Proceedings of the 4th National Conference; INDIACom-2010 Computing For Nation Development, February 25 – 26, 2010

Bharati Vidyapeeth’s Institute of Computer Applications and Management, New Delhi

An Overview of Wavelet-Like Transforms And Image Data Hiding

Sushil Kumar1 and S. K. Muttoo2

1Department of Mathematics, Rajdhani College, University of Delhi, New Delhi, India 2Department of Computer Science, University of Delhi, Delhi, India

Email: 1skazad@rajdhani.du.ac.in, 2skmuttoo@cs.du.ac.in ABSTRACT With the rapid development of Internet the Image data hiding techniques have not only been applied for solving the problems in the multimedia security but also have found applications in number of domains such as medical, legal and military image. Image data hiding is the process to insert a message into a cover image so that no distortions or perceptible artifacts are introduced. The number of transforms has been successfully applied in data hiding techniques in the last several decades. Recently Wavelet transforms have been employed in this area of research to obtain better results. Wavelet coefficients can be manipulated for hiding information in images, audio or text. In fact, Wavelet-based algorithms have been used for image compression for over a decade, and they are also emerging in the domain of source coding. But the research on applying the Wavelets to data embedding technique is still too weak. Wavelets rely on a dictionary of roughly isotropic elements occurring at all scales and locations, but lacks directional selectivity. In present paper, we provide a general overview of the application of Wavelet -like transforms in the area of data hiding and propose a secure and robust data hiding algorithm based on wavelet-like transform. We use PSNR as the measure for image quality. KEYWORDS DWT, SLT, STEGANOGRAPHY, PSNR. 1. INTRODUCTION The research on applying the Wavelet to data hiding areas such as Image Steganograhy and Watermarking has become one of the popular growing area in recent years. The wavelet transform and wavelet bases are of critical concerns to researchers. Wavelet transforms are most widely-used tool in signal processing due to its inherent mutiresolution representation akin to the operation of the human visual system. However, the research on applying the wavelets to data hiding techniques is still too weak, only a few publications deal with this topic at present. This paper focuses on this challenging research topic. The wavelet transform and wavelet bases formulate the basic wavelet theory. The Wavelet transforms provide short windows at high frequencies and long windows at low frequencies but cannot yield a discrete-time basis that is optimal with respect to time localization. In fact, Wavelets rely on a dictionary of roughly isotropic elements occurring at all scales and locations, but lacks directional selectivity. The wavelets have poor directional selectivity for diagonal features because the wavelet

filters are separable and real. Thus, the wavelet- related research resulted into development of other transforms, known as directional transforms, and namely 2-D multiscale transforms Ridgelet, Curvelet and contourlet. These transforms represent edges with different orientations with different subbands more efficiently than the Wavelet transforms. To attain the properties such as shift invariance and directional selectivity, Kingsbury [5] in 1999 introduced Complex wavelet transform replacing the tree structure of the conventional wavelet transform with a dual tree. Selesnick [10] have proposed a wavelet- like transform, known as Slantlet transform that can provide better time localization and better signal compression compared to classical Haar-Wavelet transform. Panda and Meher [7] have proved these facts through simulation. The Slantlet transform provides better time-localization due to the shorter supports of component filters. They are essentially piecewise linear filters, have desirable properties of orthogonality and two vanishing moments,have an octave-band characteristic, can exactly provide a scale dilation factor of 2,and provides a multiresolution decomposition . Selesnick [11-13] also proposed another complex wavelet transform, namely double-density dual-tree DWT based on double-density DWT and dual-tree DWT. We give a general review of Wavelet transform in the next section 2 and Wavelet-like transforms in section 3 below. 2. WAVELET TRANSFORMS The continuous wavelet transform were introduced by A. Grossmann, J. Morlet and their coworkers in 1980s. With the construction of a class of wavelet bases by two experts, female mathematicians, Daubechies and a French signal analysit, S. Mallat with a mathematician, Y. Meyer, a novel method for time-frequency analysis has been formed. Wavelets are functions generated from one single function ψ(t), called wavelets, by dilations and translations: Where a is the scaling parameter and b is the shifting parameter.

