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MATEMA^4HE TA KOMÏ^TEPHE MOДEЛЮBAHHЯ
MATEMA^4ECKOE И KOMПЬЮTEPHOE MOДEЛИPOBAHИE
MATHEMATICAL AND COMPUTER MODELLING
YAK 004
M. Candik
DISCRETE ORTHOGONAL TRANSFORMS IN DIGITAL IMAGE WATERMARKING
Basic principles of digital image watermarking based on discrete orthogonal transforms are in this paper presented. Digital image watermarking based on using of Karhunen-Loeve Transform (KLT) and Discrete Cosine Transform (DCT) is in the paper presented in more details. Experimental verification of both methods was realized on static grayscale images Einstein and Samuel.
1 INTRODUCTION
Digital watermarking represents a viable solution to the ever-increasing demand for copyright protection mechanisms [1]. A watermark is hidden information within a digital signal [2]. For image watermarking several techniques have been developed. Watermarking techniques can be divided into three main groups:
- spatial space watermarking,
- frequency space watermarking,
- parametric space watermarking.
Watermarking in spatial space uses the spatial image space for embedding of hiding information. Frequency space watermarking uses the selected discrete orthogonal transforms (DCT, DWT) for embedding of the hiding information into the spectral coefficients or into the sequence coefficients, in generally. Parametric space watermarking includes algorithms, where a watermark is embedded into the image in its parametric space (typically in digital image watermarking techniques established on fractal image coding, where a watermark is inserted into
the parameters of block's similarity or into the parameters of block's positions) [3]. To achieve maximum protection of intellectual property with watermarked media, several requirements must be satisfied. Embedded watermarks must be[3]:
- imperceptible - the watermark should be imperceptible, not to affect the viewing experience of the image or the quality of signal
- undeletable - the watermark must be difficult or even impossible to remove by a hacker, at least without obviously degrading the host signal.
- statistically undetectable - A pirate should not be able to detect the watermark by comparing several watermarked signals belonging to the same author
- robustness - The watermark should be survived by the using of the loss compression techniques and signal processing operations (signal enhancement, geometric image operations, noise, filtering, etc.)
Proposed requirements that are imposed on watermarking methods are contradictory. It is technically viable and easy to embed a watermark into an image by its invisibility, but requirements of its undetectability, undeletability and its robust create technical problems. Thus, a watermarking technique for watermark embedding is a compromise between some contradictory requirements. It is an argument why properties of watermarking methods must be analyzed.
In this paper a basic analysis that must be realized after process of watermark implementation is presented. Com-
paring of two digital watermarking methods, both based on discrete orthogonal transforms, is described. In generally, discrete orthogonal transforms (mainly DCT) is a base for loss image compression. It is possible to assume that information which may be lost during lossy compression, the same information may be modified during watermark embedding. Thus, a question is satisfying of remaining watermark requirements.
2 GENERAL ALGORITHM FOR WATERMARK
EMBEDDING
Digital image watermarking algorithms based on using of discrete orthogonal transforms may be described according to generally algorithm for watermark embedding depicted in Fig.1.
A frequently case of watermarks are binary images, dimension of watermark is in generally smaller then dimension of original image. Input parameters in process of watermark embedding are original image, watermark and user's key (style of watermark permutation). The original image is processed by discrete orthogonal transform. After image transformation we obtain a set of spectral or sequence coefficients. The set dimension of transformation (spectral) coefficients equals to the original image dimension. A first operation with the watermark is reordering of its pixels by using pseudorandom permutation. The algorithm of watermark permutation must be pseudorandom because watermark must be reconstructed. A type of used pseudorandom watermark's pixels permutation is arbitrary and makes a user's key. A watermark is a binary image. Thus, values of its brightness are 0 - black point (black pixel) and 1-white point (white pixel). For satisfying of robust requirements are these values multiplied in a multiplying unit, the multiplication is realized with some constant value. A multiplying unit doesn't need to use, in generally. Then each pixel is added to some spectral coefficients of original image. Number of used spectral coefficients equals to number of watermark's pixels.
Figure 1 - The basic algorithm of watermark embedding into an image. As a transformation is reasoning some discrete orthogonal transform. The multiplying must be not strictly used
A choosing of spectral coefficients depends on their properties. We expect that embedded watermark will be invisible and immune during operation with watermarked image. From point of view of human visual system are good spectral coefficients with the relationship in spatial space of image to highest image frequencies (details in images), but mostly operation with an image is its filtering and embedded data may be lost. Thus, a way for this requirements are spectral coefficients corresponded to central part of frequency space. After modification of spectral coefficients we compute inverse transform (of modified frequency space) and obtain watermarked image.
