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Digital Image Enhancement

Nikolas P. GalatsanosIllinois Institute of Technology, Chicago, Illinois, U.S.A.

C. Andrew SegallAggelos K. KatsaggelosNorthwestern University, Evanston, Illinois, U.S.A.

INTRODUCTION

In this entry, we provide a tutorial survey of digital image

enhancement algorithms and applications. These techni-

ques are considered throughout the image-processing li-

terature and depend significantly on the underlying appli-cation. Thus, the survey cannot address every possible

realization or application of digital image enhancement.

To address this problem, we classify the methods based

on two properties: whether the processing performed is

pointor spatialand whether it islinearor nonlinear. This

leads to a concise introduction to field of enhancement,

which does not require expertise in the area of image

processing. When specific applications are considered,

simulations are provided for assessing the performance.

OVERVIEW

The goal of digital image enhancement is to produce a

processed image that is suitable for a given application.

For example, we might require an image that is easily

inspected by a human observer or an image that can be

analyzed and interpreted by a computer. There are two

distinct strategies to achieve this goal. First, the image can

be displayedappropriately so that the conveyed informa-

tion is maximized. Hopefully, this will help a human (or

computer) extract the desired information. Second, the

image can beprocessedso that the informative part of the

data is retained and the rest discarded. This requires adefinition of the informative part, and it makes an en-

hancement technique application specific. Nevertheless,

these techniques often utilize a similar framework.

The objective of this entry is to present a tutorial

overview of digital enhancement problems and solution

methods in a concise manner. The desire to improve

images in order to facilitate different applications has

existed as long as image processing. Therefore, image

enhancement is one of the oldest and most mature fields

in image processing and is discussed in detail in many

excellent references; see, e.g., Refs. [13].

Image enhancement algorithms can be classified in

terms of two properties. An algorithm utilizes eitherpoint

or spatial processing, and it incorporates either linearor

nonlinearoperations. In this vein, the rest of this entry is

organized as follows: In Point-Processing Image En-

hancement Algorithms, both linear and nonlinear point-processing techniques are presented. In Image Enhance-

ment Based On Linear Space Processing, linear spatial

processing algorithms are presented. In Image Enhance-

ment Based On Nonlinear Space Processing, nonlinear

spatial processing algorithms are presented. Finally, we

present our conclusions.

POINT-PROCESSING IMAGEENHANCEMENT ALGORITHMS

Point-processing algorithms enhance each pixel sepa-rately. Thus, interactions and dependencies between pixels

are ignored, and operations that utilize multiple pixels to

determine the value of a given pixel are not allowed. Be-

cause the pixel values of neighboring locations are not

taken into account, point operations are defined as func-

ions of the pixel intensity.

Point operations can be identified for images of any

dimensionality. However, in the rest of this section, we

consider the two-dimensional monochromatic image de-

fined by a discrete space coordinate system n = (n1,n2)

with n1 = 0,1. . .N 1 and n2 = 0,1. . .M 1. The image

data is contained in aNMmatrix, and the discrete spaceimage f(n) is obtained by sampling a continuous imagef(x,y). (For more details on image sampling, see Chapter

7.1 in Ref. [2] or Chapter 1.4 in Ref. [4].) We also assume

that f(n) is quantized to K integer values [0,1. . .K 1].When 8 bits are used to represent the pixel values,

K= 256, we refer to them as gray levels.

The fundamental tool of point processing is the

histogram. It is defined as the function h(k) = nk, where

nkis the number of pixels with gray levelk= 0,1. . .K 1,and it describes an image by its distribution of intensity

values. While this representation does not uniquely

388 Encyclopedia of Optical Engineering

DOI: 10.1081/E-EOE 120009510

Copyright D 2003 by Marcel Dekker, Inc. All rights reserved.


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identify an image, it is quite meaningful for a variety of

applications. This is readily apparent in Fig. 1(a) and (b),

where we show an aerial image of an airport and its

corresponding histogram. From the histogram, we deduce

that the image is relatively dark and with poor contrast.

Additionally, we conclude that there are three major typesof image regions. This conclusion is derived from the two

peaks as well as the shape of the histogram in the brighter

intensities. In the image, these regions correspond to the

dark background, lighter road and building objects, and

the bright airplane features.

