Discrete cosine transform (DCT) has been an international standard in Joint Photographic Experts Group (JPEG) format to reduce the blocking effect in digital image compression. This paper proposes a fast discrete cosine transform (FDCT) algorithm that utilizes the energy compactness and matrix sparseness properties in frequency domain to achieve higher computation performance. For a JPEG image of8×8block size in spatial domain, the algorithm decomposes the two-dimensional (2D) DCT into one pair of one-dimensional (1D) DCTs with transform computation in only 24 multiplications. The 2D spatial data is a linear combination of the base image obtained by the outer product of the column and row vectors of cosine functions so that inverse DCT is as efficient. Implementation of the FDCT algorithm shows that embedding a watermark image of 32 × 32 block pixel size in a 256 × 256 digital image can be completed in only 0.24 seconds and the extraction of watermark by inverse transform is within 0.21 seconds. The proposed FDCT algorithm is shown more efficient than many previous works in computation.

Methods based on discrete cosine transform (DCT) have been proposed for digital watermarking of audio signals; however, the watermark is often vulnerable to data compression and signal processing. This paper presents an effective audio watermarking method by energy averaging of DCT coefficients such that an audio signal with watermark is robust to data processing. The method is to divide an audio signal into segments by three parameters defining the segment length, the segment sequence of watermark location, and the frequency range of DCT coefficients for watermark location. An error correcting code is also integrated to improve audio signal quality after watermarking. Experimental results show that the method is robust to data compression and many other kinds of signal processing. No original signal is required for decoding the watermark. Comparison of watermarking performance with a recent work validates that the watermarking method has better audio quality and higher robustness.

This paper proposes an image watermarking method based on the fast discrete cosine transform (DCT) algorithm for implementation in digital signal processor. A digital watermark can be effectively embedded and efficiently extracted without the host image. The keys in watermarking process include four frequency coefficients in DCT, two random permutation vectors, and a quantization matrix for normalizing the watermark and the host image. The fast DCT algorithm has been shown to reduce the complexity of two-dimensional image transformation so that embedding/decoding an image watermark can be completed in real time within 0.33 seconds. The quality of both watermarked image and extracted (retrieved) watermark remains excellent. It is shown that the watermarking method is efficient in and robust to data cropping, transmission loss, and compression/decompression.

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