)(1

)(, a

bt

atba

Unlike other transforms, Wavelet transforms makes us possible to analyse many aspects of the signal like trends, breakdown points, discontinuities in higher derivatives and self-similarity. Wavelets can be used to easily compress or enhance a signal

Proceedings of the 4th National Conference; INDIACom-2010

without much computational load. Wavelet analysis allows the use of long time intervals in situations where more precise low-frequency information is required and shorter regions where high-frequency information is required. The 1-D wavelet transform is given by :

The inverse 1-D wavelet transform is given by:

The Discrete Wavelet Transform (DWT) involves choosing scales and positions based on powers of two. The mother wavelet is rescaled or dilated by powers of two and translated by integers. It converts an input series x0, x1, ..xm, into one high-pass wavelet coefficient series and one low-pass wavelet coefficient series (of length n/2 each) given by:

where sm(Z) and tm(Z) are called wavelet filters, K is the length of the filter, and i=0, ..., [n/2]-1. In practice, such transformation will be applied recursively on the low-pass series until the desired number of iterations is reached. The multi-resolution aspect of wavelets and the flexibility to analyze signals both in time and frequency have enabled their use in a vast number of areas from medical imaging to information security. A filter bank representation is also popular for wavelet decomposition of a signal into low frequency approximate components and high frequency detail components. For a simple 2D image, the figure 2.1 shows the process of subsequent decomposition and reconstruction.

Figure 2.1: Filter-bank representation of a Discrete Wavelet Transform A wavelet transformation of a 2D signal or an image is done resulting in the LL, HL, LH and HH bands as shown in the figure2.2.

Figure 2.2: Wavelet Transform of the ‘LENA’ image (two-level decomposition) A large number of wavelet families exist, each having their specific characteristics that are suitable for different applications. Some of these families are Haar, Daubechies, Symmlets, Coiflets as shown in the Figure 2.3. 3. WAVELET-LIKE TRANSFORMS In image compression, the Wavelet transforms produce much less blocking artifacts than the DCT. They are adopted in JPEG2000. They also perform well in image de-noising. However, 2D wavelet transform is, intrinsically, a tensor-product implementation of the 1D wavelet transform, and it provides local frequency representation of image regions over a range of spatial scales, and it does not represent 2D singularities effectively. Therefore it does not work well in retaining the directional edges in the image, and it is not sufficient in representing the contours not horizontally or vertically.

An Overview of Wavelet-Like Transforms and Image Data Hiding.

Figure 2.3 3.1 SLANTLET TRANSFORM SLT has been recently proposed as an improvement over the classical DWT [8]. SLT is an equivalent form of the DWT implementation but provides better time-localization due to the shorter supports of component filters. The SLT filters are essentially piecewise linear filters, have desirable properties of orthogonality and two vanishing moments, have an octave-band characteristic, can exactly provide a scale dilation factor of 2, provides a multiresolution decomposition [10]. The SLT filterbank is implemented in form of a parallel structure, employing different filters for each scale whereas DWT is usually implemented in form of an iterated filterbank, utilizing a tree structure. SLTs can exactly provide a scale dilation factor of 2 and are less frequency selective due to shorter supports of component filters whereas DWT filters can approximately provide a scale dilation factor of 2 and provide short windows at high frequencies and long windows at low frequencies. In the compression scheme using SLT, the data is first applied to two-level filter structures G0(z), G1(z), G2(z), and G3(z). The output of these filters are down sampled by a factor of 4, which are the transform coefficients of the input data obtained after the convolution operation of the original data with the filter coefficients , as shown in figure 3.2. The transform coefficients are then thresholded using a suitable parameter. The inverse slantlet transform are performed on these thresholded coefficients to reconstruct the original data. The filter coefficients used in the slantlet filter bank as derived in by Selesnick[10] are:

Figure 3.1:(a) Two-scale iterated D2 filterbank and (b) corresponding two-scale Slantlet filterbank.

