Shape registration problem have been an active research topic in computational geometry, computer vision, medical image analysis and pattern recognition communities. Also called the shape alignment, it has extensive uses in recognition, indexing, retrieval, generation and other downstream analysis of a set of shapes. There have been a variety of works that approach this problem, with the methods varying mostly in terms of (can be called pillars of registration) the shape representation, transformation and registration criteria that is used. One such method is proposed by Huang et al. in this paper, which uses a novel combination of the three pillars, where an implicit shape representation is used to register an object both globally and locally. For the registration criteria, the proposed method uses Mutual Information based criteria for its global registration phase, while sum-squared differences (SSD) for its local phase. 

The method starts off with defining an implicit, non-parameteric shape representation which is translation, rotation and scale invariant. This makes the first step of the registration pipeline which transforms the input images into a domain where the shape is implicitly defined. The image is first partitioned into three spaces, namely $[\Omega]$ (the image domain), $[R_S]$ (points inside the shape), $[\Omega - R_S]$ (points outside the shape), and $[S]$ (points lying on the shape boundary). Using this partition, a function based upon the Lipschitz function $\phi : \Omega -> \mathbb{R}^+$ is defined as:

\begin{equation}
    \phi_S(x,y)
    \begin{cases} 
         0 & (x,y) \in S \\
         + D((x,y), S)>0 & (x,y) \in [R_s] \\
         - D((x,y), S)<0 & (x,y) \in [\Omega - R_s]
    \end{cases}
\end{equation}
    
Where $D((x,y),S)$ is the distance function which gives the minimum Euclidean distance between point $(x,y)$ and the shape $S$. 

Given the implicit representation, global shape alignment is performed using the Mutual Information (MI) objective function defined between the probability density functions of the pixels in source image and the target image sampled from the domain $\Omega$. 

\begin{equation}

MI(f_{\Omega}, g_{\Omega}^{A}) = \underbrace{\mathcal{H}[p^{f_{\Omega}}(l_1)]}_{\substack{\text{Entropy of the}\\ \text{distribution representing $f_{\Omega}$}}} + \underbrace{\mathcal{H}[p^{g_{\Omega}^{A}}(l_2)]}_{\substack{\text{Entropy of the}\\ \text{distribution representing $g_{\Omega}^{A}$} \\ \text{which is the} \\ \text{transformed source ISR using $A(\theta)$}}} - \underbrace{\mathcal{H}[p^{f_{\Omega}, g_{\Omega}^{A}}(l_1, l_2)]}_{\substack{\text{Entropy of the}\\ \text{joint distribution}\\\text{representing $f_{\Omega}, g_{\Omega}^{A}$}}}

\end{equation}

Following global registration, local registration is performed by embedding a control point grid using the Incremental Free Form Deformation (IFFD) method. The objective function to minimize is used as the sum squared differences (SSD). The local registration is also offset by using a multi-resolution framework, which performs deformations on control points of varying resolution, in order to account for small local deformations in the shape. In case where there is prior information available for feature point correspondence between the two shapes, this prior knowledge can be added as a plugin term in the overall local registration optimization term. 

The method was applied on statistically modeling anatomical structures, 3D face scan and mesh registration.