NEW INVARIANT DESCRIPTORS BASED ON THE MELLIN

TRANSFORM

S. Metari

Université de Sherbrooke, 2500 boulevard de l’Université, Sherbrooke, Québec, Canada, J1K 2R1

François Deschênes

Université du Québec en Outaouais, 283 boulevard Alexandre-Taché, Gatineau, Québec, Canada, J8X 3X7

Keywords:

Radiometric invariants, Combined invariants, Moments, Mellin transform, Point Spread Function.

Abstract:

In this paper we introduce two new classes of radiometric and combined radiometric-geometric invariant

descriptors. The ﬁrst class includes two types of radiometric descriptors. The ﬁrst type is based on the

Mellin transform and the second one is based on central moments. Both descriptors are invariant to contrast

changes and to convolution with any kernel having a symmetric form with respect to the diagonals. The

second class contains two subclasses of combined descriptors. The ﬁrst subclass includes central-moment

based descriptors invariant simultaneously to translations, to uniform and anisotropic scaling, to stretching, to

contrast changes and to convolution. The second subclass includes central-complex-moment based descriptors

invariant simultaneously to similarity transformation and to contrast changes. We apply those invariants to the

matching of geometrically transformed and/or blurred images.

1 INTRODUCTION

In pattern analysis, one important research axis is the

analysis and characterization of objects and patterns

corrupted by radiometric and/or geometric degrada-

tions. Real images can reveal geometric distortions

as well as photometric degradations. Two important

types of factors can be at the origin of those degra-

dations. First, those that originate from the imaging

system. They might be due to defects or limitations

of the imaging device such as diffraction, bad im-

age sensor, limited depth of ﬁeld, limited dynamic

range, etc. Second, the conditions under which the

image is taken, i.e. ambient illumination, weather

conditions, viewpoint position, etc. Several research

ﬁelds are interested in the characterization of the ideal

image from the acquired one by neither having re-

course to restoration nor to geometric standardiza-

tion as those two processes are often ill-posed prob-

lems (Katsaggelos, 1991). The obtained results may

thus be not unique as they depend on the problem

formulation. Moreover the algorithmic complexity

may be high. To solve those problems, we are in-

terested by invariant descriptors that are calculated

from the image functions and allow the identiﬁca-

tion and characterization of the ideal scene image re-

gardless of the photometric and/or geometric degra-

dations. In the literature we can ﬁnd four major de-

scriptor types, namely, algebraic descriptors, visual

descriptors, transform coefﬁcient descriptors and sta-

tistical descriptors. Our research work is classiﬁed in

the category of statistical descriptors.

In this paper, we introduce two new classes of in-

variant descriptors. The ﬁrst class includes radiomet-

ric descriptors and the second one contains combined

radiometric-geometric descriptors. The proposed in-

variants are inspired by a new relationship established

between the Mellin transforms of all of the original

image, the degraded image, and the convolution ker-

nel. They have a high discriminant power in matching

of geometrically transformed and/or radiometrically

degraded images and their algorithmic complexities

are low. In the rest of this paper we proceed as fol-

lows. In section 2, we give a brief synopsis of litera-

ture. In section 3, we introduce a new class of radio-

metric descriptors. Section 4 is devoted to the class of

combined radiometric-geometric descriptors. Finally,

experimental results are given in section 5.

13

Metari S. and Deschênes F. (2008).

NEW INVARIANT DESCRIPTORS BASED ON THE MELLIN TRANSFORM.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 13-21

DOI: 10.5220/0001072200130021

Copyright

c

SciTePress

2 RELATED WORK

Two main degradation types are tackled in the ﬁeld of

analysis and interpretation of degraded images based

on statistical invariant descriptors. In the case of ge-

ometric degradations, the author in (Hu, 1962) intro-

duces the ﬁrst class of statistical moment invariants

based on the theory of algebraic invariants. The pro-

posed descriptors are invariant to translation, rotation

and scaling, and are used in the recognition of de-

graded planar objects. Authors in (Flusser and Suk,

1993) have developed the so called afﬁne moment

invariants, that is image descriptors that are invari-

ant under general afﬁne transformation. These de-

scriptors are based on central moments and are used

for the recognition of patterns and objects degraded

by a general afﬁne transformation. Several other re-

searchers (Belkasim et al., 1991), (Reiss, 1991), (Teh

and Chin, 1988) tackled the subject of moment based

invariant descriptors for the recognition of geometri-

cally degraded images.

