Digital Watermarking (PDF, 743KB)

Digital watermarking
1. INTRODUCTION
With the widespread distribution of digital media contents, the protection of the
intellectual property rights has become increasingly important. Digital media offer
several distinct advantages over analog media, such as high quality, easy editing, high
fidelity copying. The ease by which digital information can be duplicated and
distributed has led to the need for effective copyright protection tools. Various
software products have been recently introduced in attempt to address these growing
concerns. It is done by hiding data (information) within digital audio, images, text
documents and video files. One way such data hiding is digital signature, copyright
label or digital watermarking that completely characterizes the person who applies it
and, therefore, marks it as being his intellectual property.
2. DEFINITION AND TERMS
Digital watermarking is the process that embeds data called a watermark into a
multimedia object such that watermark can be detected or extracted later to make an
assertion about the object. Watermarking is either “visible” or “invisible”. Perceptible
mark (“visible watermark”) of ownership or authenticity has been around for centuries
in the form of stamps, seals, signatures or classical watermarks. Nevertheless, given
current data manipulation technologies, imperceptible digital watermarks are
mandatory in most applications. A Digital watermark is distinguishing piece of
information that is adhered to the data that is intended to protect. This means that it
should be very difficult to extract or remove the watermark from the watermarked
object. Since watermarking can be applied to various types of data, the
imperceptibility constraint will take different forms, depending on the properties of the
recipient (i.e., the human senses in most cases).
3. TYPICAL WATERMARKING APPLICATIONS
A number of watermarking applications include the following:
• Copyright Protection: for the protection of intellectual property, the data
owner can embed a watermark representing copyright information in his data.
This watermark can prove his ownership in court when someone has infringed
on his copyrights.
• Fingerprinting: to trace the source of illegal copies, the owner can use the
fingerprinting technique. In this case, the owner can embed different
watermarks in the copies of the data that are supplied to different customers.
Fingerprinting can be compared to embedding a serial number that is related to
the customer’s identity in the data. It enables the intellectual property owner to
identify customers who have broken their license agreement by supplying the
data to third parties.
• Copy protection: the information stored in watermark can directly control
digital recording devices for copy protection purposes. In this case the
watermark represents a copy-prohibit bit and watermark detectors in the
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ecorder determine whether the data offered to the recorder may be stored or
not.
• Broadcast monitoring: by embedding watermark in commercial
advertisements, an automated monitoring system can verify whether the
advertisements are broadcasted as contracted. Broadcast monitoring can
protect not only the commercials but also the valuable TV products.
• Data authentication: fragile watermarks can be used to check the authenticity
of data. A fragile watermark indicates whether the data has been altered and
applies localization information as to where the data was altered.
• Indexing: indexing of video mail, where comments can be embedded in the
video content; indexing of movies and news items, where markers and
comments can be inserted that can be used by search engines.
• Medical safety: embedding the date and the patient’s name in medical images
could be useful safety measure.
• Data Hiding: watermark techniques can be used for the transmission of secret
messages. Since various governments restrict the use of encryption services,
people can hide their messages in other data.
4. TYPES OF DIGITAL WATERMARKS
Watermarks and watermarking techniques can be divided into various categories in
various ways. The watermarking can be applied in spatial domain. An alternative to
spatial domain watermarking is frequency domain watermarking. Different types of
watermarks are shown in the figure below. (Fig. 1)
Spatial
Domain
According to Working
Domain
Frequency
Domain
Text
Image
Watermarking
According to Type of
Document
Audio
Private
Video
Invisible
Robust Fragile
Public
Figure 1: Types of watermarking techniques.
According to Human
Perception
Semi-Private
Visible
Dual
2

Watermarking techniques can be divided into four categories according to the type of
multimedia document to be watermarked as follows:
• Image Watermarking
• Video Watermarking
• Audio Watermarking
• Text Watermarking.
According to the human perception, the digital watermarks can be divided into three
different types as follows:
• Visible watermark
• Invisible-Robust watermark
• Invisible-Fragile watermark
• Dual watermark.
Visible watermark is such a watermark that it appears visible to a casual viewer on a
careful inspection. The invisible-robust watermark is embedded in such a way that
alterations made to the pixel is perceptually not noticed. Also the watermark should
withstand the classical signaling operation (so called “attacks”, see section 8) and it
can be recovered only with appropriate decoding mechanism. The invisible-fragile
watermark is embedded in such a way that any manipulation or modification of the
image would alter or destroy the watermark. Dual watermark is a combination of a
visible and an invisible watermark. In this type of watermark an invisible watermark is
used as a back up for the visible watermark.
