32 TRANSFORM DOMAINTransform domain is a mathematical procedure

Categories: Math


Transform domain is a mathematical procedure done over data (image files or music files) that converts it from one domain (for example time) to another (say frequency), generally performing Fourier's or Laplace's Transforms. In this domain the data can be handled much easily in cases of lossy compression, de-noising, sharpening, etc. After editing the data is transformed back to its original domain.

DCT Steganography

The discrete cosine transform (DCT) is closely related to Discrete Fourier Transform (DFT) but it offers greater energy compaction property in comparison with DFT for digital images.

It is a real domain transform which represents an image as coefficients of different frequency of cosine which is a basis vector for this transform. In digital imaging systems, DCT is used to convert the image in spatial domain to its corresponding frequency domain.

With reference to Niels Provos [31], for each color component, the JPEG image format utilizes a discrete cosine transform (DCT) to transform successive 8 ? 8 pixel blocks of the image into 64 DCT coefficients each.

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The DCT coefficients F (a, b) of an 8 ? 8 block of image pixels f(x, y) are given by-

7 7

F(a,b) = ? C(a)C(b) ? ? f(x,y) x Z (2)

x=0 y=0

where Z = cos??(2x+1)a?/16?+sin??((2x+1)b?)/16?

Here C(x) = 1/2 when x equal 0 and C(x) = 1 otherwise. Furthermore, the following operation quantizes the coefficient which is an effective algorithm for data hiding in transform domain-

'JSteg Algorithm'-

Input: secret data, cover image

Output: stego image


while secret data left to embed


get next DCT coefficient from cover image

if DCT ? 0 and DCT ? 1 then

get next LSB from secret data

replace DCT LSB with secret data LSB

end if

insert DCT into stego image

end while

As it runs, the JSteg algorithm sequentially replaces the least-significant bit of discrete cosine transform (DCT) coefficients with secret data.

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It does not require a shared secret for improvising secrecy.

B.G. Banik [36], describes an effective algorithm which could embed secret data, embedded by Arnold Transform in frequency domain using the quantization coefficient alteration in Discrete Cosine Transform (DCT). Initially the cover image is split into blocks, then two dimensional DCT is applied on each image block and the secret data transformed using Arnold transform is induced by computing mid-band coefficients of DCT. In two dimensional DCT, 1D DCT is performed two times- first in the x direction, followed by y direction. At first the cover image is bifurcated into 8x8 pixel block and on each pixel block two-dimensional discrete cosine transform (2D DCT) is applied which results in a 8x8 DCT coefficient matrix. After the secure transferring of the cover image to the receiver, the secret data is efficiently retrieved in high quality from the Stego image.

Y.M. Behbahani [37] propose an advanced JPEG-based steganography technique, using eigen characteristics of matrices of quantized DCT coefficients. This new approach is called as Eigenvalue Steganography. It is known that conventional steganography techniques which are based upon JPEG images, embed secret data by modifying the least significant bit (LSB) of the DCT coefficients. Eigenvalue Steganography method embeds secret confidential data by incorporating the eigenvalues of quantized DCT matrices and modifying them. This method suggests the further division of quantized DCT coefficients blocks into smaller submatrices in DCT domain. After that, the method efficiently embeds the secret confidential information into these sub-divided DCT submatrices, by incorporating the proposed algorithm and modifying the eigenvalues of these submatrices in DCT domain. Mathematically in this approach, 8?8 DCT blocks of image are bifurcated into smaller non overlapping submatrices of size 2?2. Each of these 2*2 submatrices consists of attributes called eigenvalues and eigenvectors. In Eigenvalue Steganography, modifying these Eigen attributes, the sender embeds the confidential data in the DCT coefficients of the image. The recipient's end is able to retrieve the secret data by using the proper division of 8?8 blocks in DCT domain and producing corresponding 2?2 submatrices and further estimating their corresponding eigenvalues because after the embedding procedure, Eigenvalue of each submatrix of stego-image consists of a hexadecimal value of the confidential data. Experimental outcomes in the paper clarifies that the two approaches that are introduced to implement an effective Eigenvalue Steganography possess ample amount of resistance and robustness against a steganalysis scheme known as Subtractive Pixel Adjacency Matrix (SPAM) steganalyzer. This steganalysis tool fails in detecting properly the secret data from the cover image. The embedding capacity and efficiency is also improved by the proposed methods.

R.Biswas [38] describes an algorithm using DCT domain which demonstrates a new area in color image steganography in frequency domain. The proposed algorithm is the exploration of a deft method for image-secret data-steg pass based sampling along with encryption and embedding in frequency domain with a variable bit retrieval function where the secret data becomes much more secure by combining it with the steg password, such that without the knowledge of the steg password the confidential data cannot be identified. The strong amalgamation amongst the color images, secret data and the steg password, varied with a pixel dependant embedding in DCT domain generates a highly protected, robust and reliable substitution solution. A group of 8?8 quantized DCT Coefficient (QDC) is selected as the secret data carrier for the color image. The variable bit operation is applied to the proper QDCs to hide a byte of secret data, where the variable bit operation is dependent on the pixel value. Rigorous and Assiduous statistical analysis has been done and demonstrated to highlight the security features of the algorithm against varied steganalysis methods. Experimental results also reveal its huge carrier capacity and stego image quality.