2.1 BASIC MATHEMATICAL DESCRIBING OF WATERMARK EMBEDDING
Let original image I is defined according to relationship (1)
io = {I(i, j) ;0 < i < Ni;0 < j < N2},
(1)
where ig(i, j) 6 {0, 1, ...,2 - 1} is a brightness value of pixel ig(i, j);i, j, terminates position of pixel and L denotes number of bits used do describing brightness value in each picture element. The original image dimension is (N1 XN2 ).
Watermark W is a binary image (L =1) defined according to relationship (2).
W = { w(i, j) ;0 < i < M1 ;0 < j < M2 },
(2)
where w (i, j) = { 0 ;1} describes a brightness value of a watermark. The dimension of watermark W is (M/ XM^),
M1 <N1t M2<M2.
Let watermarked image is denoted as Iw . Then process of inserting of watermark W into the original image I may be described according to relationship (3).
9:1 x W ^ Iw; Iw = I + YW,
(3)
where 9 is embedding procedure, y is a two-dimensional inserting mask, that includes operations of superposition, permutation etc.
3 A GENERAL ALGORITHM FOR WATERMARK
EXTRACTION
Analogue, we describe a general algorithm of watermark extraction. Input parameters in process of watermark extraction are original image, watermarked image and user's key (algorithm of watermark permutation). A general algorithm for watermark extraction is depicted in Fig.2.
For the original image we compute discrete orthogonal transformation; on the original image we compute discrete orthogonal transformation too. We compare used spectral coefficients in both images. Because watermarked image
may be modified (noise corruption, attack of data, etc.), may be also changed coefficients, that weren't modified in process of watermark embedding. In this case is an advantage to use a threshold unit which provides selection of modified spectral coefficients. A next step of watermark extraction is reordering of used spectral coefficients. After this manner we obtain extracted watermark.
8:Ix Iw ^ W' ; W' = v( W°),
(4)
7 7
F(u, V) = ÇM-ÇM £ £ f(j, k) cos
•( 2j + 1 ) u n " 16
j = 0 k = 0
x cos
•( 2 k + 1 ) V n ■ 16
(5)
where
F(.,.) denotes DCT coefficients,
f(.,.) are values of brightness (pixel's values) in current block,
C ( u ) = 1/72 for u=0; C(u)=1 for u=1,2...7,
C (V) = 1/72 for v =0; C(v) =1 for v =1,2...7.
3. For obtained spectral coefficients we use its reordering according to ZIG-ZAG algorithm, depicted in Fig.3.
Figure 2 - The basic algorithm of watermark extraction from watermarked image. The threshold unit must be not strictly used
3.1 BASIC MATHEMATICAL DESCRIBING OF WATERMARK EXTRACTION
A process of watermark extraction may be described according to follow relationship (4)
where 5 is extraction procedure, V (.) describes inverse permutation of reordered watermark W°, which is obtained after comparing of spectral spaces both input images I and Iw . Because watermarked image may be processed, embedded watermark W may be, in generally, corru W • .
4 THE ALGORITHM OF DCT WATERMARKING
The algorithm of digital image watermarking based on two-dimensional discrete cosine transform (DCT) is a basic and the most processed watermarking method. Its large distribution is given by large using of digital images, mainly in JPEG format, which uses this orthogonal transformation.
The algorithm of digital watermarking based on DCT is follow:
1. Original image is divided to non-overlapped blocks, that cover all image. The dimension of each block is 8 x8.
2. For each block we compute DCT transform defined in equation (5)
Figure 3 - Reordering of spectral coefficients according to Zig-zag algorithm
Zig-zag algorithm provides a transposition of spectral coefficients of a two-dimensional block into one-dimensional vector. Obtained spectral coefficients sequence includes elements that are equivalent to decreasing frequency parts in actual image block. Thus, the first element depends on mean brightness value of block, the second element depends on low-frequency properties of image blocks, etc., the last coefficient depends on high-frequency properties of image blocks (details in image).
4.Because dimensions of image I and watermark W are different, it is necessary to compute, how many watermark's pixels are used in current DCT block.
