adaptive nonlinear filters for 2d and 3d image enhancement

7/21/2019 Adaptive Nonlinear filters for 2D and 3D Image Enhancement

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Signal Processing 67 (1998) 237254

Adaptive nonlinear filters for 2D and 3D image enhancement

Sebastien Guillon, Pierre Baylou, Mohamed Najim*, Naamen Keskes

Equipe Signal and Image ENSERB and GDR ISISCNRS, BP 99, 33 402 Talence Cedex, France

De&partement Image CSTJFELF Aquitaine, 64 018 Pau Cedex, France

Received 25 June 1997; received in revised form 2 February 1998

Abstract

Unsharp masking method is a popular approach for image enhancement, in which a highpass version of an image is

added to the original one. This method is easy to run, but is very sensitive to noise. Suppressing noise is generally

performed with lowpass filters, and leads to edge blurring. So, an approach which is a combination of a nonlinear

lowpass and highpass filters is proposed. These filters are based on an adaptive filter mask. We demonstrate that this

approach performs noise reduction as well as edge enhancement. It also improves the contrast enhancement in

comparison with other methods. These results are illustrated by processing blurred and noisy images. The method is then

extended for 3D data processing and used on 3D seismic images. 1998 Elsevier Science B.V. All rights reserved.

Zusammenfassung

Die Methode zur Unscharfeverdeckung ist ein beliebter Ansatz zur Bildverbesserung, bei dem eine Hochpa{versiondes Bildes dem Original hinzuaddiert wird. Die Methode ist einfach zu handhaben, aber sehr empfindlich auf Rauschen.

Die Rauschunterdruckung wird im allgemeinen mit Tiefpa{filtern durchgefuhrt und fuhrt zur Kantenverschmierung.

Somit wird ein Ansatz vorgeschlagen, der in einer Kombination eines nichtlinearen Tiefpa{- und Hochpa{filters besteht.

Diese Filter beruhen auf einer adaptiven Filtermaske. Wir zeigen, da{ dieser Ansatz sowohl eine Rauschreduktion als

auch eine Kantenverbesserung vollbringt. Er verbessert im Vergleich zu anderen Methoden auch die Kontrastverstarkung.

Diese Ergebnisse werden anhand der Verarbeitung unscharfer und verrauschter Bilder illustriert. Die Methode wird

sodann auf die Verarbeitung von 3D-Daten erweitert und auf seismische 3D-Bilder angewandt. 1998 Elsevier

Science B.V. All rights reserved.

Resume

La methode Unsharp Masking, tres utilisee en rehaussement de contraste, consiste a ajouter a limage une versionpasse-haut delle-meme. Cependant cette methode, facile a mettre en oeuvre, est tres sensible au bruit. Par ailleurs, la

reduction de bruit est generalement realisee au moyen de filtres passe-bas, ce qui engendre un flou au niveau des contours.

Nous proposons dans cet article de combiner deux filtres adaptatifs non-lineaires passe-bas et passe-haut afin de realiser

dans le meme traitement un lissage et un rehaussement de contraste. Nous mettons en evidence la robustesse de la methode

en traitant des images floues et bruitees. Enfin le filtre est etendu au cas tridimensionnel et est applique au traitement de

blocs sismiques 3D. 1998 Elsevier Science B.V. All rights reserved.

Keywords: Image enhancement; Edge sharpening; Adaptive filtering; Nonlinear filtering

* Corresponding author. Tel.: 33 56 84 66 74; fax: 33 56 84 84 06; e-mail: [email protected].

0165-1684/98/$19.00 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 5 - 1 6 8 4 ( 9 8 ) 00 0 4 2 - 5


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1. Introduction

The purpose of image enhancement techniques is

to remove noise and to sharpen details in order to

improve the visual appearance of an image. A large

variety of enhancement filters have been proposed

linear filters, homomorphic filters, L-filters, andthe like. In this paper we propose to extend the

classical unsharp masking (UM) method [6,7] by

using nonlinear filters.

The UM method operates by adding a fraction of

the highpass filtered version of the input image to

the original one (see Fig. 1). This operator is sensi-tive to noise due to the presence of the linear

highpass filter which cannot discriminate signal

from noise. Moreover it perceptually enhances

image more in dark areas than in lighter ones.

Various schemes have therefore been proposed

in order to improve the performances of the UM

filter. In [8,1012,17,18], the authors propose the

use of quadratic filters instead of a simple linear

highpass filter. These filters perform detail and edge

enhancement in accordance with the characteristics

of human vision. Indeed, enhancement is greater in

bright areas than in dark ones in order to respect

Webers law [6] which states that the just noticeable

brightness difference is proportional to the average

background brightness. The perceived noise thendecreases. In [9,13] Ramponi proposes to use

a simple cubic operator which only performs

a sharpening action if the processing mask is located

across the edge of an object. Based on the same

idea, DeFigueiredo has proposed exponential Vol-

terra filters capable of sharpening edges in noisy

images [1].

