Image Steganography Scheme using Chaos and Fractals with the Wavelet Transform

Transcription

1 International Journal of Innovation, Management and Tehnology, Vol. 3, o. 3, June Image Steganography Sheme using Chaos and Fratals with the Wavelet Transform Yue Wu and Joseph P. oonan bstrat This paper introdues a novel image steganographi method for embedding seret image information into digital images. The proposed algorithm uses a fratal image as the over image, takes a random-like sequene generated by a haoti map as the referene for embedded position and employs a wavelet transform to realize the embedding proedure. The fratal over image that an be parametrially generated implies a over image ould be unique in the world. The haoti logisti map guarantees that the sequene of embedding positions is ompletely unknown for an adversary and the embedding proedure is almost random-like. The wavelet transform ensures the seret information is only embedded within edges with the least visual distortion. The proposed sheme is tested over different over images and seret images and simulations results demonstrate its effetiveness and robustness. Index Terms Steganography, hao fratal, logisti map, wavelet transform. I. ITRODUCTIO The term steganography [] originates from the Greek word steganos meaning overed or proteted and graphein meaning to write. owaday it often refers to the siene of invisible ommuniation. Generally speaking, steganography an be lassified into two ategories: image watermarking and image data hiding. Image watermarking tehniques are aimed to provide opyright protetion [-5] and authentiation [6, 7]. Image data hiding tehniques is aim to amouflage a seret message within a arrier message suh that the seret message an be sent without making any notie. This study fouses on data hiding. There are two kinds of image steganography for image data hiding: spatial-domain based and transform-domain based methods. Spatial-domain based methods embed messages in the intensity of pixels of images diretly [8, 9]. mong them, the least signifiant bit (LSB) method is the most well-known [-]. This method embeds the fixed-length seret bits in the same fixed-length LSBs of pixels. This tehnique is simple and fast but it may ause notieable distortion when the number of embedded bits for eah pixel exeeds three []. While transform-domain based methods first transform a over image into a new domain and then messages are embedded in as the transform oeffiients Manusript reeived pril, ; revised May 9,. Y. Wu is with the Department of Eletrial and Computer Engineering, Tufts University, M 55, US ( ywu3@ee.tufts.edu). J. P. oonan is with the the Department of Eletrial and Computer Engineering, Tufts University, M 55, ( jnoonan@ees. tufts.edu). [3, 4]. The proposed method is one of the transform-domain based steganographi tehniques. Unlike those related works [5, 6] fousing on hiding the seret message with oding sheme we diretly used the wavelet oeffiients to embed the message. In addition, we onstrut a series of over images via fratal iterative funtion whih guarantee that ) a over image an easily regenerated with a given set of parameters; ) a over image is almost unique and novel; and 3) a over image is inaessible for an adversary without knowing the orret set of parameters. Moreover, based on the fat of the image has different wavelet oeffiients when using different wavelets. We use the wavelet funtion as a part of our shared key to further enhane the seurity level. The remainder of the paper is organized as follows: in Setion II, bakgrounds related to the proposed method are reviewed; in Setion III, the proposed steganographi tehnique is disussed in details; in Setion IV, omputer simulation results and related analysis are presented; In Setion V, we onlude this paper and make disussions on the future work. II. BCKGROUDS Simply speaking, the proposed algorithm uses a fratal image as the over image, takes a random-like sequene generated by a haoti map as the referene for embedded position and employs a wavelet transform to realize the embedding proedure. More details about the proposed algorithm are disussed in Setion III. In the rest of the setion, we will briefly review the wavelet transform, the haoti map and the fratal image.. Wavelet Transforms wavelet transform is used to divide a ontinuous-time/ disrete time funtion into wavelets. Unlike the Fourier transform, whih only onstrut a frequeny representation of a signal, the wavelet transform is able to onstrut a time-frequeny representation of a signal. Therefore, the wavelet transform offers very good time and frequeny loalization. Mathematially, the Continuous Wavelet Transform (CWT) of a ontinuou square-integrable funtion f ( x ) at a ertain sale s > and a translation value R an be defined as Eq. (), where ( x) is defined in Eq. () [7]. Here ( x) is alled the mother wavelet and ( x) is alled a daughter wavelet at sale s and translation of ( x). To reover the original funtion f ( x ), inverse wavelet transform is defined in Eq. (3), where C is defined in Eq. (4) 85