For radiometric degradations, very little research

work is interested in this topic. The research work

of Flusser and his research group (Flusser et al.,

1995), (Flusser and Suk, 1997), (Flusser and Suk,

1998) is the ﬁrst signiﬁcant contribution in this do-

main. They have developed new classes of radiomet-

ric descriptors invariant to blur degradations. The pro-

posed descriptors are based on central moments. They

haveseveral application ﬁelds, such as the recognition

of blurred images, the recognition and classiﬁcation

of 1-D degraded signals and the template matching

on satellite image functions. Authors in (Stern et al.,

2002) developed two new moment based methods for

the recognition of motion blurred images.

In extension to previous work, Flusser and his

research group (Flusser and Zitová, 1999), (Flusser

et al., 2003), (Suk and Flusser, 2003) have developed

new classes of combined invariant descriptors, that is

descriptors that are simultaneously invariant to both

geometric and radiometric degradations. The pro-

posed descriptors are based on central and complex

moments and are used in the recognition of afﬁne

transformed and blurred images, in template match-

ing on blurred and rotated images, etc. In this line

of thoughts, the authors in (Van Gool et al., 1996)

propose a new class of combined afﬁne radiometric

invariants. These descriptors are used for the recog-

nition of afﬁne transformed and photometrically de-

graded gray level images. Authors in (Mindru et al.,

1999) introduce the so called generalized color mo-

ments for the characterization of the multispectral na-

ture of data in a limited area of the image. The pro-

posed descriptors are used in the recognition of pla-

nar color patterns regardless of the viewpoint and il-

lumination. A more detailed survey can be found

in (Flusser, 2006) and (Flusser, 2007). In what fol-

lows we introduce our ﬁrst class of radiometric fea-

tures.

3 RADIOMETRIC INVARIANTS

In this section, we propose a new set of Mellin-

transform based descriptors invariant simultaneously

to uniform scaling, to contrast changes and to blur

degradations that can be modelled by any convolu-

tion kernel h having a symmetric form with respect

to the diagonals, i.e. h(x,y) = h(y,x). Then, in-

spired by those invariant descriptors we introduce a

new central-moment based descriptor which is simul-

taneously invariant to translations, to uniform scaling,

to contrast changes and to convolution. Note that the

symmetry constraint of the convolution kernel is not

a severe limitation for the applicability of our invari-

ant descriptors since the majority of convolution ker-

nels used to model optical blur are symmetric with

respect to the diagonals (e.g., Gaussian and pillbox

ﬁlters (Chaudhuri and Rajagopalan, 1999)) as well

as those used to approximate the Atmospheric Point

Spread Function APSF which models the atmospheric

veil on images (Metari and Deschênes, 2007a).

3.1 Radiometric Invariant based on the

Mellin Transform

The Mellin integral transform of a function f(x, y) is

deﬁned as (Zayed, 1996):

M ( f (x,y))(s,v) =

+∞

Z

0

+∞

Z

0

x

s−1

y

v−1

f(x,y)dxdy, (1)

with s,v ∈ C. The idea behind the elaboration of

our ﬁrst radiometric invariant feature is based on two

properties of the Mellin transform and Mellin convo-

lution. The ﬁrst one mentions that the Mellin convo-

lution in R

+

is equivalent to ordinary convolution in

R (Korevaar, 2004): Let g, f and h be three functions

deﬁned and integrable on the reals, the ordinary con-

volution product of f with h is given by:

g(x) = ( f ∗ h)(x) =

+∞

Z

−∞

f(x− t)h(t)dt. (2)

