Suppose S1 and S2 are simple connected surfaces with a boundary in R3. We may use conformal map φ1:S1→ D1 and φ2:S2→ D2, where D1 and D2 are unit discs in R2 with center (0,0). We may simply pick a rotation γ:D1 → D2. Then f = φ2-1。γ。φ1 → S1 to S2.
▲The conformal morphing process from a pillow into a doll with γ = rotation( 0° ).
▲The conformal morphing process from a pillow into a doll with γ = rotation( 90° ).
Conformal Morphing with Beltrami Surface Reconstruction
The effect of morphing between two different facial expressions of a human face is not always fine, especially when the variation of boundaries of two patches is huge. In order to morph between different facial expressions of a human face well, some facial features must be kept during the morphing process. Under the assumption that the facial expressions are similar to another, we can keep the facial features matched using canonical boundary.
Conformal Morphing with Matching Technique
The ratio of facial features differ from every person. In order to morph one’s face into another well, in other words, keeping the facial features matched during the morphing process, matching of the feature points is important.
We may choose an appropriate matching function f: D1→D2 such that f matches their facial features well. Then f = φ2-1。f。φ1$ maps S1 to S2 which preserves the continuity of facial features.
We may apply the conformal morphing with matching technique to make a series of motion picture with two pictures only.
▲The morphing process from a girl’s face into another.
▲The conformal morphing process from a smile face into sad face and into a ferocious motion.
▲The conformal morphing process simulating the motion of eye blinking.