October 13, 2017

arXiv:1710.04207v1 [eess.IV] 11 Oct 2017

Algebraic Image Processing
Enrico Celeghini
Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze
I50019 Sesto Fiorentino, Firenze, Italy.
Departamento de Fı́sica Teórica, Atómica y Óptica and IMUVA.
Universidad de Valladolid, 47011 Valladolid, Spain.
e-mail: celeghini@fi.infn.it.

Abstract
We propose an approach to image processing related to algebraic operators
acting in the space of images. In view of the interest in the applications in optics
and computer science, mathematical aspects of the paper have been simplified as
much as possible. Underlying theory, related to rigged Hilbert spaces and Lie
algebras, is discussed elsewhere.
OCIS codes 100.2000, 100.2980, 110.1085, 110.1758

1

Introduction

The fundamental problem in image analysis is that the information contained in an image
is immense, too much in order that the human mind can manage it. So in science we
have to disregard a large part of the contained information and isolate the required one,
quite more restricted, in function of our specific interests.
We suggest that this work to clean the signal from forgettable elements and to highlight
the relevant information could be improved by the mathematical theory of operators
acting in the space of images.
Same application of these ideas, samehow similar to adaptive optics, are described in
the following. Yet, while in adaptive optics the perturbations to be removed are related
to the phases of a complex function f (r, θ) our approach, that we could call Algebraic
Image Processing (AIP), acts on the module of the image |f (r, θ)|. So, we suggest that

the image |f (r, θ)|2 registered by an experimental apparatus can be assumed not as the
final result of the optical measure but possibly as an intermediate step to be further
elaborated by a computer program based on operators acting in the space of images. AIP
is indeed a soft procedure and operates on a set of pixels to obtain another set of pixels
independently from the causes of the effects we wish to eliminate.
Moreover AIP is a general theory of the connections of images between them and thus
can be applied also outside the cleaning of images to any manipulation of images.
We restrict ourselves in this paper to practical aspects relevant in optics and computer
science. Underlying mathematics, related to rigged Hilbert spaces and Lie groups, is
considered in detail in [1] and, at higher theoretical level, in [2–4]. We recommend to the
reader interested in the theory to look there and references therein.
Features of the proposed approach are:
1. It can be used on-line and also off-line if, at later time, more accuracy is required.
2. It is a soft procedure, cheap and without mechanical moving parts.
3. It allows to consider together images of different origin and frequencies like optical
and radio images.
The relevant operative points are:
• The vector space of images on the disk has as a basis the Zernike functions [5].
• The space of images can be integrated with the operators in the space of images i.e.
the operators that transform Zernike functions into Zernike functions.
• This space of images and its operators define the unitary irreducible representation
+
+
D1/2
⊗ D1/2
of the Lie algebra su(1, 1) ⊕ su(1, 1) .
• Every operator of AIP can be written as a polynomial in the generators of the
algebra su(1, 1) ⊕ su(1, 1)
• and can be computed applying this polynomial to the Zernike functions.
In sect.2 we summarize the vector space of images. In sect.3 we introduce the operators
acting on this space, their algebraic structure and their realization. In sect.4 we give a
description of a possible modus operandi that could be realized in automatic or semiautomatic way by a computer program. Color images can also be introduced by means
of an additional factorization of a color code.

2

Vector Space of Images

Radial Zernike polynomials Rnm (r) can be found in [5] and are real polynomials defined
for 0 ≤ r ≤ 1 such that Rnm (1) = 1, where n is a natural number and m an integer, such
that 0 ≤ m ≤ n and n − m is even.

2

As the interest in optics is focused on real functions defined on the disk, starting from
Rnm (r) it is usual to introduce, with 0 ≤ θ < 2π, the functions
Zn−m (r, θ) := Rnm (r) sin(mθ)

Znm (r, θ) := Rnm (r) cos(mθ)

and, because only smooth functions are normally considered, to take into account only
low values of n and m, summarized in a unique sequential index [6].
However, in a general theory, functions are not necessarily smooth and formal properties are relevant. We thus came back to the classical form of Born and Wolf in the
complex space [5]:
Znm (r, θ) := Rn|m| (r) eimθ
with n natural number, m integer with n − m even and −n ≤ m ≤ n. The symmetry can
be improved writing n and m in function of two arbitrary independent natural numbers
k and l [7]
n=k+l

m=k−l

(k = 0, 1, 2, ...; l = 0, 1, 2, ...)