All these techniques present a drawback in having

a fixed filter mask for the entire image, which can be

Fig. 1. Unsharp masking (UM) method.

considered as a nonstationary process. In

[2,14,16,19,20], the authors propose to use a filter

mask that varies according to the local image

gradient. The sharpening effect of the edges is

achieved, but the method requires a high computa-

tion cost due to the large number of iterations

needed.In using simultaneously a quadratic filter and

a varying filter mask, we propose in [3,5] a family

of adaptive quadratic filters extending the unsharp

masking operator. The operator formed by a paral-

lel architecture composed of an adaptive linear

smoothing and an adaptive quadratic sharpeningcomponents, performs noise reduction and edge

enhancement.

The use of quadratic filters leads to a stronger

enhancement of bright pixels than dark ones. Al-

though this treatment is relevant for human percep-

tion, it could be detrimental for a post-processing.

For instance, in seismic images black and white

pixels represent positive and negative reflections,

and it is important to enhance them similarly. In

this case we propose to simply use a linear adaptive

UM method.

The proposed approach is developed in two steps.

We first define the filter mask as a set of coefficients

which quantifies the contribution of each pixel of

the support in order to compute the output value.Depending on the type of images we want to process

(natural, periodic or pseudo-periodic texture, seis-

mic, etc.), we propose two different ways to compute

these coefficients: locally adapted and recursively

adapted. The second step consists in combining the

pixel values according to this set of coefficients.

The aim of this paper is to provide a robust

enhancement filter in comparison with other 2D

methods, and a 3D extension of this filter in order

to process seismic images.

The paper is organized as follows. In Section 2

the derivation of the best filter mask according to

a performance criterion is presented. In Section 3,

the contrast enhancement filter as a combination of

a lowpass and a highpass filter is described. Thetwo lowpass and highpass components are then

developed and their properties discussed. In Sec-

tion 4, results of image enhancement are presented

in comparison to other approaches. Section 5 is

devoted to the 3D extension of the filters, and its

238 S. Guillon et al./Signal Processing 67 (1998) 237254


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application to the restoration of seismic images.

Concluding remarks can be obtained in Section 6.

2. Filter mask

Let us define the support S as the set of pixelslikely to contribute in the filtering of the current

pixel (x,y). Without loss of generality, we consider

thatS is a rectangular window of size KK

:

S"(i,j)3!K

!1

2 ;

K!1

2 !

K!1

2 ;

K!1

2 . (1)When scanning the image, the pixels included in the

mask may belong to different regions. The principle

of adaptive filtering then consists in adjusting

the contribution of each pixel of S according to

their similarity with the current pixel. For this

purpose, we define the filter mask M as a set of

coefficientsm

(in the range [0,1]) associated with

the support S:

M"m3[0,1] (i,j)3S. (2)

Each coefficient m

is viewed as a level of confidence

for the pixel (x#i, y#j) to belong to the sameregion as the current pixel (x,y), i.e. m

P1 for

pixels similar to the current pixel, respectively

mP0 for the others.

In order to compute the set of coefficients M,

a criterion must be defined to estimate the similaritybetween two pixels.

A first criterion could be defined in comparing

the two corresponding grey levels. The deviation

between the two pixel values is calculated, and

a value between 0 and 1 is assigned to m

such that

m"1 when the deviation is zero, andm

"0 for

large deviation. This method is referred to a local

adaptive approach. This estimation is fairly appro-

priate for natural images, for which each pixel to be

filtered needs a specific mask.

A second approach, devoted to periodic or

pseudo-periodic textured images, is also presented.

For this kind of images, the filter mask is stable or

slow spatially-varying. Then we propose a recursive

approach, in which the mask M evolves iteratively

according to the minimization of a criterion defined

later.

2.1. Local adaptive mask

The simplest way to define the similarity betweentwo pixels is to compare their grey levels. The

approach is based on the observation that variations

of gray levels inside a nontextured region are smaller

than those between two different regions. Then, for

a given location (i,j) ofS, we estimate a continuity

value f(x#i, y#j)!f(x, y), and the coefficientsm

are computed by

m"(f(x#i,y#j)!f(x,y)), (3)

where is a positive decreasing function, such that(d)P0 as d increases.

A large number of such functions have been

proposed in the bibliography for edge-preserving

smoothing techniques [2]: inverse gradient filter[20], improved inverse gradient filter [19] and

adaptive Gaussian weighted filter [16]. Hence a pos-

sible choice for is

m"exp!

(f(x#i,y#j)!f(x,y))

2 , (4)where is a parameter controlling the width of thecurve or the effective degree of similarity between

the two positions. Its choice will be discussed in

Section 4. The function has proved, through a large

number of simulations, to give better results than

those presented in [19,20]. Intuitively, from Eq. (4)

we see that pixels belonging to the same region as

the pixel (x,y) will have larger weighting coefficientsthan those located outside the region.

