Time Invariant Curvelet Denoising Birgir Bjorn Saevarsson, Johannes R. Sveinsson and Jon Atli Benediktsson University of Iceland Department of Electrical and Computer Engineering Hjardarhaga 2-6, 107 Reykjavik ICELAND E-mail: {birgirsa, sveinsso, benedikt}@hi.is ABSTRACT The purpose of this paper is to develop a method for denoising images corrupted with additive white Gaussian noise (AWGN). The noise degrades quality of the images and makes interpretations, analysis and segmentation of images harder. In the paper the use of the time invariant discrete curvelet transform for noise reduction is con- sidered. The discrete curvelet transform is a new image representation approach that codes image edges more efficiently than the wavelet transform. Edges are very important in image perception and with fewer coefficients to represent edges, a better denoising scheme can be achieved. By making the curvelet transform time invariant greatly reduces the energy of the error resulting in better denoising. 1. Introduction Almost every kind of data contains some kind of noise. For digital images, noise reduction is often a required step for many sophisticated exploring methods such as remote sensing of digital images. The normal representation of a digital image is a matrix of pixels where each pixel measures the brightness of an object. The pixel values for a normal 8-bit grayscale images lie between 0 and 255. Denoising is the process of reducing the noise in the digital images. Denoising usually consists of three stages: 1) Transform the noisy image to a new space, i.e., find a representation which discriminates the image from the noise. 2) Manipulate the coefficients in the new space, i.e., keep the coefficient where the signal to noise ratio is high, reduce the coefficient where the signal to noise ratio is low. 3) Transform the manipulated coefficients back to the original space. In [1], Donoho and Candes proposed the ridgelet trans- form and showed that it is superior to other denoising methods such as Fourier and wavelet denoising at han- dling one dimensional discontinuity, i.e., straight edges. Because regular images have curved edges the ridgelet transform is not sufficient to handle linear discontinuities in images. This is why Donoho and Candes proposed the curvelet transform by utilizing the ridgelet transform [2]. Here a time invariant version of the discrete curvelet transform is proposed by implementing cycle spinning on two of the three subbands of the curvelet transform. The biggest problem with image denoising is the edges of images. By performing cycle spinning the largest error of a denoised image is reduced resulting in lower energy of the error which gives better denoising result [3]. The paper is organized as follows. First, in Section 2, the curvelet transform is introduced. The time invariant dis- crete curvelet transform is discussed in Section 3 and the denoising method is described in Section 4. Experimental results are given in Section 5 and finally, conclusions are drawn in Section 6. 2. CURVELET TRANSFORMATION The discrete curvelet transform for a 256 × 256 image is performed as is shown in Fig. 1. As can be seen in Fig. 1 the discrete curvelet transform can be performed in three steps: 1) The 256 × 256 image is split up in three subbands. 2) Tiling is performed on subbands ∆ 1 and ∆ 2 . 3) Discrete ridgelet transform is performed on each tile. 2.1 Subband Filtering The subband filtering for a 256 × 256 image I is done as follows. The image is split up in three subbands s =0, 1, 2 and undecimated discrete wavelet transformation is used to implement the subband filtering. The 6-tap Daubechies undecimated discrete wavelet transform is used [4]. Fig. 2 shows the the relationship between subbands and the frequency domain where s =0 indicates the basis subband, s =1 indicates the bandpass I ∆ ∆ P 0 1 2 Tiling Tiling each tile transform Ridgelet Ridgelet transform each tile 0 1 2 c c c 256 x 256 256 x 256 16 x 16 32 x 32 256 x 256 256 x 256 512 x 512 512 x 512 1024 x 1024 1024 x 1024 256 x 256 Fig. 1. The flowchart for the discrete curvelet transform. Proceedings of the 6th Nordic Signal Processing Symposium - NORSIG 2004 June 9 - 11, 2004 Espoo, Finland ©2004 NORSIG 2004 117