and introducing a multiplicative factor. The Zernike functions we consider here are thus
√
|k−l|
Vk,l (r, θ) := k + l + 1 Rk+l (r) ei(k−l)θ
(1)
and depend from two natural numbers k and l and two continuous variables r and θ.
The functions Vk,l (r, θ) are an orthonormal basis in the space L2 (D) of square integrable
complex functions defined on the unit disk D as:
Z 2π Z 1
1
dθ
dr 2 Vk,l (r, θ)∗ Vk′ ,l′ (r, θ) = δk,k′ δl,l′
2π 0
0
(2)
∞ X
∞
X
1
∗
2
2
Vk,l (r, θ) Vk,l (s, φ) = δ(r − s ) δ(θ − φ)
2π k=0 l=0
and have the symmetries
Vl,k (r, θ) = Vk,l (r, θ)∗ = Vk,l (r, −θ).
Every function f (r, θ) ∈ L2 (D) can thus be written
f (r, θ) =

∞ X
∞
X

fk,l Vk,l (r, θ)

(3)

k=0 l=0

where
fk,l

1
=
2π

Z

2π

dθ
0

Z

1

dr 2 Vk,l (r, θ)∗ f (r, θ)

(4)

0

is the component along Vk,l (r, θ) of the function f (r, θ). In this paper we consider only
normalized states, so that the Parseval identity gives for each state:
3

1
2π

Z

0

2π

dθ

Z

0

1
2

2

dr |f (r, θ)| =

∞ X
∞
X
k=0 l=0

|fk,l |

2

= 1.

(5)

Restrictions on the values of k and l can be used as filters. k + l > h (< h) is a
high-pass (low-pass) filter in r, while |k − l| > h (< h) is high-pass (low-pass) filter in θ
and a combination of the restrictions can be also considered. Of course, using filters, the
results must be multiplied for adequate factors to obtain normalized states.

3

Operators and Algebra in the disk space

Now let us consider the differential applications :
A+
A−
B+
B−


e+iθ
d
:=
−(1 − r 2 )
2
dr

−iθ
e
d
:=
+(1 − r 2 )
2
dr

d
e−iθ
−(1 − r 2 )
:=
2
dr

e+iθ
d
:=
+(1 − r 2 )
2
dr

r
k+l+2
1
+ r(k + l + 2) + (k − l)
r
k+l+1
r
k+l
1
+ r(k + l) + (k − l)
r
k+l+1
r
1
k+l+2
+ r(k + l + 2) − (k − l)
r
k+l+1
r
k+l
1
+ r(k + l) − (k − l)
r
k+l+1

(6)

that, by inspection, are the rising and lowering recurrence applications on the Zernike
functions Vk,l (r, θ) :
A+ Vk,l (r, θ) = (k + 1) Vk+1,l (r, θ) ,
B+ Vk,l (r, θ) = (l + 1) Vk,l+1(r, θ) ,

A− Vk,l (r, θ) = k Vk−1,l (r, θ) ,
B− Vk,l (r, θ) = l Vk,l−1(r, θ) .

(7)

A± and B± establish the recurrence relations but are not operators because each
application of A± or B± modifies the parameters k or l to be read by the following applications. To obtain the true rising and lowering operators, we need indeed to introduce
the operators R, DR , Θ, DΘ , K, L:
R Vk,l (r, θ) := r Vk,l (r, θ) ,

dVk,l (r, θ)
,
dr
dVk,l(r, θ)
DΘ Vk,l (r, θ) :=
,
dθ
L Vk,l (r, θ) := l Vk,l (r, θ) ;

DR Vk,l (r, θ) :=

Θ Vk,l (r, θ) := φ Vk,l (r, θ) ,
K Vk,l (r, θ) := k Vk,l (r, θ) ,

4

and rewrite eqs.(6) as operators:

r
K +L+2
e+iΘ
1
A+ :=
−(1 − R2 )DR + R(K + L + 2) + (K − L)
,
2
R
K +L+1

 r
1
K +L
e−iΘ
+(1 − R2 )DR + R(K + L) + (K − L)
,
A− :=
2
R
K +L+1

r
1
e−iΘ
K +L+2
2
−(1 − R )DR + R(K + L + 2) − (K − L)
B+ :=
,
2
R
K +L+1

 r
e+iΘ
1
K +L
2
B− :=
+(1 − R )DR + R(K + L) − (K − L)
.
2
R
K +L+1

(8)

Now, as A± and K are operators, we can apply them iteratively on Vk,l (r, θ) and in
particular we can calculate the action of the commutators:
[A+ , A− ] Vk,l (r, θ) = −2(k + 1/2) Vk,l (r, θ) ,

[K, A± ] Vk,l(r, θ) = ± Vk±1,l (r, θ) .

