【正文】 s method offers the advantages of smoothness and adaptation. However, as Coifman and 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 nonTI 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 termbyterm wavelet and Bui extended this neighbouring wavelet thresholding idea to the multiwavelet case. They claimed that neighbour multiwavelet denoising outperforms neighbour single wavelet denoising for 外文文獻原文 5 some standard test signals and reallife 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 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 puting integrals over different orientations and locations. A ridgelet is constant along the lines x1cos_ + x2sin_ = constant. In the direction orthogonal to these ridges it is a have been successfully applied in image denoising recently. In this paper, we bine the dualtree plex wavelet in the ridgelet transform and apply it to image denoising. The approximate shift invariance property of the dualtree plex wavelet and the good property of the ridgelet make our method a very good method for image results show that by using dualtree plex ridgelets, our algorithms obtain higher Peak Signal to Noise Ratio (PSNR) for all the denoised images with di_erent noise anization of this paper is as follows. In Section 2, we explain how to incorporate the dualtree plex 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. IMAGE DENOISING BY USING COMPLEX Ridgelets Discrete ridgelet transform provides nearideal sparsity of representation of both smooth objects and of objects with edges. It is a nearoptimal method of denoising for Gaussian noise. The ridgelet transform can press 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 deposition. 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 very di_erent output wavelet coe_cients. In order to overe this problem, Kingsbury introduced a new kind of wavelet transform, called the dualtree plex wavelet transform, that exhibits approximate shift invariant 外文文獻原文 6 property and improved angular resolution. Since the scalar wavelet is not shift invariant, it is better to apply the dualtree plex wavelet in the ridgelet transform so that we can have what we call plex ridgelets. This can be done by replacing the 1D scalar wavelet with the 1D dualtree plex wavelet transform in the last step of the ridgelet transform. In this way, we can bine the good property of the ridgelet transform with the approximate shift invariant property of the dualtree plex wavelets. The plex ridgelet transform can be applied to the entire image or we can partition the image into a number of overlapping squares and we apply the ridgelet transform to each square. We depose the original n _ n image into smoothly overlapping blocks of sidelength R pixels so that the overlap between two vertically adjacent blocks is a rectangular array of size R=2 _ R and the overlap between two horizontally adjacent blocks is a rectangular array of size R _ R=2 . For an n _ n image, we count 2n=R such blocks in each direction. This partitioning introduces a redundancy of 4 times. In order to get the denoised plex ridgelet coe_cient, we use the average of the four denoised plex ridgelet coe_cients in the current pixel location. The thresholding for the plex ridgelet transform is similar to the curvelet thresholding [10]. One difference is that we take the magnitude of the plex ridgelet coe_cients when we do t。s book, A Friendly Guide to Wavelets but an introductory level knowledge of how basis functions work is necessary to understand the underlying principles of the wavelet theory. Therefore, this information will be presented in this section. T