# Computational Inversion Using MCMC and NURBS

Zenith Purisha^{*} and Samuli Siltanen

This research focuses on nonlinear inverse problems arising from shape optimization and reverse engineering. It is based on Bayesian inversion implemented via Markov Chain Monte Carlo methods and using Non-Uniform Rational B-Splines for representing the unknowns. This approach has not been studied much so far. The advantage is that industrial production devices, such as computer numerical control (CNC) machines, use NURBS for describing the shapes of objects to be manufactured. \newline Most modern factories use CNC machines, and relatively cheap (less than USD 5000) devices are available for small-scale production. Having the result of reconstruction immediately in industrially producible form can save time in research and development work and enable new kinds of production possibilities for start-up companies located anywhere in the world, including developing countries. \newline With the increasing availability of computer power, Monte Carlo techniques are more widely used. Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, cellular structures, and also some research mentioned above. \newline In recent years, statisticians have been increasingly drawn to Markov Chain Monte Carlo methods to simulate nonstandard multivariate distributions. The Gibbs sampling algorithm is one of the best known of these methods. A considerable about of attention is now being devoted to the Metropolis-Hasting algorithm. In this research, we will use both methods and then compare the results. \newline The data are reconstructed by NURBS. NURBS (Non-Uniform Rational B-Splines) are mathematical representations for describing and modeling curves and surfaces which are basically piecewise polynomial functions. The idea of writing inversion method directly in terms of NURBS has the following benefits: less parameters help create more robust algorithms, and the results are readily in a form used by Computer-Aided Design (CAD) software. \newline NURBS have several important qualities that make them ideal choice for modeling. NURBS provide the flexibility to design a large variety of shapes (standard analytic shapes and free-form curves and surfaces) by manipulating the control points and the weights. The amount of information required for a NURBS representation of a piece of geometry is much smaller than the amount of information required by common faceted information and also the evaluation is reasonably fast and computationally stable. And the other interesting and important quality is that NURBS are invariant under scaling, rotation, translation, and shear as well as parallel and perspective projection.

Mathematics Subject Classification: 49N45

Keywords: Nonlinear inverse problems; Bayesian inversion; Markov chain; Monte Carlo; NURBS

Minisymposion: High Resolution Imaging with Geometric Priors