An invisible robust private watermarking scheme requires at least the original or
reference image for watermark detection. Type I systems, extract the watermark from
the tested, possibly distorted image and use the original image as a hint to find where
the watermark could be in possibly distorted image. Type II systems also require a
copy of the embedded watermark for extraction and just yield a “yes” or “no” answer
to the question: does the tested image contain the watermark? It is expected that this
kind of scheme will be more robust than the others since it conveys very little
information and requires access to the secret material. An invisible robust semiprivate
watermarking does not use the original image for detection but answers the
same “yes/no“ question. An invisible robust public watermarking (also referred to as
blind watermarking) remains the most challenging problem since it requires neither
the secret original image nor the embedded watermark. Indeed such system really
extracts the watermark from the watermarked image. Public watermarks have much
more applications than the others, actually the embedding algorithms used in public
systems can always be used into a private one improving robustness at the same time.
Regarding to the key material necessary for watermark extraction one distinguishes:
• Public key
• Private key
• Asymmetric, which has the property that any user can read the watermark,
without being able to remove it.
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5. WATERMARK REQUIREMENTS FOR STILL IMAGES
A lot of watermarking techniques have been proposed in the open literature in the
recent years. In order to be effective, a watermark should have the main characteristics
outlined below:
• Perceptual Transparency: the embedding algorithm must embed the
watermark in such a way that this does not affect the quality of the host image.
If the humans cannot distinguish the original data from the data with the
inserted watermark than a watermark-embedding procedure is truly
imperceptible. Even the smallest modification in the host image may become
apparent when the original data is compared with watermarked data. Usually
the users of the watermarked data do not have access to the original data. Thus,
they cannot perform this comparison. It may be sufficient that the
modifications in the watermarked data go unnoticed as long as the data are not
compared with the original image.
• Payload of the Watermark: the amount of information that can be stored in a
watermark depends on the application. For the copy protection purposes, a
payload of one bit is usually sufficient. Roughly 70 bits of information should
be embedded in an image. This does not include extra bits added for error
correction codes, which could be also used in watermarking schemes.
• Robustness: The watermark must be difficult to remove. The watermark
should be resilient to standard manipulations of unintentional as well as in
intentional nature. It should be robust against various common signal
processing techniques (compression, quantization, etc.) and common
geometric distortions (cropping, rotation, etc.). Secondly, it should be
statistically unremovable, that is, a statistical analysis should not produce any
advantage from the attacking point of view. However, both the type of
manipulations and the attacker expected computational power heavily depend
on the application.
• Unambiguousness: the retrieval of the watermark should unambiguously
identify the owner.
6. STRUCTURE OF A TYPICAL WATERMARKING SYSTEM
Every watermarking system consists at least of two different parts: watermark
embedding unit and watermark detection and extraction unit.
A. Embedding process:
Figure 2 shows an example of embedding unit for still images. Let us denote an
original image by I, a watermark by W={w1, w2, ..., wL}, where L is the length of the
watermark sequence, and the watermarked image by I.’ The embedding function . E
takes an image I and a watermark W and it generates a new image which is called
watermarked image I’. The original image can be before the watermark embedding
transformed in the frequency domain or the watermark embedding can be performed
in spatial domain. It depends on the selected watermarking technique. If the
embedding is performed in frequency domain , the inverse transform should be
applied to obtain the watermarked image. Mathematically,
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E ( I , W ) = I '
(1)
for the spatial domain technique or:
E ( f , W ) = I '
(2)
for the frequency domain technique, where f are coefficients of the applied
transformation.
Figure 2 illustrates the embedding process.
Original
image (I)
Spatial or
Transform
Domain
Watermark
sequence (W)
Embedding (E)
Figure 2: The embedding process.
Watermarked
image (I’)
Attacks
When the watermarked image is obtained and placed on Internet or transmitted over
the communication channel possibly, the attacks occur (image J is generated).
B. Extraction process:
A detector function D (see Fig. 3) takes an image J (J can be a watermarked or unwatermarked
image, and possibly altered) whose ownership is to be determined and
recovers a watermark W’ from the image. In this process the original image I can also
be included. Mathematically,
D ( J,
I)
= W '
(3)
The extracted sequence W’ could be than compared with the owner watermark
sequence by a comparator function Cδ and a binary output decision generated. If there
is match the output is 1 , otherwise it is 0.That is represented as follows.