By carrying out the following change of variables

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

14

x = ln(x

′

) and t = ln(t

′

), we obtain:

(g◦ ln)(x

′

) = ( f ∗ h)(ln(x

′

)),

=

+∞

Z

0

f(ln(x

′

) − ln(t

′

))h(ln(t

′

))

dt

′

t

′

,

=

+∞

Z

0

( f ◦ ln)(

x

′

t

′

)(h◦ ln)(t

′

)

dt

′

t

′

,

= (( f ◦ ln) ∗

Mel

(h◦ ln))(x

′

), (3)

with ∗

Mel

denoting the Mellin convolution product.

Thus, Mellin convolution in R

+

is equivalent to ordi-

nary convolution in R.

The second one reveals that the Mellin transform

of the Mellin convolution product of two functions is

equal to the product of the Mellin transforms of these

functions (Davies, 2002): Let f and h be two func-

tions deﬁned and integrable on the positive reals, the

Mellin transform of the Mellin convolution product of

f and h is given by:

M (( f ∗

Mel

h)(x))(s) =

+∞

Z

0

x

s−1

+∞

Z

0

f(t)h

x

t

dt

t

dx,

=

+∞

Z

0

f(t)

t

+∞

Z

0

x

s−1

h

x

t

dxdt,

=

+∞

Z

0

f(t)

t

+∞

Z

0

(tx)

s−1

h(x)tdxdt,

=

+∞

Z

0

t

s−1

f(t)dt

+∞

Z

0

x

s−1

h(x)dx,

= M ( f(x))(s)M (h(x))(s). (4)

It results from the above mentioned properties the

new relationship shown in equation (6). Let us con-

sider the functions g, f and h related by the following

relationship:

g(x,y) = f (x,y)∗ h(x,y), (5)

where ∗ is the ordinary convolution operator.

The Mellin transform of g can thus be obtained as

follows:

M (g) = M ( f ∗ h) = M ( f)M (h). (6)

Proof: From equation (3), we have:

(g◦ ln)(x

′

) = (( f ◦ ln) ∗

Mel

(h◦ ln))(x

′

). (7)

Applying the Mellin transform to equation (7), we ob-

tain:

M ((g◦ ln)(x

′

))(s) = M ((( f ◦ ln) ∗

Mel

(h◦ ln))(x

′

))(s),

M ((g◦ ln)(x

′

)

| {z }

g(x)

)(s) = M (( f ◦ ln)(x

′

)

| {z }

f(x)

)(s)

× M ((h◦ ln)(x

′

)

| {z }

h(x)

)(s),

M (g(x))(s) = M ( f(x))(s)M (h(x))(s). (8)

In addition to this, as we will show below, Mellin

transform of order (s,v) of a symmetric ﬁlter with re-

spect to the diagonals is equal to its Mellin transform

of order (v,s). The theorem and the proof of the pro-

posed invariant descriptor are thus given by:

Theorem 1: The radiometric invariant descriptor

k

f

(s,v) of the image function f(x,y) is deﬁned as:

k

f

(s,v) =

M ( f(x,y))(s,v)

M ( f(x,y))(v,s)

, s,v ∈ N. (9)

The proposed descriptor k

f

(s,v) is invariant to uni-

form scaling, to contrast changes and to convolu-

tion with any kernel h having a symmetric form with

respect to the diagonals, i.e. k

η( f∗h)(rx,ry))

(s,v) =

k

f(x,y)

(s,v), with η is a positive constant number and

r denotes the uniform scaling factor.

