Correspondences Points
George Triangulation
Me Trangulation
George Triangulation Midpoint
Me Trangulation Midpoint
In this part, I define the correspondences between two images to morph them together. The correspondences are defined by selecting points on the two images that correspond to each other, using the Correspondence Tool provided. I defined 62 correspondences on each face, and I used the Delaunay triangulation algorithm to create a mesh of triangles that connect the corresponding points. Then, I compute the midpoint triangulation by averaging the coordinates of the corresponding points. The images below show the correspondences I used and the triangulation of the two faces.
Correspondences Points
George Triangulation
Me Trangulation
George Triangulation Midpoint
Me Trangulation Midpoint
In this part, I morph each images into the midpoint triangulation by computing the affine transformation that maps the original triangulation to the midpoint triangulation. I compute the transformation matrix \( \mathbf{A} \) by solving a linear system of equations for each triangle. Here, I find the transformation from midpoint triangulation to the original triangulation to avoid holes in the morphed image. The process below shows how I compute the transformation matrix \( \mathbf{A} \) for each triangle.
For each triangle, we have the equation: \[ \mathbf{A} \mathbf{p}_i' = \mathbf{q}_i' \] Where \( \mathbf{p}_i' \) represents \( \mathbf{p}_i \) padded by a 1 at the bottom such that an affine transformation would make sense. Breaking down the matrix \( \mathbf{A} \), we have the equation: \[ \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p_{x_i} \\ p_{y_i} \\ 1 \end{bmatrix} = \begin{bmatrix} q_{x_i} \\ q_{y_i} \\ 1 \end{bmatrix} \] This gives us the system of equations: \[ a p_{x_i} + b p_{y_i} + c = q_{x_i} \] \[ d p_{x_i} + e p_{y_i} + f = q_{y_i} \] Expanding this for all 3 points, we have: \[ a p_{x_1} + b p_{y_1} + c = q_{x_1} \] \[ d p_{x_1} + e p_{y_1} + f = q_{y_1} \] \[ a p_{x_2} + b p_{y_2} + c = q_{x_2} \] \[ d p_{x_2} + e p_{y_2} + f = q_{y_2} \] \[ a p_{x_3} + b p_{y_3} + c = q_{x_3} \] \[ d p_{x_3} + e p_{y_3} + f = q_{y_3} \] In matrix form, this system can be written as: \[ \begin{bmatrix} p_{x_1} & p_{y_1} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{x_1} & p_{y_1} & 1 \\ p_{x_2} & p_{y_2} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{x_2} & p_{y_2} & 1 \\ p_{x_3} & p_{y_3} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{x_3} & p_{y_3} & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ d \\ e \\ f \end{bmatrix} = \begin{bmatrix} q_{x_1} \\ q_{y_1} \\ q_{x_2} \\ q_{y_2} \\ q_{x_3} \\ q_{y_3} \end{bmatrix} \] Since this is a linear system, we can solve it to find \( a, b, c, d, e, f \) to construct the transformation matrix \( \mathbf{A} \).
Using this transformation matrix, I can compute the morphed image by applying the inverse transformation to the midpoint triangulation. For each triangle, I first get all the points in the triangle, apply the transformation matrix to each point, and then fill in the pixels in the bounding box of the triangle with the transformed points. Here, I use scipy's RegularGridInterpolator to interpolate the pixel values in the bounding box of the triangle. The images below show the morphed images of George and me.
George - Original
Me - Original
Midpoint (Morphed)
In this part, I wrote a function that generates a morph sequence of the two images. The function takes in the two images, the corresponding points, triangulation, and the ratios for the morphing. Then, it computes 45 frames of the morph sequence by linearly interpolating the two images and the triangulation. The image below (animated-PNG) show the morph sequence of George and me. A gif version is available here (and in loop).
For this part, I used the Danes dataset to compute the "mean face" of the population. The dataset contains 240 images of 40 Danish people in different orientation and lighting, alongside with their correspondences. I computed the mean face by taking the average of the points and the triangulation of the mean points, and then I morphed each image into the triangulation. The left image below shows the mean face for all 240 images, and the right two images show the mean face for the front and side views. The second row shows the mean image for male and female images separately. Since there are much more male images than female images, I also computed the mean face by equally weighting the male average and female average images, as shown below.
Averaged Danes
Averaged Front
Averaged Side
Male Averaged
Female Averaged
Male and Female, Equally Weighted
Finally, I morph my own photo to the mean face of the Danes dataset, and vise versa. To that end, I had to match the correspondences of my face to the Danes dataset. Morphing my face to the mean face of the Danes dataset, I got the image on the left below. Morphing the average Danes face to my face, I got the image on the right.
Me to Danes Average
Average Danes to Me
For this part, I applied the morph function to extrapolate from the mean face of the Danes dataset. To make the images above more natural, I took the midpoint image of my face shape and average Danes face shape, resulting in the image below. For bells and whistles, I also computed the average of the smiling Danes face and morphed my face to the smiling Danes face with 50% appearance. Then, I morphed the smiling Danes face shape to my face shape, making the average Danes face look sad.
Me and Average Danes Midpoint
Smiling Me
Smiling Me with Midpoint Appearance
Saddened Average Dane