Conformal Interpolations of Locally Defined Rotation, Scaling, and Translation Operations

In image morphing applications, deformation is not desired since it looks strange. To avoid deformation, the angles must be locally preserved. By deforming features, we can obtain funny-looking images, however, the shapes of the features will not look preserved if angles are not preserved. The angle-preserving operations are called conformal maps and locally they expect to be reduced into a concatenation of rotation, uniform scaling, and translation operations. This conceptual framework appears to give us a straightforward framework to construct general conformal maps. We can simply define desired rotation, uniform scaling, and translation in some select points and interpolate them by preserving their conformal property. Unfortunately, this does not work if these local operations are defined by using 3x3 matrices. Our approach is based on the well-known conformal property of maps in complex domains. Namely, any map in a complex domain is angle-preserving. Based on this well-known property of complex numbers, we show that if the local operations are defined as affine operations on complex numbers then any interpolation on complex numbers will give us conformal mapping of the whole domain. This is very useful since we can make extreme exaggeration of some local features and the resulting maps are still guaranteed to be conformal. Using this property, we have implemented an extended and conformal version of feature-based image metamorphosis.