An Improving Algorithm Of Finite Ridgelet Transform Using MAP And Its Application In Low Bit Rate Image Compression

Fourier Analysis and Wavelets Analysis are two influential tools in digital image processing.Wavelets have better locally focusing ability in time domain as well as in frequency domain than Fourier transform,However,2-D Wavelets have only horizontal, vertical,and diagonal direction as a tensor product of 1-D Wavelets.As a result,2-D wavelets lose its advantages when processing images with straight edges.In 1998,Candes presented the theory flame of Ridgelet transform in his Ph.D paper[2].After that,Donoho constructed orthogonal ridgelet[3]as well as the corresponding ridgelet transform in 1999.Ridgelet theory is an non-adaptive high dimension representation method which can give the "best" approximation of multi-variable functions with linear singularities.As the Radon transform used by the ridgelet couldn't be realized in digital field,the application of ridgelet is blocked.The Finite Ridgelet Transform proposed by Minh N.Do in 2003^[3] is widely used in image compression and denoising.FRIT could present images with straight lines efficiently.Unfortunately the "wrap around" effect of FRIT influenced the results of its application in image processing.A novel ordering is introduced by Minh N.Do and Martin Vetterli to overcome the "wrap around" effect,using Fourier slices.However,we found it more clearly to explain the effect in time domain considering the different slopes and intercepts of the FRAT lines.First,In this paper,the "wrap around" effect is made more clearly,as we interpret it in the time domain and considering the different slopes and intercepts of the FRAT lines.Second,In this paper,we made a classification of the FRAT coefficients via MAP trying to diminish the influence of the "wrap around" effect,finding more effective image representation method.The MAP method is applied to the coefficients in the Finite Radon Transform(FRAT) in order to select the fitted FRAT columns for 1-DCT and 1-DWT.Third,it can be proved that the discarded rows r(k,l₀),1≤k≤p could be replaced by the low frequency of the wavelets coefficients.In order to keep the continuous of the ranks' coefficients in Radon domain,we select the dots which have the longest distances from Fourier coordinate origin,whose "wrap around" effect is worst.Experiments show that this algorithm can concentrate the coefficients energy more greatly.It performs even better than the other methods in approximating images with straight edges.Last but not least,the denoising problem of additive white Gaussian noise (AWGN) using discrete ridgelet transform(DRIT) is modeled.Based on analysis of statistical characteristics of statistical characteristics of DRIT coefficients,a denoising method is proposed.Denoising experiments are carried out on abundant images containing strong linear singularities and texture components with varying levels of AWGN and the results show that this method achieves prominent improvement in terms of signal to noise ratio and visual quality.We have also compared the compression performance where the singularity line varies its orientation using truncated Gaussian image.In each case,we keep only a few most significant coefficients.Results showed that FRITMAP outperforms all the other methods.It can be very effective in compressing images with straight edges.