G0(z)= ( -√10/20 – √2/4) + ( 3 √10/20 + √2/4)z-1

+(-3 √10/20 – √2/4)z-2 +(√10/20 + √2/4)z-3 G1(z)= ( 7√5/80 – 3√55/80) + (- √5/80 - √55/80)z-1 +(-9 √5/80 +√55/80)z-2 + (-17√5/80 + 3√55/80)z-3

+(-17√5/80 - 3√55/80)z-4 + (9√5/80 + √55/80)z-5 + (-√5/80 -√55/80)z-6 +(-7√5/80 - 3√55/80)z-7

G2(z)= (1/16 + √11/16) +(3/16 +√11/16)z-1 +(5/16 + √11/16)z-2 + (7/16 + √11/16)z-3

+(7/16 - √11/16)z-4 + (5/16 - √11/16)z-5

+ (3/16 - √11/16)z-6 +(1/16 - √11/16)z-7

G3(z)=z-3 G2(1/z)

Proceedings of the 4th National Conference; INDIACom-2010

Figure 3.2: The Slantlet filters In the SLT each filter has its length in power of 2 and the filterbank gives a reduction of (2i –2) samples or supports for scale I and the reduction in support approaches one thirds as i increases. The detailed description of this procedure is available in [10]. 3.2 RIDGELET TRANSFORM Since 2D wavelet transform of images exhibits a large number of wavelet coefficients to represent image edges, a new transform, known as Ridgelet transform (RGT), was developed in an attempt to break this limit [1,3]. A basic model for calculating Ridgelet coefficients is to view Ridgelet analysis as a form of wavelet analysis in the Radon domain. The RGT are found to useful in solving the problem of sparse approximation of smooth objects with straight edges. However, in image processing, edges are typically curved rather than straight and Ridgelets alone cannot yield efficient representation. But, if one uses a sufficient fine scale to capture curved edges, such edges get almost straight, therefore Ridgelets are deployed in a localized manner. The continuous Ridgelet transform CRT is defined by CRTf(a,b, θ)= a-1/2 ∫ ( (t-b)/a) Rf(θ , t)dt

R where the Radon transform, Rf(θ , t) is given by Rf( ,t) = ∫ f(x, y)δ ( x cos θ + y sin θ -t) R2 dx dy

Here, (θ , t)ε [0,2π ]x R and δ is the Dirac distribution. Radon transform can be obtained applying the 1D inverse Fourier transform to the 2D Fourier transform restricted to radial lines going through the origin. The computation o f Ridgelet transform can thus be given as: 1. Compute f2, the 2D-FFT of f 2. Compute 1D inverse FFT of each angular lines, 3. Apply the 1D scalar wavelet transform on the resulting

angular lines, and 4. Obtain the ridgelet coefficients

3.3 CURVELET TRANSFORM Starck, Candes and Donoho [2] developed the continuous curvelet transform (CVT) in polar coordinates to represent the curves more efficiently. Infact, CVT is based on multiscale ridgelets combined with a spatial bandpass filtering operation. The CVT uses the directional filter bank to capture the directional curvature information. It is known that the continuous form of CVT is rotation invariant and the expansion to the CVT can produce perfect reconstruction. However the implementation of the CVT in the discrete form is not a trivial issue. The CVT of a n image can be obtained as follows: a. Decomposing the original image into a sequence of

subbands by using a pyramidal tree structured filter-bank b. Windowing each subband into blocks of appropriate size (,

so that windowed parts of smooth lines look straight), depending on the centre frequency of the subband, and

c. applying the discrete Ridgelet transform on these blocks.