Basic Point Transformations

Point transformations are represented by the expression

gn Tfn 1

wheref(n) is the input image,g(n) is the processed image,

andTis an operator that operates onlyat the pixel location

n. For linear point operators, Eq. 1 becomes

gn afn b 2

Linear point transformsstretchor shrinkthe histogramof an image. This is desirable when the available range of

intensity values is not utilized. For example, assume that

A minnfnand B maxnfn. The linear transfor-mation in Eq. 2 can map the gray levels A and B to gray

levels 0 and K 1. Using simple algebra, the transforma-tion is given by

gn K 1

BA

fn A 3

The effect of stretching the histogram of an image is

shown in Fig. 2(a), where the histogram of Fig. 1(a) is

modified. The resulting histogram appears in Fig. 2(b).As can be seen from the figure, the image in Fig. 2(a) is

both more pleasing and more informative that the image

in Fig. 1(a). This is especially noticeable on the right

half of the image frame.

Other transformations are also utilized for image en-

hancement. For example, the power-law transform des-

cribes the response of many display and printing devices.

It is given byg(n) = c( f(n))g, where the exponent is called

thegamma factor, and it leads to a process called gamma

correction.[1] (A television or computer monitor typically

has a voltage-to-intensity response that corresponds to

1.5 g 3). As a second example, the log transformcompresses the dynamic range of an image and is given asg(n) = clog(f(n) + 1). This transform is often employed to

display Fourier spectra. In Fig. 3, e.g., we show (a) an

image of the vocal folds obtained by a flexible endoscope,

(b) the magnitude of the two-dimensional discrete Fourier

transform (DFT) of the image, and (c) the log-transformed

magnitude of the Fourier transform. (For this image,

c = 1.) Note that the actual magnitude in (b) provides little

information about the spectrum when compared to the

log-transformed image in (c). This image is revisited in a

later section, as it facilitates noise removal.

Fig. 1 Representing an image with its histogram: (a) original

aerial image and (b) the corresponding histogram. While the

histogram does not completely describe the image, it does

suggest that the image contains three region types.

Digital Image Enhancement 389

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Histogram Processing

One of the standard methods for image enhancement is

histogram equalization. Histogram equalization is simi-

lar to the stretching operation in Eq. 3. However, instead

of utilizing the entire dynamic range, the goal of his-

togram equalization is to obtain a flathistogram. This is

motivated by information theory, where it is known that a

uniform probability density function (pdf ) contains the

largest amount of information.

[5]

Fig. 3 Visualizing the Fourier spectrum: (a) image of vocal

chords obtained by a flexible endoscope, (b) magnitude of Fourier

spectrum, and (c) log-transform of Fourier spectrum magnitude.

Note that inspecting the magnitude values in (b) provides little

insight into the shape of the spectrum, as compared to the visual

representation in (c).

Fig. 2 Stretching the histogram of an image: (a) aerial image

after histogram stretching and (b) the corresponding histogram.

The stretching procedure increases the contrast of the image and

makes objects easier to discern.

390 Digital Image Enhancement


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The method of histogram equalization is well des-

cribed in Chapter 2 of Ref. [2], where the normalized

histogram provides the foundation of the method. This

histogram provides the probability of occurrence of gray

level k in the image f(n) and is expressed as

pfk nk=n 4

wheren =NM, the total number of pixels in the image. Due

to the definition of a histogram in Eq. 4, we know that

XK1k 0

pfk 1 5

where the function pf(k) can be viewed as the pdf off(n).

The cumulative density function (cdf) is therefore equal to

Pfr Xr

k 0

pfk 6

where it should be clear that

pfk Pfk Pfk1, k 0,1 . . . K 1 7

To explain the process of histogram equalization, we

first consider the case of continuous intensity values.

In other words, we denote by pf(x) and Pf(x) the con-

tinuous pdf and cdf of a continuous random variable, x,

respectively. These two functions are related by pf(x) =

dPf(x)/dx. Furthermore,Pf 1(x) exists and Pf(x) is non-

decreasing. Assume that the sought after transformation

is given by

g Pff 8

The enhanced imageg has a flat histogram because the

cdf ofg is given by

Pgx Prg x PrPff x

Pr f P1f x

Pf P1

f x

x 9

and, therefore, its pdf by

pgx dPgx=dx 1 10

To flatten the histogram of a digital image, the

following procedure is employed. First,Pf(k) is computed

using Eq. 6. Then, Eq. 8 is applied at each pixel. This

implies that the function Pfcan be applied on a pixel-by-

pixel basis to the image f(n) according to

gn Pffn 11

Finally, Eq. 3 is used to stretch the histogram.

