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How can one construct the $SO(2)$ representation of a rotation of a flattened $N\times N$ image, i.e. a vectorized matrix $\mathbf{x}\in\mathbb{R}^{N^2}$? The mapping $T$ should rotate the original image for generic $N$ and rotation angle $\theta$, such that $T_\theta:\mathbf{x}\to \mathbf{A}(\theta)\mathbf{x}\in\mathbb{R}^{N^2}$ with $\mathbf{A}(\theta)\in\mathbb{R}^{N^2\times N^2}$. In practice, the real rotated image will not be identical to the output due to edge effects and interpolation of the pixels. Therefore, we wish to minimize $||\mathbf{A}(\theta)\mathbf{x}-\mathbf{y}||^2$, where $\mathbf{y}\in \mathbb{R}^{N^2}$ is the flattened, rotated image.

Researcher_Witty
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  • The group of rotations does not act on the set of $N\times N$ images, only on the circular part in the center. Or did you mean something else? I would imagine that people who are into computer graphics have figured this out. Yes, your error metric leads to many problems, including (anti)aliasing of individual pixels. People who seem to know what they are doing told me that doing a rotation as a combination of stretches and shears like here helps with such problems. – Jyrki Lahtonen Nov 10 '21 at 14:16
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    Although accustomed to image processing, I find it difficult to understand what you mean ; you should provide a graphical example. Otherwise the answers that will be given could be at many light-years from what you desire. – Jean Marie Nov 10 '21 at 14:19

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If I understand correctly, you wish to find a representation of $SO(2)$ into $GL(\mathbb{R}^{N\times N})$. I warn that there is no unique action of $SO(2)$ on $\mathbb{R}^{N\times N}$ but the most standard way I could think of the consider the irreducible representation of $SO(2)$ on $\mathbb{R}^{N}$, then take tensor product $\mathbb{R}^N\otimes\mathbb{R}^N$, i.e., the action is given by $$ g\cdot (v w^T) = (gv)(gw)^T \quad g\in SO(2), v,w\in\mathbb{R}^N. $$ This corresponds to the taking $N$-th symmetric power $S(g)$ of the matrix $g$ which is an $N\times N$ matrix and then conjugation: $$ g\cdot X = S(g) X S(g)^{T}, \quad g\in SO(2), X\in\mathbb{R}^{N\times N}. $$

Levent
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