Remark.In order to reduce the computation cost

due to the Gaussian term in the above equations,

note that the possible continuity values are limited

to 255 values, for an 8-bit coded image. In practice

a table can be used to estimate the m

values.

2.2. Recursive adaptive mask

The idea of adapting recursively the filter mask is

due to Salembier who derived an adaptive approach

S. Guillon et al. /Signal Processing 67 (1998) 237254 239


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for rank order-based filters [15] to process periodic

or quasi-periodic textured images. We propose here

another adaptation process of this method. The

method, proposed by Salembier to define an

adapted unweighted filter mask, consists in defining

a search area, and assigning a weighting coefficient

to each possible location. The current filter masksupport is obtained by thresholding the set of

coefficients if a coefficient is greater than the

threshold, the corresponding location is con-

sidered as belonging to the unweighted filter

mask. An adaptation process is used to optimize

(according to a criterion) the filter mask support bymodifying the set of continuous values of the search

area.

In the approach we propose, a weighted filter

mask is used instead of a binary one. All the pixels

(x#i, y#j) belonging to a predefined support S,

are taken into account according to their weight

m

no arbitrary thresholding is necessary to

obtain a binary mask. The algorithm then operates

in two steps: it first computes the coefficients m

,

and secondly implements the linear filtering. The

structure of the filter is given in Fig. 2 where the

coefficients are adaptively modified by minimizing

a prediction error.

The predicted value p(x,y) we propose is

a weighted mean over S*

"S(0,0):

p(x,y)"

*m

f(x#i,y#j)

*m

. (5)

The optimality criterion we consider in this paper

is the mean square error (MSE):

J"E[(p(x,y)!f(x,y))], (6)

whereE[)] is the mathematical expectation. Similar

results are obtained with a mean absolute error(MAE) criterion.

The minimization of the prediction error is per-

formed with a steepest descent algorithm [21],

chosen for the simplicity of the LMS algorithm,

and its ability to deal with nonstationary signals.

The minimization process is then reached by com-

puting an estimate of the gradient of the criterion

J with respect to the set of parametersM. If (m

)

is the current set of coefficients at step k, the next

estimate at step k#1 is computed with the

Fig. 2. Filter structure.

following formula:

m "m!

J

m, (7)

where is a parameter which controls the conver-gence of the algorithm, and is a function main-

taining them

values in the range [0,1] in order to

satisfy the definition of the filter mask M(Eq. (2)).

The function can be a simple clamping or an

affine correction as follows:

m "

!min

max!min, (8)

where

"m

! J

m

, (9)

min"min

, (10)

max"max

, (11)

In the derivation of the LMS algorithm, the

gradient estimate of the criterion involved in Eq. (6)is computed by simply approximating the expecta-

tion with the instantaneous value. This leads to the

following estimate:

J

m

"E[(p(x,y)!f(x,y))]

m

"2E(p(x,y)!f(x,y))p(x,y)

m

K2(p(x,y)!f(x,y))p(x,y)

m

, (12)



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with

p(x,y)

m

"f(x#i, y#j)

*m!

*m

f(x#r, y#s)

(*

m

) "!

p(x,y)!f(x#i, y#j)

*

m

. (13)

So we obtain the following updating algorithm:

m "m#2(p(x,y)!f(x,y))

p(x,y)!f(x#i, y#j)

*

m

. (14)As Eq. (14) is a nonlinear function of the coefficients

m

, we cannot easily carry out its convergence

properties. However, two well-known properties of

the LMS algorithm [21] are still valid in this case:

the coefficients m

exhibit an exponential con-

vergence, with a time constant inversely pro-portional to the parameter :

J1

; (15)

the final misadjustment of the coefficients in-

creases with , and the mean square estimationerror e

is proportional to :

E[e

]J. (16)

So the choice of results from a compromisebetween the convergence speed (necessary in the

case of non-stationary images) and the misadjust-

ment of the coefficients m

.

One can observe in the updating equation (14)

that as p(x,y) is a predicted value of f(x,y), the set

Mevolves in order to improve the estimationp(x,y)

of f(x,y). Indeed ifp(x,y) underestimatesf(x,y) (i.e.p(x,y)(f(x,y)),

then the values m

for which f(x#i,y#j)'

p(x,y) are increased;

ifp(x,y) overestimates f(x,y) (i.e. p(x,y)'f(x,y)),

then the values m

for which f(x#i,y#j)(p(x,y) are increased.

By copying this procedure, one can derive several

updating equations which increase coefficients of

pixels similar to the current pixel, and decrease

other coefficients. For example, by introducing

a similarity index r

(in the range [!1,1]) between

the two pixels (x,y) and (x#i,y#j), which is max-

imum for two similar values and minimum for twodifferent values, we propose the general following

updating equation:

m "[m

#r

]. (17)

In the last section we will provide an example of

such an updating function for the restoration of

seismic images.

