Pose from Homography

Given camera intrinsics $K$ and a homography $H$ from a planar calibration board to the image, we can decompose $H$ to recover the camera pose (rotation $R$ and translation $\mathbf{t}$) relative to the board.

Problem Statement

Given: Intrinsics $K$ and homography $H$ such that $\mathbf{p}\sim H\cdot [{\mathbf{P}}_{xy},1{]}^T$ for board points at $Z=0$.

Find: Rigid transform $T_{C,B}=[R|\mathbf{t}]$ (board to camera).

Assumptions:

• The board lies at $Z=0$ in world coordinates
• $K$ is known (or has been estimated)
• The homography $H$ was computed from correct correspondences

Derivation

Extracting Rotation and Translation

Recall from the Homography chapter:

$$H\sim K[{\mathbf{r}}_1{\mathbf{r}}_2\mathbf{t}]$$

where ${\mathbf{r}}_1,{\mathbf{r}}_2$ are the first two columns of the rotation matrix and $\mathbf{t}$ is the translation.

Removing the intrinsics:

$$K^{-1}H=\lambda [{\mathbf{r}}_1{\mathbf{r}}_2\mathbf{t}]$$

Let $[{\mathbf{a}}_1{\mathbf{a}}_2{\mathbf{a}}_3]=K^{-1}H$. The scale factor $\lambda$ is recovered from the constraint that ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ have unit norm:

$$\lambda =\frac{2}{\Vert {\mathbf{a}}_1\Vert +\Vert {\mathbf{a}}_2\Vert }$$

Then:

$${\mathbf{r}}_1=\lambda {\mathbf{a}}_1,{\mathbf{r}}_2=\lambda {\mathbf{a}}_2,\mathbf{t}=\lambda {\mathbf{a}}_3$$

The third rotation column is:

$${\mathbf{r}}_3={\mathbf{r}}_1\times {\mathbf{r}}_2$$

Projecting onto SO(3)

Due to noise, the matrix $R_{\text{approx}}=[{\mathbf{r}}_1{\mathbf{r}}_2{\mathbf{r}}_3]$ is not exactly orthonormal. We project it onto $\text{SO}(3)$ using SVD:

$$R_{\text{approx}}=U\Sigma V^T$$

$$R=U{V}^T$$

If $\det (R)=-1$, flip the sign of the third column of $U$ and recompute.

Ensuring Forward-Facing

If $t_z<0$, the board is behind the camera. In this case, flip the sign of both $R$ and $\mathbf{t}$:

$$R\leftarrow -R,\mathbf{t}\leftarrow -\mathbf{t}$$

This resolves the sign ambiguity inherent in the scale factor $\lambda$.

Accuracy

The pose from homography is an approximate estimate because:

1. The homography itself is subject to noise (DLT algebraic error minimization, not geometric)
2. The SVD projection onto SO(3) corrects non-orthogonality but introduces additional error
3. Distortion (if not corrected) biases the homography

Typical rotation error: 1-5°. Typical translation direction error: 5-15%. These estimates are refined in non-linear optimization.

OpenCV equivalence: cv::decomposeHomographyMat provides a similar decomposition, returning up to 4 pose candidates. calibration-rs returns a single pose by resolving ambiguities via the forward-facing constraint.

API

#![allow(unused)]
fn main() {
let pose = estimate_planar_pose_from_h(&K, &H)?;
// pose: Iso3 (T_C_B: board to camera transform)
}

Usage in Calibration Pipeline

In the planar intrinsics calibration pipeline, pose estimation is applied to every view after $K$ has been estimated:

1. Compute homography $H_k$ per view
2. Estimate $K$ from all homographies (Zhang's method)
3. Decompose each $H_k$ to get pose $T_{C,B}^{(k)}$
4. Use poses as initial values for non-linear optimization