ISSN 1054-6618, Pattern Recognition and Image Analysis, 2009, Vol. 19, No. 3, pp. 491–496. © Pleiades Publishing, Ltd., 2009. INTRODUCTION Adequate image information representation is one of the most challenging problems in modern blind ste- ganography [1]. Due to concomitant restrictions only a few of adaptive blind steganography methods ensure the amount of hidden information (embedding capac- ity) no more than 10–15% of the image volume. Higher capacity implies the overlapping of embedded bit-planes in the process of message data embedding into image. In our opinion the accurate formalization of successive embeddings of message bit-planes is needed for efficient solution. In fact in blind steganog- raphy task we face a novel problem of reversible signal combining while no message no source image are a priory known and no any data are accessible at the receiving end, except for data extracted from the image containing the message. Message hiding in itself is minor problem which should be overcome as needed. To solve the mentioned problems and increase the hidden information amount up to 30–50% of image volume we have developed a model that describes the image as a peculiar tool for coded information storage and transmission. According to our model an image possesses its own hardware-independent digital mem- ory which is able to store the coded information inde- pendently of prescribed image transformations (bias- ing, stretching, packing according to intensity and other) and keeps the imaging and processing marks, distortions by transmission or is filled up with data codes adjusted algorithm accordingly. The cells of virtual memory are juxtaposed with pixels in one-to-one relationship. But unlike the memory cells of usual computer memory the virtual memory cells contain no bits but trits [2–3]. Virtual memory cells consist of equal number of trits like the cells of computer memory. The storage elements of virtual memory are classified into fixed (read-only) trits, containing invariable values, and modifiable (read-write) trits, containing bits of the message. The methods of data embedding into virtual memory differ from LSB-method due to modifiable trits occupies mainly most significant trit-planes and fixed trits assemble in the least significant ones. The total num- ber of trits depends on image content and usually exceeds the number of bits in source computer image representation, say, twice. At that embedding capacity estimated as the number of modifiable trits specifies the total volume of hidden and perceptible message data which does not exceed 70–90% of image volume. The model construction basing on juxtaposition of virtual memory cells to pixel values considered in computed intensity ranges and also the steganographic applications have been previously presented in [4–7]. In this paper we focus on algebraic properties of image transformations. INVARIANT IMAGE REPRESENTATION According to our model a virtual memory consists of definite number of trit-planes that depends on image content. Virtual memory cells contain the pixel values of certain image representation produced by isomorphic image transformation where the transfor- mation is considered as isomorphic relative to inten- sity order if it does not change the result of the poste- rior image intensity packing by means of numbering of histogram intensities with sequential numbers and substitution of source pixel intensities with that num- bers. Let Hu be an image representation in virtual mem- ory produced by isomorphic transformation H of Algebraic Description of Data Embedding Basing on Idempotent Image Transformations 1 M. V. Kharinov St. Petersburg Institute for Informatics and Automation Russian Academy of Sciences, Chetyrnadtsataya liniya 39, St. Petersburg, 199178 Russia e-mail: khar@iias.spb.su Abstract—In this paper, we develop an algebraic description for previously proposed model of image con- taining the invariant image representation and embedded data in its own digital hardware-independent mem- ory (named virtual memory) independently of prescribed image transformations. The properties of idempo- tent image transformations used for generation of invariant representation and also for data embedding are established. The statements are illustrated with image processing examples. DOI: 10.1134/S1054661809030158 MATHEMATICAL THEORY OF PATTERN RECOGNITION Received February 9, 2009 The article is published in the original.