# 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.

## 1 Answer

Every linear transformation (invertible or not) preserves the notion of "parallel affine subspaces of the same dimension" (such as parallel lines), while homographies generally do not. That is, every composition of linear transformations is (or may be viewed as) a homography, but not every homography arises in this way. Particularly, composing linear transformations gives no homography in which parallel lines map to intersecting lines, which is normally what one means by "perspective transformation".

• Thank you. I.e. 3 transformations (rotation-V, scale-Σ, rotation-U) can't describe any perspective transformation, but I can use SVD as Homogeneous Linear Least Squares approach to find any perspective transformation also known as homography, isn't it?
– Alex
Commented Jan 4, 2016 at 15:27
• 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