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______________________________________________________PROCEEDING OF THE 25TH CONFERENCE OF FRUCT ASSOCIATION Grayscale and Color Basis Images Valery Gorbachev, Elena Kaynarova Anton Makarov, Elena Yakovleva High School of Print and Media of St. Petersburg State University St. Petersburg State University of Industrial Technology and Design St. Petersburg, Russia St. Petersburg, Russia {a.a.makarov, e.s.yakovleva}@spbu.ru {valery.gorbachev, helenkainarova}@gmail.com Abstract—Orthogonal transformation of digital images can be orthogonal transformations, for example, DCT-SVD [16]– represented as a decomposition over basis matrices or basis [18]. SVD (Singular Value Decomposition) is conversion of images. Grayscale and color basis images are introduced. For a rectangular matrix into a block-diagonal matrix. Embedding particular case of DWT (Discrete Wavelet Transform) obtained basis wavelet images have a block structure similar to frequency can be performed using the coefficients of the orthogonal bands of the the DWT coefficients. A steganographic scheme for SVD transform, which operates with frequency wavelet blocks frequency domain watermarking based on this representation is LL, LH, HL, HH. DWT transform is not necessarily one- considered. Presented example of detection algorithm illustrates level [19], [20]. A useful calculation tool is IWT (Integer how this representation can be used for frequency embedding Wavelet Transform) [21]. In this transform the accuracy of techniques. calculations is limited, then the brightness values of the image pixels and transform coefficients are recorded in the same I. INTRODUCTION integer encoding. This eliminates the loss of rounding and A digital image has various representations and some of creates reversibility at least in part of integer encoding. them are required by applications. Many useful representations The paper is organized as follows. Firstly, the orthogonal are produced by orthogonal transforms that are powerful tools transformation and decomposition over the basis image are of image processing. Well known examples are JPEG and considered. Then we introduce grayscale basis images par- JPEG2000 lossy compression formats based on DCT (Discrete ticulary for wavelets and present an example of detection Cosine Transform) and DWT (Discrete Wavelet Transform). algorithm for the case of DWT coefficient watermarking. For the image compression problem block based DCT and Finally, we discuss RGB basis images. DWT techniques are developed [1]–[4] and generalized to non- separable transforms [5], [6] and irregular transforms [7]–[9]. Orthogonal transform produces scattering of digital data, a II. BASIS IMAGES process that redistributes pixel energy of transformed image. It Orthogonal transform of digital image is performed with an is useful for protection of hidden data in steganography, when orthogonal matrix. a message is embedded into image. Hidden data are scattered among all digital cover image and become more robust to lossy data compression and some statistical attacks [10], [11]. Orthogonal matrix The orthogonal transform of images may be considered as A matrix of real numbers is said to be orthogonal if (for a decomposition over matrices known as basis matrices [12]. details see [22]) Being some kind of grayscale images, the basis matrices look attractive and they are often reproduced by textbooks [13]. We UUT = I, will also call these matrices basis images. In this paper we focus on the following items: color and wavelet basis images, I orthogonal transform by matrix of basis images and their where is the identity matrix. It implies that quantum analogues. For color images the solution is directly U T U = I. achieved by considering three-dimensional orthogonal trans- form but for wavelets the solution is not so simple. The reason u uT is that in practice, DWT is calculated by algorithms using Columns of this matrix m and rows n are orthonormal signal processing techniques instead of orthogonal transforms. vectors Nevertheless these algorithms can be used to calculate wavelet u ,u = δ , basis images. So it was found for various wavelets that the m n mn T T basis has a block structure similar to DWT coefficients [14], um,un = δmn, [15]. The watermarking schemes, which use wavelet orthogo- where x, y denotes scalar product of two vectors x and y, nal transform, are also available. The technique uses two and δmn is the Kronecker symbol. ISSN 2305-7254 ______________________________________________________PROCEEDING OF THE 25TH CONFERENCE OF FRUCT ASSOCIATION Representation of matrix where the scalar product of matrices is denoted by A, B := A B F = {Fmn} M × N mn mn. Let be a real rectangular matrix, m,n that corresponds to a grayscale image. We introduce two 