# Can 3 transformations (V, Σ, U) of SVD to describe a perspective transformation?

As known SVD (Singular value decomposition) is a factorization of the form M = UΣV∗. https://en.wikipedia.org/wiki/Singular_value_decomposition

SVD of the linear map T can be easily analysed as a succession of three consecutive moves:

• isometry V∗
• endomorphism Σ
• isometry U

Or the same for the two-dimensional real shearing matrix M:

• rotate V∗
• scale Σ
• rotate U

Also known that SVD used to Homography Estimation, for 4 pairs of points (source->destination): https://cseweb.ucsd.edu/classes/wi07/cse252a/homography_estimation/homography_estimation.pdf

A homography is an isomorphism of projective spaces - or also called perspective (projective) transformation: https://en.wikipedia.org/wiki/Homography

But can 3 transformations (rotation-V, scale-Σ, rotation-U) to describe a perspective (projective) transformation?

Or is there SVD used only as Homogeneous Linear Least Squares approach and is not related to geometry of homography directly.

• SVD also gives a composition of linear transformations, so also preserves parallel subspaces. What is true, however, is that every homography of $n$-dimensional projective space comes from an invertible linear transformation $T$ on an $(n + 1)$-dimensional vector space $V$, and certainly you can do SVD on $T$. Again, though, I'm not sure that's what one would call "perspective projection". (If it matters, the phenomenon of parallel lines mapping to intersecting lines happens when passing from $V$ to the projective space $\mathbf{P}(V)$, not from applying a homography itself.) Commented Jan 4, 2016 at 15:40