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Groupwise Congealing Algorithm

Learned-Miller et al. [] originally introduced their `congealing' algorithm for registering a set of hand-written digits. The aim was to avoid the arbitrary selection of a co-ordinate frame, by repeatedly registering each image with an evolving ``average" model. Given the current set of transformed images (initially the original images), for each pixel position, $i$, the probability density function of intensities, $v$, at that position across the set of images, $p_i(v)$ was estimated. The objective function was then the sum of entropies of these distributions across the whole image, $F=-\sum_i \int p_i(v) {\bf log} p_i(v) dv$. A set of image deformations were optimised to minimise this. In later work on registering sets of 3-D medical images [], the objective function was approximated by $\sum_j \sum_i {\bf log}p_i(v_{ij})$, where $v_{ij}$ is the value of pixel $i$ in deformed image $j$. During optimisation, each image was warped so as to bring pixels with similar intensities into correspondence across the set. This later approach was implemented.