3) The sums of the diagonal elements are orthogonal matrices U = {Umn} and V = {Vpk} of the size M × M and N × N respectively. Then taking into account akk = I, that the matrix F = UUT FVVT ,wefind k T F = UGV , akk(x, y)=δxy, (1) G = U T FV, k where G is a M × N matrix. akp(x, x)=δkp. Let us assume that F is an image in a spatial domain (that is x G the image as we see it). Matrix is usually called a frequency Analyzing these properties we came to the conclusion that F representation of or an image in frequency domain. The the basis images are orthonormal. This observation allows frequency domain image may look senseless, however the us to consider the orthogonal transform (2) as a standard orthogonal transform is reversible and the original image can decomposition over the orthonormal basis. It is obvious that always be retrieved. the first equation in (2) takes the form Using the matrix form of the representation (1) F = Gkp akp, (3) F = U G V T = (u ⊗ v ) G , xy xk kp py k p xy kp k,p k,p k,p where Gkp = F, akp. we get a decomposition over tensor products of rows and columns of the matrices U and V , denoted by ⊗. Thus, A. Generation of basis images T if uk is column vector uk =(U1k,U2k,...,UMk) and v =(V ,V ,...,V )T u ⊗ v := u vT There are at least two ways to get basis images. The first is p 1p 2p Np , then k p k p and to use its definitions. In this case the orthogonal matrix has to (uk ⊗ vp)xy = Uxk Vyp. U = V M = N be given. The second way follows from orthogonal transform Here and later we assume and , as this is of the basis images. more interesting case. Then the decomposition produced by Let us focus on the second approach. Let F = aab in the the orthogonal transformation takes the form representation (3). Then we find the basis image representation F = (uk ⊗ up)Gkp, of the form Gkp = δkaδpb. It means that the matrix G has one k,p non-zero pixel, it is equal to 1 and its position is (a, b). So, T T (2) G = (ux ⊗ uy )Fxy. the orthogonal transform of a basis image is a binary matrix x,y of unit brightness. We denote such unit matrix as Grayscale basis images eab = {δkaδpb}, We introduce the matrices where k, p =1,...,N. Then the following relations are valid akp := uk ⊗ up, a = Ue U T , T T ab ab dxy := ux ⊗ uy , T (4) dab = U eabU. that we call basis images. There are N 2 basis images of size So together with the unit vectors ek the unit matrices eab N × N, every image pixel is a product of two items of the form a standard basis and the orthogonal transform of the orthogonal matrix U: basis is a set of basis images aab. Indeed, with the help of the akp(x, y)=UxkUyp. standard basis any matrix can be presented in the following form Properties of basis images G = Gkpekp. Being the tensor products of columns and rows of orthogo- k,p nal matrix, the basis images have properties that follow from Then we get the decomposition given by (3), using the orthogonality. We focus on the basis images akp,asforU = V orthogonal transform and taking into account (4). the properties of dxy are the same. 1) The matrix product of two basis images is another basis Example of WHT basis images image The 2 × 2 orthogonal Walsh-Hadamard Transform (WHT) +1 akp amn = aknδpm. matrix known also as Hadamard matrix consists of and −1, 2) The scalar product 1 11 H = √ . akp,amn = δkmδpn, 2 1 −1 ---------------------------------------------------------------------------- 110 ---------------------------------------------------------------------------- ______________________________________________________PROCEEDING OF THE 25TH CONFERENCE OF FRUCT ASSOCIATION In optics this matrix describes so called 50% beam splitter, Using Z, we can present any single particle operator F as a linear optical element often used in experiments to split the follows beam into two parts. Four basis images akp, denoted as tensor F = |kp|k|F|p. product of columns, have the following form k,p 1 11 1 1 −1 a = ,a= , Operator F can be written as a matrix, using Q and (5), 11 2 11 12 2 1 −1 then the right part of this equation takes the form (3). As a 1 11 1 1 −1 result we find that some of representations of single particle a = ,a = . 21 2 −1 −1 22 2 −11 operators can be considered as basis grayscale images. The determinant of every matrix equals to 0 and the matrices III. WAVELET BASIS IMAGES are non invertible. The matrices can be generated from a unit Basis images can be generated by DWT. In calculation the matrix by WHT: DWT techniques do not use matrix methods and the basis 10 1 11 wavelet images can be achieved by transform of standard basis. H : e = = a . 11 00 2 11 11 Wavelet coefficients This equation illustrates relations between the basis images The DWT coefficients have a block structure due to orthogo- and the standard two-dimensional basis.

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