Cδ ( W ',
W ) = {
0,
1, c ≤ δ
otherwise
where Cδ is the correlator, ( x = Cδ
( W ',
W ) ), c is the correlation of two watermarks and
δ is certain threshold.
A watermark must be detectable or extractable to be useful. Depending on the way the
watermark is inserted and depending on the nature of the watermarking algorithm, the
method used can involve very distinct approaches. In some watermarking scheme, a
watermark can be extracted in its exact form, a procedure we call watermark
extraction. In other cases, we can detect only whether a specific given watermarking
signal is present in an image, a procedure we call watermark detection. It should be
noted that watermark extraction can prove ownership whereas watermark detection
can only verify ownership
(4)
J
5

Test
image (J)
Original
image (I)
Spatial or
Transform
Domain
Spatial or
Transform
Domain
7. WATERMARKING ALGORITHMS
Extraction (D)
Extracted
watermark (W’)
Original
watermark (W)
Figure 3: The extraction process.
Comparator
function (Cδ)
The watermarking algorithms (techniques) can be performed either in spatial domain
or in the transform domain. The spatial-domain techniques directly modify the
intensities or color values of some selected pixels. One common technique in spatial
domain is Least Significant Bits (LSB) technique where the watermark is embedded in
the least significant bits of some randomly selected pixels. This technique is very easy
for implementation and very fast, but its robustness is very low. The transform-domain
techniques modify the values of some transformed coefficients. That means that the
transform domain coefficients should be modified to embed the watermark and finally
the inverse transform should be applied to obtain the marked image. The transform
techniques commonly used for watermarking purposes are respectively: the Discrete
Cosine Transform (DCT), the Discrete Fourier Transform (DFT) and the Discrete
Wavelet Transform. There are also approaches dealing with the Complex Wavelet
Transform (CWT), the Fourier-Mellin Transform (FMT) and others.
8. OVERVIEW OF ATTACKS
In practice, a watermarked object may be altered either on purpose or accidentally, so
the watermarking system should be able to detect and extract the watermark. The
distortions are limited to those that do not produce excessive degradations, since
otherwise the transformed object would be unusable. These distortions also introduce a
degradation on the performance of the system. The best known watermarking attacks,
which may be intentional or unintentional depending on the application are:
• Additive Noise. This may originate in certain applications from the use of D/A
and A/D converters or from transmission errors. However, an attacker may
introduce perceptually shaped noise with the maximum unnoticeable power.
This will typically force to increase the threshold at which the correlation
detector works.
• Filtering. The filtering attacks are linear filtering: high pass, low pass filtering,
Gaussian and sharpening filtering, the local median, midpoint and trimmed
mean filtering, the Wiener filtering etc. Low-pass filtering, for instance does
not introduce considerable degradation in watermarked images or audio, but
can dramatically affect the performance, since spread-spectrum-like
watermarks have non negligible high-frequency spectral contents.
x
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• Compressions. This is generally an unintentional attack which appears very
often in multimedia applications. Practically all the audio, video and images
that are currently being distributed via Internet have been compressed. If the
watermark is required to resist different levels of compression, it is usually
advisable to perform the watermark insertion task in the same domain where
the compression takes places. For instance, DCT-domain image watermarking
is more robust to JPEG compression than spatial-domain watermarking. Also
DWT-domain watermarking is robust to JPEG2000 compression.
• Statistical Averaging. An attacker may try to estimate the watermark and then
“unwatermark” the object by subtracting the estimate. This is dangerous if the
watermark does not depend substantially on the data. Note that with different
watermarked objects it would be possible to improve the estimate by simple
averaging. This is a good reason for using perceptual masks to create the
watermark.
• Multiple Watermarking. An attacker may watermark an already watermarked
object and later make claims of ownership. The easiest solution is to timestamp
the hidden information by a certification authority.
• Geometrical Attacks. Geometric attacks do not pretend to remove the
watermark itself but to distort it through spatial alterations of the watermarked
image. With such attacks watermarking detector loses the synchronization with
the embedded information. The common geometrical attacks are rotation,
scaling, change of aspect ratio, translation and shearing etc.
• Cryptographic Attacks There are two categories of cryptographic attacks: brute
force attack, whose aim is to find the secret information trough an exhaustive
search and Oracle attack, which can be used to create a non-watermarked
image when watermark detector device is available.
• Protocol Attacks. The aim of protocol attack is to attack the concept of the
watermarking application. The copy attack belongs to this group and the aim
of this attack is to predict the watermark from the watermarked image and to
copy the predicted watermark to the target data. The estimated watermark is
then adapted to the local features of the stego data to satisfy the
imperceptibility requirements.
