discrete wavelet transform matrix for vectorized image
For only one level on both direction, the classical DWT schemes are applied separately (rows then columns, or colums then rows). 

For more levels, there exist two schemes:


one interleaving decompositions: row, col, row, col, etc. (also called non-separable/square/non-standard/isotropic/Mallat),
one on all rows, then all colums (also called separable/rectangle/standard/anisotropic/tensor). 


The second one only deserves the "fully separable" mention.

References to related questions:


2D DWT Image Issue
What is the correct order of operations for a 2D Haar wavelet decomposition?
2D DWT computation order