Efficient 2-D DCT Computation from an Image Representation Point of View

Abstract

Discrete Cosine Transform (DCT) constitutes a powerful tool in signal processing, since its first introduction (Ahmed et al., 1974). It belongs to a class of mathematical operations that includes the well-known Fast Fourier Transform (FFT), having as basic operation taking a signal and transforming it from one type of representation to another. More precisely, DCT transforms a signal from the spatial domain to the frequency space, with minimum information redundancy, since its kernel functions (cosines) comprise an orthogonal basis. The main advantage of the DCT transform is its high energy compactness and thus the resulted DCT coefficients fully describe the signal in process. This benefit in conjunction with its implementation simplicity has inspired the scientists to use DCT as the basic transform in the well known image compression standard calling JPEG (ISO/IEC, 1994). Particularly, a 2-D version of the DCT is used to compute the projection of an image in the orthogonal basis of cosines functions, by resulting to a set of coefficients that constitutes the DCT image domain. According to the JPEG standard these coefficients are being compressed in a next step by applying a specific quantization table and an entropy coding procedure. Besides the usage of the 2-D DCT as part of image compression algorithms, it is widely used as feature extraction or dimensionality reduction method in pattern recognition applications (Sanderson & Paliwal, 2003; Er et al., 2005; Jadhav & Holambe, 2008; Liu & Liu, 2008), in image watermarking and data hiding (Qi et al., 2008; Choi et al., 2008; Alturki et al., 2007; Wong et al., 2007) and in various image processing applications (See et al., 2007; Krupinski & Purczynski, 2007; Abe & Iiguni, 2007). From the above it is obvious that the applicability range of the 2-D DCT is wide and increases continuously. This fact has motivated many researchers to investigate and develop algorithms that accelerate the computation time needed to calculate the DCT coefficients of an image. As a result of this massive research, many algorithms that present high computation rates were proposed (Zeng et al., 2001; Diab et al., 2002; Wu et al., 2008; Shao et al., 2008; Plonka & Tasche, 2005) and many hardware implementations were presented (Nikara et al., 2006; Tan et al., 2001) through the years.