quad-trees from Kendall and Wilson (2003), and amend the CFTP-Ising construction to segment using the Ising quad-tree model.
- Investigate optimal settings of parameters, behaviour at object boundaries. Compare reconstruction speeds for original and quad-tree Ising models. Consider effect of conditioning at all levels using quad-tree decimation of original image.

This could lead on to a PhD in many possible different ways, depending on the inclinations and strengths of the student:

[**Theory**] Adapt Finch (2010) work to establish weak limit for small lambda of Ising quad-tree conditioned on having an interface.

[**Theory**] Develop Finch (2010) work to understand interfaces, using estimates analogous to those of Gielis and Grimmett (2002).

[**Implementation**] Study "adaptive-resolution CFTP", in which one does just enough work to capture reconstruction at desired level of detail (which might even depend on image itself!). This is a new idea, based on work by Ambler and Silverman (2010).

[**Theory**] Study applicability of ideas to generalize the so-called "mass transport" principle to the quad-tree case.

[**Implementation**] Consider evidence for "cut-off" for Ising quad-tree, based on the theoretical work of Lubetzky and Sly (2014).

#### References

1. Ambler, G. K., & Silverman, B. W. (2010). Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding. In N. H. Bingham & C. M. Goldie (Eds.), *Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman* (pp. 64–90). Cambridge: Cambridge University Press.

2. Finch, S. (2010). The Random Cluster Model on the Tree. *PhD Thesis*, Warwick.

3. Gielis, G., & Grimmett, G. R. (2002). Rigidity of the interface in percolation and random-cluster models. *Journal of Statistical Physics*, 109(1-2), 1–37.

4. Huber, M. (2015). Perfect Simulation. Boca Raton: Chapman and Hall/CRC.

5. Kendall, W. S. (2005). Notes on Perfect Simulation. In Wilfrid S Kendall and F. Liang and J.-S. Wang (Ed.), *Markov chain Monte Carlo: Innovations and Applications* (pp. 93–146). Singapore: World Scientific.

6. Kendall, W. S. (2015). Introduction to CFTP using R. In V. Schmidt (Ed.), *Stochastic Geometry, Spatial Statistics and Random Fields* (pp. 405–439). Springer.

7. Kendall, W. S., & Wilson, R. G. (2003). Ising models and multiresolution quad-trees. Advances in Applied Probability, 35(1), 96–122.

8. Lubetzky, E., & Sly, A. (2014). Universality of cutoff for the Ising model. arXiv, 1407.1761, 24–27.