3.4 CONTOURLET TRANSFORM The Contourlet transform (CTT) is an efficient directional multiresolution image representation, which differs from the wavelet transform in that the CTT uses non-separable filter banks developed in the discrete form. Thus, the CTT is a true 2D transform, and it overcomes the difficulty in exploring the geometry in digital images due to the discrete nature of the image data. However, CTT has adverse property of showing other types of artifacts due to the discrete approach. Contourlet has the properties Multiresolution (representing images from a coarse level to fine-resoltution level), Localization (Basis elements can be localized in both the spatial and the frequency domains), Critical sampling (representation form a basis, or a frame with small redundancy), Directionality (representation of basis elements oriented at variety of directions) and Anisotropy (capturing of smooth contours in images). The first three properties are also provided by separable wavelets. However, CTT is not shift invariant. A new derivative of CTT called Non-subsampled Contourlet Transform (NSCT) was proposed. Navas et al [8] have proposed a novel blind reversible data hiding algorithm for telemedicine applications using NSCT. The method proposed by them has the advantages of high capacity, imperceptibility and robustness to attacks.

An Overview of Wavelet-Like Transforms and Image Data Hiding.

Contourlet coefficients

Figure 3.4: Contourlet decomposition of level 2 of image: new3.tif M.N. Do and M. Vetterli [4] developed the Contourlet in the discrete form which is defined on the regular grids instead of the polar coordinates and more “digital-friendly”. They proposed the Pyramidal Directional Filter-Bank (PDFB), which overcomes the block-based approach of CVT by a directional filter bank, applied on the whole scale. The CTT has been developed to offer the directionality and anisotropy to image representation that are provided by separable wavelet. CTT is a multiscale and directional decomposition of a signal using a combination of a modified Laplacian Pyramid (LP) and a Directional Filter Bank (DFB). The DFB is similar to the local Radon decomposition in curvelet transform. It is constructed from two building blocks: the first is a two-channel quincunx filter bank [4] with fan filters that divides a 2D spectrum into two categories: horizontal and vertical, and the second is shearing operator where a -450 direction edge becomes a vertical edge. The Contourlet have the 2D frequency partition on the centric-squares, whereas curvelets have the 2D frequency partition on the centric-circles. Further, the Contourlet construction provides a space domain multiresolution scheme that offers flexible refinement for both the spatial resolution and the angular resolution. 4. PROPOSED METHOD There are two processes in the proposed Data hiding method: embedding process and extracting process. 4.1 EMBEDDING PROCESS Input: Image, text Output: Stego image 1. Decompose the cover image on applying 2-D

DWT/Slantlet/CTT. 2. Encode the message into compreesed binary form using

self-synchronizing codes, T-codes .

3. Embed the message in the transform coefficients based on modified LSB method.

4. Apply inverse 2-D DWT/Slantlet/CTT to obtain the stego.

LSB algorithm: The basic idea of LSB embedding is to embed the message bit at the rightmost bits of pixel value so that the embedding method does not affect the original pixel value greatly. The formula for the embedding is as follows:

x’ = x - x mod 2k + b Where k is the number of LSBs to be substituted.

The extraction of message from the high frequency coefficients is given as:

b = x mod 2k 4.2 EXTRACTING ALGORITHM The extraction algorithm is the reverse process of the embedding algorithm. The steps are summarized as follows: Input: Stego image, encoded-key Output: Original message 1. Apply 2-D DWT/Slantlet/CTT on the stego-image. 2. Obtain the selected embedded coefficients and extract the

embedded bits using Modified LSB method. 3. Decodet the secret message into the original message

using T-codes 5. EXPERIMENTAL RESULTS The performance of the proposed algorithm is evaluated by using number of gray images. To assess the visual quality of an image after embedding, we use the PSNR as the measure of distortion due to data hiding. The PSNR is defined as follows:

PSNR = 10 log10 (2552/ MSE),

MSE=(1/N) 2 ∑∑ (xij – x’ij ) 2 , Where xij’s denote the original pixel values, xij’ denote the modified pixel values, and N is the modified dimension of image. Experiments have been performed using MATLAB. The experimental results based on proposed method in terms of PSNR are given in the table 5.1.

Proceedings of the 4th National Conference; INDIACom-2010

Table 5.1: Comparison of Proposed method with existing Haar Wavelet method based on PSNR.

