  A consistent description of images on the disk and of their transformations
is given as elements of a vector space and of an operators algebra. The vector
space of images on the disk $\mathbb{D}$ is the Hilbert space $L^2(\mathbb{D})$
that has as a basis the Zernike functions. To construct the operator algebra
that transforms the images $L^2(\mathbb{D})$ must be complemented and the full
rigged Hilbert space $RHS(\mathbb{D})$ considered. Only this rigged Hilbert
space allows indeed to write the operators of different cardinality we need to
build the ladder operators on the Zernike functions that by inspection, belong
to the representation $D_{1/2}^+ \otimes D_{1/2}^+$ of the algebra $su(1,1)
\oplus su(1,1)$. Consequently the transformations of images are operators
contained inside the universal enveloping algebra $UEA[su(1,1) \oplus
su(1,1)]$. Because of limited precision of experimental measures, physical
states can be always described by vectors of the Schwartz space
$\S(\mathbb{D})$, dense in the $L^2(\mathbb{D})$ space where the manipulation
of images is performed.