3. Contrast enhancement filter

In order to improve the performances of the UMmethod, we propose to combine a lowpass and

a highpass filter (see Fig. 3). The lowpass component

g

is used to smooth data in homogeneous areas,

while the highpassg

is chosen as an edge detector

in order to sharpen details in the image. The filter

outputg(x,y) is given by

g(x,y)"g

(x,y)#g

(x,y), (18)

where

g

(x,y)"

h

f(x#i,y#j), (19)

g

(x,y)"

w

f(x#i,y#j). (20)

The coefficient is a factor driving the edge en-hancement effect. The operator we propose is then

defined by the choice of the coefficients h

and

w which are different combinations of the m

Fig. 3. Enhancement filter structure.



6/18

presented in the previous section. As reported

in [10], to preserve the expected output of a uniform

luminance input, the following conditions must

hold:

h"1 and

w"0. (21)

In the following we first give the expression of thelowpass and highpass filters, and finally propose an

adaptive estimation of the scaling factor .

3.1. Lowpassfilter

As the purpose of lowpass filtering is to reduce

noise in the image without smoothing details or

edges, we consider the following lowpass filter

g

(x,y):

g

(x,y)"

h

f(x#i, y#j ),

with h"

m

m

. (22)

The condition (21) can be easily checked:

h"

m

m

"1. (23)

It results from the definition of the mask M, thatg

(x,y) is a weighted mean over the set of pixels the

most similar to (x,y) (pixels having highm

values).

So the lowpass filter effectively performs noise

reduction without edge smoothing. This filter be-

longs to the class of edge preserving smoothing

filters (EPSF) widely studied in [2,14,16,19,20].

The performances of the lowpass filter can be

improved by defining the following submask S

:

S"(i,j)3S m

', (24)

where 3[0,1] is a threshold, and by performing

the new lowpass filter:

g

(x, y)"

h

f(x#i,y#j),

with h"

m

m

. (25)

According to the choice of the threshold , g

(x,y)

becomes a weighted mean restricted over a class

S

of pixels the most similar to the current pixel.

This threshold could be set arbitrary to a constant

value (for example "0.5), or be estimated adap-

tively by taking

"mN"1

Card(S)

m

. (26)

With this value the class S

is the set of pixels

having the highest levels of confidence.

3.2. Highpassfilter: aplacian like enhancement

(E)

In this section, we propose a new highpass filter

defined by a linear combination of the coefficients

of the filter maskM. We show that it is equivalentto a Laplacian filter, but that it exhibits an improved

noise robustness.

The highpass filter g

(x,y) we propose is defined

by

g

(x,y)"

w

f(x#i,y#j),

with w"m

!mN ,

mN"

m

.(27)

The set of coefficientsW"(w

) verifies Eq. (21):

w"

(m!mN)"0. (28)

To point out the behaviour of the highpass com-

ponent, we write g

(x,y) as follows:

g

(x,y)"

(m!mN)f(x#i,y#j)

"

(m!mN )f(x#i,y#j)

#

(m!mN)f(x#i,y#j)

"

m!mN f(x#i,y#j)

!

m!mN f(x#i,y#j), (29)



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Fig. 4. Laplacian behavior.

with

S"(i,j)3S m

*mN

set of pixels similar to current pixel,

S"(i,j)3S m

(mN

set of pixels different from current pixel.

Sog

(x,y) can be interpreted as the local difference

of pixel values belonging to two different regions

the ones belonging to S

which are similar to the

current pixel, and the others belonging to S

which

are different from the current pixel.

The filter response can be studied in the following

cases:

homogeneous area

If the current pixel (x,y) is located on a quasi

uniform area, then we can state that

f(x#i,y#j)+f(x,y)NmK1 (i,j)3S

NmNK1

Nm!mN K0 (i,j)3S

Ng

(x,y)+0. (30)

The output of the highpass filter is then negligible.

presence of an edge

If the pixel (x,y) is located across an edge (Fig. 4),

the two classes S

and S

are characterized by

(i,j)3SNm

K1,

(31)(i,j)3S

Nm

K0.

So considering that Card(S)"N, Card(S

)"n

and Card(S

)"N!n, we have mNKn/N and

g

(x,y)K

1!n

Nf(x#i,y#j)

!

0!

n

Nf(x#i,y#j)

Kn(N!n)

N

f(x#i, y#j)

n

!

f(x#i, y#j)

N!n

Kn(N!n)

N (fM

!fM

), (32)

with fM

and fM

the mean values of festimated

overS

andS

, respectively. Thus we easily show

that if (x,y) is located at the bottom of the edge,then g

(x,y)(0 asfM

(fM

. If (x,y) is located at

the top of the edge, then g

(x,y)'0 asfM'fM

.

And finally if (x,y) is located along the edge then

g(x,y)"0 as fM"f

M.