Images DWT Method

SLT Method

CTT Method

Lena.jpg 32.8191 60.1595 59.9006

Peppers.png 32.1525 60.0936 59.8681

Cameraman.tif 24.7191 60.0989 59.9265

Barbara.png 31.1263 60.1161 59.8646

Pool.bmp 27.6687 60.2708 59.8138

Flower.jpg 36.8154 60.0955 59.8544

The PSNR in case of Slantlet and Contourlet are far better than the conventional Haar wavelet method 6. CONCLUSION We have reviewed the wavelet-like transforms in this paper an d proposed a data hiding technique based on them.. The proposed has the advantages of high capacity, better imperceptibility than conventional DCT/ Haar wavelet method. Experimental results have proved that the proposed method outperforms the earlier works in terms of visual quality and embedding capacity. 7. REFERENCES [1]. E.J.Candes, and D.L. Donoho, ‘Ridgelets: a key to higher

dimensional intermittency’, Phil. R.Soc. London A., 357(1999), 2495-2509.

[2]. E.J.Candes, and D.L. Donoho, ‘Curvelets, multiresolution representation, and scaling laws’, Proc. Of Wavelet Applications Signal Image Processing VIII ( Proc. SPIE), San Diego, CA, vol. 4119, 2000, pp. 1-12.

[3]. M.N. Do and M. Vetterli, ‘The finite ridgelet transform for image representation’, IEEE Trans. On Image Processing, 12(2003), 16-28.

[4]. M.N. Do and M. Vetterli,’The Contourlet transform: An efficient directional multiresolution image representation ‘, IEEE Trans. On Image Processing, 14(12)(2005), 2091-2106.

[5]. N.G. Kingsbury, ‘Inage processing with complex wavelets’, Philos. Trans. R. Soc. London A., Math. Phy. Sci., 357(1760)(1999), 2543-2560.

[6]. S. Mallat, ‘A Wavelet tour of signal processing’, Academic Press, 1998

[7]. G. Panda and S.K. Meher, “ An efficient approach to signal compression using slantlet transform”, IETE Journal of Research, Vol. 46, No. 5, September 2000, pp. 299-307

[8]. K.A.Navas, Nisha Mary Mathew, and Sasikumar M.,’ Contourlet based data hiding in medical images’, 2009

[9]. D.Y.Po, and M.N.Do, ‘Directional multiscale modeling of images using the contourlet transform’, IEEE Trans. On Image Processing, 15(6)(2006), 1610-1620.

[10]. Ivan W. Selesnick, “ The Slantlet Transform”, IEEE transactions on signal processing,Vol. 47, No. 5, May 199, pp. 1304-1312.

[11]. I. W. Selesnick, “The double density DWT”, in: Wavelets in Signal and Image Analysis: From theory to Practics, A. Petrosian and F. G. Meyer, (Eds), Norwell, MA: Kluwer, 2001.

[12]. I.W.Selesnick, “The Double-density dual-tree DWT”, IEEE Trans. On Signal Processing, 52(5)(2004), 1304-1314.

[13]. I.W.Selesnick, R. Baraniuk, and N.G. Kingsbury, “The dual-tree complex wavelet transform: A coherent framework for multiscale signal and image processing”, IEEE Signal Proc. Magazine, (2005) 123-151.

[14]. Sushil Kumar and S.K. Muttoo,”Distortionless Data Hiding based on Slantlet Transform”, in MINES 2009, Wuhan, China

[15]. Titchener, M.R., ”Generalised T-codes: extended construction algorithm for self- synchronization codes”, IEE Proc. Commun., Vol. 143, No.3, 1996, pp. 122-128.

DWT SLT stego-image stego-image

stego-image stego-image

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An Overview of Wavelet-Like Transforms and Image Data Hiding.

stego-image stego-image

stego-image stego-image

stego-image stego-image

stego-image stego-image

stego-image stego-image

Figure 5.1: Stego images of proposed method based on a)DWT and b)SLT : lena.if, peppers.png, Barbara.png, lena.jpg, pool.bmp, cameraman.png, F;lower.jpg