In Fig. 4(a), we show the result of histogram equa-

lization for the airport image in Fig. 1(a). In Fig. 4(b), we

show the histogram of the image in Fig. 4(a). By com-

paring the images in Figs. 2(a) and 4(a), we observe that

histogram equalization is more effective in bringing out

the salient information of the image than linear histogram

stretching. Furthermore, it is important to remember that

although the theory of histogram equalization strives for a

uniform histogram, this is not always achieved in practice.

The reason for this discrepancy is that the algorithmassumes that the histogram is a continuous function, which

is not true for digital images.

Fig. 4 Equalizing the histogram of an image: (a) aerial image

after histogram equalization and (b) the corresponding his-

togram. Histogram equalization often assists in the analysis of

images, as it makes objects distinct. This is evident in the line

features in the top left portion of the frame.


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While histogram equalization attempts to maximize the

information content of an image, some applications may

require a histogram with arbitrary shape. For example, it

might be desirable to match the histograms of two images

prior to comparison. Such an objective has been studied

within the context of mapping a random variable with agiven distribution to another variable with desired dis-

tribution, and it is discussed in Refs. [1,3].

IMAGE ENHANCEMENT BASED ONLINEAR SPACE PROCESSING

Linear filtering is the basis for linear spatial enhancement

techniques. Linear spatial filtering can also be represented

by Eq. 1. However, in this case, the value ofg(n) depends

not only on the value f(n) of but also on the values of f

in the neighborhood of n. The inputoutput relation in

Eq. 1 is therefore written as

gn1; n2 X1

m1 1

X1m2 1

fm1,m2 hn1,n2;m1,m2

12

where h is the function that describes the effect of the

linear system T. An interesting subcategory of linear

filtering isspace-invariantlinear filtering. In such a case,

the inputoutput relation in Eq. 1 is written as

gn1,n2

X1

m11 X1

m21

fm1,m2 hn1 m1,n2 m2

fn1,n2* hn1,n2 13

where the functionh is called the impulse responseof the

system and the operation in Eq. 13 is called convolution

and represented by *. Although the summation in Eq. 13

is over an infinite range, these limits are finite in practice

as images have finite support.

A useful property of space-invariant filtering is that the

input output relation remains constant over the entire

image. Thus, the value of the impulse response depends

only on the distance between the input and output pixels

and not on their spatial location.

Another useful property of linear space-invariant

filtering is that it can be performed both in the spatialand in the Fourier frequency domains. The DFT of the

discrete impulse response is given by

Hk1,k2 XM1

n1 0

XN1n2 0

hn1,n2

exp

"j

2p

Mn1k1

2p

Nn2k2

!#

Ffhn1,n2g 14

where the function H(k1,k2) is called the frequency res-

ponse , and k1 = 0,1. . .

M 1, k2 = 0,1. . .

N 1 are the

discrete frequencies. Utilizing the DFT, inputoutput re-

lationships of linear and space-invariant systems are

expressed according to the convolution theorem in either

the spatial or frequency domains

Hk1,k2 Fk1,k2 Ffhn1,n2*

fn1,n2g

hn1,n2 fn1,n2 F1fHk1,k2 * Fk1,k2g 15

where the operator F1 represents the inverse DFT.[4]

Thus, linear space-invariant filtering is performed either

by convolving the input image with the impulse response

of the filter or by multiplying on a point-by-point basis the

Fourier transform of the image with the frequency res-

ponse of the filter. The ability to perform linear filtering in

both the spatial and the Fourier domains has a number of

advantages. First, it facilitates the design of filters because

it helps separate the part of the signal that should be re-

tained from the part that should be attenuated or discarded.

Second, it helps improve the speed of computation byutilizing a fast Fourier transform (FFT) algorithm for

computing the DFT.

