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I've been going through a set of slides about a modified version of the Procrustes problem. The whole problem is described by trying to find a transformation that satisfies $$A_i = sRB_i + T$$ where $R\in SO(3), T\in R^3,s\in R^+$ Meaning that R is a rotation matrix and T stands for translation. The scaling parameter $s$ is omitted. $A_i$ and $B_i$ and known point correspondences. This is formulated as the following minimization problem.

$$\min_{R,T}\sum_{i}^N \lVert A_i -RB_i + T\rVert^2$$

The slides says that after differentiating for $T$ we obtain

$$T = \frac{1}{N}\sum_{i}^NA_i-R\frac{1}{N}\sum_{i}^NB_i = \overline A-R \overline B$$ Where $ \overline A$ and $ \overline B$ are so called centroids. Two centroids are then subtracted from the problem as follows $$\min_{R}\lVert A -RB\rVert_F^2$$ where $A=(A_1-\overline A, ...,A_n-\overline A)$ and $B=(B_1-\overline B, ...,B_n-\overline B)$. I have two questions related to this:

  1. How did they obtain the derivative for T?
  2. Why is the $T$ in the original $\min_{R,T}$ with a plus sign? I would expect something like $\min_{R,T}\sum_{i}^N \lVert A_i -(RB_i + T)\rVert^2$ as in trying to minimize the differences between the projected point and its correspondence.
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