2 International Journal of Innovation, Management and Tehnology, Vol. 3, o. 3, June and Ψ ( μ) is the Fourier transform of ( x). Eqs. () to (4) define a reversible transformation as long as the so-allled admissibility riterion C < is satisfied [8]. In 989 Fast Wavelet Transform (FWT) is developed and provided a good way of alulating wavelet transforms of a given signal. W (, s ) = f() x () x dx () x ( x) = ( ) s s () ( x) f ( x) = W ( ) d ds C s (3) Ψ( μ) C d = μ (4) μ One dimensional wavelet transforms an be easily extended to two dimensional ases like images. In two dimension a two-dimensional saling funtion φ ( x, y) and H V three two-dimensional wavelet ( x, y), ( x, y) and D ( x, y) are required for the horizontal, vertial and diagonal diretions respetively. Eah of above four funtions is the produt of two one-dimensional funtions as Eqs. (5) (8) show, where φ ( d) is the one dimensional saling funtion along diretion d ; ( d) is the one dimensional wavelet mother funtion along diretion d. In this paper, various two dimensional wavelet funtions are used to deompose images. φ( x, y) = φ( x) φ( y) (5) H ( x, y) = ( x) φ( y) (6) V ( x, y) = φ( x) ( y) (7) D ( x, y) = ( x) ( y) (8) B. Chaoti Maps Generally speaking, the haoti map refers to any map that exhibits some sort of haoti behavior: any slight hange in the initial onditions yields widely diverged outomes whih makes impossible for long-term predition. lthough haoti behavior an be observed in many dynami system we fous on the one dimensional logisti map in this paper. Mathematially, a logisti map an be defined as an iterative funtion in Eq. (9), where r is the parameter and x is used as the initial value. x = rx ( x ) n+ n n (9) Fig.. The haoti logisti map. ormally, parameter r is set to (3.58, 4) and x is set to (,) for keeping the haoti behavior. In Fig., the x oordinate value represents the parameter value r and the y oordinate value represents the generated x sequenes. It is lear to see that for r (3.6, 4), the map generates random-like x sequenes. It is worthwhile to note that sequenes with initial value falling in the region around r = 3.83 are periodi. In order to avoid trapping the sequene into osillation, we simply use r (3.9, 4) in this paper. The purpose of using the logisti map in the paper is to generate a random-like but deterministi sequene for embedded positions. Therefore, the logisti map an be substituted by any haoti map. C. Fratal Images The term fratal was oined by Benoit Mandelbrot in 975 and was derived from the Latin fratus meaning broken or fratured. Mathematially, a fratal is based on an equation undergoing iteration whih is a form of reursive feedbaks. fratal usually has the following harateristis as desribed in [9]: ) It has a fine struture at any arbitrarily small sale. ) It is too irregular to be easily desribed in traditional Eulidean geometri language. 3) It is self-similar or at least approximately similar. 4) It has a simple and reursive definition. 5) Its Hausdorff dimension is higher than its topologial dimension. In this paper, we use the fratal images generated by Ultra Fratal [] as over image eah of whih is uniquely defined by a set of fratal parameters suh as type, formula, sale, loation, olor spae et. In other word the fratal image an be perfetly restored as long as this set of parameters is known. In Ultra Fratal this set of parameters an be saved as a partiular file, whih an be transported very easily. Therefore, the over image an be perfetly restored. III. THE STEGOGRPHIC LGORITHM The proposed steganographi algorithm relies on the wavelet transform and embeds seret data as the wavelet oeffiients in a over image. Here, we assume a over image is a RGB olor image. The flowhart of the proposed steganographi algorithm is skethed in Fig. The details of eah stage will be disussed in the rest of this setion.. Stage I: Transforming Image in Wavelet Domain This stage is the start of the steganography. The olor over image C is deomposed in the YCr olor spae and only the omponent C is used for future proessing. This hoie of omponent is heuristi and is seleted based on our experiment results. This hoie is supposed to ahieve a higher ovariane between the stego image S and the over image C than using Cr or Y omponent. Later both the C and the seret image I are transformed using wavelet W and W respetively, and beome W ( C ) 86

