Complex Ridgelets for Image Denoising1 IntroductionWavelet transforms have been successfully used in many scientific fields such as image compression, image denoising, signal processing, computer graphics,and pattern recognition, to name only a few.Donoho and his coworkers pioneered a wavelet denoising scheme by using soft thresholding and hard thresholding. This approach appears to be a good choice for a number of applications. This is because a wavelet transform can compact the energy of the image to only a small number of large coefficients and the majority of the wavelet coeficients are very small so that they can be set to zero. The thresholding of the wavelet coeficients can be done at only the detail wavelet decomposition subbands. We keep a few low frequency wavelet subbands untouched so that they are not thresholded. It is well known that Donohos method offers the advantages of smoothness and adaptation. However, as Coifmanand Donoho pointed out, this algorithm exhibits visual artifacts: Gibbs phenomena in the neighbourhood of discontinuities. Therefore, they propose in a translation invariant (TI) denoising scheme to suppress such artifacts by averaging over the denoised signals of all circular shifts. The experimental results in confirm that single TI wavelet denoising performs better than the non-TI case. Bui and Chen extended this TI scheme to the multiwavelet case and they found that TI multiwavelet denoising gave better results than TI single wavelet denoising. Cai and Silverman proposed a thresholding scheme by taking the neighbour coeficients into account Their experimental results showed apparent advantages over the traditional term-by-term wavelet denoising.Chen and Bui extended this neighbouring wavelet thresholding idea to the multiwavelet case. They claimed that neighbour multiwavelet denoising outperforms neighbour single wavelet denoising for some standard test signals and real-life images.Chen et al. proposed an image denoising scheme by considering a square neighbourhood in the wavelet domain. Chen et al. also tried to customize the wavelet _lter and the threshold for image denoising. Experimental results show that these two methods produce better denoising results. The ridgelet transform was developed over several years to break the limitations of the wavelet transform. The 2D wavelet transform of images produces large wavelet coeficients at every scale of the decomposition.With so many large coe_cients, the denoising of noisy images faces a lot of diffculties. We know that the ridgelet transform has been successfully used to analyze digital images. Unlike wavelet transforms, the ridgelet transform processes data by first computing integrals over different orientations and locations. A ridgelet is constantalong the lines x1cos_ + x2sin_ = constant. In the direction orthogonal to these ridges it is a wavelet.Ridgelets have been successfully applied in image denoising recently. In this paper, we combine the dual-tree complex wavelet in the ridgelet transform and apply it to image denoising. The approximate shift invariance property of the dual-tree complex wavelet and the good property of the ridgelet make our method a very good method for image denoising.Experimental results show that by using dual-tree complex ridgelets, our algorithms obtain higher Peak Signal to Noise Ratio (PSNR) for all the denoised images with di_erent noise levels.The organization of this paper is as follows. In Section 2, we explain how to incorporate the dual-treecomplex wavelets into the ridgelet transform for image denoising. Experimental results are conducted in Section 3. Finally we give the conclusion and future work to be done in section 4.2 Image Denoising by using ComplexRidgelets Discrete ridgelet transform provides near-ideal sparsity of representation of both smooth objects and of objects with edges. It is a near-optimal method of denoising for Gaussian noise. The ridgelet transform can compress the energy of the image into a smaller number of ridgelet coe_cients. On the other hand, the wavelet transform produces many large wavelet coe_cients on the edges on every scale of the 2D wavelet decomposition. This means that many wavelet coe_cients are needed in order to reconstruct the edges in the image. We know that approximate Radon transforms for digital data can be based on discrete fast Fouriertransform. The ordinary ridgelet transform can be achieved as follows:1. Compute the 2D FFT of the image.2. Substitute the sampled values of the Fourier transform obtained on the square lattice with sampled values on a polar lattice.3. Compute the 1D inverse FFT on each angular line.4. Perform the 1D scalar wavelet transform on the resulting angular lines in order to obtain the ridgelet coe_cients.It is well known that the ordinary discrete wavelet transform is not shift invariant because of the decimation operation during the transform. A small shift in the input signal can cause