We present two examples of linear enhancement. In

the first example, we show enhancement by linear fil-

tering in the spatial domain, i.e., by using convolution. In

Fig. 5(b), we show the result of convolving the image in

Fig. 5(a) by the uniform 3 3 mask

h

1=9 1=9 1=9

1=9 1=9 1=9

1=9 1=9 1=9

264

375

16

The image in Fig. 5(a) has been corrupted by additive

Gaussian noise. In Fig. 5(c), we show the result of the

same approach when a 5 5 uniform mask is used.Clearly, this type of filtering removes noise although at

the expense of blurring image features. In the second

example, image enhancement is performed in the Fourier

domain by point-by-point multiplication. The regular

noise pattern in the image in Fig. 3(a) manifests itself as a

periodic train of impulses in both directions of the Fourier

domain.[1] This is observed in the log-transformed spec-

trum of this image in Fig. 3(c). We selected a filter with

frequency response

Hk1,k2 1 for N1 k1 M1;N2 k2M2

0 elsewhere

17

where the parameters Mi, Ni for i = 1,2 are selected such

that the largest area around the central impulse in Fig. 3(c)

that does not include any of the other satellite impulses

is maintained. The magnitude of G(k1,k2) = F(k1,k2)H(k1,k2) is shown in Fig. 6(a). In Fig. 6(b), we show

F1fGk1; k2g, the enhanced image.Another very popular method for image enhancement

is the Wiener filter. This method is optimal in a mean-



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squared error sense.[5]

When the noise is zero mean anduncorrelated with the image, the frequency response of

the Wiener filter is defined as

Hk1,k2 Sfk1,k2

Sfk1,k2 Swk1,k2 with

k1 0,1 . . .M 1,k2 0,1 . . .N1

18

whereSf(k1,k2) andSw(k1,k2) are thepower spectraof the

image and noise, respectively. The filtered image is then

given by

fn1,n2 F1fHk1,k2 Gk1,k2g 19

In general, the power spectra are not known and haveto be estimated from the observed data. Finding these

quantities is not a trivial problem and is investigated in

the literature (e.g., iterative Wiener filter[6]). In Fig. 7(a),

we show the result of Wiener filtering the noise-degraded

image in Fig. 5(a) when the power spectra are estimated

from the original images. In Fig. 7(b), we show the result

of Wiener filtering the image in Fig. 5 (a) when the power

spectra are estimated from the observed data using the

periodogram method. Noise-filtering techniques have

been studied extensively in the literature, often as a spe-

cial case of image restoration techniques, when the degra-

dation system is represented by the identity.

[7]

Fig. 5 Linear filtering for noise removal: (a) original image corrupted by additive Gaussian noise, (b) image processed with a 3 3averaging operations, and (c) image processed with a 5 5 averaging operation. The averaging procedure reduces the high-frequencycontent of the image and attenuates noise.


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IMAGE ENHANCEMENT BASED ONNONLINEAR SPACE PROCESSING

Nonlinear filtering allows for the preservation of image

features and the removal of impulsive noise. Unlike linear

enhancement methods, the output of these operators is not

defined as a linear sum of the input samples. Instead,

input images are filtered with highly configurable and

adaptive procedures. Most often, the adaptation is based

on information from the unprocessed image frame. In this

case, the inputoutput relationship is expressed as

gn1,n2 X1

m1 1

X1m2 1

fm1,m2

hn1,n2;m1,m2,f, . . . 20

where additional parameters are allowable.

The number of algorithms described by Eq. 20 is

immense.[810] In the remainder of this section, however,

we consider three specific types of nonlinear filtering

methods. These types describe a majority of common en-

hancement algorithms, and we define them as order-sta-

tistic, transform-mapping, and edge-adaptive enhance-ment techniques.

Order-Statistic Filtering

Order-statistic filters attenuate intensity values based on

their rank within a local processing window. To construct

this type of procedure, the following approach is fol-

lowed. First, the processing window is defined. This

requires a definition for the values ofm1and m2in Eq. 20

for which h(n1,n2; m1,m2,f,. . .) is always zero. Thus, it is

analogous to the kernel of a linear filter in that it

determines the neighboring pixels that contribute to the

filtered result. Having selected the filter parameters, the

next step for an order-statistic filter is to sort all of the

pixels within the processing window according to their

intensity value. The filter then extracts the intensity value

that occupies the desired rank in the list and returns it as

the output. The spatial location of the returned value

varies across the image frame and is controlled by the

value of f(n1,n2).