• Cropping. This is a very common attack since in many cases the attacker is
interested in a small portion of the watermarked object, such as parts of a
certain picture or frames of video sequence. With this in mind, in order to
survive, the watermark needs to be spread over the dimensions where this
attack takes place.
• Random Geometric Distortions. The Unzign and Stirmark have shown
remarkable success in removing data embedded by commercially available
programs.
• Printing-Scanning. This process introduces geometrical as well as noise-like
distortions.
Also to investigate the watermark robustness against various attacks some alterations
have to be made to the watermarked image. For this purpose some benchmarking
software packages are used. The popular benchmarking software are for example
Stirmark , Checkmark, Unzign, etc (see Section 10).
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9. PRACTICAL EXAMPLE
In this example we will watermark the tested image ( in this case the Lena image) with the
pseudorandom sequence used as a watermark. To perform one simple additive algorithm
for digital image watermarking in the DCT domain, we will do the following steps:
1. Initialization
- read original image;
- generate watermark vector of length N (e.g., N=1000).
2. Embedding
- apply 2D DCT transform on the entire image;
- find first N largest coefficients;
- generate watermarked coefficients v’ by v '= v * ( 1+
alpha * w)
where w is
the corresponding watermark component, and alpha is the scaling factor to
control the strength of watermark (e.g., alpha =0.1);
- apply 2D IDCT and truncate pixel value to [0, 255] to obtain watermarked
image;
- display the marked image, visualize the difference between marked and
unmarked image, check visual quality measure
3. Distortion
- generate a distorted version of watermarked image;
- the possible distortions are JPEG compression, gaussian filtering, wiener
filtering etc.
4. Detection
- image registration of the test image with respect to the original unmarked
image is required before applying detection;
- apply 2D DCT on test image;
- identify the N largest coefficients, substract the corresponding value of the
original unmarked ones, and compute the detection statistic with the
watermark vector which is suspected to have been put in the test image
This program can be written in MATLAB or C/C++ programming language. Here is
presented one example of the program written in MATLAB environment.
%-------------------------------------------------------------------------------
% WATERMARKING IN THE DCT DOMAIN
%-------------------------------------------------------------------------------
% 1. Initialization
I=imread('lena.jpg'); % read the original image
figure(4),imshow(I); % show the original image
W=randn(1000,1); % generate the pseudorandom watermark, N=1000
% 2. Embedding
V=dct2(I); % perform the 2D DCT of the original image
[a1,a2]=size(V);
V1=[];
alpha=0.1; % the strength factor alpha
for i=1:a2 % prepare it for sorting, making an one-
p =V(:,i); % dimensional array V1 from the two dimensional
V1=[V1,p']; % array V
end;
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[Vsort,ind] = sort(V1); % sorting the array V1 to find N largest DCT
[a3,a4]=size(Vsort); % coefficients
k=1;
for i=a4:-1:(a4-999) % embedding the watermark
Vsort(i)=Vsort(i)*(1+w(k)*alpha);
k=k+1;
end;
Vw=zeros(1,a4); % re-ordering the sorted array
for i=1:a4
p=ind(i);
Vw(p)=Vsort(i);
end;
Vw1=[]; % preparing for IDCT
for i=0:(a2-1) % making the two-dimensional array from one-
col=Vw((i*256+1):(i*256+256)); % dimensional array,
Vw1=[Vw1,col'];
end;
In=idct2(Vw1); % performing the IDCT, In is the new watermarked image
In=uint8(In); % truncate pixel value to range [0,255]
figure(5),imshow(In); % show the watermarked image
Idiff=(double(I)-double(In)); % find the pixel difference between original and
% image watermarked in spatial domain
Id=256.-Idiff; % for better visibility
Id=uint8(Id); % truncate pixel value to range [0,255]
figure(6),imshow(Id); % show the difference image
imwrite(In,'watermatked.jpg'); % save the watermarked image as JPEG file
% 3. Attacks
J =imnoise(In,'gaussian',0,0.005);% add a Gaussian white noise of mean value 0 and
% variance 0.005
figure(8),imshow(J); % show the distorted image
K = wiener2(In,[5 5]); % perform the Wiener filtering with window size 5x5