3.1.1 Invariance to Convolution

In the case of a shift invariant imaging system the ac-

quired image g(x,y) is the result of the convolution

product of a clear image f(x,y) with a kernel h(x, y)

where models the Point Spread function PSF of the

imaging system (cf. equation (5)). Applying the pro-

posed invariant k(s,v) to equation (5) and using the

Mellin transform property (equation (6)), we obtain:

k

g

(s,v) =

M (g(x,y))(s,v)

M (g(x,y))(v,s)

,

=

M ( f(x,y))(s,v)M (h(x,y))(s,v)

M ( f(x,y))(v, s)M (h(x,y))(v,s)

,

the ratio

M (h(x,y))(s,v)

M (h(x,y))(v,s)

= 1 according to the following

proof:

M (h(x,y))(s,v) =

+∞

Z

0

+∞

Z

0

x

s−1

y

v−1

h(x,y)dxdy,

by making the following change of variables x = y and

y = x, we obtain:

=

+∞

Z

0

+∞

Z

0

∂(x,y)

∂(y,x)

| {z }

=1

y

s−1

x

v−1

h(y, x)dydx,

NEW INVARIANT DESCRIPTORS BASED ON THE MELLIN TRANSFORM

15

where

∂(x,y)

∂(y,x)

denotes the jacobian of the transforma-

tion. If we suppose that h is a symmetric kernel with

respect to the diagonals, i.e. h(x,y) = h(y,x) then we

have:

M (h(x,y))(s,v) =

+∞

Z

0

+∞

Z

0

x

v−1

y

s−1

h(x,y)dxdy,

= M (h(x,y))(v,s). (10)

Thus,

k

g

(s,v) =

M ( f(x,y))(s,v)

M ( f (x,y))(v,s)

= k

f

(s,v). (11)

3.1.2 Invariance to Uniform Scaling and to

Contrast Changes

Following a global contrast change and a uniform

scaling of the image function f(x,y), the Mellin trans-

form M ( f

′

(u,w)) of the resulting image function

f

′

(u,w) is related to Mellin transform M ( f(x,y)) of

the original image function f(x,y) by what follows:

M ( f

′

(u,w)) =

+∞

Z

0

+∞

Z

0

u

s−1

w

v−1

f

′

(u,w)dudw,

=

+∞

Z

0

+∞

Z

0

(rx)

s−1

(ry)

v−1

ηf (x,y)rdxrdy,

= ηr

s+v

M ( f(x,y)), (12)

with η a positive constant number and r the uniform

scaling factor. Applying k(s,v) to a function f

′

(u,w),

we obtain:

k

f

′

(u,w)

(s,v) =

ηr

s+v

M ( f(x,y))(s,v)

ηr

v+s

M ( f(x,y))(v, s)

,

= k

f(x,y)

(s,v). (13)

Thanks to the above proofs we can conclude that the

proposed feature is simultaneously invariant to uni-

form scaling, to contrast changes and to convolution

with any diagonal-symmetric kernel.

3.2 Invariant Descriptor based on

Central Moments

Inspired by previous invariants, let us now propose a

new set of radiometric invariants based on central mo-

ments. Speciﬁcally, if we replace each occurrence of

the Mellin transform of order (s, v) by one central mo-

ment of order (s+v) in equation (9), we obtain a new

set of image descriptors invariant simultaneously to

vertical and horizontal translations, to uniform scal-

ing, to contrast changes as well asto blur degradations

that can be modelled by any kernel having a symmet-

ric form with respect to the diagonals.

Theorem 2: The descriptor P

f

(s,v) is given by:

P

f

(s,v) =

µ

f(x,y)

sv

µ

f(x,y)

vs

, x,y ∈ R

+

, (14)

where s,v ∈ N, µ

f(x,y)

sv

is the (s + v)th order central

moment of a function f(x,y) and is deﬁned as:

µ

f(x,y)

sv

=

+∞

Z

−∞

+∞

Z

−∞

(x− x)

s

(y− y)

v

f(x, y)dxdy, (15)

with x = m

f

10

/m

f

00

, y = m

f

01

/m

f

00

, representing the co-

ordinates of the mass center of the function f(x,y).

The functional m

f

sv

is the geometric moment of order

(s+ v) of the image function f(x,y) and is given by:

m

f

sv

=

+∞

Z

−∞

+∞

Z

−∞

x

s

y

v

f(x, y)dxdy, s, v ∈ N. (16)

Since the descriptor P(s,v) is based on central mo-

ments, it is trivial to prove that it is invariant to both

horizontal and vertical translations. The proofs of the

descriptor invariance with respect to uniform scaling,

contrast changes and to convolution are identical to

the ones of the descriptor k(s,v). Note that without

loss of generality we suppose that the gravity center

of the image function coincides with the origin (0,0).

