New steerable pyramid steganography algorithm resistant to the Fisher Linear Discriminant steganalysis

This paper describes a new steganography algorithm based on a steerable pyramid transform of a digital image and the steganalysis of the existence of secret messages hidden by this new method. The data embedding process uses the elements of a Lee and Chen steganography algorithm which is adapted to the steerable pyramid transform domain. This article describes the Fisher Linear Disriminant (FLD) analysis and its steganalysis application, too. The main part of the paper is the description of the conducted research and the results of FLD steganalysis of stegoimages produced by the new steganography algorithm.


Introduction
Steganography is a modern and dynamically developing part of information security which protects information by its hidding. Secure communication with steganography has been known since ancient times when slaves' heads were shaved and tatooed to hide some information [1,2]. When the hair grew up, the message could be delivered. Today steganography has got many different ways to hide information.
Almost all communication channels are digital and many different kinds of data are saved in a digital form i.e.: music, pictures and movies. Digital recording has very useful features for steganography. The first of them is high quality which assures that the information may be equally reconstructed even when it is partially destructed during transmission. The second one is a high time unvariability, even if the digital media were played many times. And, finally, digital data in many cases are redundant, i.e. wave music consists of frequencies not heard to human ear which is utilised during mp3 compression. The redundant data may be replaced by secret information using the steganography algorithm, too.
Our research is concentrated on developing the most secure and efficient steganography method [3,4,5,6] with digital image as a cover of the secret information. To reach this purpose, we have prepared a new steganography algorithm which hides data into digital images. This algorithm was developed during the research done for the PhD dissertation [7] and was described there in more detail. This algorithm possesses interesting features e.g. low cover image destruction and resistance to statistical steganalysis with Fisher Linear Discriminant.

Steerable pyramid transformation
The proposed stego-algorithm is based on a steerable pyramid transformation of a digital image [8,9,10]. In this linear multiresolution the image decomposition is subdivided into a collection of subbands localised in scale and orientation. The subbands have octave bandwidth and orientation bandwidth of 2π/m where m is an integer. Each next scale level is computed recursively using convolution and decimation operations.
This transform has some advantages compared to the wavelet transform which is the most known and frequently used method of multiresolution decomposition. The advantages of this representation are that [10]: L lowpass filters (L 1 are the narrowband filters) and blocks with the sybmol B bandpass filters. The diagram represents the first scale level of the decomposition only. We can get higher scale levels of the decomposition after the recursive including the dashed line around the part of the diagram into the extension point where the input image is after lowpass filtering and decimation. The recursion constraint is: We can describe the system response in the frequency domain as follows: where a indicates the aliasing term which is eliminated by L 1 filter. This elimination is achieved thanks to that L 1 (ω) = 0 for |ω| > π/2. The decomposition subbands are polar-separable in the Fourier domain. The frequency tiling of the single stage for two of the badbass filters are shown in the Fourier magnitude spectrum in Fig. 1b. The subbands of the two-level and four-direction transform of the sample image are shown in Fig. 1c.
The bandpass filters of the transformation are highly constrained. Their Fourier transform radial component must obey a recursive system diagram requirement and the angular component is constrained by the property of steerability [11]. The steerability term means that we can synthesize any direction filter by linear combination of a few basis filters. Each filter from the basis is a rotated copy of each other.
In our case, the steerable basis is a set of n + 1 nth-order directional derivatives of a circular symmetric function. The Fourier magnitude of the i-th oriented bandpass filter can be written in the polar-separable form: where θ = tan −1 (ω y /ω x ), θ i = 2π/m and ω = | − → ω |.
A directional derivative operation in the spatial domain corresponds to the multiplication by a linear ramp function in the frequency domain. We can write a derivative operator in the x direction in the polar coordinates as follows: −jω x = −jω cos (θ) .
As the factor ω is reduced by a radial portion of the function derivative operator is thus cos (θ). We can get higher order directional derivatives by multiplication a ramp raised to a power and the angular portion of the filter is cos (θ) n , where n is the order of the derivative. The steerable pyramid transform has another important feature for steganography. This is a flat system response, which can be described response as follows: It means that the output image after the decomposition into subbands and reverse composition is the same as the input image.

