Image Alignment

Image alignment between two images \(I_k\) and \(I_{k-1}\) arises in SLAM problems where the objective is to estimate relative rigid body transformation between the above two images.

Let’s consider a more simpler problem of sparse image alignment where we use only specific pixels (pixels with enough gradient) in image \(I_{k-1}\).

Using a direct approach

\[\delta I(T_{k,k-1}u_i) = I_k\left( \pi \left(T_{k,k-1} \cdot \rho_i \right) \right) - I_{k-1}\left( \pi \cdot \rho_i \right)\]

Our objective is to find the following;

\[T_{k,k-1} = \underset{T_{k,k-1}}{\text{arg min}}\frac{1}{2}\sum_{i \in \mathcal{R}}\lVert\delta I(T_{k,k-1}, u_i)\lVert^2\]

The above is a non-linear optimization problem which requires us to linearize the problem by introducing a small perturbation around the current estimate.

If we let \(\xi \in \mathcal{R}^6\) denote small perturbation on the tangent space then \(T(\xi) = \exp(\left[\xi\right]_x)\) is a small perturbation on the manifold \(SO(3)\).

Introducing the above perturbation and substituting we have,

\[\delta I(\xi, u_i) = I_k\left( \pi \left(T_{k,k-1} \cdot \rho_i \right) \right) - I_{k-1}\left( \pi \left( T(\xi) \cdot \rho_i \right) \right)\]