Poisson Noise Removal From Images Using the Fast Discrete Curvelet Transform Sandeep Palakkal * and K.M.M. Prabhu † Department of Electrical Engineering Indian Institute of Technology Madras Chennai – 600 036, India Email: * sandeep.dion@gmail.com, † prabhu@ee.iitm.ac.in Abstract—We propose a strategy to combine the variance sta- bilizing transform (VST), used for Poisson image denoising, with the fast discrete Curvelet transform (FDCT). The VST transforms the Poisson image to approximately Gaussian distributed, and the subsequent denoising can be performed in the Gaussian domain. However, the performance of the VST degrades when the original image intensity is very low. On the other hand, the FDCT can sparsely represent the intrinsic features of images having discontinuities along smooth curves. Therefore, it is suitable for denoising applications. Combining the VST with the FDCT leads to good Poisson image denoising algorithms, even for low intensity images. We present a simple approach to achieve this and demonstrate some simulation results. The results show that the VST combined with the FDCT is a promising candidate for Poisson denoising. Index Terms—variance stabilizing transform, fast discrete curvelet transform, nonsubsampled contourlet transform, Poisson denoising. I. I NTRODUCTION Poisson images occur in many situations where image acquisition is performed using the detection of particles, e.g., photons [1]. The signal-to-noise ratio (SNR) of such images is signal dependent and varies across the image plane. Variance stabilizing transforms (VST) such as the Anscombe VST [2] offer a pragmatic solution for Poisson noise removal. The VST stabilizes the variance of the Poisson image to a constant, and the resulting image tends to homoskedastic Gaussian as the intensity of the original image tends to infinity. This makes it possible to employ Gaussian-based denoising methods, which are, unlike Poisson denoising methods, well developed and widely employed. After denoising, the estimated image can be obtained by inverting the VST. A drawback of the VST is that when the intensity level of the input image is very low (i.e., at low SNRs), its performance deteriorates [3]. Multiscale VSTs (MS-VST), proposed in [3], combine the VST with the lowpass filters involved in various multiscale multidirection (MS-MD) transforms such as wavelets. The lowpass filters average out the noise to an extent thereby improving the SNR. Moreover, the MS-MD transforms sparsely capture the intrinsic geometry of an image. Due to these facts, MS-VST- based Poisson denoising methods yield good performance. In [3], MS-VSTs were developed for the wavelet, ridgelet and first generation curvelet transforms [4], and the curvelet was shown to yield the best performance. Recently, we have proposed [5] an MS-VST Poisson image denoising algorithm, based on the nonsubsampled contourlet transform (NSCT) [6]. We have shown that the NSCT, when combined with the VST, yields excellent Poisson denoising performance for low intensity images. Its performance is quite comparable to that of the curvelet, and it outperforms the wavelet by a large margin. The NSCT has less redundancy and computational complexity than the curvelet. In this paper, we propose a strategy to combine the MS-VST with the second generation curvelet transform, specifically the fast discrete curvelet transform (FDCT) [7]. The FDCT is far lessredundant than the NSCT. We apply our proposed FDCT- based MS-VST to denoise Poisson images and demonstrate a few preliminary results. We note that more experiments are to be conducted to arrive at a complete conclusion on the performance of the proposed method. However, the prelimi- nary results show that the FDCT-based MS-VST is indeed a promising candidate for Poisson noise removal. This paper is organized as follows. Section II briefly discusses the MS-VST. In Section III, we provide a brief overview of the implementation of FDCT. The proposed FDCT-based MS-VST is developed in Section IV. The steps involved in Poisson noise removal by using the MS-VSTs is outlined in Section V. Some simulation results are presented in Section VI, and Section VII concludes the paper. II. MULTISCALE VARIANCE STABILIZING TRANSFORM Let X =(X i ) i∈Z 2 , be an observed N × N Poisson noisy image. Each pixel value X i is an independent Poisson random variable, i.e., X i ∼ P (λ i ). The mean λ i of the Poisson variable X i represents the true intensity of the i-th pixel, and the variance σ 2 i = λ i of X i can be considered as noise [1]. The SNR at i-th pixel is λ i , and therefore, it is intensity dependent. The denoising problem aims to estimate the underlying true intensity profile (λ i ) i∈Z 2 of the image. The performance of the VST is improved for low intensity images by preprocessing the input image using a lowpass filter. Let Y be the filtered Poisson image and h be the impulse response of the filter, which is assumed to be of finite length (FIR). Define τ k = ∑ i (h[i]) k , for k =1, 2,..., and assume that the image is locally homogeneous, i.e., λ j−i = λ, for all i within the support of h. The VST for a filtered image is