With reference to [42], Ahmed ElSayed in his paper elaborates a highly robust secret data hiding model in a cover image by utilizing the DCT encryption and the low frequency Curvelet domain. This approach under DCT domain comes with high security because it is consists of four secret keys namely the data hiding key, the encryption key and the two shuffling keys plus utilizing only the frequency component which are low in the Curvelet domain. The usage of low these frequency components of Curvelet transform in image steganography provides numerous advantages compared to other steganography approaches such as: 1) Reduction in computation time and 2) These Curvelet transform are modeled to manage the discontinuities in curves by using only a small number of coefficients, so hiding secret data in the low frequency components will not affect the coefficients of edges which in turn results in better quality of the stego-image. The proposed algorithm is divided into two parts namely the sender's algorithm and the receiver's algorithm. Both the algorithms are implemented in the curvelet domain.

Achmad Solichin [46] proposed the combination of the cryptographic method called Data Encryption Standard (DES) algorithm and the steganographic method under Discrete Cosine Transform (DCT) domain to generate a secure digital data model for communication. The developed security application can be utilized to secure document data in formats like Word, Excel, PowerPoint or PDF. Data is encrypted with DES algorithm and is further embedded in cover image using basic DCT algorithm. This combination of cryptography and steganography resulted in improved security level of the confidential data and along with that the stego-image quality is also not disrupted with stego-image having average PSNR value of 46.9 db. Furthermore, the experimental analysis determine that the average computational time is 0.75 millisecond/byte, an average size increases 4.79 times and a success rate is 58%. It also portrays that several other cryptographic methods like AES (Advanced encryption standard), Blowfish, RSA etc. can be combined with steganographic techniques to improve data security for communication over vulnerable networks. It also proposes that the level of data security also depends upon the resolution of the cover image which in case of DES and DCT combined is specified to be 1024x720 or more. The computation time for embedding data and retrieval process can be optimized by using different combinations of cryptographic and steganographic methods. In this case the two methods resulted in a computation time of 0.75 milliseconds/bytes. The obvious advantage of using them in collaborative way is existence of multiple layers of security.

Furthermore Xianhua Song [47] proposed a steganography method based upon integer DCT and affine transformation. Integer DCT is found to be an appropriate domain for image steganography because it is both lossless and invertible, but the modification of the DCT coefficients can damage its Laplacian-shape like distribution. To establish the security of the method, affine transformation is used to retrieve the Laplacian-shape-like distribution of the integer DCT coefficients. Experimental outcomes portray that the proposed method can ensure the stego image visually and practically undetectable secret data even with very large payload. The histogram based analysis detection also fails in detecting the steganographic approach applied here. The confidential secret data bits are embedded in the integer DCT coefficients, thus causing its histogram distribution to appear fluctuant and skeptical for attackers. The method deploys two levels of embedding schemes for secret data. The first embedding scheme deals with the integer DCT coefficients and the second one deals with the affine transformation which is effective in a manner that inverse operations like scaling and rotation will aid in retrieving the secret data completely without any disruption.

DWT Steganography

As per digital signal processing in Discrete wavelet transform (DWT) wavelets are discretely sampled. It has advantage over Fourier transform in terms of temporal resolution because it captures both the frequency and the location information. It has also come into top priority for image steganography due to its accurate and parallel behavior with human vision system (HVS).

Abdelwahab [19] in his paper proposes an introductory wavelet-based image data hiding approach. The introduced method ensembles to high security and robustness against several steganalysis approaches including image processing operations such as JPEG (Joint Photographic Experts Group) compression, image cropping, image blurring, image sharpening, median image filter, and noise addition to the stego-image. After the application of the discrete wavelet transform on an image, four different sub-images are acquired:

  1. LL: It contains the low frequency components of the image. Also it is a coarser approximation to the original image containing the comprehensive information about the image. It is obtained by applying the low-pass filter on both x and y coordinates.
  2. HL and LH: They are obtained by applying the high-pass filter on one coordinate and the low-pass filter on the other coordinate.
  3. HH: It consists of the high frequency component of the image in the diagonal direction. It is obtained by applying the high-pass filter on both x and y coordinates.

The embedded secret image data can be identified when extracted effectively. The stego-image has high Peak Signal to Noise Ratio (PSNR) value and it is perceptually similar to the original cover image, which maintains the secrecy of the hidden image data and it remains unnoticeable. The proposed method does not require the original cover image to extract the embedded secret image. The secret image is hidden inside a cover image using two secret keys to generate a stego-image which in turn incorporates high robustness.

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32 TRANSFORM DOMAINTransform domain is a mathematical procedure. (2019, Nov 22). Retrieved from http://studymoose.com/32-transform-domaintransform-domain-is-a-mathematical-procedure-example-essay

32 TRANSFORM DOMAINTransform domain is a mathematical procedure
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