While the order-statistic filter can be described using

Eq. 20, it is traditionally expressed as

gn1,n2 Ranki fn1 m1,n2 m2 ,m1,m22 M21

where g(n1,n1) is the filtered result, Ranki is the de-

signated rank,f(n1,n1) is the original image frame, and M

is the processing window. For simplicity, we assume that

the processing window is square although this needs not

be the case. Nevertheless, note that the output of the filter

is always equal to an intensity value within the local pro-

cessing window. This is in stark difference to linear me-

thods, where the output is defined as a weighted sum of

the input intensities.

Choosing the rank of the filter is an important design

parameter, and it depends largely on the application.

One common choice is to utilize the median, or middle,

intensity value within the sorted processing window. The

resulting median filter is well suited for removing im-

pulsive noise that is additive and zero mean, and it can

be realized using one of two available methods. In the

first approach, the pixels in the processing window are

all extracted from the original image frame, as is

denoted in Eq. 21. In the second approach, a recursive

procedure is utilized and pixels in the window are ex-

tracted from the previously filtered result when avail-

able. Intensity values at the unprocessed locations are

Fig. 6 Linear filtering in the Fourier domain: (a) the central

part of the Fourier transform in Fig. 3(c) and (b) the inverse

transform of (a).



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extracted from the original image, and the procedure is

expressed as

gn1,n2 Rankifn1 m1,n2 m2,

gn1 m01,n2m

02,

m1,m22 M1,m01,m

022 M2 22

where M1 are locations in the processing window that

have not been filtered, and M2 are locations that have

been previously filtered. The advantage of the second

approach is that it provides better noise attenuation

given the same processing window. This is illustrated in

Fig. 8, where the image in (a) is corrupted by salt-and-

pepper noise. The nonrecursive and recursive median

filters then filter the noisy image, and the results are

shown in (b) and (c), respectively. Notice that while the

recursive approach provides better noise attenuation,

both are adept at removing noise.

Another choice for the rank is to utilize the maximum

(or minimum) value within the ordered list. This opera-

tor removes the spatially small and dark (or bright) ob-

jects within the image frame, and it is a fundamental

operator in the theory of mathematical morphology.[1113]

This field is concerned with the study of local image

structure and is originally cast within the context of

Boolean functions, binary images, and set theory. When

extended to grayscale images, however, the basicdilation

and erosion operators correspond to the maximum and

minimum order-statistic filters, respectively. Moreover,

concatenating the dilation and erosion produces additio-

nal processing methods. For example, dilation followed

by erosion is called a close, while erosion followed by

dilation is called an open. (The open and close operators

can also be concatenated.)

Morphological filters have many interesting properties.

For example, the open and close operators are idempotent,

which means that an image successively filtered by an

open (or close) does not change after the first filter pass.

Additionally, the dilation and erosion operators are se-parable. Combining these properties with the fact that

finding the minimum (or maximum) value in a list is

computationally efficient, the morphological operators are

well suited for a variety of enhancement applications in-

cluding those subject to computational constraints.

The filtering characteristics of the morphological op-

erators are illustrated in Fig. 9. The image in Fig. 8(a)

is filtered with an erosion (minimum value) and dilation

(maximum value). Results appear in (a) and (b), respec-

tively. From the figures, we see that erosion enlarges the

dark image regions and removes small bright features,

while dilation enlarges the bright image regions and re-

moves small dark features. (The processing window

establishes the definition of small.). The open and

close operators are appropriate when feature size should

be preserved. These filters are illustrated in (c) and (d),

respectively, and the open operation removes small and

dark image regions, while the close operation attenuates

small bright image regions. Finally, the openclose and

closeopen operators are shown in (e) and (f) and remove

both bright and dark small-scale features. Note, however,

that the results in (e) and (f) are not identical.

Modifications to the general order-statistic filter are

also useful. For example, when the sample values within

Fig. 7 Wiener filtering: (a) result of filtering with the actual power spectra and (b) filtering utilizing periodogram estimates. Finding

the power spectra from the noisy image is nontrivial.


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the processing windows are known with unequal cer-

tainty, a weighted operator can be constructed.

[14]

In thismethod, a weight is assigned to each location within the

window. The weights are normalized so that the sum is

equal to the number of pixels in the processing window,

and pixel vales within the widow are sorted according to

intensity. Although unlike the traditional order-statistic

filter, the assigned rank is not equal to the number of

intensity values preceding it in the sorted list. Instead,

the assigned rank is equal to the cumulative sum of

the weights. The intensity value that is closest to (but

greater than) the desired rank is then chosen as the

filtered result.