Globally the highpass filter is then equivalent to

a classical Laplacian operator, but with much less

noise sensitivity as we will see further (Section 4).

More generally, g

(x,y) is equivalent to a direc-

tional second order derivative considered ortho-

gonally to the edge. For example, a support S of

size 33 located across an edge gives the followingfilter mask:

(33)

withaa parameter which depends on the way the

filter mask is implemented.



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Fig. 5. Filter structure with an adaptive.

3.3. Scaling factor

As introduced in Eq. (18), is a factor drivingthe enhancement effect. Choosing a constant value

for has the drawback of enhancing sharpedges more than smoothed ones because the en-

hancement is proportional to the sharpness of thedetected edge. This implies an overshooting effect

on the image (false black or white lines along edges),

which could result in an unpleasant enhancement

effect. This can be solved by applying a local

adaptation of depending on the sharpness of theedge. As the output of the highpass filter measures

this sharpness, we use the value g

(x,y) to control

(Fig. 5):

"

(g(x,y)). (34)The enhancement function ()) must be continuous( : P) and satisfy

lim

(x)"0: a small g

value is considered

as a false detection and no enhancement is

required. lim

(x)"0: a high g

value corresponds

to a sharp edge, and no enhancement is

required.

Lagrangian functions (Eq. (35)) satisfy all these

constraints:

"(g

(x,y))"Cg

(x,y)exp(!g

(x,y)). (35)

We can also use a simple piecewise linear approxi-

mation of this function (Fig. 6).

Threshold

controls noise removal and thre-

shold

controls the overshooting effect. The

simulations presented in the last section are carried

out with "5 and

"50 (for 8-bit coded

images).

Fig. 6. Adaptive.

3.4. Iterative application of the enhancementfilter

The new proposed filter performs simultaneously

edge enhancement and region smoothing. However,

according to the level of noise and blur in theimage, a single processing of the image could be not

sufficient to achieve enough enhancement. We sug-

gest in this case to apply iteratively the enhancement

filter.

Let g(x,y,n) be the filtered image at iteration n,

then the filtering process becomes

g(x,y,0)"f(x,y),

g(x,y,n#1)"g

(x,y,n)#g

(x,y,n) n*0.(36)

This process is equivalent to that proposed in [14],but needs a smaller number of iterationsless than

five iterations are sufficient in practice. We provide

an example in the next section.

4. Application to image enhancement and filtering

In this section we will describe some practical

examples which demonstrate the performances of

these new filters to restore blurred or noisy images.

Simulations are performed with different classes ofimagessynthetic, natural and textured.

4.1. Application of the localfilter

In this section, the coefficients are calculated

from the deviation between the current pixel value

and the values of the pixels included in support

S (Eq. (4)). The parameter has to be set at first.



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4.1.1. Choice of In order to state how to choose the parameter

involved in the filter mask, we first study thebehavior of the new filter on a noisy region with or

without the presence of an edge.

First, if the pixel (x,y) is located on a uniform

area (mean valuef) corrupted by a white Gaussiannoisev(x,y) of variance:

f(x,y)"f#v(x,y). (37)

Then the set of coefficients of the filter mask is

defined by

m"exp!

(f(x,y)!f(x#i,y#j))

2

"exp!(v(x,y)!v(x#i,y#j))

2 . (38)

Assuming thatv(x,y) is a Gaussian random variable,

m

is a realisation of a random variable m whose

density function p

(m) can be expressed by

p

(m)"

2m

!ln mm3[0,1] with"

.

(39)

Then, the mean mN and the variance

of these

coefficients can be obtained:

mN"E[m]"

mp

(m)dm"

2#(40)

and

"E[(m!mN)]"

mp

(m)dm!mN

"

4#!

2#. (41)

We can state from relations (40) and (41) that if we

take '2, then mN'0.8 and )0.04, which

implies

NmKmN (i,j)3S

Ng

(x,y)"

m

) g(x#i,y#j)

m

K1

Card(S)

g(x#i,y#j) (42)

Ng

(x,y)"

(m!mN ) ) g(x#i,y#j)K0.

This means that if verifies '2, then the en-hancement filter is equivalent to a smoothing filter,

and the output variance is given by

"

Card(S). (43)

In the second case in which the pixel (x,y) islocated across an edge, the choice of depends onthe height h of this edge. If a and b are the grey

levels on the two sides of the edge, and iff(x,y)"awe can state that

(i,j)3S f(x#i,y#j)Ka N m

"m

K1,

(i,j)3S f(x#i,y#j)Kb

N m"m

Kexp!

(b!a)

2

"exp! h2

. (44)

The enhancement effect of the filter is effective only

if the two classesS

andS

are well separated, i.e. if

mm

, which implies

exp!(b!a)

2 1 h. (45)

Finally, we can state that the choice of the

parameter must satisfy the following condition:

2((h

, (46)

whereis the noise variance, andh

the smallest

height of edge we want to enhance.

