ImageCompression Using Real Fourier Transform, Its Wavelet Transform And Hybrid Wavelet With DCT

Abstract

This paper proposes new image compression technique that uses Real Fourier Transform. Discrete Fourier Transform (DFT) contains complex exponentials. It contains both cosine and sine functions. It gives complex values in the output of Fourier Transform. To avoid these complex values in the output, complex terms in Fourier Transform are eliminated. This can be done by using coefficients of Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST). DCT as well as DST are orthogonal even after sampling and both are equivalent to FFT of data sequence of twice the length. DCT uses real and even functions and DST uses real and odd functions which are equivalent to imaginary part in Fourier Transform. Since coefficients of both DCT and DST contain only real values, Fourier Transform obtained using DCT and DST coefficients also contain only real values. This transform called Real Fourier Transform is applied on colour images. RMSE values are computed for column, Row and Full Real Fourier Transform. Wavelet transform of size N2xN2 is generated using NxN Real Fourier Transform. Also Hybrid Wavelet Transform is generated by combining Real Fourier transform with Discrete Cosine Transform. Performance of these three transforms is compared using RMSE as a performance measure. It has been observed that full hybrid wavelet transform obtained by combining Real Fourier Transform and DCT gives best performance of all. It is compared with DCT Full Wavelet Transform. It beats the performance of Full DCT Wavelet transform. Reconstructed image quality obtained in Real Fourier-DCT Full Hybrid Wavelet Transform is superior to one obtained in DCT, DCT Wavelet and DCT Hybrid Wavelet Transform.