Other modifications to the order-statistic filter combine

nonlinear and linear filters. For example, an alpha-trimfilter sorts the pixels in the processing window according

to intensity value. Instead of choosing a single value from

the sorted list, however, a number of pixels are extracted

(e.g., the middle five values) and, subsequently, processed

by a linear filter. The result is an operation that is less

sensitive to impulsive noise than a linear filter but is not

constrained to intensity values within the original image

frame. As a second example, the linear combination of

several morphological operators can be considered. With

this filtering procedure, the original image is filtered with

several morphological operators (e.g., an openclose and

Fig. 8 Median filtering: (a) image corrupted with salt and pepper noise, (b) image processed with nonrecursive median filter and 3 3processing window, and (c) image processed with recursive median filter and 3 3 processing window. Note that the nonrecursivemethod does not remove all of the noise.



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Fig. 9 Morphological filtering of salt and pepper image: (a) erode, (b) dilate, (c) open, (d) close, (e) open close, and (f) close open.

The openclose and closeopen operators are able to remove much of the noise in Fig. 8(a).


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close open) and the results are then combined in a

weighted sum. As in the previous method, this exploits the

performance of the order-statistic filter while allowing

intensity values that do not appear in the original image.

Transform Mapping

A second form of the nonlinear enhancement algorithms

is the transform-mapping framework. In the approach, an

image frame is first processed with a transform operator

that separates the original image into two components.

One is semantically meaningful, while the other contains

noise. A nonlinear mapping then removes the noise, and

the enhanced image corresponds to the inverse transform

of the modified data. This is expressed as

gn1,n2 X1

m1 1

X1m2 1

xm1,m2t1

n1,n2;m1,m2,f, . . . 23

where

xm1,m2

P1m11

P1m21

fn1,n2tm1,m2;n1,n2,f, . . ., xm1,m22 S

0, xm1,m2=2S

8>:

24

S is the set of semantically meaningful features, and t

and t 1 are the forward and inverse transforms res-

pectively. In most applications, the transform operators

are linear.

Fig. 11 Denoising with the wavelet transform: The noisy

image in Fig. 5(a) is transformed with the Haar wavelet, and

all transform coefficients smaller than a threshold are set equal

to zero.

Fig. 10 Denoising with the wavelet transform: (a) three-level transform, wheretL(n) and tH(n) are the low-pass and high-pass filters,

respectively, and (b) the resulting transform coefficients. Most of the image content appears in x4(n), which corresponds to the upper left

of (b).



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Selecting the transform operator and separating the

noise from meaningful features are important problems

within the transform-mapping method. One common

choice is to couple the discrete wavelet transform with a

thresholding operation. The wavelet operation processes

the image with a filter bank containing low-pass and high-pass filters.[1517] This is illustrated in Fig. 10, where the

filter bank and transform coefficients are shown in (a) and

(b), respectively. As can be seen from the figure, the

discrete wavelet transform compacts the image features

into a sparse set of significant components. These cor-

respond to the low-frequency transform coefficients (at the

upper left of the decomposition) as well as the significant

coefficients within the high-frequency data. Coefficients

with small amplitudes are identified as noise and removed

by hard thresholding, or setting all coefficients with mag-

nitudes less than a threshold equal to zero.

An example of the wavelet and hard threshold ap-proach appears in Fig. 11. In the figure, the image in

Fig. 5(a) is transformed with the discrete wavelet trans-

form. (The Haar wavelet is utilized.) Small transform

coefficients are then set equal to zero, and the inverse

discrete wavelet transform is calculated. Note that in-

creasing the threshold decreases the amount of noise but

Fig. 12 Edge-adaptive smoothing: (a) estimated locations of edges in Fig. 5(a), (b) processing the image with an adaptive 3 3averaging operation, and (c) processing the image with an adaptive 5 5 averaging operation. The filters are disabled in the vicinityof edges.


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also smoothes the image data. This smoothing is a result

of removing some of the salient image features.