Steganographic algorithm
Our steganographic algorithm is composed of a steerable transformation of the cover image and Lee and Chen steganography method [12] hidding secret message bits through the average values of image pixel blocks modification. The Lee and Chen algorithm was adapted to a new domain as can be seen in Fig. 2. The original method was prepared for grayscale images in the spatial domain. In our case some elements of this algorithm were used for RGB colour space images in the transform domain.
Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 03/11/2022 04:07:17 U M C S At the beginning of the steganography process the cover image is decomposed into a steerable pyramid transformation. The parameters of the transformation, i.e. the number of the scale decomposition levels and the number of different directions of image filtering are parts of stegokey which is needed to extract the hidden message. After that we get a set of different scale subbands and we can choose one to hide information in it. This is the first step shown in the algorithm diagram and the chosen subband index is the second part of the stegokey. The final part of the algorithm is the message embedding. The average value for each coefficient block falls into one of the stegotable subranges and since then the block represents bit value ascribed to this subrange. If the next message bit which we want to hide in the block is equal to the value ascribed to this block, the algorithm modifies the average value into the center value of the stegotable subrange. If these values are different the algorithm changes the average value of the block to the center value of the neighbour stegotable subrange with the equal bit value. The hidden message is destruction resistant unless the average value is modified too much and remains in the same stegotable subrange.
The message is repeated many times during the hidding process to utilise all the coefficient blocks. If all the blocks are modified, the steganography process is harder to detect in a simple manner. The redundance is very important for the message extraction process, too, because the composition of the output image from many transform subbands made distortions into the average values of the coefficients. When the message is extracted by the message recipient, it is the averaged form of many instances to assure right extraction. The bit messages hidden in the three-colour band subbands are averaged, too. Thanks to that we get the simplest error correction based on the Hamming distance [13].
To assure the security of the secret information hidden into the image, the message is preprocessed by a bit permutation with a secret key which is the next part of the stegokey. The message may be encrypted and supplemented with the correction bits but it does not assure the steganographic security. By steganographic security we mean that the message remains undetectable. When the modification is detected by a steganaliser, he may try to destroy the embedded message or to stop the secret communication channel.
This new steganographic algorithm was tested with the RGB images of 256 × 256 pixels resolution and allows us to hide up to about 300 bits of secret information. The distortions made during the steganographic process are quite small SN R ≈ 39dB and in many cases lower those made by other steganographic tools like EzStego [14] or F 5 [15]. The embedding methods result in the resistance of the hidden information (it must be short, about 70 bits) to JP EG compression with a quality factor of 100%, too. It means that the input image is loss compressed with the ratio of 3:1 to 4:1 [7].

Fisher Linear Discriminant Steganalysis
Many steganalysis methods are built as the answers to the particular steganographic algorithms and they detect typical different steganography methods data modification [16,17,18]. We have developed a new information hidding algorithm. However the corresponding steganalysis method does not exist yet. If we want to test the hidden data security we must perform the blind steganalysis process (blind means that we do not know what steganographic technique was used). We should carry out blind steganalysis along with the statistical analysis because the information hidding disturbs the stastistics of the image [18].
Many steganalysis researchers believe that a good method for blind steganalysis was developed by Farid [19,20]. This technique is based on Fisher Linear Discriminant Analysis (FLD) [21]. The Farid research shows that this algorithm is useful for steganalysis images stegoed by the classic steganography tools like Jpeg-Jsteg [22] and OutGuess [23].
The FLD algorithm classifies the objects into separate classes on the basis of their chosen features [24]. In the simplest case it recognises the objects belonging to two classes (Fig. 3).
The steganalysis target is to find that the particular image belongs to the class of stego-modified images or the class of untouched images. Classification features in the Farid's steganalysis method are higher-order image statistics.
At the beginning of the method, the statistic features from the training sets are collected. The first set includes the natural images set and the second modified ones. Then the algorithm looks for the deviation from these models to determine a threshold between two feature classes. Then this threshold is used to classify novel images.
As there exist strong high-order statistical regularities within a wavelet-like decomposition of natural images [21], the statistics are collected from the coefficients of the image decomposition with Quadrature Mirror Filters (QMFs) [25,26]. The decomposition splits the frequency space of the image into multiple scales and three orientations: vertical, horizontal and diagonal. The first statistic set is composed of mean, variance, skewness and kurtosis of subband coefficients at each scale and orientation. The second statistics set is based on errors in an optimal linear predictor of a coefficient value. The subband coefficients are corelated to their spatial and scale neighbours. This fact allows to construct a predictor for the magnitude of coefficients in a subset of neighbours. The predictors for vertical (V ), horizontal (H) and diagonal (D) bands and scale i can be described as follows: We can express the predictor in a matrix form (example for the vertical band): The coefficients are determined by a minimizing quadrature error function: This minimization is calculated by differentiating with respect to w, setting the result equal to zero:

69
The solvation vector is: and the error in the linear predictor can be described as: The second set of statistics: mean, variance, skewness and kurtosis is collected from this linear prediction error. This calculation is performed for all decomposition subbands for all scales and orientations. These statistics are combined with the coefficient statistics and results with 24(n − 1) statistics feature vector which is used for discrimination between classes.
When the features are collected fom two training sets, the F LD calssification process begins. At the beginning the classification algorithm calculates withinclass means for the two classes from the testing sets. These means are defined as: where: x i , i = 1, . . . , N x and y j , j = 1, . . . , N y are the examples from the two classes in the training sets. The second step is the determination of betweenclass mean: Then the within-class S w and between-class S b scater matrices are calculated: where: the i-th column of matrix M x contains the zero-meaned ith example given by x i − µ x . Similarly j-th column of matrix M y is defined.
Let e be the maximal generalized eigenvalue-eigenvector of S w and S b . At the end the training examples are projected onto the one-dimensional linear subspace defined by e (i.e. x T i e and y T j e). During this process the within-class scatter is minimized and the between-class scatter is maximized.
When the projection axis is determined, as can the seen in Fig. 3, a novel example from the testing set is classified by projection onto the same subspace (i.e. z T e).

Research results
Farid in [21] describes the results of steganalysis with the two-class F LD. The steganalysis program was trained with the training set consisting of 1800 unmodified images cropped into 640 × 480 pixel area and random subset of the 1800 stegoed images with a varing message size and the same size. Steganography was performed with Jsteg, OutGuess, EzStego and by substitution of LSB.
The classification accuracy for the tested tools and the message size 256 × 256 were quite large: 94% for the Jsteg stegoed images and 92.8% for OutGuess without statistical correction (option of the tool). For EzStego and LSB the message size was 194 × 194 and the accuracy was lower: 45.2% and 42.3% respectively.
In our case, the new steganography algorithm described above was tested by hidding binary messages within the RGB images of resolution 256 × 256. This type of stegoimage was used to verify steganalysis resistance, too.
Two training sets were prepared, too. The first of them included 400 untouched images and the second was the same set of images but modified steganographically. Different values of steganographic algorithm parameters were set during the preparation of the image set. For example, the number of stegotable subranges or the size of coefficient block used to hide one message bit.  The conducted steganalysis of images stegoed with the new stego-algorithm has shown that the Farid method with the mentioned assumptions (i.e. an image size and quantities of image sets) has not detected any steganographical modifications.
The result of the classification of stegoed images as the images after modification amounts to 0.67% of the tested images only and the result of the classification of untouched images as the images without modification amounts to 98.67% of the tested images. These very good results come from the fact that the features of images from both classes overlap which can be seen on the scatter plots in Fig. 5. It means that the right discrimination is impossible in this case.

Conclusions
We have developed a new steganographic method with a digital image as an information cover and based on a steerable pyramid transform. The information embedding part of the algorithm is a steganography method developed by Lee and Chen tailored to the transform domain. The new method hides about 300 bits of information into 256 × 256 size colour image with minimal cover image distortions (SN R ≈ 39dB) and prevents from the resistance of short (about 70 bits) hidden messages to the JPEG compression with the quality factor 100% and noise adding [7].
This method is resistant to statistic steganalysis with FLD proposed by Farid. This fact results from a minimal data modification and an overlapping class of features taken from untouched and stegoed images.
Our future research will be concentrated on testing the FLD classification with larger image sets and the utilisation of different size covers. Different statistical steganalitic methods, for example Support Vector Machines (SVM) will be verified too.