3 International Journal of Innovation, Management and Tehnology, Vol. 3, o. 3, June and W ( I ). Here W and W are determined by the key and they may or may not be neessarily the same. One benefit of this design is to ahieve a larger key spae. Unless the orret wavelet mother funtion W is used, one annot extrat the exat embedded seret information from a over image; unless the orret wavelet mother funtion W is used, one annot restore the seret information from extrated seret information in the spatial domain. Stage I Stage II Seret Image Ι W(Ι) DWT W(S) Logisti Map Seleting Positions Energy Equalization Embedding Data IDWT Stage III W(C) S DWT IYCr C Cover Image C YCr CCr CY Stego Image S Fig.. The flowhart of the proposed steganographi sheme. B. Stage II: Seret Data Embedding This stage is the ore of the steganography. It embeds the transformed seret image W ( I ) in the transformed over image W ( C ). First, the initial value x and the parameter value r of the logisti map in Eq. (9) are determined by the key. Based on the number of oeffiients in W ( I ), the logisti map generates an equal length haoti sequene Seq. However, this sequene Seq has to be quantized before used for extrating wavelet oeffiients in W ( C ). The number of quantization sales is alulated via Eq. (), where #(.) omputes the number of elements and dt. rounds the number towards to zero. Then Seq is quantized with respet to the bin numbers of the equal-length bins in (,), namely k k B =,,, B,,, B, k = = Consequently, Seq is translated to a random-like sequene Seq omposed of only numbers from to. However Seq is still not the sequene whih an be used to extrat wavelet oeffiient in W ( C ). Seq is alulated for partial sum and the Seq is obtained aording to Eq. (). Seq is used to extrat wavelet oeffiients in W ( C ). It is worthwhile to note that W ( C ) is omposed of four regions (see Fig. ), namely approximation region (passed by the low-pass filter along y and followed by the low-pass filter along x ), horizontal-details region (passed by the low-pass filter along y and followed by the high-pass filter along x ), vertial-details region VH(passed by the high-pass filter along y and followed by the low-pass filter along x ), and diagonal-details (passed by the high-pass filter along y and followed by the high-pass filter along x ). We keep region but hoose to substitute oeffiients in, and by the wavelet oeffiients of W ( I ). This is the reason why Eq. () ontains a fator of 3/4. In fat, the region an be onsidered as the DC omponent of the image C. ny hange in this DC omponent may lead to big hanges in the future. However,, and are more or less about the edge strength of C. Fratal over images normally have a large number of edges and rely on the DC omponent of the image to have the fratal-like patterns. 3 #( C ) = () 4 #( W ( I)) k Seq ( k) = Seq ( k) () i= The seleted oeffiients in W ( C ) are uniquely determined by Seq and we named the set of the hosen oeffiients as P. Eq. () is used as the ratio of energy in set X to that in set Y, where E(.) represents the energy funtion. EX ( ) x RXY (, ) = EY ( ) = () y We then ompute L = R( P, W ( I)) and lift wavelet oeffiients orrespondingly, Consequently, the lifted energy LE( W ( I )) approximately math the energy of the original EP ( ). Finally, we all these modified wavelet oeffiients W ( S ). C. Stage III: Transforming Image Bak to Spatial Domain Parallel to the Stage I, the main task of the Stage III is to inverse transform everything in the wavelet domain to the spatial domain. W ( S ) is inverse wavelet transformed to the spatial domain and beomes the image S. The S is used to replae the original omponent C for the olor over image C. Eventually, the omponent set { C, S, C } is onverted Y Cr from the YCr olor spae to the RGB olor spae. IV. SIMULTIO RESULTS D DISCUSSIO Our omputer simulation is run in the MTLB R9a environment under Window XP operation system with 3GB memory and Core Quad.6GHz CPU. The key we used in the simulation has the format of { x, r, width, height, db #, db #}, where x and r are used in C I the logisti map for generating the embedding/ extrating positions; width and height are used to reform the rebuilt 87

