Translation Invariant Wavelet Denoising of Poisson Data

Abstract

In this paper, we propose a novel image denoising algorithm to reduce Poisson noise based on state-of-the-art wavelet domain statistical models. We propose an efficient method to estimate the model parameters from the observations, and solve the optimization problem in orthonormal and translation invariant (TI) wavelet domains. It is observed that TI wavelet transforms produce better estimation performance than orthogonal wavelet transforms. We briefly explain this observation from the viewpoint of approximation theory and minimax estimation theory. I. Reduction of Poisson Noise In many applications, due to the imperfection of imaging systems, the observed image is a degraded version of the original scene. In this paper, we consider the Poisson imaging problem: the observations {yi} are independently Poisson distributed with intensity {fi}. Image denoising aims at obtaining a good estimate of the original image f from the degraded observations y. In image processing literature, Poisson denoising problems have been approached in the spatial domain, see [1, 2, 3, 4, 5] for examples. More recently, methods based on transform-domain models gained in popularity. For instance, wavelet representations are often more suitable to distinguish signal and noise due to energy compaction. Signal singularities such as edges and textures are often modeled based on wavelet representations. Multiscale estimation of Poisson processes has been addressed in [6] and extended in [7]. There, Poisson intensities are estimated in a coarse-to-fine manner using Haar wavelets. Each wavelet coefficient corresponds to a dyadic size square tile in the spatial domain image. The intensity of each tile is related to that at the next finer resolution using a Beta distribution. The estimation methods in [6, 7] are also extended to TI wavelet domain. However, the methods described in [6, 7] are not applicable to other types of wavelets. In this section, we propose a novel and efficient Poisson noise reduction algorithm based on state-of-the-art statistical wavelet models. We consider statistical models for f , and compute the maximum a posteriori (MAP) estimator: f̂ = arg min f [ − log p(y|f) + Φ(f) ] , (1) where Φ(f) = − logπ(f) penalizes unlikely estimates, and π(f) is the prior for f . II. Regularization in wavelet domain Research in image compression has given rise to sophisticated wavelet domain statistical models. Based on these models, we formulate the regularization penalty in the wavelet domain. We use the notation Φ(f̃) to denote the wavelet domain regularization penalty, where f̃ is the wavelet domain representation of the original image. A Choice of the penalty function The models considered for Φ(f̃) are listed below. The first two models are classical ones and are used for comparison purposes. The third one is a more accurate model that accounts for the spatial inhomogeneity of wavelet coefficients [8, 9]. 1. Gaussian model. Assume the wavelet coefficients within a subband j are iid N(0, σ f̃ ,j ). The corresponding penalty takes the form Φ(f̃) = 1 2 ∑ j,i f̃ 2 j (i)/σ 2 f̃ ,j , where f̃j(i) denotes the signal coefficient in subband j at location i. This quadratic penalty is used for instance in [10]. 2. Laplacian model. The statistics of wavelet coefficients within a subband j are modeled as iid Laplacian [11]. The penalty function Φ(f̃) is thus the l norm. 3. EQ model [8]. It is assumed that wavelet coefficients are independent Gaussian with zero mean and slowly varying variance. Indeed it has been verified [9] that wavelet coefficients normalized by their local standard deviation estimated from a small neighborhood approximately follows the N(0, 1) distribution. Under this model, the regularization penalty takes the form Φ(f̃) = 1 2 ∑ j,i f̃ 2 j (i)/σ 2 f̃ ,j (i), where σ f̃ ,j (i) is the local signal variance in subband j, at location i. The three models described above make different assumptions about the wavelet coefficients in the detail subbands. For the approximation subband (coarse scale), coefficients are modeled as stationary Gaussian with positive mean. The difference between each coefficient and the average of its four neighbors is iid Gaussian. As a benchmark, we also consider a spatial domain Gaussian Markov random field (MRF) model which assumes that the difference between each pixel and the average of its four neighbors is iid Gaussian. B Gradient descent algorithm Gradient descent algorithms can be applied to solve the optimization problem (1). Under the Poisson imaging model yi ∼ Poisson(αfi), where α is a positive constant controlling signal-to-noise ratio (SNR), the cost function to be minimized in (1) takes the form

4 Figures and Tables

Cite this paper

@inproceedings{Liu2001TranslationIW, title={Translation Invariant Wavelet Denoising of Poisson Data}, author={Juan Liu and Pierre Moulin}, year={2001} }