Let = N1/8 and ^ = N2/8 describes number of DCT blocks in horizontal and vertical direction of image I, respectively. Thus, original image I contains ^ x ^ DCT blocks. In watermark W will be strikes
(M1/h2) x (M2/h2) pixels for each DCT block. The process of pixel's selection in watermark W is free, only number of pixels depends on dimension of watermark W and on dimension of original image I, too.
Figure 4 - Example of a possibility of spectral coefficients modification based on Zig-zag algorith.
One of way is watermark dividing into hi X ^ watermark sub-blocks, each sub-block to convert into one-dimensional vector and this vector add to selected spectral coefficients in Zig-zag vector (see Fig.4).
5.Watermark will be permuted, i.e. pixels of watermark W will be reordered. A watermark with reordered pixels is
denoted W° .
6.For satisfying of robust requirements will be permuted watermark multiplied with a constant k(k> 0), i.e.
WP = kW°P
(6)
Thus, we obtain binary image WP (permuted watermark), its brightness values are 0 - black picture elements, k - white picture elements.
7.Selected pixels of watermark Wp is superposed into selected DCT spectral coefficients of image block. Superposition is the addition of value 0 or k to values of selected spectral coefficients.
8.Inserting of watermark pixels will be realized for each DCT image block.
9.After inserting watermark to (DCT) frequency space of original image we acquire modified DCT spectrum of original image, its dimension is ( N1 X N2 ).
10.We realize the same dividing of this spectrum into non-overlapped blocks, that cover all image, and that was used in step 1. For each block we compute inverse DCT transform according to follow relationship:
7 7
1
F(i, j) =4 £ S C(u) • C(v)F(u,v)cos
u =0v = 0
(2i+ 1 )un " 16
X cos
( 2j + 1 ) v n ■ 16
(7)
where describing of parameters is the same as in equation (5).
ll.After computation we obtain watermarked image Iw .
4.1 EXTRACTION OF WATERMARK
We retry that input parameters in process of watermark extraction are original image I, watermarked image Iw
and user's key (algorithm of pseudo-random watermark permutation). A process of watermark extraction may be described in follow steps:
1.Original image I is divided to non-overlapped blocks, which cover all image. The dimension of each block is
8 X 8 .
2.For each block we compute DCT transform defined in (5).
3.Watermarked image Iw is divided to non-overlapped blocks, that cover all image. The dimension of each block is 8 X 8 .
4.For each block we compute DCT transform according to relationship (5).
5.In both image spectral spaces we retrieve spectral coefficients that were modified in process of watermark embedding. Thus, we compare spectral spaces of both images (we compute difference of values of these spectral coefficients).
6.We redistribute computed differences into two-dimensional space that is corresponded with pixel's distribution of permuted watermark. Output of this computational step is a matrix, which corresponds with permuted watermark, dimension of this matrix equals to dimension of watermark, or permuted watermark, respectively.
7.Obtained matrix is processed by inverse permutation, i.e. reordering of values in its space.
8. Because watermarked image could be modified, we use thresholding of values. The level of threshold is from interval < 0;k >. Thresholding makes a comparing of proposed matrix values with some constant value (threshold). Matrix values that are smaller then threshold value are interpreted as a black pixel (0-value) and matrix values that are greater than threshold value are interpreted as white pixel (value 1, or value k ).
9.After thresholding we obtain watermark W . Its quality depends on threshold value and on type of processing with watermarked image.
5 THE ALGORITHM OF THE KLT WATERMARKING
The algorithm is in generally, the same, as algorithm based on DCT transform. Karhunen-Loeve transform (KLT) is the optimal transform among all orthogonal transforms. It is also known as Hotteling or principal components transform [4]. This transform has the best energy compaction property, and it de-correlates the image data most efficiently. Moreover, the mean square error distortion due to discarding transform coefficients is the last. The realization of Karhunen-Loeve transform can be described by following steps:
1. Dividing of original image I into non-overlapped blocks that cover all image. The blocks havedimensions
K1 X K2 .
2.Each block is transformed into vector u of dimension 1 x KXK2
3.For each block we compute covariance matrix
Cu = E{[u - E(u)][u - E(u)]T} , (8)
where operation E[.] represents a mean value.