4 COMBINED RADIOMETRIC

GEOMETRIC INVARIANTS

In this section, we introduce a new class of combined

invariant features. We start by introducing the com-

bined invariant based on central moments.

4.1 Combined Descriptor based on

Central Moments

In this subsection, we propose a new set of central-

moment based descriptors invariant simultaneously to

horizontal and vertical translations, to uniform and

anisotropic scaling, to stretching, to contrast changes

and to convolution with any diagonal-symmetric ker-

nel.

Theorem 3: The combined descriptor B(s,v) is

given by:

B

f

(s,v) =

µ

f(x,y)

sv

µ

f(x,y)

vs

×

µ

f(x,y)

v+n,s+n

µ

f(x,y)

s+n,v+n

, x,y ∈ R

+

, (17)

with s,v ∈ N and n ∈ N

∗

. One can easily notice that

B

f

(s,v) is a combination of the radiometric invari-

ant descriptor P

f

(s,v) for different orders (s,v), i.e.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

16

B

f

(s,v) = P

f

(s,v) × P

f

(v + n, s + n). We chose to

build the combined descriptor B

f

(s,v) based on the

radiometric descriptor P

f

(s,v) because of its invari-

ance to horizontal and vertical translations, to uniform

scaling, to contrast changes as well as to convolution.

In what follows, we show the invariance of B

f

(s,v) to

anisotropic scaling and to stretching.

4.1.1 Anisotropic Scaling Invariance

Anisotropic scaling is given by (Jahne, 2005):

u = rx,

w = ty,

with r and t the directional scale factors. Following

an anisotropic scaling of the function f(x,y), µ

f

′

(u,w)

sv

is related to µ

f(x,y)

sv

by what follows:

µ

f

′

(u,w)

sv

=

+∞

Z

−∞

+∞

Z

−∞

u

s

w

v

f

′

(u,w)dudw,

=

+∞

Z

−∞

+∞

Z

−∞

(rx)

s

(ty)

v

f(x,y)rdxtdy,

= r

s+1

t

v+1

µ

f(x,y)

sv

. (18)

Applying the combined invariant B(s,v) to a function

f

′

(u,w), we obtain:

B

f

′

(u,w)

(s,v) =

r

s+1

t

v+1

µ

f(x,y)

sv

r

v+1

t

s+1

µ

f(x,y)

vs

r

v+n+1

t

s+n+1

µ

f(x,y)

v+n,s+n

r

s+n+1

t

v+n+1

µ

f(x,y)

s+n,v+n

,

= B

f(x,y)

(s,v). (19)

The above proof allows us to conclude that the com-

bined feature is invariant to anisotropic scaling.

4.1.2 Stretching Invariance

To prove the invariance of B

f

(s,v) to stretching, we

just haveto replace the factor t by

1

r

in the aboveproof

(cf. equations (18) and (19)).

All of these proofs allow us to conﬁrm the invari-

ance of our combined descriptors B(s,v) to horizontal

and vertical translations, to uniform and anisotropic

scaling, to stretching, to contrast changes and to con-

volution.

4.2 Combined Feature based on Central

Complex Moments

In this section, we introduce a new class of combined

features based on central complex moments. The pro-

posed features are invariant simultaneously to simi-

larity transformation and to contrast changes. Note

that the similarity transformation includes horizontal

and vertical translations, rotation and uniform scaling

(Zisserman and Hartley, 2003). We start by giving

some mathematical deﬁnitions.

Central Complex Moment: The central complex

moment κ

f

pq

of order (p+q) of the function f(x,y) is

deﬁned as:

κ

f

pq

=

+∞

Z

−∞

+∞

Z

−∞

((x− x

c

) + i(y− y

c

))

p

× ((x− x

c

) − i(y− y

c

))

q

f(x,y)dxdy, (20)

with p,q ∈ N, (x

c

,y

c

) the coordinates of the centroid

of the image function f(x,y) and i the imaginary unit.

In polar coordinates, the central complex moment κ

f

pq

is given by:

κ

f

pq

=

+∞

Z

0

2π

Z

0

r

p+q+1

e

i(p−q)θ

f(r, θ)dθdr. (21)

Theorem 4: The invariant descriptor Z

f

(p,q) of the

image function f(x,y) is given by:

Z

f

(p,q) =

κ

f(x,y)

pq

κ

f(x,y)

qp

×

κ

f(x,y)

q+n,p+n

κ

f(x,y)

p+n,q+n

, n ∈ N

∗

. (22)

Proofs of the invariance of Z(p,q) to the above men-

tioned geometric-radiometric degradations are given

in what follows.

4.2.1 Translation Invariance

The invariance of the feature Z(p,q) to horizontal and

vertical translations is trivial to prove as the central

complex moment is invariant to translations by deﬁ-

nition.

4.2.2 Rotation - Contrast Change Invariance

Let g be a rotated (around the origin) and contrast

changed version of the image function f , i.e. g(r,θ) =

ηf(r, θ + α) where α is the angle of rotation and η is

a positive constant number. The central complex mo-

ments κ

g

pq

of the image function g is related to the

central complex moment κ

f

pq

of the original image

function f by the following equation:

κ

g

pq

= ηe

−i(p−q)α

κ

f

pq

. (23)

NEW INVARIANT DESCRIPTORS BASED ON THE MELLIN TRANSFORM

17

Applying Z(p,q) to the image function g(x, y), we ob-

tain:

Z

g

(p,q) =

ηe

−i(p−q)α

κ

f

pq

ηe

−i(q− p)α

κ

f

qp

×

ηe

−i(q− p)α

κ

f

q+n,p+n

ηe

−i(p−q)α

κ

f

p+n,q+n

,

=

κ

f

pq

κ

f

qp

×

κ

f

q+n,p+n

κ

f

p+n,q+n

= Z

f

(p,q). (24)

4.2.3 Uniform Scaling Invariance

Let g be a scaled version of the original image func-

tion f, i.e. g(r,θ) = f(βr,θ), where β is the parameter

of the uniform scaling. The central complex moment

κ

g

pq

is related to κ

f

pq

by the following equation:

κ

g

pq

= β

−(p+q+2)

κ

f

pq

. (25)

Applying the descriptor Z(p,q) to the image function

g, we obtain:

Z

g

(p,q) =

β

−(p+q+2)

κ

f

pq

β

−(q+p+2)

κ

f

qp

×

β

−(q+p+2(n+1))

κ

f

q+n,p+n

β

−(p+q+2(n+1))

κ

f

p+n,q+n

,

=

κ

f

pq

κ

f

qp

×

κ

f

q+n,p+n

κ

f

p+n,q+n

= Z

f

(p,q). (26)

According to the above proofs, we conclude that

the proposed feature Z(p,q) is invariant simultane-

ously to similarity transformation (translations, rota-

tion, uniform scaling) and to contrast changes.

5 EXPERIMENTAL RESULTS

In this section, we carry out a number of tests in order

to express the discrimination power of our invariant

descriptors. We start by applying the invariant based

on the Mellin transform. Then, we show results of

the radiometric descriptor based on central moments.

Finally, we show the experimental results related to

the combined invariants.

5.1 Mellin-Transform based Descriptor

To validate our invariant descriptor in practical sit-

uations we carried out tests on real images. Figure

1 shows real images (a,c-f) which were taken with a

digital camera (Canon 1D professional) with different

parameter settings. The radiometric invariant feature

k(s,v) is applied to real images in Figure 1. Experi-

mental results are given in Table 1.