Alternatives to hard thresholding may improve the

image enhancement procedure. For example, the soft

threshold method attempts to preserve image content with

the rule

xm1,m2

Tfn1,n2 b, Tfn1,n2> b

Tfn1,n2 b, Tfn1,n2 < b

0; otherwise

8>:

25

where T is the forward transform, and b is the thres-

hold. Other examples include the use of spatial cor-

relations within the transform domain, recursive hy-

pothetical test, Bayesian estimation, and generalized

cross-validation.[18]

Edge-Adaptive Approaches

Edge-adaptive methods focus on the preservation of

edges in an image frame. These edges correspond tosignificant differences between pixel intensities, and

retaining these features maintains the spatial integrity

objects within the scene. The general structure of the

approach is to limit smoothing in the vicinity of potential

edge features. For example, one can disable smoothing

with an edge map. This is illustrated in Fig. 12, where (a)

are the location of edges estimated from the Fig. 5(a), (b)

is the result of filtering with a 3 3 averaging operationat locations that do not contain an edge, and (c) is the

result of adapting a 5 5 averaging operation with thesame procedure.

A second approach to edge-adaptive smoothing is

realized as

hn1,n2;m1,m2,f, . . .

1

Zexp

n1 m12 n2m2

21gs2fn1,n2

1s2

( ) 26

where Z is a normalizing constant, s2

is the variance ofthe filter, sf

2(n1,n2) is an estimate of the local variance at

f(n1,n2), and g is a tuning parameter. In this procedure,

the variance estimate in Eq. 26 responds to the presence

of edges, and it reduces the amount of smoothing to

preserve the edge features. This is evident in Fig. 13,

where the combination of Eqs. 20 and 26 processes the

noisy frame in Fig. 5(a). Results shown in (a) and (b)

correspond to tuning parameters of 0 and 0.001, respec-

tively. (In the example,s2 = 4.) This illustrates the impact

of adaptivity. When the parameter is zero, the filter does

not respond to the edge features and results in linear

enhancement. When the parameter is 0.001, the filter

preserves edges.

A final example of the edge-adaptive approach is the

anisotropic diffusion operator. This operator attempts to

identify edges and smooth the image simultaneous-

ly.[19,20] It is realized numerically with the iteration

gtDtn1,n2 gtn1,n2

DtX

i fN,S,E,Wgci,tn1,n2rigtn1,n2

27

Fig. 13 Edge preservation with an adaptive Gaussian kernel: (a) image processed with linear Gaussian operator and (b) image

processed with adaptive procedure defined in Eq. 26. Local variance estimates control the filter and preserve edges.



14/15

where ci,s(n1,n2) is the diffusion coefficient of the ith

direction at location (n1,n2), ri is the derivative ofgs(n1,n2) in the ith direction, Dt< 0.25 for stability, and

g0(n1,n2) =f(n1,n2). The directional derivatives are de-

fined as the simple difference between the current pi-

xel and its neighbors in the North, South, East, and

West directions.Construction of the diffusion coefficient determines

the performance of the algorithm. When the coefficient is

constant (i.e., spatially invariant), then the diffusion

operation is isotropic and is equivalent to filtering the

original image with a Gaussian kernel. When the coef-

ficient varies relative to local edge estimates, however,

object boundaries are maintained. For example, a dif-

fusion coefficient defined as

ci,sn1,n2 exp rigsn1,n2

k 2

( ) 28

where k is the diffusion coefficient; the amount of

smoothing is limited when rigs(n1,n2) becomes muchlarger than k.[19] Alternatively, the coefficient

ci,sn1,n2 exp riGn1,n2 * gsn1,n2

k

2( ) 29

utilizes a filtered representation of the current image to

estimate edges, where G(n1,n2) is a Gaussian operator

and * denotes two-dimensional convolution. Other co-

efficients are discussed in Ref. [21].

An example of diffusion is shown in Fig. 14. In the

figure, the image in Fig. 5(a) is processed with 27. The

diffusion coefficient in Eq. 28 is utilized, k is defined as

the standard deviation of rig0(n1,n2), and Dt= 0.24.

Results produced by the diffusion operator after 5 and25 iterations appear in (a) and (b), respectively. As can be

seen from the figures, additional iterations increase the

amount of smoothing. Nevertheless, the diffusion method

smoothes the noisy image while preserving edges.