4.1.2. Application

In order to illustrate the behavior and the noise

robustness of this enhancement filter, we comparethe response of the UM and LLE methods. We first

consider a blurred edge (h"110), and we compare

results obtained by the highpass component: LLE

(33,"20) and Laplacian (33). In Fig. 7, onecan observe that these two components are similar,

and finally the edge enhancement is equivalent

(Fig. 8).

We then degrade the synthetic edge by adding

a white Gaussian noise of variance"16. Resultsof the highpass components are provided in Fig. 9

the support S is of size 55, and "40 for theLLE. These results show the noise robustness of the

LLE operator in comparison with the Laplacian.



10/18

Fig. 7. Comparison of the Laplacian behavior.

Fig. 8. Comparison of edge enhancement.



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Fig. 9. Comparison of the Laplacian behavior on a noisy edge.

As expected, the new enhancement filter allows

noise smoothing in uniform areas in the same time

than edge enhancement (Fig. 10).In the second application, we propose to enhance

the image of boat presented in Fig. 11. If we exercise

the quadratic operators proposed by Mitra and

Ramponi [8,10] on this image, we observe noise

amplification in background area (such as the sky).

The cubic operator proposed by Ramponi [9] is

more robust against noise, but one can observe thatsome continuous edges are discarded. In Figs. 12

and 13 we provide the results obtained with the

Mitra quadratic operator, the Ramponi cubic

filter and the LLE operator, respectively. In

these images, the contrast enhancement effect of

the LLE operator is easily perceived without exhi-

biting noise amplification in background areas.

An adaptive scaling factor allows enhancementwithout overshoot although the image obtained

with an adaptive is perceptually not as good asthat obtained with a constant , the absence ofovershoot allows to use this image for a post-

processing.

4.2. Application of the recursive filter

Some examples of enhancement of periodic orquasi-periodic textured images are presented topoint out the interest of the recursive filter mask.

Indeed, on such images, the optimal filter mask will

be constant or slow spatially-varying. An adaptive

evaluation of the filter mask M will then be more

robust than a local oneif the pixel to be filtered is

corrupted by noise, the coefficients of the filter mask

will not be affected due to the high memory effect ofM.

The image to be filtered is the canvas image

presented in Fig. 14. If we process such an image by

a quadratic or a cubic operator (Fig. 15: Ramponis

cubic filter [9] with d"0.0003, Mitras quadratic

filter [8] with"512), we observe a noise amplifi-cation which gives unpleasant filtered images. On

the contrary, results obtained with a 33 LLEfilter appear more convincing in terms of contrastenhancement (Fig. 16: LLE method) the texture

is well enhanced without noise amplification. The

image is scanned along columns so that the

coefficients quickly converge.



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Fig. 10. Comparison of edge enhancement on a noisy edge.

Fig. 11. Boat image.

In the second application, we propose to restore

the canvas image corrupted by a zero-mean Gaus-

sian white noise with standard deviation "60.Classically additive Gaussian noise is removed by

applying low pass filters, but at the expense of edge

blurring. As the LLE operator combines a lowpass

filter to remove noise and a highpass filter to restore

edges, restoration with this operator will be more

robust. We have iteratively filtered the noisy imagewith "0.4, "0.01 and S"[55] (Fig. 17).After four iterations the image becomes nearly

binary with a good restoration of its main features,

whose perception are difficult in the original

image.

5. 3D extension and application to seismic images

The main purpose in seismic data analysis is the

detection of geological horizons separating homo-

geneous layers of rocks, sediments, etc. A 2D seismic

image (Fig. 19) is composed of seismic traces,

sinusoid-like waveforms, which are a record of

reflected waves arising from impedance contrasts

between strata. If a cycle is correlated laterally

across many seismic traces on an image, it is called

a seismic horizon.

Filtering seismic images then consists in restoring

these seismic horizons. As a seismic image is often



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Fig. 12. Image enhancement.

Fig. 13. Local zoom.

Fig. 14. Canvas image.

extracted from a 3D bloc of data, the filtering

process can be implemented over a 2D or 3D

support. In the case of 3D processing, the filter

maskM is naturally extended as a window of size

KKK (corresponding support S).

S"(i,j,k)3!K

!1

2 ;

K!1

2 !

K!1

2 ;

K!1

2 !

K!1

2 ;

K!1

2 , (47)M"m3[0,1] (i,j, k)3S. (48)

The outputg(x,y,z) of the filter then becomes

g(x,y,z)"g

(x,y,z)#g

(x,y,z), (49)

with the componentg

and g

defined by a simple

3D extension:

g

(x,y,z)"

m

f(x#i,y#j,z#k)

m

,



14/18

Fig. 15. Quadratic enhancement.

g

(x,y,z)"

(m!mN )f(x#i,y#j,z#k),

with mN"1

Card(S)

m

. (50)

In the following, we propose local adaptive and

recursively adaptive methods, specifically adapted to

seismic images, due to the way of computing the set

of coefficients M. To simplify the presentation, theequations are given in the 2D case, the 3D case being

a simple extension by adding the third coordinate z.

