Face Morphing with Delaunay Triangulation

A morph between two faces needs to do two things simultaneously: warp the geometry of one face toward the other, and blend the pixel colors. Doing either alone looks wrong: color blending without warping produces a ghost double-exposure; shape warping without blending leaves the wrong texture in the wrong place. The pipeline does both, parameterized by a single scalar α ∈ [0, 1].

Correspondence and Triangulation

~50 landmark points are manually annotated on each face: eyes, nose tip, mouth corners, jawline, hairline. For the morph at parameter α, the intermediate landmark set is the linear interpolation:

$\mathbf{p}_\alpha = (1 - \alpha)\,\mathbf{p}_A + \alpha\,\mathbf{p}_B$

Delaunay triangulation is computed on $\mathbf{p}_\alpha$ and the same triangulation topology is applied to both source point sets. Delaunay is chosen because it maximizes the minimum angle across all triangles, preventing degenerate sliver triangles whose affine transforms produce visible seams at triangle edges.

Affine Warp Per Triangle

For each triangle in the intermediate mesh, we need to find the affine transform that maps it back to the corresponding triangle in each source image. An affine map $T: \mathbb{R}^2 \to \mathbb{R}^2$ has the form $T(\mathbf{x}) = A\mathbf{x} + \mathbf{t}$, or in homogeneous coordinates:

$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b & t_x \\ c & d & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$

Given source triangle vertices $\{(x_1,y_1), (x_2,y_2), (x_3,y_3)\}$ and destination vertices $\{(x_1',y_1'), (x_2',y_2'), (x_3',y_3')\}$, the 6 unknowns $(a, b, c, d, t_x, t_y)$ are solved from the 6 linear equations:

$\begin{bmatrix} x_1 & y_1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & x_1 & y_1 & 1 \\ x_2 & y_2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & x_2 & y_2 & 1 \\ x_3 & y_3 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & x_3 & y_3 & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ t_x \\ c \\ d \\ t_y \end{bmatrix} = \begin{bmatrix} x_1' \\ y_1' \\ x_2' \\ y_2' \\ x_3' \\ y_3' \end{bmatrix}$

The warp is applied inverse: for every pixel in the output triangle, apply the inverse transform to find the source coordinate, then sample with bilinear interpolation. This avoids holes from forward-mapping. The final pixel at each location is the weighted blend:

$I_\alpha(\mathbf{x}) = (1-\alpha)\, I_A(T_A^{-1}(\mathbf{x})) + \alpha\, I_B(T_B^{-1}(\mathbf{x}))$

Morph Sequence

Five frames from a 46-frame morph sequence. The eyes shift early; they’re constrained by many nearby landmarks. The jawline transitions more gradually because it spans a larger area with fewer triangle boundaries between source and target positions.

Population Mean Face

Given a dataset of 40+ Danish faces with annotated landmarks, the mean face is computed by:

1. Averaging all landmark coordinates to get $\bar{\mathbf{p}}$
2. Warping each face to the mean shape using the same per-triangle affine procedure
3. Averaging the warped pixel values

$\bar{I}(\mathbf{x}) = \frac{1}{N}\sum_{i=1}^N I_i(T_i^{-1}(\mathbf{x}))$

Left: the population mean. Center: my face warped to the mean’s geometry, same texture, average proportions. Right: caricature at α = 1.5. Instead of interpolating toward the mean, the landmarks are extrapolated away from it: $\mathbf{p}_{1.5} = \mathbf{p}_\text{me} + 0.5\,(\mathbf{p}_\text{me} - \bar{\mathbf{p}})$. Whatever made my face geometrically distinctive gets amplified. The effect exaggerates the features that differ most from the population average.

Caricature quality is sensitive to correspondence quality; a poorly placed landmark gets extrapolated into a more severe artifact than the original misplacement. This is a good diagnostic: caricatures reveal which correspondences were lazily annotated.