# Exact Unbiased Inverse of the Anscombe Transformation and its Poisson-Gaussian Generalization

##### Mäkitalo, Markku (2013)

Mäkitalo, Markku

Tampere University of Technology

2013

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http://urn.fi/URN:ISBN:978-952-15-3039-5

##### Tiivistelmä

Digital image acquisition is an intricate process, which is subject to various errors. Some of these errors are signal-dependent, whereas others are signal-independent. In particular, photon emission and sensing are inherently random physical processes, which in turn substantially contribute to the randomness in the output of the imaging sensor. This signal-dependent noise can be approximated through a Poisson distribution. On the other hand, there are various signal-independent noise sources involved in the image capturing chain, arising from the physical properties and imperfections of the imaging hardware. The noise attributed to these sources is typically modelled collectively as additive white Gaussian noise. Hence, we have three common ways of modelling the noise present in a digital image: Gaussian, Poisson, or Poisson-Gaussian.

Image denoising aims at removing or attenuating this noise from the captured image, in order to provide an estimate of the underlying ideal noise-free image. For simplicity, denoising algorithms often assume the noise to be Gaussian, and ignore the signal-dependency. However, in an image corrupted by signal-dependent noise, the noise variance is typically not constant and varies with the expectation of the pixel value. Thus, for the removal of signal-dependent noise, we can either design an algorithm specifically for the particular noise model, or use an indirect three-step variance-stabilization approach. In the indirect approach, the noisy image is first processed with a variance-stabilizing transformation (VST), which renders the noise approximately Gaussian with a known constant variance. Then the transformed image is denoised with a Gaussian denoising algorithm, and finally an inverse VST is applied to the denoised data, providing us with the final estimate of the noise-free image.

For the Poisson and Poisson-Gaussian cases, the most well-known VSTs are the Anscombe transformation and the generalized Anscombe transformation, respectively. The former of these was introduced in the late 1940s, and has been in wide use ever since in all areas of applied statistics where the Poisson distribution is relevant.

In addition to a suitable VST, the above indirect denoising process requires a properly designed inverse VST. Unfortunately, the importance of the inverse VST is often neglected. The simple inverse VST obtained by taking the direct algebraic inverse of the VST generally leads to a biased estimate. Usually the issue of bias is addressed by using an asymptotically unbiased inverse, but also this inverse remains highly inadequate, especially for low-intensity data (e.g., for an image captured in a dark environment or with a very short exposure time).

In this thesis, we address this inadequacy by first proposing an exact unbiased inverse of the Anscombe transformation for the Poisson noise model, and then generalizing it into an exact unbiased inverse of the generalized Anscombe transformation for the Poisson-Gaussian noise model. We show that its role in the denoising process is significant, and that by replacing a traditional inverse with an exact unbiased inverse, we obtain substantial gains in the denoising results, especially for low-intensity data. Moreover, in combination with a state-of-the-art Gaussian denoising algorithm, the proposed method is competitive with the best Poisson and Poisson-Gaussian denoising algorithms. We provide extensive experimental results, and supplement them with rigorous mathematical considerations about the optimality and accuracy of the proposed inverses. In addition, we construct a closed-form approximation for both of these exact unbiased inverses.

In practical applications involving noise removal, identifying a suitable noise model does not guarantee accurate denoising results per se, but for the best results we must also be able to produce reasonable estimates of the noise model parameters. Hence, we conclude the thesis by investigating the effect of inaccurate parameter estimation on variance stabilization; based on the theoretical results, we also devise a novel way of estimating Poisson-Gaussian noise parameters from a single image using an iterative variance-stabilization scheme.

Image denoising aims at removing or attenuating this noise from the captured image, in order to provide an estimate of the underlying ideal noise-free image. For simplicity, denoising algorithms often assume the noise to be Gaussian, and ignore the signal-dependency. However, in an image corrupted by signal-dependent noise, the noise variance is typically not constant and varies with the expectation of the pixel value. Thus, for the removal of signal-dependent noise, we can either design an algorithm specifically for the particular noise model, or use an indirect three-step variance-stabilization approach. In the indirect approach, the noisy image is first processed with a variance-stabilizing transformation (VST), which renders the noise approximately Gaussian with a known constant variance. Then the transformed image is denoised with a Gaussian denoising algorithm, and finally an inverse VST is applied to the denoised data, providing us with the final estimate of the noise-free image.

For the Poisson and Poisson-Gaussian cases, the most well-known VSTs are the Anscombe transformation and the generalized Anscombe transformation, respectively. The former of these was introduced in the late 1940s, and has been in wide use ever since in all areas of applied statistics where the Poisson distribution is relevant.

In addition to a suitable VST, the above indirect denoising process requires a properly designed inverse VST. Unfortunately, the importance of the inverse VST is often neglected. The simple inverse VST obtained by taking the direct algebraic inverse of the VST generally leads to a biased estimate. Usually the issue of bias is addressed by using an asymptotically unbiased inverse, but also this inverse remains highly inadequate, especially for low-intensity data (e.g., for an image captured in a dark environment or with a very short exposure time).

In this thesis, we address this inadequacy by first proposing an exact unbiased inverse of the Anscombe transformation for the Poisson noise model, and then generalizing it into an exact unbiased inverse of the generalized Anscombe transformation for the Poisson-Gaussian noise model. We show that its role in the denoising process is significant, and that by replacing a traditional inverse with an exact unbiased inverse, we obtain substantial gains in the denoising results, especially for low-intensity data. Moreover, in combination with a state-of-the-art Gaussian denoising algorithm, the proposed method is competitive with the best Poisson and Poisson-Gaussian denoising algorithms. We provide extensive experimental results, and supplement them with rigorous mathematical considerations about the optimality and accuracy of the proposed inverses. In addition, we construct a closed-form approximation for both of these exact unbiased inverses.

In practical applications involving noise removal, identifying a suitable noise model does not guarantee accurate denoising results per se, but for the best results we must also be able to produce reasonable estimates of the noise model parameters. Hence, we conclude the thesis by investigating the effect of inaccurate parameter estimation on variance stabilization; based on the theoretical results, we also devise a novel way of estimating Poisson-Gaussian noise parameters from a single image using an iterative variance-stabilization scheme.

##### Kokoelmat

- Väitöskirjat [3963]