4.We compute eigenvalues and eigenvectors of covariance matrix Cu . The base vectors of Karhunen-Loeve transform are given by the orthonormalized eigenvectors of the covariance matrix Cu , that is,
Cu§k = h^k,
0 < k <( K1K2 - 1),
(9)
where ^ is corresponding eigenvalue of the eigenvector ^k. Eigenvalues of the covariance matrix, which are all non-negative, can be ordered according their magnitude. The larger is the magnitude, the more energy the eigenvalue contains. We create matrix U, where a first column contains eigenvector of maximal eigenvalue, second column contains eigenvector with a second largest eigenvalue, etc. After reordering of columns it is necessary to save, how columns were reordered.
5.The Karhunen-Loeve transform is defined according to relationship (10):
Figure 5 - Used original images for digital watermarking: a) EINSTEIN; b) SAMUEL
As watermarks we use for our experiments binary images Wmark1 and Wmark2, both of dimensions M1=M2=64, resolution of watermarks was 1bit/pixel. Used watermarks are depicted in Fig. 6.
M
Y = Uu
(10)
6. Inverse transformation is defined according to relationship (11),
= UTY,
(11)
where parameter T describes a matrix transpose.
7. We create two-dimensional block of dimensions K1 x K2 from the vector u of dimension 1 x (K2K1) .
Thus, after theoretical description we focus on practical aspects of its implementation into watermarking process. Dimensions of non-overlapped blocks of divided image I must be not strictly 8 x 8, but this dimension is a good way to properties compare of both approaches in digital watermarking. Increasing of its dimensions gives more difficult computing and long time for KLT transform computation. Selection of spectral coefficients modifying is the same, as in DCT watermarking method.
6 EXPERIMENTAL PART
Our experiments we are realized on static grayscale images EINSTEIN and SAMUEL in pgm (portable gray-map) format. Their dimensions were N=N2=256, resolutions of images were 8 bits/pixel, i.e 28=256 levels of brightness. Used images are depicted in Fig.5.
Figure 6 - Used watermarks: a) Wmark1; b)Wmark2
Embedding of watermarks into original images was realized by multiplying factor k =4, i.e. black pixels in watermark are represented by zero brightness level and white pixels in watermark are described by brightness level k.
6.1 SELECTED PROPERTIES OF BOTH METHODS
Quality of extracted watermark and watermarked images were analyzed with objective and subjective criteria of image quality. As objective criteria of image quality were used MAE (Mean Absolute Error) and MSE (Mean Square Error). Mean Absolute Error is defined in (12)
1
N1 N2
MAE = HI(*) - Iw(*),
1 2 x = 1 * = 1
and Mean Square Error is defined in (13):
(12)
MSE =
1
N1 N2
12 X = 1, = 1
2 .
(13)
A practical criterion for describing of watermark modification is the number of detected errors (number of changes in pixel's values) in extracted watermark. Amount of extracted modified watermark's pixels is denoted as 8 . Relative amount 8r of modified watermark's pixel is defined according to relation (14).
er m1m2
■100%.
(14)
Next objective criteria of image quality used for watermarked images was Peak-Signal-to-Noise-Ratio defined according to relationship (15)
PSNR = 10log
2552
N1 N2
■•(15)
N1N2
J] J] [I(x,y) -Iw(x,y)]2
10 x = 1 y = 1
Quality of watermarked image depends mainly on used multiplying factor k. Higher values of multiplying factor generate smaller quality of watermarked images, across they improve robustness of embedded watermark. Second parameter with relation to duality of watermarked image is used watermark. Modifying of spectral coefficients (and image pixels) depends on amount of white pixels in watermark, because only white pixels create changes in image by embedding of watermark.
Obtained quality of watermarked images is described in Tab.1, effect of thresholding to quality of extracted water-
mark is described in Tab. 2. Threshold levels are denoted with "/". Because we had not analyzed effect of lossy operations with watermarked images (i.e. robust analysis), we were focused to lower thresholding levels and survey speed of errors elimination in extracted watermarks. We retry that threshold levels are defined in interval (0 ;k) . Modifications of watermarks which are produced by process of embedding and extracting of watermarks are accrued mainly in rounding of computed values and in produced error during computation.
Experimental comparing of both approaches of digital watermarking shows that higher objective quality of watermarked images (smaller errors MAE, MSE and higher PSNR) produces using of Karhunen-Loeve transform in watermark embedding, only one case - embedding of watermark Wmarkl with the multiplying factor k =2 gives in DCT approach higher watermarked image quality (by using another original images and another watermarks presented effect was not notable).