5.1. Seismic filter mask

In Section 2, we defined various criteria of sim-

ilarity between two pixels in order to compute thecoefficients m

. Here we propose to improve the

similarity criterion for seismic images. Indeed twopixels could be said to be similar if they belong to

the same seismic horizon, which means that their

corresponding seismic traces are highly correlated.

A seismic trace is a column of the image, and its

representation is a sinusoid-like waveform (Fig. 18:

intensity evolution along a column of the image).

So for each position (x,y) we define the vector

seismic trace by

(x,y)"[f(x!k,y)]

. (51)

Fig. 16. LLE enhancement.

The similarity of two pixels is then locally esti-mated by measuring the correlation factor oftheir two corresponding traces, which gives in the

2D case:

m"

1

2#

1

2(

(x,y),

(x#i,y#j))

"1

2#

1

2

(x,y) )

(x#i,y#j)

(x,y) )

(x#i,y#j). (52)



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Fig. 17. Iterative enhancement.

Fig. 18. Seismic trace.

Thus we could assume that m3[0,1], and that

m"1 only if

(x,y)"

(x#i,y#j) ('0). As

it takes into account the vicinity of the pixels, the

estimation of the coefficients m

is more robust to

noise than the one proposed in Section 2.1.

A second approach for computing these coeffi-

cients can be derived from the recursive schemeexposed in Section 2.2. Indeed we assume that the

generic updating equation expressed by

m "[m

#r

], (53)

wherer is a measurement of similarity betwen two

pixels respecting

r"#1, if (x,y) and (x#i,y#j) are similar,

r"!1, if (x,y) and (x#i,y#j) (54)

are widely different.

As the correlation factor (Eq. (52)) of two seismictraces satisfies these conditions, we propose to define

the updating equation by

m "m#

(x,y) )

(x#i,y#j)

(x,y) )

(x#i,y#j).(55)

Thus we obtain an adaptive estimation of the set of

coefficients M which tends to increase the weights



16/18

of the positions the most correlated to the current

position. The parameterdrives the memory of thefilter mask.

5.2. Application

The two proposed schemes we have presented

are used to process the seismic image shown in

Fig. 19. We first process this image with a 3D

locally adapted mask, whose weights are evaluated

with the correlation factor of the seismic traces

(Eq. (52)),S"[555], a seismic trace of size 11(N"5) and an enhancement factor "0.1. Theresult is presented in Fig. 20.

To evaluate the filtering effect of our approach

on this class of images, we should focus our attention

on the main features of these images: seismic hor-

Fig. 19. Seismic image.

Fig. 20. Seismic filtered image with a local mask.

izons, channels (located in the upper left of the

image) and faults (right side of the image). Important

features for expert interpretation are preserved while

an efficient cancellation of noise is performed.

We compare this result with the filtering using

a recursive mask, where the coefficients evolve

according to the correlation factor of the seismictraces (Eq. (55)). We scan the image in the column

direction so that the filter mask is slow spatially-

varying. The image is processed with an

S"[555], a vector seismic trace of size 11, anupdating coefficient"0.01, and an enhancementfactor"0.3. The result is given in Fig. 21.

The evaluation of the result is still done by

focusing our attention to the features of this image:

seismic horizons appear more regular than those

resulting from the local filter (Fig. 20), but the

channel and the fault are not as well preserved as

with the local filter.

Finally, the two approaches (local and recursive)

can be combined if we are able to detect the position

of the two main features (fault and channel). If we

noted(x,y) the probability function of such a detec-

tion (d(x,y)P1 if a feature is detected), then the

output image is obtained by

g(x,y)"g

(x,y) ) d(x,y)#g

(x,y) ) (1!d(x,y)), (56)

with g

and g

respectively being the local andrecursive filtering.

From Fig. 22, one can observe that the main

features are well preserved, and horizons regularized.

This method combines the advantages of the two

previous ones.

Fig. 21. Seismic filtered image with a recursive mask.



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Fig. 22. Seismic filtered image combining the local and recursive

mask.

The detection d(x,y) we used is based on

a measure of dispersion of local gradients [4] not

described in this paper.

6. Conclusion

In this paper new nonlinear image enhancement

filters have been presented. They are based on the

choice of a locally-adapted filter mask or a re-cursive adaptive process for computing the mask

coefficients. The filter mask defines the set of sur-

rounding pixels most closely correlated to the cur-

rent pixel. Then two new enhancement techniques,

called Laplacian like enhancement (LLE1 andLLE2), have been introduced as an improvement of

the classical unsharp masking (UM) method.

We have shown, through tests on synthetic and

real images, the improvements brought by these

algorithms: noise reduction and edge enhancement

are performed simultaneously. Comparisons with

other techniques have also been provided. Finally,

these new approaches, including a 3D extension,

have been applied to practical applications dealing

with seismic images restoration.

