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.
Highlights
Image compression is storing images using lesser number of bits than its original size
This paper focuses on Real Fourier Transform which is an orthogonal transform, obtained by combining cosine and sine coefficients of Fourier transform
Wavelet transform is generated from orthogonal Real Fourier Transform by using the algorithm in [4] and hybrid wavelet is generated by following the procedure in [5] and compression of image is studied using all these three transforms
Summary
Image compression is storing images using lesser number of bits than its original size. It separates an image into different frequency components. High frequencies are located in bottom right corner Elimination of these high frequency elements gives transformed image with few low frequency components. If image is reconstructed from such lesser number of transformed, low frequency elements, it gives compressed image without losing much data contents in original image. Wavelet transform coding is preferred over simple orthogonal transform in image compression due to its multi-resolution property. It provides enhanced image quality even at higher compression ratios [2]. Hybrid transformation techniques have come into picture which combines properties of two different transforms [3] It gives compressed image with visually perceptible image quality.
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