Image Denoising Using Digital Image Curvelet

Image reconstruction is one of the most important areas of image processing. As many scientific experiments result in datasets corrupted with noise, either because of the data acquisition process or because of environmental effects, denoising is necessary which a first pre-processing step in analyzing such datasets. There are several different approaches to denoise images. Despite similar visual effects, there are subtle differences between denoising, de-blurring, smoothing and restoration. Although the discrete wavelet transform (DWT) is a powerful tool in image processing, it has three serious disadvantages: shift sensitivity, poor directionality and lack of phase information. To overcome these disadvantages, a method is proposed which is based on Curvelet transforms which has very high degree of directional specificity. Allows the transform to provide approximate shift invariance and directionally selective filters while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses where as these properties are lacking in the traditional wavelet transform.Curvelet reconstructions exhibit higher perceptual quality than Wavelet based reconstructions, offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curve linear features. The Curvelet reconstruction does not contain the quantity of disturbing artifacts along edges that we see in wavelet reconstruction. Digital Implementations of newly developed multiscale representation systems namely Curvelets, Ridgelet and Contourlets transforms are used for denoising the image. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained from well established methods based on the thresholding of Wavelet Coefficients. Keywords: Curvelets Transform, Discrete Wavelet Transform, Ridgelet Transform, Peak signal to Noise Ratio (PSNR), Mean Square Error (MSE)

Tomographic reconstruction with non-linear diagonal estimators

In tomographic reconstruction, the inversion of the Radon transform in the presence of noise is numerically unstable. Reconstruction estimators are studied where the regularization is performed by a thresholding in a wavelet or wavelet packet decomposition. These estimators are efficient and their optimality can be established when the decomposition provides a near-diagonalization of the inverse Radon transform operator and a compact representation of the object to be recovered. Several new estimators are investigated in different decomposition. First numerical results already exhibit a strong metrical and perceptual improvement over current reconstruction methods. These estimators are implemented with fast non-iterative algorithms, and are expected to outperform Filtered Back-Projection and iterative procedures for PET, SPECT and X-ray CT devices

The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement