IMAGE RESTORATION VIA HUMAN PERCEPTION

AND LIE GROUPS

∗

Vittoria Bruni

, Elisa Rossi

and Domenico Vitulano

Dept. SBAI University of Roma Sapienza, Via A. Scarpa 16, 00161 Rome, Italy

Istituto per le Applicazioni del Calcolo, C.N.R., Via dei Taurini 19, 00185 Rome, Italy

Keywords:

Image Restoration, Lie Group Transformations, Semi-transparent Blotches.

Abstract:

This paper presents a novel and user’s independent model for image restoration based on Lie group transfor-

mations. They allow to deﬁne a redundant set of transformations from which automatically select the ones

that better invert the physical formation of the defect. Hence restoration consists of gradually reducing the

visual perception of the defect. Extensive experimental results on original photographs, from Alinari Archive,

affected by semi-transparent blotches show the potential of the proposed approach in removing degradation in

different contexts without altering the original content of artworks.

1 INTRODUCTION

The growing demand for archived material as well as

the improvement of computer technologies has led

to the need of reliable and useful tools for digital

and (semi)-automatic removal of degradation, such as

noise, blotches, line-scratches, tear, moire, shake and

ﬂicker (Kokaram, 1998).

The model proposed in this paper focuses on a

generalization of the plethora of existing restoration

methods for the aforementioned kinds of degradation

— see next section for a short review. In order to

make more concrete the content of the paper, from

now on we will focus on the class of semitransparent

degradationswhose main peculiarity is that part of the

original information often survives in the degraded

area (Crawford et al., 2007). The main goal is the

deﬁnition of a general framework that is as much as

possible independent of a priori speciﬁc assumptions

on the degradation under exam. The framework is re-

quired to select suitable restoration transformations

from a redundant set only accounting for the semi-

transparency of the degraded region. To this aim, the

visual perception of degradation cannot be neglected

(Winkler, 2005). Image defects are detected by hu-

man eye ’at ﬁrst glance’ even in complicated contexts.

This means that degradation represents an ’anomaly’

in any natural image. Hence, the reduction of the vi-

∗

Authors thank Alinari Archive for kindly providing im-

ages.

sual contrast of the degraded region (visual anomaly)

should decrease the visual contribution of the de-

graded area without creating new artifacts. A local

contrast-based restoration process that embeds trans-

formations in a Lie group gives us the opportunity of

deﬁning a redundant set of transformations that also

contains the inversion of the unknown degradation

process. In addition, it allows to develop a restoration

algorithm that automatically selects the more suit-

able transformations for points having the same visual

contrast.

This approach is two-fold innovative:

i) The combination of Human Visual System (HVS)

and Lie algebra allows the proposed model to

have not a precise target to converge — as it

usually happens in Lie algebra based approaches

(see, for instance, (Drummond and Cipolla, 2000;

Porikli et al., 2006; Mansouri and Mukherjee,

2004; Mukherjee and Acton, 2007)). The exact

ﬁnal solution is not known in advance and the

model is only required to force the contrast of the

ﬁnal solution to be in a suitable range of values

according to ’typical’ contrasts of the surround-

ing clean image — blotch has to be invisible. To

this aim, the modiﬁed successive mean quantiza-

tion transform (SMQT) (Nilsson et al., 2005) is

employed for contrast-based classiﬁcation of de-

graded pixels;

ii) The rich set of Lie groups transformations over-

comes the search of the solution of the restoration

Bruni V., Rossi E. and Vitulano D..

IMAGE RESTORATION VIA HUMAN PERCEPTION AND LIE GROUPS.

DOI: 10.5220/0003860600660074

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 66-74

ISBN: 978-989-8565-03-7

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

problem through simple translation and shrinking

operations, that are commonly used by existing

competitors.

The reminder of the paper is the following. Next

section contains the motivations of the work and a

short state of the art of image restoration methods.

Section 3 brieﬂy introduces Lie group transforma-

tions. Section 4 presents the proposed restoration

model. Finally, Section 5 contains results and con-

cluding remarks.

2 MOTIVATION OF THE WORK

Although semitransparent degradation includes de-

fects like blotches, scratches and so on, for the sake

of clarity, we will focus on the former.

Blotches are usually caused by dirt or moisture

on archived material that partially or completely ob-

scures some image regions. They appear as irregu-

larly shaped regions with a darker average intensity

and a slight variation in color, as shown in Fig.1.

Hence, the lack of distinctive features, like shape and

color, makes their detection and restoration not trivial

tasks. Blotches must be restored using clean infor-

mation from the immediate vicinity of the blotch, al-

though this is not essential. However, part of the orig-

inal information still survives even after the degrada-

tion process. This is due to its physical formation

that can be modeled by the spreading and penetration

of water droplets into material. During the spread-

ing process, the radius of the wet region grows to an

equilibrium point. From this point on, the liquid is

absorbed depending on the porosity of the considered

medium. In ideal conditions, the central pores absorb

more than the external pores, since they come in con-

tact with the liquid earlier. For complete spreading

and absorption processes, one can expect a smooth

transition to the unaffected area; on the contrary a

spurious edge is evident.

Hence, the degraded image J at the point x =

(x,y)

can be modeled as

J(x) = T (I(x)),

where T is a proper composition of transformations

and I is the original image. The goal should be to

ﬁnd the inverse of T in order to reconstruct the orig-

inal image I. Unfortunately, the evolution of a drop

involves different parameters, such as drop geometry

and the regularity of paper surface, that are unknown

in real applications. Hence T is unknown as well as

its inverse T

−1

. The proposed model employs the

projective Lie group as a redundant set of transforma-

tions where automatically select the best T

−1

. The

selected transformations are not global but they are

adapted to the local properties of the damaged area,

according to a contrast-based classiﬁcation. External

information is involved just in the deﬁnition of the

admissible range values for degraded pixels and for

comparing global measures like the inner contrast be-

tween the damaged area and its surrounding regions,

making the model quite independent as of external

features.

2.1 Blotch Restoration: A Short State of

the Art

A good restoration method is required to preserve all

the original content of the artwork for historical and

artistic purposes. Inpainting methods or texture syn-

thesis approaches (Bertalmio et al., 2000; Bertalmio

et al., 2003; Criminisi et al., 2004; Efros and Free-

man, 2001; Kokaram, 2002) are not appropriate for

the restoration of partially missing data regions, since

they completely discard the original information that

is still contained in those regions.

On the contrary, existing approaches that exploit

the semi-transparency property, like (Crawford et al.,

2007; Stanco et al., 2005; Greenblatt et al., 2008;

Bruni et al., 2011), make implicit or explicit assump-

tions on the physical model that causes the degrada-

tion so that their restoration consists of ’ad hoc’ op-

erations. For example, in (Stanco et al., 2005) semi-

transparency is modeled as a linear dependence be-

tween the intensity values of the degraded and orig-

inal region, assuming similar statistic features inside

and outside the degraded one. A non linear model

closer to the visual appearance of degradation is used

in (Greenblatt et al., 2008): it uses ﬂattening to em-

phasize blotches darkness and enhancement to ex-

ploit the local image statistics. Afﬁne point-wise

transformations are employed in (Crawford et al.,

2007) and (Bruni et al., 2011). While the former in-

volves a minimization algorithm that emphasizes the

propagation of information from the outside-in, the

latter relies on a precise model of both physical and

visual characteristics of the speciﬁc degradation kind.

3 A SHORT REVIEW ABOUT LIE

ALGEBRA

A ﬁnite Lie group G is a set of elements equipped with

a group multiplication ∗ : G× G → G such that:

i) ∀g

∈ G (g

∗ g

) ∗ g

= g

∗ (g

∗ g

);

ii) ∃ e ∈ G, identity element, such that ∀ g ∈ G e∗g=

g∗ e = g;

IMAGE RESTORATION VIA HUMAN PERCEPTION AND LIE GROUPS

Figure 1: Examples of semi-transparent blotches in real photographs.

1 2 3 4 5 6 7 8

0.02

0.04

0.06

0.08

0.1

GROUPS

CONTRAST

0 50 100 150 200 250

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

z=(m+M+2∆)/2

Figure 2: Left: Contrast curve γ of B. Right: Plane section of a blotch (light gray), the surface of the i− th group (piecewise

curve) and the corresponding paraboloid (dark and solid line).

iii) ∀ g ∈ G ∃ g

−1

∈ G : g∗ g

−1

= g

−1

∗ g = e

and a differentiable manifold of ﬁnite dimension, i.e.

a space locally diffeomorphic to R

, if n is the dimen-

sion of G. Moreover the group operation ∗ and the

inverse map (G → G, g 7→ g

−1

) are C

∞

with respect

to the differentiable structure of the manifold. So a

Lie group G has algebraic properties coming from

the group structure and geometric properties coming

from the differentiable structure and they are related.

Finally, every ﬁnite Lie group can always be viewed

as a matrix group.

A Lie algebra g is a vector space endowed with a bi-

linear operation,

[ , ] : g × g → g, (X,Y) 7→ [X,Y],

called Lie bracket, antisymmetric and satisfying Ja-

cobi identity.

If G is a Lie group, its tangent space at identity,

g, which is a vector space, has a Lie algebra struc-

ture. Hence, g is a vector space of the same size of G

endowed with a Lie bracket. If G is a matrix group,

the Lie bracket is the matrix commutator, i.e. [X,Y] =

XY −YX. Since G is a differentiable manifold, there

exists a correspondence between its tangent space at

identity (the Lie algebra g) and G itself, that is the ex-

ponential map, exp : g → G. Let X ∈ g be a tangent

vector at e in G; locally there exists the integral curve

of X, a smooth curve starting from the identity with

tangent vector X, i.e. γ : [0, T] → G such that γ(0) = e

and

γ(0) = X; we deﬁne exp(X) = γ(1). The exponen-

tial map is a local diffeomorphism around the origin

of g and it gives a natural way to move from g (vector

space) to G (manifold). For the matrix group, it corre-

sponds to matrix exponential: exp(X) =

∑

∞

n=0

. For

a complete treatment of Lie groups and Lie algebras

see (Varadarajan, 1974) and (Helgason, 1962).

3.1 Lie Group Transformations

Most of the matrix Lie groups can be used to describe

transformations in the plane or in the space. For in-

stance, rotations in the plane are represented by the

group

R =



cosθ sinθ

−sinθ cosθ



,θ ∈ R



The dimension of the group can be thought as

the number of free parameters needed to describe the

transformations. In case of plane rotation we just

need one parameter, θ, so the dimension of SO

R is

1. We can think about its Lie algebra elements, which

are tangent vectors at the identity, as the inﬁnitesimal

transformations (rotation of an ’inﬁnitesimal angle’)

of the points.

In this paper we are interested in using projec-

tive transformations in the space and they can be

described as a group matrix, P

, acting on the space

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

points expressed in homogeneous coordinates, with

the convention that the fourth value in the coordinates

is always scaled back to 1. Projective transformations

are characterized by 15 parameters, that is the dimen-

sion of P

, described by the following basis of its Lie

algebra representing translations, scaling, shear and

projections:



0 0 0 1

0 0 0 0





0 0 0 0

0 0 0 1

0 0 0 0





0 0 0 0

0 0 0 1

0 0 0 0





0 0 0 0

0 0 1 0

0 −1 0 0

0 0 0 0





0 0 −1 0

0 0 0 0

1 0 0 0

0 0 0 0





0 1 0 0

−1 0 0 0

0 0 0 0





1 0 0 0

0 0 0 0





0 0 0 0

0 1 0 0

0 0 0 0





0 0 0 0

0 0 1 0

0 0 0 0





0 1 0 0

0 0 0 0





0 0 1 0

0 0 0 0





0 0 0 0

0 0 1 0

0 0 0 0



= G

with T the transpose operator.

Hence, every real linear combination of

,...,G

is an inﬁnitesimal projective transforma-

tion in the space that corresponds to a transformation

of the group P

thanks to the exponential map.

The inﬁnitesimal transformation of a generic point

p =

= G





j = 1, ..., 15 whose afﬁne

coordinates L

respectively are

−y

−z

−x





3.2 Surfaces Distance Minimization by

Projective Transformation

The relation between Lie algebras and Lie groups al-

lows us to deﬁne an iterative procedure able to map a

given surface S

to another one, S

, in R

. ∀p ∈ S

, let

be the unit normal at S

in the point p and d

the

distance between p and S

along n

. Hence,

∑

p∈S

is the distance between S

and S

Let L

, for j = 1,..., 15, be the inﬁnitesimal pro-

jective transformation L

applied to the point p. The

goal is to estimate 15 real parameters, α

,...,α

such that the inﬁnitesimal projective transformation

∑

j=1

, projected onto the normal direction n

minimizes the distance between S

and S

, i.e.

(α

,... , α

) = min

∑

p∈S

−

∑

j=1



· n



Therefore,

α = (α

... α

)

is such that

α =

−1

where the matrix

A and the column vector

b are

∑

p∈S



· n





· n



and

∑

p∈S



· n



The exponential map transforms the inﬁnitesimal

transformation G =

∑

j=1

into a projective trans-

formation T of the group i.e.,

T = exp(G) =

∞

∑

n=0

Finally, S

is updated applying T to its points: S

(1)

T(S

). The minimization process can be then iterated

using the couples of curves (S

(1)

), and so on.

For the numerical computation of exp(G) applied

to generic point x, a 4th order Runge Kutta algorithm

can be used — see (Drummond and Cipolla, 2000)

for details. It is equivalent to cut the 4th order series

expansion of the matrix exponential and apply it to

the point x, that is

T (x) ≈



I + G+



(x),

but it directly manages afﬁne coordinates.

4 THE PROPOSED MODEL

Lie group transformations offer a redundant set of

transformations as well as a direct and fast method for

IMAGE RESTORATION VIA HUMAN PERCEPTION AND LIE GROUPS

Figure 3: Top: Degraded image, its detection mask, D

, D

and D

masks. Bottom: Restored images after: processing D

(left), processing D

(middle) and ﬁnal masking (right).

Figure 4: Top: Original degraded image, its detection mask, D

, D

and D

masks. Bottom: Restored images after: processing

(left) and ﬁnal masking (right).

selecting them once the ﬁnal target curve is known.

Unfortunately, in case of digital restoration the ﬁnal

clean image is unknown. That is why some basic rules

of HVS are used for deﬁning suitable ranges of inten-

sity values for the damaged area to be not visible with

respect to its neighborhood. Despite the wide ﬂexibil-

ity of Lie transformations, the minimization process

is global. In other words, at each step the parameters

{α

}

j=1,...,15

are the same for each point. Hence, if on

one hand global transformations preserve the original

information contained in the degraded region, on the

other they forget that pixels may have been subjected

to a different amount of degradation.

In order to ﬁnd a tradeoff between preservation of

original information and model ﬂexibility,it is neces-

sary to classify damaged pixels accounting for their

visual importance and restore them accordingly.

4.1 Preprocessing of the Damaged Area

Let I be the analyzed image, B its damaged area and

let E be a sufﬁciently small neighborhood of B such

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

Figure 5: Original (left) and restored (right) images.

that we can assume that E and B share the same in-

formation. In order to classify the points in B accord-

ing to their contrast properties, the SMQT algorithm

(Nilsson et al., 2005) is used. It groups pixels having

comparable visual contrasts. More precisely, SMQT

builds a binary tree using the following rule: given a

set of data D and a real parameter L (number of lev-

els), split D into two subsets,



x ∈ D|D(x) ≤



and



x ∈ D|D(x) >



where D is the mean value of D. D

and D

are

the ﬁrst level of the SMQT. The same procedure is re-

cursively applied to D

and D

and so on until the L

level, that is composed of 2

subsets. We set L = 3

in order to obtain a log

quantization of the dam-

aged pixels (i.e. 2

= log

(256) groups, B

,... , B

where [0,255] is the gray scale range) and will assume

,··· , B

ordered from the darkest to the brightest

pixel. We compute the inner contrast C

of each

, with σ

the standard deviation and µ

the mean of

. Let γ = {C

,··· ,C

} be the discrete contrast curve

of B, as in Fig. 2 (Left). Darker pixels in B, say D

are those sets B

whose inner contrast is greater than k

(where k is the least detection threshold given by the

Weber’s law, see (Winkler, 2005)). On the contrary,

brighter pixels in B, say D

, are those B

whose inner

contrast curve has positive ﬁrst derivative. Let D

the union of the remaining groups.

Summing up, SMQT quantization of the degraded

region B combined with its contrast properties allows

to split B into three regions: the darkest, D

, the

brightest, D

and the central one, D

, where most of

information lives. The whole range of values of dam-

aged pixels has been then split into these three inter-

vals: [b

], ]b

[ and [b

]. Let M and m be re-

spectively the maximum and the minimum value for

E (except for outliers). [m, M] is the range of admis-

sible values for the ﬁnal solution. In fact, a natural

scene component is required to not exceed the range

of values of the surrounding information in terms of

visibility bounds. [m, M] is proportional to the global

contrast of the whole image I, i.e.

M− m

where µ

the mean value of I. Since we are dealing with a lo-

cal degradation, we can think that any transformation

of the degraded region does not inﬂuence too much

, so that it can be considered constant. That is why

from now on we will just deal with ranges instead of

contrasts.

The second goal of preproccesing is to understand

if each group of pixels must be moved. In fact, even

though the damaged area B appears darker than the

neighborhood E, it is important to check if the inter-

section between E and B values contains points that

are darker than the blotch, as it is the case for the

blotch on the dome in Fig. 1 (leftmost). To this aim, it

is necessary to check if the darker region of E, corre-

sponding to those pixels whose value is in the interval

IMAGE RESTORATION VIA HUMAN PERCEPTION AND LIE GROUPS

], masks D

. Setting the just noticeable thresh-

old ε = .33 (Pappas et al., 2005), if their contrast ra-

tio is included in [1− ε,1 + ε], then the same infor-

mation lives outside the blotch, and D

must be left

unchanged; otherwise D

must be transformed and

mapped into the range [m,M − δ], where δ is such

that [M − δ,M] and is proportional to [b

]. It is

worth stressing that we choose to move D

values in

[m,M− δ] and not in [m,M] in order to preserve even-

tual original information.

With respect to D

, i.e. the brightest region of

B, we check if it is masked by the region in E cor-

responding to the values in the interval [b

]. D

mostly corresponds to the transition area from E to B,

according to the physical properties of the damaged

area. Hence, the ratio between the inner contrasts is

included in [1− ε,1+ ε] and with high probability we

don’t need to transform it.

Finally, D

contains most of degradation and it must

be mapped into the interval [m, M]. However, because

of some rigidity of the model, it will be better to shift

the interval, i.e. [m+ ∆, M + ∆]. ∆ measures the dark-

ness of the damaged area and it is well represented by

the difference between the mean values of B and E.

4.2 Processing of the Damaged Area

The surfaces distance minimization described in sec-

tion 3.2 requires the deﬁnition of suitable surfaces

living in the transformation domains, [m+ ∆, M + ∆]

and [m,M − δ], of D

and D

. To preserve the origi-

nal information, HVS perception mechanisms can be

embedded in the restoration process. In particular,

we aim at processing in the same way points that

are equally perceived by human eye: points having

the same context (same visual contrast), have to con-

verge to the same target surface. To this aim, we ap-

ply the SMQT algorithm to D

. Each group in D

corresponds to a surface (deﬁned by interpolation).

The target surface of the i-th group is deﬁned as a

paraboloid cut by the plane

z =

m+ M + 2∆

and whose vertex is proportional to the mean value

of the group, according to the values interval. Each

surface converges to the corresponding paraboloid, as

shown in Fig. 2 (Right).

The iterative minimization process stops when the

target surface has been reached in agreement with vis-

ibility bounds. More precisely, let S be the initial sur-

face, S

the solution at the n-th iteration and P the

target paraboloid; S

is the ﬁnal solution if

∑

x∈S

(x) − P(x)|

∑

x∈S

P(x)

≤ k , (1)

where k is the least detection threshold given by We-

ber’s law. The ﬁrst member of previous equation cor-

responds to Weber’s contrast evaluated at the points

of the analysed surfaces. For D

we use the same rule

with

z =

m+ M − δ

4.3 Masking Reﬁnement

The iteration algorithm tends to preserve the original

information at the expense of some rigidity (solutions

tend to be dark). As a result, we need to stress the

range values relative to D

in order to have sufﬁciently

bright solutions and a sufﬁciently large range values

to avoid oversmoothed solutions. This requires a ﬁnal

masking operation.

