UNCONSTRAINED OPTIMISATION FOR CONFORMAL DIFFEOMORPHIC IMAGE REGISTRATION

다국어 초록 (Multilingual Abstract)

Image registration is the process of transforming a source image so that its appearance approximates a target image. It can be used to study the similarities and differences between the images. There are a variety of different methods of transforming the image that differ principally by the type of allowable transformation. In computational anatomy, the most common approach is to find a diffeomorphism between the two images using a method known as Large Deformation Diffeomorphic Metric Mapping. However, it may be that a smaller set of allowable transformations yields additional information about the relationship between the images. Motivated by the fact that many biological transformations seem to be approximately conformal, in this study we consider conformal image registration. Conformal maps are locally equivalent to similarity transformations (they linearize to rotations, translations, and scalings). In order to avoid having to enforce conformality via a constraint at every point, we represent conformal maps by truncated Taylor series, that is, by complex polynomials. The coefficients of the Taylor series are determined by gradient descent on the discrepancy between the target and the transformed source images. Numerical examples illustrate the ability of the method to perform conformal registration and the convergence of the method during gradient descent and with respect to the number of terms in the truncated Taylor series.