How are the two compressed coordinates $z_1$ and $z_2$ obtained from the original 3D vector in the face measurement example?

Conditions

• The original sample $\mathbf{x}$ lies in $\mathbb{R}^3$.
• Principal directions $\mathbf{e}_1$ and $\mathbf{e}_2$ define the 2D coordinate system.
• Coordinates are obtained by dot products with those directions.

Reasoning, step by step

1. Start from the original three-measurement description of a face in 3D feature space.
2. Introduce a new two-dimensional coordinate system attached to the extracted principal plane.
3. Compute the first compressed coordinate $z_1$ as the dot product of $\mathbf{x}$ and $\mathbf{e}_1$.
4. Compute the second compressed coordinate $z_2$ as the dot product of $\mathbf{x}$ and $\mathbf{e}_2$.
5. Represent the 3D sample by the pair of scalar projections instead of three original measurements.

Example

The video shows the formula $[z_1, z_2]^T = [\mathbf{x} \cdot \mathbf{e}_1, \mathbf{x} \cdot \mathbf{e}_2]^T$ in the right panel, replacing the left panel's $[x_{nose}, x_{eye}, x_{jaw}]^T$.

Common misconceptions

• Assuming the compression simply deletes one of the original coordinates (e.g., dropping the jaw width).
• Believing the dot product calculates the Euclidean distance rather than the projection length.

Watch the explanation

Explore next

• Why does the video claim a 33.3% storage reduction in the face example?
• How is the reconstruction from 2D coordinates back to 3D performed?