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

Question

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.

Answer 1

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".