So, defining A3 := K + 1/2 , we find that {A+ , A3 , A− } are on the Vk,l (r, θ) the
generators of one algebra su(1, 1):
[A+ , A− ] = −2A3 ,

[A3 , A± ] = ±A± .

(9)

Analogously
[B+ , B− ] Vk,l (r, θ) = −2(l + 1/2) Vk,l (r, θ) ,

[L, B± ]Vk,l (r, θ) = ± Vk,l±1(r, θ)

exhibit that {B+ , B3 := L + 1/2 , B− } are on the Vk,l (r, θ) the generators of another
Lie algebra su(1, 1):
[B+ , B− ] = −2B3

[B3 , B± ] = ±B± .

(10)

Finally, as Ai and Bj commute on the Vk,l (r, θ), the algebra is completed by
[Ai , Bj ] = 0 .

(11)

Thus in the vector space of images on the disk, eqs.(9 -11) define a differential realization
of the 6 dimensional Lie algebra su(1, 1) ⊕ su(1, 1) .
We can now calculate on Vk,l(r, θ) the Casimir invariants of the two su(1, 1):


1
1
2
CA Vk,l (r, θ) =
{A+ , A− } − A3 Vk,l (r, θ) = Vk,l (r, θ),
2
4


1
1
2
{B+ , B− } − B3 Vk,l (r, θ) = Vk,l (r, θ).
CB Vk,l (r, θ) =
2
4

As the Casimir of the discrete series Dj+ of su(1, 1) is j(1 − j) [8] with j = 1/2, 1, ..,
the space {Vk,l (r, θ)} is isomorphic to {|k, li|k, l = 0, 1, 2 . . . }, the unitary irreducible
5

+
+
representation D1/2
⊗ D1/2
of the group SU(1, 1) ⊗ SU(1, 1), where indeed the action of
the generators of the algebra is:

A+ |k, li = (k + 1) |k + 1, li ,
A3 |k, li = k |k, li ,
A− |k, li = k |k − 1, li ,

B+ |k, li = (l + 1) |k, l + 1i ,
B3 |k, li = l |k, li ,
B− |k, li = l |k, l − 1i .

(12)

Now we move from the generators of the algebra su(1, 1) ⊗ su(1, 1) to their products that belong to the associated universal enveloping algebra (UEA). It is the algebra
β1 β2 β3
UEA[su(1, 1) ⊕ su(1, 1)] constructed on the ordered monomials Aα+1 Aα3 2 Aα−3 B+
B3 B−
(where αi and βj are natural numbers) submitted to the relations (9-11). So every
operator O ∈ UEA[su(1, 1) ⊕ su(1, 1)] can be written as
X
X
β1 β2 β3
O=
Oᾱ,β̄ =
cᾱ,β̄ Aα+1 Aα3 2 Aα−3 B+
B3 B− .
(13)
ᾱ,β̄

ᾱ,β̄

where cᾱ,β̄ are arbitrary complex functions of ᾱ = (α1 , α2 , α3 ) and β̄ = (β1 , β2 , β3 ). Since
{Vk,l (r, θ)} is a differential representation of the algebra su(1, 1) ⊕ su(1, 1), it is also a
differential representation of the UEA[su(1, 1) ⊕ su(1, 1)].

+
+
Because the representation D1/2
⊗ D1/2
is unitary and irreducible the set of unitary
2
operators acting on the space L (D), {O[L2 (D)], is isomorphic to the set of operators
+
+
acting on D1/2
⊗ D1/2
. Therefore all invertible operators that transform disk images into
disk images can be written in the form (13) and belong to the UEA[su(1, 1) ⊕ su(1, 1)].

In concrete: all transformations of an arbitrary image in whatsoever other image can
be realized by means of iterated applications of the operators (8).

4

Applications to image processing

In physics a fundamental point of image processing is that every image is the result
of a measure and each measure has a measure error. This implies that all numbers in
the preceding sections –that in mathematics are unlimited– in physics can be considered
always finite, because the limited level of accuracy. All problems related to the rigged
Hilbert space are thus irrelevant as, in finite dimensions, rigged Hilbert spaces and Hilbert
spaces are equivalent (see [1], [3] and [4]).
Let us start depicting the procedure that, starting from one image |f (r, θ)|2 , by means
of an operator O of the UEA, gives us another image |g(r, θ)|2 .

As each image |f (r, θ)|2 does not depend from the phases, it is completely determined
by |f (r, θ)|. Thus we look for the components along Vk,l (r, θ) of |f (r, θ)|:
Z 2π Z 1
1
dθ
dr 2 Vk,l (r, θ)∗ |f (r, θ)|
(14)
fk,l =
2π 0
0
6

where, as |f (r, θ)| is real, fl,k = (fk,l )∗ .