Results in Table 1 show that for any order (s,v) the

numerical values obtained by applying the invariant

(a) (b) (c)

(d) (e) (f)

Figure 1: a, c-f: real images of the same scene with different

blur degrees. b- foreign image.

descriptor k(s,v) to images (a) and (c-f) are almost

identical but are different from the numerical value

obtained with image (b). Obtained results conﬁrm the

invariance of the descriptor k(s,v) to both convolution

and contrast changes.

Table 1: Results of the application of the radiometric invari-

ant descriptor k(s,v) to real images of Figure 1.

k(6, 7) k(4, 12) k(7,2) k(10, 6)

Image a 1.55926 32.92205 0.09745 0.17464

Image b 0.87445 0.277227 1.41440 1.92620

Image c 1.55775 32.62386 0.09713 0.17579

Image d 1.55791 32.70831 0.09841 0.17529

Image e 1.55093 31.83843 0.10427 0.17771

Image f 1.54944 31.62592 0.10479 0.17840

With an aim of comparing our invariant feature to the

state of the art and to the most related feature, we car-

ried out tests on images using the radiometric feature

C(s,v) developed by Flusser and Suk in (Flusser and

Suk, 1998). Results given in Table 2 provide an ex-

ample of those tests using images in Figure 1. We

may observe that for any order (s,v) the distance be-

tween the invariant descriptors of two degraded im-

ages of the same scene is smaller with the proposed

feature k(s,v) than the ones provided by C(s,v). For

instance, let us consider the order (6,7). Following

a data normalization, i.e., by dividing all descriptor

values (related to the original image and to its de-

graded versions) by the descriptor value of the orig-

inal image, the standard deviation of our results is

equal to 0.0029, while the one obtained with C(s,v)

is equal to 0.0523. Computing times (relative to im-

age (a) of Figure 1) of our invariant feature is equal to

0.047 s while the one with C(s,v) is equal to 31.078 s.

Note that computing time is evaluated in seconds. We

hence notice that, for any order (s,v), computing time

of our invariant feature is quite low. Finally notice

that k(s,v) can be computed for any values of (s,v),

including even values.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

18

Table 2: Results obtained using Flusser and Suk feature.

C

f

(6,7) C(4,12) C

f

(7,2) C(10,6)

[10

−44

] [10

−31

]

Image a -7.7130 0 (n/a) 5.7757 0 (n/a)

Image b 4367.96 0 (n/a) 155.93 0 (n/a)

Image c -7.0798 0 (n/a) 5.7263 0 (n/a)

Image d -6.9568 0 (n/a) 5.5082 0 (n/a)

Image e -7.7330 0 (n/a) 5.7717 0 (n/a)

Image f -7.1183 0 (n/a) 5.5061 0 (n/a)

5.2 Radiometric Descriptors based on

Central Moments

In what follows, we apply the invariant descriptor

P(s,v) to images in Figure 2.

(a) (b) (c)

(d) (e) (f)

Figure 2: First row: a- original image, b- foreign image

(lena), c- contrast change of translated image. Second row:

d- translated images, e- translated and blurred image (ﬁlter

size = 7 × 7), f- contrast change of translated and blurred

image.

From Table 3, we notice that the numerical values of

the invariant descriptor P(s,v) relative to images (a)

and (c-f) are almost identical but are different from the

one of image (b) as expected. Obtained results con-

ﬁrm the invariance of the descriptor P(s,v) to contrast

changes, to blur degradations as well as to horizontal

and vertical translations.

Table 3: Results obtained by applying the invariant descrip-

tor P(s,v) to images of Figure 2.

P(4,3) P(11,1) P(9, 2) P(7,2)

Image a 0.01999 -0.40269 2.76839 6.74357

Image b -3.30720 0.30403 -0.40323 -0.36359

Image c 0.01706 -0.40277 2.77025 6.87348

Image d 0.01999 -0.40269 2.76839 6.74357

Image e 0.02040 -0.39920 2.81426 6.89165

Image f 0.01868 -0.40375 2.76167 6.80592

5.3 Combined Descriptor based on

Central Moments

In this subsection, we apply the combined invariant

B(s,v) to images in Figure 3. Note that Image (f) in-

cludes at the same time, vertical and horizontal trans-

lations, stretching, uniform scaling, contrast change

and blur degradation. Experimental results are given

in Table 4.

