Digital Image Correlation (DIC) and Digital Volume Correlation (DVC) are popular techniques to measure displacement fields from 2D and 3D image pairs, respectively. These image registration techniques face a considerable challenge, namely, their ill-posedness. In fact, the limited available information (i.e., gray levels) leads to an unavoidable compromise between the measurement uncertainty and the spatial resolution.

A method for overcoming this limitation is to assume the displacement field to be continuous over the entire region of interest (ROI). Hence, it can be decomposed over basis functions that fulfill this constraint, such as those used in the Finite Element (FE) method. The imposed inter-dependence (coupling) between all degrees of freedom leads to the so-called global DIC and DVC methods [1, 2]. These differ from their local counterparts [3, 4] that do not assume any continuity in the sought displacement field.

Additionally, regularization techniques can be employed to further circumvent the ill-posedness of the registration. In particular, in the context of experimental mechanics, it is natural to seek a displacement field that best registers the images while also being mechanically admissible. Such is the goal of the so-called "mechanical regularization" based on the equilibrium gap method [5, 6]. This regularization constrains the displacement field to one that locally follows a linear elastic behavior.

Unfortunately, this approach by itself is not capable of applying the adequate regularization to each type of boundary present in the analysis. In fact, the guiding principle is only valid for the bulk and free-surfaces of the studied sample. For such reason, in reference [6], the authors proposed an approach that mimics the bulk, as if those boundaries had an elasticity of their own in addition to the bulk (as a kind of "surface tension"). However, the link between both models (bulk and surfaces) is relatively poor.

Here is presented an extension to the mechanical regularization that distinguishes the roles that different boundaries (Neumann or Dirichlet) play and treats them accordingly. Moreover, it does not require further (nor custom) developments other than those already postulated by the equilibrium gap. As such, it provides a single framework for handling the object boundaries both in 2D as in 3D without any modification. Consequently, objects with arbitrarily complex boundaries can be studied in their entirety using DIC or DVC, accordingly.
The proposed complete mechanical regularization opens many possibilities in the fields of DIC and DVC. First, since the technique is an extension to the bulk mechanical regularization, it naturally inherits all its benefits; such as providing lower levels of uncertainty, even when dealing with images of poor quality. Moreover, it provides fast convergence, which, in practical terms, allows even complex cases to be treated both robustly and fast. This behavior is desirable even in simple cases that may not "require" regularization, since they can be swiftly solved using lesser iterations.

References:

1. G. Besnard et al. "Finite-element displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands". In Experimental Mechanics (2006).
2. S. Roux et al. "Three-dimensional image correlation from X-ray computed tomography of solid foam". In Composites Part A: Applied Science and Manufacturing (2008).
3. H. Schreier et al. Image Correlation for Shape, Motion and Deformation Measurements. Springer US, 2009.
4. B. K. Bay et al. "Digital volume correlation: Three-dimensional strain mapping using X-ray tomography". In Experimental Mechanics (1999).
5. D. Claire et al. "A finite element formulation to identify damage fields: the equilibrium gap method". In International Journal for Numerical Methods in Engineering (2004).
6. Z. Tomicevic et al. "Mechanics-Aided Digital Image Correlation". In The Journal of Strain Analysis for Engineering Design (2013).