Analogous, as objective watermarked image quality, subjective image quality criteria to qualification of embedding process are used. Examples of unwatermarked (original) image and watermarked images in Fig. 7 are depicted. For our examples we choose watermark Wmark2, because this watermark produces more errors by its embedding into an image - watermark Wmark2 contains more white pixels (than watermark Wmarkl) which cause them. The multiplying factor k =4 was used by embedding of watermarks.
Таблица 1-Quality of watermarked images
QUALITY OF WATERMARKED IMAGES
MAE MSE PSNR
DCT method KLT method DCT method KLT method DCT method KLT method
EINSTEIN Wmark1 k = 2 0,0022 0,0065 0,0022 0,0115 74,682 67,510
k = 4 0,0700 0,0147 0,0702 0,0459 59,665 61,508
k = 6 0,1011 0,0224 0,1059 0,0985 57,883 58,196
Wmark2 k = 2 0,1888 0,0923 0,1888 0,1624 55,371 56,026
k = 4 0,5827 0,2004 0,7031 0,6342 49,660 50,109
k = 6 0,8848 0,3030 1,4210 1,3784 46,605 46,737
SAMUEL Wmark1 k = 2 0,0023 0,0066 0,0023 0,0116 74,593 67,505
k = 4 0,0708 0,0148 0,0710 0,0460 59,617 61,503
k = 6 0,1021 0,0222 0,1070 0,0989 57,838 58,177
Wmark2 k = 2 0,1902 0,0926 0,1902 0,1620 55,338 56,035
k = 4 0,5870 0,2039 0,7086 0,6368 49,627 50,091
k = 6 0,8917 0,3073 1,4321 1,3839 46,571 46,720
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ISSN 1607-3274 "Радюелектрошка. 1нформатика. Управл1ння" № 2, 2003
Таблица 2 Quality of extracted watermarks
QUALITY OF EXTRACTED WATERMARKS
Wmark 1 Wmark2
DCT method KLT method DCT method KLT method
EINSTEIN l = 0 MAE 0,0491 0,0959 0,1901 0,2808
8 201 393 779 1150
8r er 4,91 9,59 19,02 28,08
l = 0,15 MAE 0,0022 0,0024 0,0881 0,0999
8 9 10 361 409
8r 0,22 0,24 8,81 9,99
l = 0,30 MAE 0,0015 0 0,0713 0,0034
8 6 0 292 14
8r 0,15 0 7,13 0,34
l = 0,45 MAE 0 0 0,04565 0,0002
8 0 0 187 1
8r 0 0 4,57 0,02
SAMUEL l = 0 MAE 0,0447 0,0957 0,1901 0,2795
8 183 392 779 1145
8r 4,47 9,57 19,02 27,95
l = 0,15 MAE 0,0007 0,0027 0,0867 0,1030
8 3 11 355 422
8r 0,07 0,27 8,67 10,30
l = 0,30 MAE 0 0 0,0706 0,0020
8 0 0 289 8
8r 0 0 7,06 0,20
l = 0,45 MAE 0 0 0,0457 0
8 0 0 187 0
8r 0 0 4,57 0
Figure 7 - Quality of watermarked images
a) original image EINSTEIN ;
b) used watermark Wmark2;
c) watermarked image, DCT method;
d) watermarked image, KLT method;
e) original image SAMUEL;
f) used watermark Wmark2;
g) watermarked image, DCT method;
h) watermarked image, KLT method.
Figure 8 - Histogram's modifications by watermark embedding:
a) histogram of the original image EINSTEIN;
b) differemce between unwatermarked and watermaked image histograms, image EINSTEIN, watermark Wmarkl, DCT method;
c) differemce between unwatermarked and watermaked image histograms, image EINSTEIN, watermark Wmarkl, KLT method;
d) differemce between unwatermarked and watermaked image histograms, image EINSTEIN, watermark Wmark2, DCT method;
e) differemce between unwatermarked and watermaked image histograms, image EINSTEIN, watermark Wmark2, KLT method;
f) histogram of the original image SAMUEL;
g) differemce between unwatermarked and watermaked image histograms, image SAMUEL, watermark Wmarkl, DCT method;
h) differemce between unwatermarked and watermaked image histograms, image SAMUEL, watermark Wmarkl, KLT method;
i) differemce between unwatermarked and watermaked image histograms, image SAMUEL, watermark Wmark2, DCT method; j) differemce between unwatermarked and watermaked image histograms, image SAMUEL, watermark Wmark2, KLT method.
