How is the reconstruction of a 3D point from its 2D compressed coordinates mathematically formulated?

Conditions

• A compressed coordinate vector $\mathbf{z} \in \mathbb{R}^2$ is available.
• Reconstruction uses principal component directions $\mathbf{v}_1, \mathbf{v}_2$.
• The output is an approximation $\hat{\mathbf{x}}$ in $\mathbb{R}^3$.

Reasoning, step by step

1. Take the 2D compressed coordinates $z_1$ and $z_2$.
2. Multiply $z_1$ by the first principal component direction vector $\mathbf{v}_1$.
3. Multiply $z_2$ by the second principal component direction vector $\mathbf{v}_2$.
4. Sum the two weighted vectors to obtain the reconstructed 3D point $\hat{\mathbf{x}}$.
5. Alternatively, represent this as the matrix-vector product $\mathbf{W}\mathbf{z}$.

Example

The video displays the formula $\hat{\mathbf{x}} = \mathbf{W}\mathbf{z} = z_1\mathbf{v}_1 + z_2\mathbf{v}_2$ while animating blue arrows lifting yellow 2D points to cyan 3D points.

Common misconceptions

• Assuming the reconstruction perfectly recovers the original 3D point $\mathbf{x}$.
• Believing the reconstruction matrix $\mathbf{W}$ maps from $\mathbb{R}^3$ to $\mathbb{R}^2$ instead of the reverse.

Watch the explanation

Explore next

• Why is the reconstructed point written as $\hat{\mathbf{x}}$ instead of $\mathbf{x}$?
• What is the reconstruction error in PCA and how is it visualized?