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Mathematical Problems in Engineering
Volume 2017, Article ID 7401845, 7 pages
Research Article

A Fast DCT Algorithm for Watermarking in Digital Signal Processor

1Department of Computer Science and Information Engineering, Chang Jung Christian University, Tainan City 701, Taiwan
2Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan City 701, Taiwan

Correspondence should be addressed to S. M. Yang; wt.ude.ukcn.liam@gnayms

Received 3 November 2016; Accepted 23 January 2017; Published 9 February 2017

Academic Editor: Xinkai Chen

Copyright © 2017 S. E. Tsai and S. M. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 of block 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.