4 International Journal of Innovation, Management and Tehnology, Vol. 3, o. 3, June seret image; db # and db # C I refer the used Daubehies wavelets for the over/stego image C/ S and the seret image I respetively. Fig. 3 shows the used over images for simulation. They are all generated from Ultra Fratal []. The image an be saved as a parameter file and thus an be easily transported and perfetly restored in other omputers installed Ultra Fratal. Here all over images are at size of (a) Leaf (b) Feather () Fragment (d) Storm Fig. 3. The over image generated by Ultral Fratal. Fig. 4 shows the test seret images in the simulation. The image of Tufts Logo is of size of 8 8 while the image of ameraman is at size of (PSR) and the root-mean-square error (RMSE) of the embedding results are two ommon measurements for evaluating the quality of reonstruted signal. RMSE is defined by the Eq. (3), where C and S represents the over image and the stego image respetively; m and n refer to the width and height of the image. PSR an be defined based on RMSE as Eq. (4) shows. TBLE I: PSR D RMSE OF STEGO IMGES Tufts Logo Cameraman Cover Image PSR RMSE PSR RMSE Leaf Feather Fragment Storm The PSR and the RMSE test on the stego image using the proposed method are shown in Table I. It is worthwhile to note that what we exatly alulated is RMSE ( C, S ). From Table I, it is easy to see that the over image with a larger portion of homogenous regions produes a lower RMSE and thus a higher PSR. This is beause no wavelet oeffiient has been replaed in the approximation region, whih an be onsidered as the DC omponent the image. Component Color Image Cover Image (a) Tufts Logo (b) Cameraman Fig. 4. The test images used in the simulation. We test all four over images and they produe more or less the similar results. nd thus only the results of fratal over Leaf are shown in Fig. 5 as an example. Here, we use Haar wavelet for both the over image and the seret image. From the point view of human visual inspetion, the seret image and the over image are very lose to eah other. The differenes of two images onentrate on the edge pixel whih are very hard to be notie without omparisons. strong benefit of using this type of fratal images is to largely redue the risk of known over image, beause all used over images are synthesized by a set of fratal parameters. If a over image is not aessible by any unauthorized user, it is then very diffiult to differentiate a true fratal image and a fratal image with seret information. Consequently, the seurity of seret information in the stego image is largely enhaned. m n RMSE( C, S) = ( C( i, j) S( I, j) ) (3) mn i = j = MX PSR log = RMSE (4) In signal proessing, the peaks of the signal-to-noise ratio Stego Image bsolute Differene Fig. 5. The omparison of the over image and the seret image. We also investigated the influene of using different types of wavelet funtions for both the over image and the seret image. These results are shown in Fig. 6, axis indiates the used wavelet funtion and db refers to the Daubehies wavelet family. We tested the Daubehies wavelets (built-in funtions in MTLB) on both the over image C and the seret image I. x Reall we used W for the image C and W for the image I. Here tree image is used as the over image and tufts logo is used as the seret image. The red solid urve is obtained by fixing W = db while keep hanging W ; the 88

5 International Journal of Innovation, Management and Tehnology, Vol. 3, o. 3, June blue dotted urve is obtained by fixing W = db while keep hanging W. We used the ovariane of the over image C and the stego image S to measure the quality, namely a high ovariane value implies a better quality. Generally speaking, the proposed method works well regardless to the seleted wavelet. However, different wavelets do produe different performanes and a periodi-like urve is observed when hanging the wavelet funtion for W. The reason behind this phenomenon is still unknown. Fig. 6. The influene of using different wavelet funtion. V. COCLUSIOS new steganographi method based on wavelet transforms and fratal images are proposed. This new method is able to embed the seret image into a fratal over image without produing notieable hanges. s long as the user has the orret key, the seret image an be extrated from the stego image without knowing the over image. lthough theoretially the attaker is able to sense the positions of the seret image information by omparing the stego image and the over image, it is very safe for the proposed method. Beause the over image itself is onsidered to be seure for it is synthesized and an be uniquely expressed as a set of fratal parameters. In other word the only way for an attaker to obtain the over image is to use the exatly set of fratal parameters and generate the over image. Moreover, the wavelets used to transform the over image and the seret image provide another level of seurity. Even in the worst ase, the attaker raked the over image and knew the seret image position but the seret image annot be derypted without knowing the used wavelets. 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