A Fast DCT Algorithm for Watermarking in Digital Signal Processor

Abstract

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.

An Effective Watermarking Method Based on Energy Averaging in Audio Signals

“…To guard against digital signal processing such as MP3 compressing, decompression, low pass filter, time-scaling, and resampling, integration with error correcting code is shown to achieve smaller bit error rate. The performance of the watermarking method is compared with a previous work [19,20], and the results validate that the method is effective on all music types and robust to signal processing.…”

“…where a = 0.488, b = 0.463, c = 0.416, d = 0.192, e = 0.098, f = 0278, and A * = [ I I 0 0 I −I ] with I the 2×2 identity matrix. It has been shown that the fast DCT (FDCT) algorithm needs only 12 multiplications [20].…”

“…A fast DCT algorithm has recently been developed to achieve higher computation efficiency [19][20][21]. By using the algorithm, a watermarking method based on energy averaging of DCT coefficients is presented in this paper for digital audio signals.…”

An Efficient Image Watermarking Method Based on Fast Discrete Cosine Transform Algorithm

“…The original image S is divided into nonoverlapping pixel blocks of size 8×8 and then transformed into frequency domain. Each block is processed independently by two-dimensional (2D) DCT into matrix F. The fast DCT algorithm [18] is applied to decompose the 2D DCT into a pair of 1D DCTs for computation efficiency. For a digital image S, the 2D DCT matrix F in frequency domain and its inverse transform back to spatial domain S can be written in matrix form by…”

“…The 2D spatial data matrix S in (2b) is the linear combination of the base images, which are obtained by the outer product of the column and row vectors of M. By exploiting the redundancy in DCT coefficients, it can be shown that the fast DCT algorithm can reduce the complexity of a 2D DCT of 8 × 8 block from S to F in (2a) in only 24 multiplications, and the inverse transform from F to spatial domain S in (2b) is simply the transpose of the DCT matrix M [18]. Watermarking is performed first by decomposing a true color image into RGB components, and the watermark should be embedded in the main composition of the original image to maintain image quality after watermarking.…”