figure(9),imshow(K); % show the distorted image
imwrite(In,'watermarked_75.jpg','Quality',75); % perform the JPEG compression with
% quality factor 75
M=imread('watermarked_75.jpg'); % read the compressed image
figure(10),imshow(M); % show the compressed image
% 4. Detection
Vt=dct2(J); % perform the 2D DCT of the tested image
[a1,a2]=size(Vt); % prepare it for sorting, making an one-dimensional
V1t=[]; % array V1t from the two dimensional array Vt
for i=1:a2
p=Vt(:,i);
V1t=[V1t,p'];
end;
[Vtsort,indt] = sort(V1t); % sorting the array V1 to find N largest DCT
% coefficients
[a3,a4]=size(Vtsort);
We=[]; % initialization of the extracted watermark vector
for i=a4:-1:(a4-999) % extraction the watermark
we=(Vtsort(i)-Vsort(i))/(Vsort(i)*alpha);
We=[We,we'];
end;
C=corr2(w,We'); % perform the correlation between the original and
% extracted watermark sequence
% 5. Results
Cj=-0.9368 % correlation value for test image with Gaussian noise
Ck=-0.8283 % correlation value for Wiener filtered test image
Cm=-0.9703 % correlation value for test image, JPEG compressed
% with 75 quality factor
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The Figure 4 presents the original Lena image. On the Figure 5 the watermarked Lena
image is given and on the Fig. 6 the difference image can be observed. For the greater
strength factor alpha, the perceptual difference is more visible (Fig. 7)
Figure 4: The original Lena image
Figure 5: The watermarked Lena image
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Figure 6: The difference between the original
and watermarked Lena image (alpha=0.3)
Figure 7: The difference between the original
and watermarked Lena image (alpha=0.5)
Figure 8: The watermarked Lena image with
Gaussian noise
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10. INTERNET RESOURCES
Figure 9: The Wiener filtered watermarked
Lena image
Figure 10: The JPEG compressed watermarked
Lena image with quality factor 75
The companies and products for the digital watermarking are:
• Alpvision
• Alpha Tec Ltd., Greece (EIKONAmark)
• AT&T - security software.
• Blue Spike Inc.
• CRL
• datamark
• DCT Digital Copyright Technologies
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• digital-watermark.com
• Digimarc Corp
• Intertrust
• Macrovision Corporation
• MediaSec Technologies LLC
• Philips
• Signum Technologies..
• Verance
The most popular benchmarking software are:
• StirMark
• Certimark
• Checkmark
• Optimark
• Unzign
References:
• S. Katzenbeisser, Information Hiding (Boston, London: Artech House, 2000
• F. Hurtung and M. Kutter, Multimedia Watermarking Techniques, Proceedings of
IEEE Special Issue on Identification and Protection of Multimedia Information 87,
No. 7, 10691107 (July 1999).
• F. A. P. Petitcolas, R. J. Anderson, Evaluation of copyright marking systems, in IEEE
Multimedia Systems, Florence, Italy, 7-11 Jun 1999
• S. Voloshynovskiy, S. Pereira, V. Iquise and T. Pun, Attack Modelling: Towards a
Second Generation Watermarking Benchmark, Signal Processing, Special Issue:
Information Theoretic Issues in Digital Watermarking, May 2001
• F. A. P. Petitcolas, Attacks on Copyright Marking Systems, Information Hiding:
Second International Workshop, D. Aucsmith, Editor, Lecture Notes in Computer
Science 1525, Springer-Verlag, Portland, OR (April 1517, 1998), pp. 219239.
• I. J. Cox, J. Kilian, T. Leighton, T. Shamon, A secure robust watermark for
multimedia, in Information Hiding, Cambridge, UK. Springer Lecture Notes. No.
1174, pp. 185-206, 1996.
• Alexander Herrigel, Joseph Ó Ruanaidh, Holger Petersen, Shelby Pereira, Thierry
Pun: Secure Copyright Protection Techniques for Digital Images. in Information
Hiding vol. 1525, pp. 169-190, Springer-Verlag 1998:
• G. Langelaar, I. Setyawan, and R. Lagendijk, Watermarking digital image and video
data, IEEE Signal Processing Magazine, vol. 17, No. 5, 20-46, September 2000
13

• J. K. Su, B. Girod, On the robustness and imperceptibility of digital fingerprints, in
Proc. IEEE Intl. Conf. Multimedia Comp. Sys., vol. 2, pp. 530-535, (Florence, Italy),
Jun., 1999.
• A. N. Skodras, C. A. Christopolos, T. Ebrahimi: JPEG2000: The upcoming still image
compression standard, Proceedings of the 11 th Portuguese Conference on Pattern
Recognition (RECPA00D 20; invited paper), pp. 359-366, Porto, Portugal, May 11th -
12th 2000.