CONCLUSION

In this entry, a tutorial survey of image enhancement

methods was presented. This is a very extensive topic;

therefore, only certain approaches are presented at a

rather high level. The list of provided references, con-

sisting mainly of general purpose books, provides more

details and also directs the interested reader to additional

approaches not covered here. In this entry, we only

address grayscale image. Color images play an important

role in most applications. On one hand, most of the ap-

proaches presented here can be extended to enhance color

images by processing separately each of the planes used

for the representation of the color image (e.g., RGB,

CMY, or YIQ). On the other hand, the correlation among

color planes can be used to develop additional enhance-

ment techniques. Furthermore, mapping a grayscale image

Fig. 14 Smoothing with anisotropic diffusion: (a) image produced after 5 iterations of Eq. 27 and (b) image produced after 25

iterations of Eq. 27. Both experiments utilize the diffusion coefficient in Eq. 28 and illustrate the edge preserving properties of the

diffusion operation.


D


15/15

to a color (or pseudo-color) image can be utilized as an

enhancement technique by itself, e.g., in visualizing image

data. The interested reader can find out more about color

image processing in a number of books appearing in the

references, and more specifically in Refs. [22,23].

REFERENCES

1. Gonzalez, R.C.; Woods, R.E. Digital Image Processing;

Addison-Wesley, 2002; 3.

2. Handbook of Image and Video Processing; Bovik, A.C.,

Ed.; Academic Press: San Diego, 2000; 1.

3. Pratt, W. Digital Image Processing; John Wiley and Sons,

2001; 3.

4. Dungeon, D.; Mersereau, R. Multidimensional Digital

Signal Processing; Prentice Hall, 1984; 1.

5. Jain, A.Fundamentals of Digital Image Processing; Pren-

tice Hall, 1988; 1.6. Digital Image Restoration; Katsaggelos, A., Ed.; Springer

Series in Information Sciences, Springer Verlag, 1991; 1.

7. Banham, M.R.; Katsaggelos, A.K. Digital image restora-

tion. IEEE Signal Process. Mag. 1997, 14 (2), 2441.

8. Kuosmanen, P.; Astola, J.T. Fundamentals of Nonlinear

Digital Filtering; CRC Press, 1997; 1.

9. An Introduction to Nonlinear Image Processing; Dough-

erty, E.R., Astola, J.T., Eds.; Tutorial Texts in Optical

Engineering, SPIE Press, 1994; 1.

10. Pitas, I.; Venetsanopoulos, A. Nonlinear Digital Filters:

Principles and Applications; Kluwer International Series in

Engineering and Computer Science, Kluwer Academic

Publishers, 1990; 1.

11. Serra, J. Image Analysis and Mathematical Morphology;

Academic Press, 1982; 1.

12. Serra, J. Image Analysis and Mathematical Morphology,

Vol. 2: Theoretical Advances; Academic Press, 1988; 1.

13. Soille, P. Morphological Image Analysis: Principles and

Applications; Springer Verlag, 1999; 1.

14. Arce, G.R.; Paredes, J.L.; Mullen, J. Nonlinear Filtering

for Image Analysis and Enhancement. In Handbook of

Image and Video Processing; Bovik, A.C., Ed.; Academic:

San Diego, 2000; 81 100.

15. Kaiser, G.A Friendly Guide to Wavelets; Springer Verlag,

1997; 1.

16. Mallat, S.A Wavelet Tour of Signal Processing; Academic

Press, 1999; 2.

17. Prasad, L.; Iyengar, S.S.; Ayengar, S.S. Wavelet Analysis

with Applications to Image Processing; CRC Press, 1997; 1.

18. Wei, D.; Bovik, A.C. Wavelet Denoising for Image En-

hancement. In Handbook of Image and Video Processing;

Bovik, A.C., Ed.; Academic: San Diego, 2000; 117123.

19. Perona, P.; Malik, J. Scale-space and edge detection usinganisotropic diffusion. IEEE Trans. Pattern Anal. Mach.

Intell. 1990, 12 (7), 629639.

20. Geometry Driven Diffusion in Computer Vision. Computa-

tional Imaging and Vision; ter Haar Romeny, B.M., Ed.;

Kluwer Academic Publishers, 1994; 1.

21. Acton, S. Diffusion-Based Edge Detectors. InHandbook of

Image and Video Processing; Bovik, A.C., Ed.; Academic:

San Diego, 2000; 433447.

22. Sharma, G.; Trussell, H.J. Digital color imaging. IEEE

Trans. Image Process. 1997, 6(7), 901932.

23. Plataniotis, K.N.; Lacroix, A.; Venetsanopoulos, A.Color

Image Processing and Applications; Springer Verlag,

2000; 1.


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