Because of the measure errors, the relevant values of k and l are limited to the kM and
lM such that the Parseval identity eq.(5) is satisfied in the approximation appropriate to
the accuracy of the experimental result |f (r, θ)|2 . We thus have:
|f (r, θ)| ≈

lM
kM X
X

fk,l Vk,l (r, θ).

(15)

k=0 l=0

Let us describe now how a transformation O allows to obtain from the image |f (r, θ)|2 a
new image |g(r, θ)|2 .
The operator O will be
X
X
β1 β2 β3
O=
Oᾱ,β̄ =
cᾱ,β̄ Aα+1 Aα3 2 Aα−3 B+
B3 B− .
ᾱ,β̄

(16)

ᾱ,β̄

where in physics the sums on ᾱ and β̄ can be assumed to be finite.
We thus write
Oᾱ,β̄ |f (r, θ)| =

kM X
lM
X

β1 β2 β3
fkl cᾱ,β̄ Aα+1 Aα3 2 Aα−3 B+
B3 B− Vk,l (r, θ) ,

k=0 l=0

then –by means of iterated applications of operators (8)– we calculate
β1 β2 β3
Aα+1 Aα3 2 Aα−3 B+
B3 B− Vk,l (r, θ)

finding the coefficients gk,l that satisfy
β1 β2 β3
Aα+1 Aα3 2 Aα−3 B+
B3 B− Vk,l (r, θ) = gk+α1 −α3 ,l+β1 −β3 Vk+α1 −α3, l+β1 −β3 (r, θ).

Thus we have
kM ,lM

Oᾱ,β̄ |f (r, θ)| =

X

fkl cᾱ,β̄ gk+α1 −α3 , l+β1 −β3 Vk+α1 −α3 , l+β1 −β3 (r, θ) ,

k,l=0,0

that, combined with eq.(16), allows to obtain:
g(r, θ) = O |f (r, θ)|
from which |g(r, θ)|2, the transformed image under O of |f (r, θ)|2, is obtained.

Analogous procedure can be applied to obtain the operator O from |g(r, θ)| and
|f (r, θ)|.
To conclude, let us consider now a possible application to improve, by means of AIP,
an image obtained by a flawed instrument. We start observing a null signal that, with a
7

perfect tool, would give |f (r, θ)| = |V0,0 (r, θ)|. A defective instrument, on the contrary,
will give a perturbed image |f (r, θ)|2 such that
|f (r, θ)| =

lM
kM X
X

fk,l Vk,l (r, θ) =

lM
kM X
X

fk,l

k=0 l=0

k=0 l=0

l
Ak+ B+
V0,0 (r, θ)
k! l!

(17)

∗
where fk,l (= fl,k
) are the parameters, obtained from eq.(14), that characterize the distortion of the null image made by the instrument.

So that the operator that eliminates the defects of the instrument is
kM X
lM
X

fk,l

k=0 l=0

l
Ak+ B+
k! l!

!−1

.

If the observation of the object we are interested in gives, with gk,l = (gl,k )∗ ,
|g(r, θ)| =

kM ′ lM ′
X
X

gk,l Vk,l (r, θ) =

kM ′ lM ′
X
X

gk,l

k=0 l=0

k=0 l=0

l
Ak+ B+
V0,0 (r, θ)
k! l!

the final cleaned image will be


!−1
kM ′ lM ′
lM
kM X
l
k
l
X
X
X
Ak+ B+
A+ B+


fk,l
V0,0 (r, θ)
gk,l
k!l!
k!l!
k=0 l=0
k=0 l=0

(18)

2

,

(19)

formula that can be easily calculated in series as all operators commute.

References
[1] Celeghini E, 2017 J of Phys: Conf Series 880 012055
[2] Reed M, Simon B, 1980 Methods of Modern Math Phys, (Acad Press, London)
[3] Celeghini E, Gadella M, delOlmo M A, 2016 J Math Phys 57 072105
[4] Celeghini E, Gadella M, delOlmo M A, 2017 to be published
[5] Born M, Wolf E, 2005 Principles of Optics (Cambridge UP 7th ed, Cambridge Ma)
[6] Noll R J, 1976 J Opt Soc Am 66 207
[7] Dunkl C F, 2001 Enciclopaedia of Mathematics, Suppl 3 pg 454 (Kluwer, Dordrecht)
[8] Bargmann V, 1947 Ann. Math. 48 568

8