• N. Terzija, Digital Image Watermarking in The Wavelet Domain, Technical Report,
Faculty of Engineering Sciences, Gerhard-Mercator-Universität Duisburg, Dec. 2002,
http://www.fb9dv.uni-duisburg.de/members/ter/trep_1202.pdf
11. THE LIST OF THE USED MATLAB FUNCTIONS
1. imread: Read image from graphics file
A = imread('filename.fmt'); % reads a grayscale or truecolor image named filename with
format fmt into the variable A.
2. imshow: Display an image
imshow(A); % displays the intensity image A
3. dct2: Compute two-dimensional discrete cosine transform
B=dct2(A); % returns the two-dimensional discrete cosine transform of A. The matrix B is
the same size as A and contains the discrete cosine transform coefficients B(k1,k2).
4. idct2: Compute two-dimensional inverse discrete cosine transform
A=idct2(B); % returns the two-dimensional inverse discrete cosine transform (DCT) of A.
5. sort: Sort elements in ascending order
[B,IX] = sort(A); % also returns an array of indices IX, where size(IX) == size(A). If A is
a vector, B = A(IX). Simple example :
A =[1 6 2 4];
B =[1 2 4 6];
IX=[1 3 4 2];
6. randn: Normally distributed random numbers and arrays
Y = randn(m,n); or Y = randn([m n]); % returns an m-by-n matrix of random entries,
distributed with mean 0, variance 1, and standard deviation 1.
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7. size: Array dimensions
[m,n] = size(X); % returns the size of matrix X in separate variables m and n.
8. figure: Create a figure graphics object
9. double: Convert to double-precision
double(x); % returns the double-precision value for X. If X is already a double-precision
array, double has no effect.
10. uint8: Convert to unsigned integer
i = uint8(x); % converts the vector x into an unsigned integer in the range [0, 255]. x
can be any numeric object (such as a double).
11. imwrite Write image to graphics file
imwrite(A,filename,fmt); % writes the image in A to filename in the format specified by
fmt. A can be either a grayscale image (M-by-N) or a truecolor image (M-by-N-by-3).
filename is a string that specifies the name of the output file. Empty image data is not
allowed. The possible values for fmt are determined by the MATLAB file format registry.
12. imnoise: Add noise to an image
J = imnoise(I,'gaussian',m,v); % adds Gaussian white noise of mean m and variance v to
the image I. The default is zero mean noise with 0.01 variance.
13. wiener2: Perform two-dimensional Wiener filtering
J = wiener2(I,[m n]); % filters the image I using pixel-wise adaptive Wiener filtering,
using neighborhoods of size m-by-n to estimate the local image mean and standard
deviation. If you omit the [m n] argument
12 DISCRETE COSINE TRANSFORM
The Discrete Cosine Transform (DCT) represents an image as a sum sinusoids of varying
magnitudes and frequencies. DCT is the most popular transform used in the area of image
compression. DCT has the property that the most of the visually significant information
about the image is concentrated in just few coefficients of the DCT. It is the basic
transform for of the international standard lossy image compression algorithm known as
15

JPEG (the Joint Photographic Expert Group). For an image I with the size M × N the
DCT is defined in two dimensions as:
− M 1N
−1
∑∑
⎛ x + 1/
2 ⎞ ⎛ y + 1/
2 ⎞
C( m,
n)
= k1(
n)
k2
( m)
I(
x,
y)
⋅ cos⎜π
n ⎟ ⋅ cos⎜πm
⎟
⎝ N ⎠ ⎝ M
(5)
⎠
y=
0 x=
0
where C(m,n) are DCT coefficients; n, x are defined from 0 to N-1; m, y are defined from
0 to M-1 and
k ( n)
1/
N
1 = , for n=0, or (6.1)
k ( n)
2 / N
1 = , otherwise (6.2)
k ( m)
1/
M
2 = , for m=0, or (6.3)
k ( m)
2 / M
2 = , otherwise (6.4)
The inverse discrete cosine transform, which uses the same kernel as the DCT, is defined
as:
− M 1N
−1
∑∑
⎛ x + 1/
2 ⎞ ⎛ y + 1/
2 ⎞
I( x,
y)
= C(
n,
m)
⋅ k1(
n)
⋅ k2
( m)
⋅ cos⎜π
n ⎟ ⋅ cos⎜πm
⎟
⎝ N ⎠ ⎝ M
(7)
⎠
m=
0 n=
0
16