Some basic statistical properties of watermarked image are visible on watermarked and unwatermarked image histograms. In Fig. 8 are depicted histograms of both used original images and accrued differences between unwater-marked and watermarked images. Depicted dependencies show that DCT approach in image watermarking creates changes in a few levels of brightness in image, on the other hand KLT approach creates important changes in reduced amount of levels of brightness. Using of watermark with a small amount of white pixels (pixels of modifications) decreases possibilities of resolution and detection of used method, because obtained dependencies in histogram differences are similar.
Second interested parameter in a watermarked image is distribution of errors that are caused by watermark implementation. Fig. 9 depicts examples of reflected errors distributions.
Fig.9 shows that distribution of errors depends on number of white pixels in used watermark; higher amount
of white pixels produces more errors. Intensity of errors depends on used multiplying factor that is used in process of watermark embedding.
Analogous, quality of extracted watermark must be analyzed. Objective criteria MAE and MSE are identical in case of binary images (watermarks). Practical criterion for extracted watermark quality is amount of erroneous watermark's pixels (8 ), or relative amount of modified watermark's pixels (8r) defined in (14). Quality of obtained watermarks is described in Tab.2.
Process of thresholding in process of watermark is depicted in Fig. 10. Errors in watermark are produced during computation, mainly in computation of inverse orthogonal transforms.
We remark that watermark is embedded into an image in its permutation form and first step of extraction's process is obtaining of permuted watermark. Examples of permuted watermarks are depicted in Fig.ll (permutation of the original watermark, between any changes).
Figure 9 - Distribution of errors in watermarked images:
a)original image EINSTEIN, watermark Wmarkl, DCT method;
b) original image EINSTEIN, watermark Wmarkl, KLT method;
c)original image EINSTEIN, watermark Wmark2, DCT method;
d) original image EINSTEIN, watermark Wmark2, DCT method;
e)original image SAMUEL, watermark Wmarkl, DCT method;
f) original image SAMUEL, watermark Wmarkl, KLT method;
g)original image SAMUEL, watermark Wmark2, DCT method;
h) original image SAMUEL, watermark Wmark2, KLT Metod.
M
M M
Figure 10 - Process of watermark thresholding:
a) l=0; b) l=0,01; c) l=0,04; d) l=0,25.
M
Figure 11 - Example of watermark permutation:
a) permuted watermark Wmarkl;
b) original watermark Wmarkll;
c) permuted watermark Wmark2;
d) original watermark Wmark12.
CONCLUSION
In this paper some basic principles of digital watermarking are presented. Presented methods use discrete orthogonal transforms for watermark embedding and watermark extracting too. The paper demonstrates basic properties of digital watermarking based on discrete cosine transform and Karhunen-Loeve transforms. Practical implementation of watermark requires next analyze - analyze of robustness. A robustness analyze is a large group of tests, which are focused to immunity of embedded watermark in next image processing operations, mainly loss operations of watermarked image (loss compression techniques, image quantization, noise, image filtering [5], etc.).
In proposed algorithms the original unmarked image is required for watermark extraction. They are also more sophisticated algorithms in digital image watermarking that not need original unmarked image for watermark extraction.
REFERENCES
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2. Katzenbeisser,S. - Petitcolas, F.A.P: Information Hiding Techniques for Steganography and Digital Watermarking. ARTEC HOUSE, INC., 2000. ISBN 1-58053-035-4
3. Candik, M., Levicky, D., Klenovicova, Z.: Fractal image coding with digital watermarks. In: Radioengineering, Vol. 9, No. 4, 2000, Prague, pp. 22-26, ISSN 1210-2512.
4. Candik, M.: Digital Watermarking using Karhunen-Loeve Transform. In: Fine Mechanics and Optics, No. 5/2003, Vol. 48, Olomouc, (Czech Republic) pp. 135-137.
5. Lukac, R., Candik, M.: The Influence of Noise Corruption to Image Watermarking in DCT Domain. In: Journal of Electrical Engineering, No.3-4, Vol.52, 2001, Bratislava, pp. 8187, ISSN 1335-363.
Надшшла 05.08.2003
В статье представлена основная информация о принципе внедрения цифровых "водяных знаков", основанном на дискретных ортогональных преобразованиях. Представлена сравнительная характеристика методов внедрения "водяных знаков" в цифровые изображения на основе преобразования Карунена-Лоэва. Экстремальная проверка реализована на статических изображениях в тонах серого "Эйнштейн" и "Сэмюэль".