From a computational point of view, the use of

a locally-adapted filter mask is not more costly

than other UM algorithms (parallel implementation

is possible), while the adaptive process for comput-

ing the coefficients leads to a more costly algorithm.

Notations

f()) input image

g()) output image

(x,y) position of a pixel on an

image

(x,y,z)

S

position of a voxel on a 3Dbloc of data

filtering support

(i,j) position of a pixel on a 2D

support

(i,j,k) position of a voxel on a 3D

support

M"(m

)(or (m

)) set of levels of confidence

(respectively 2D or 3D)

( )M) mean operator

E[)] mathematical expectation

Acknowledgements

The authors wish to thank ELF Aquitaine for

having provided a large number of seismic images,

and for partially supporting this work. They are

also indebted to the academic partners of the joint

project CNRS ELF Aquitaine. They also thank

the anonymous reviewers for their helpful comments

that aided a lot in making this paper more clear and

complete.

References

[1] R.J.P. de Figueiredo, S.C. Matz, Exponential nonlinear

Volterra filters for contrast sharpening in noisy images, in:

Proc. ICASSP96, Vol. 4, pp. 22632266.

[2] J.M.H. Du Buf, T.G. Campbell, A quantitative comparison

of edge-preserving smoothing techniques, Signal Processing

21 (4) (1990) 289301.

[3] S. Guillon, P. Baylou, New noncausal adaptive quadratic

filters for image filtering and contrast enhancement,

in: Proc. EUSIPCO96, Trieste, Italy, Vol. 1, 1996, pp.

583586.

[4] S. Guillon, P. Baylou, N. Keskes, Detection and/or deter-

mination method of features related to singular points,

Submitted for a French patent No. 9708602, INPI (Institut

National de la Protection Industrielle), July 1997.

[5] S. Guillon, P. Baylou, M. Najim, Robust nonlinear contrast

enhancement filters, in: Proc. ICIP96, Lausanne, Switzer-

land, Vol. 1, 1996, pp. 757760.



18/18

[6] A.K. Jain, Fundamentals of Digital Image Processing,

Prentice-Hall, Englewood Cliffs, NJ, 1989.

[7] J.S. Lee, Digital image enhancement and noise filtering by

use of local statistics, IEEE Trans. Pattern Anal. Machine

Intell. 2 (1980) 165168.

[8] S.K. Mitra, H. Li, I. Lin, T. Yu, A new class of nonlinear

filters for image enhancement, Proc. ICASSP91, pp.

25252528.[9] G. Ramponi, A simple cubic operator for sharpening an

image, Proc. IEEE Workshop on NSIP, Thessaloniki,

Greece, 2022 June 1995, pp. 963966.

[10] G. Ramponi, P. Fontanot, Enhancing document images

with a quadratic filter, Signal Processing 33 (1) (1993) 2324.

[11] G. Ramponi, G.L. Sicuranza, Quadratic digital filters for

image processing, IEEE Trans. ASSP 36 (6) (June 1988)

12631285.

[12] G. Ramponi, G.L. Sicuranza, Image sharpening using

a polynomial filter, Proc. ECCTD93, Davos, Switzerland,

pp. 14311436.

[13] G. Ramponi, N. Strobel, S. Mitra, T.H. Yu, Nonlinear

unsharp masking methods for image contrast enhance-

ment, Journal of Electronic Imaging 5 (3) (July 1996)

353366.

[14] P. Saint-Marc, J.S. Chen, G. Medioni, Adaptive smoothing:

a general tool for early vision, IEEE Trans. Pattern Anal.

Machine Intell. 13 (6) (June 1991) 514529.

[15] P. Salembier, Adaptive rank order based filters, Signal

Processing 27 (1) (1992) 125.

[16] M. Spann, A. Nieminen, Adaptive gaussian weighted

filtering for image segmentation, Pattern Recognition Let-

ters 8 (1988) 251255.[17] S. Thurnhofer, Quadratic Volterra filters for edge enhance-

ment and their application in image processing, Ph.D.

Thesis, University of California, Santa Barbara (1994).

[18] A. Vanzo, G. Ramponi, G.L. Sicuranza, An image en-

hancement technique using polynomial filters, in: Proc.

ICIP94, Austin, TX, 1316 November 1994.

[19] D. Wang, O. Wang, A weighted averaging method for

image smoothing, Proc. IEEE Conf. Com. Vision and

Pattern Recognition, Miami Beach, Florida, USA, 2226

June 1986, pp. 981983.

[20] D. Wang, A. Vagnucci, C.C. Li, Gradient inverse weighted

smoothing scheme and the evaluation of its performances,

CGIP 15 (1981) 167181.

[21] B. Widrow, S.D. Stearns, Adaptive Signal Processing,

Prentice-Hall, Englewood Cliffs, NJ, 1985.


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