Speciﬁcally, let S

be the output of the minimiza-

tion algorithm applied to D

. LetV be the set of pixels

in S

whose value is greater than M− ε, that is

V = {p ∈ S

| v(p) > M − ε},

where v(p) is the value of the pixel p in S

and V

its

complementary set in S

. The ratio betweenV andV

is the inner contrast of S

. If this ratio is included in

[1 − ε, 1 + ε], we don’t need masking; otherwise, we

have to replace the values of some pixels in V with

their original ones.

The main idea is to replace those pixels that orig-

inally were sufﬁciently bright and that became too

bright in the minimization algorithm. More precisely,

let

H = {p ∈ V | v

orig

(p) > τ}

and

orig

= {v

orig

(p) | p ∈ V}

where v

orig

(p) is the original value of the pixel

p in D

and τ is a suitable constant related to the

properties of V

orig

, then S

(H) = V

orig

(H). Note that

τ =

orig

+ ε, where

orig

is the mean value of V

orig

4.4 Algorithm

Let I, B and E respectively be the initial image, the

degraded area and its neighborhood.

Step 1. Split B into the groups D

, D

and D

using

SMQT;

Step 2. Estimate the range of the solution [m, M]

from the surrounding information;

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

Figure 6: Blotch on a textured image. Top to bottom - left to right: degraded, restored image with the present method and

restored using methods in (Greenblatt et al., 2008), (Stanco et al., 2005), (Crawford et al., 2007) and (Bruni et al., 2011).

Step 3. Check if D

has to be processed; if so, go to

step 4, otherwise go to step 7;

Step 4. Apply SMQT to D

;

Step 5. Compute paraboloids including the solution

whose vertices are set according to the mean am-

plitudes of the groups computed in step 4 and the

output of step 2;

Step 6. For each group in step 4, apply the iterative

procedure in Section 3.2 where the target surfaces

are the ones computed in step 5, until eq. (1) is

satisﬁed;

Step 7. Apply steps 4-5-6 to D

;

Step 8. Perform masking reﬁnement.

Fig. 3 shows the results of some steps of the algo-

rithm. In this case D

must be processed: the initial

contrast ratio between D

and its corresponding exter-

nal area is 2.5. On the contrary, the one between D

and its corresponding external region is 0.8; hence it

is not necessary to process it.

For the blotch in Fig. 4, both D

and D

don’t

need to be processed — contrast ratios respectively

are 1.1 and 0.9.

5 EXPERIMENTAL RESULTS

AND CONCLUDING REMARKS

The proposed approach has been tested on selected

images from the photographic Alinari Archive in Flo-

rence, affected by semi-transparent defects. Some re-

sults are shown in Figs. 5 and 6. As it can be ob-

served, the visual appearance of the recovered images

is very good: no artifacts appear, the texture of the

background is well recovered as well as eventual de-

tails of the original image (see, for example, the edges

of the dome).

The use of a selective algorithm avoids annoying

halo effects at the border of the defect along with

over-smoothing in the inner part of the restored re-

gion. As a result, the restored region is not still per-

ceived as an anomaly on the image. The convergence

process is different for each group of points so that

it could happen that some groups converge after one

or two iterations while others require longer conver-

gence time. In that way the over-smoothing is avoided

and the preservation of the inner information is guar-

anteed via the visibility based stopping criterion in eq.

(1).

Moreover, the preprocessing of the damaged area

and masking procedures allow the detection mask to

be not precise (it can be larger than the degraded re-

IMAGE RESTORATION VIA HUMAN PERCEPTION AND LIE GROUPS

gion) and to manage complicated cases where the de-

graded area intersects a darker region of the image

as in Fig. 5, so that restoration does not create arti-

facts in correspondence to not degraded pixels. This

is a great advantage, since the detection mask heavily

inﬂuences restoration results of available restoration

frameworks.

It is also worth highlighting that even though the

proposed algorithm involves iterative procedures, it

uses simple and fast operations and 4/5 iterations on

average to converge.

Future research will be oriented to reﬁne the pro-

posed model to make it more ﬂexible and adaptive to

different amount of degradation while faithfully pre-

serving original image information.

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