(a) (b) (c)

(d) (e) (f)

Figure 3: a- clear image, b- resized and blurred image, c-

foreign image, d- translated and blurred image, e- stretched

and translated image, f- contrast change of geometric de-

graded and blurred image (ﬁlter size = 11× 11).

According to Table 4, we notice that for any order

(s,v), the numerical values of the original image (Fig-

ure 3.a) and its degraded versions (Figure 3.b,d-f) are

almost identical but are different from the one of the

foreign image (Figure 3.c). The combined invariant

descriptor B(s,v) was tested for several orders and

the obtained results express its invariance to the above

mentioned radiometric and geometric degradations.

Table 4: Experimental results of the application of the com-

bined invariant B(s,v) to images of Figure 3.

B(2,3) B(3,6) B(8,5) B(5,15)

Image a 9.8918 0.0830 12.021 -0.3110

Image b 9.8666 0.0835 11.936 -0.3001

Image c -0.0007 109.26 0.0419 3.2301

Image d 9.8277 0.0841 11.815 -0.2860

Image e 10.379 0.0782 12.725 -0.3207

Image f 9.7072 0.0816 12.626 -0.3204

5.4 Invariant Descriptor based on

Central Complex Moments

In this subsection we evaluate the discrimination

power of the descriptor based on central complex mo-

ments. For this purpose we apply Z(s,v) to an image

NEW INVARIANT DESCRIPTORS BASED ON THE MELLIN TRANSFORM

19

which includes at the same time vertical and horizon-

tal translations, rotation, uniform scaling and contrast

change (cf. Figure 4.b). Note that the invariant feature

(a) (b) (c)

Figure 4: a- original image, b- contrast change of similarity

transformed image, c- foreign image.

values shown in Table 5 correspond to the multiplica-

tion of the real and imaginary parts of Z(s,v). This

representation was chosen since it has experimentally

proven to provide a high discrimination power. It is

obviously not unique, other representations can also

be used. As can be seen from the table, for all of the

orders, the values of Z(s,v) related to images (a) and

(b) are almost identical but are different from the one

related to the foreign image (c), as expected.

Table 5: Results of the application of the combined invariant

Z(s, v) to images of Figure 4.

Z

2,11

Z

3,7

Z

6,5

Z

8,5

Image a 0.04807 0.00905 0.00691 0.04281

Image b 0.05276 0.00944 0.00645 0.03912

Image c -0.02320 -0.00229 -0.00105 0.15769

Obtained results express the invariance of Z(s,v) to

similarity transformation and to contrast changes.

Note that interested readerscan ﬁnd in (Metari and

Deschênes, 2007b) a detailed comparisons made be-

tween the three above mentioned descriptors and the

most widely used descriptors in the literature.

6 CONCLUSIONS

In this paper, we introduced two new classes of invari-

ant features. The ﬁrst class includes two radiometric

features. The ﬁrst one is based on the Mellin trans-

form and the second one is based on central moments.

Both radiometric descriptors are invariant to radio-

metric degradations that can be modelled by convo-

lution as well as to global contrast changes and uni-

form scaling. The second class includes two com-

bined invariant features. The combined descriptors

based on central moments is invariant simultaneously

to all of the following transformations : both horizon-

tal and vertical translations, uniform and anisotropic

scaling, stretching, contrast changes and degradations

that can be modelled by convolution. The combined

features based on central complex moments are in-

variant simultaneously to similarity transformation

(translations, rotation, uniform scaling) and to con-

trast changes. These invariant features have been val-

idated experimentally. All of the results conﬁrm that

they provide a high discrimination power at a low

computing cost. They can be used for the recognition

and classiﬁcation of geometrically and/or radiometri-

cally degraded images.

ACKNOWLEDGEMENTS

This work is partially supported by the Natural Sci-

ences and Engineering Research Council of Canada

(NSERC) research funds.

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