Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

Image denoising can be described as the problem of mapping from a noisy image to a noise-free image. The best currently available denoising methods approximate this mapping with cleverly engineered algorithms. In this work we attempt to learn this mapping directly with a plain multi layer perceptron (MLP) applied to image patches. While this has been done before, we will show that by training on large image databases we are able to compete with the current state-of-the-art image denoising methods. Furthermore, our approach is easily adapted to less extensively studied types of noise (by merely exchanging the training data), for which we achieve excellent results as well.

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Image denoising: Can plain neural networks compete with BM3D?

We apply basic statistical reasoning to signal reconstruction by machine learning -- learning to map corrupted observations to clean signals -- with a simple and powerful conclusion: it is possible to learn to restore images by only looking at corrupted examples, at performance at and sometimes exceeding training using clean data, without explicit image priors or likelihood models of the corruption. In practice, we show that a single model learns photographic noise removal, denoising synthetic Monte Carlo images, and reconstruction of undersampled MRI scans -- all corrupted by different processes -- based on noisy data only.

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Noise2Noise: Learning image restoration without clean data

We propose a novel despeckling algorithm for synthetic aperture radar (SAR) images based on the concepts of nonlocal filtering and wavelet-domain shrinkage. It follows the structure of the block-matching 3-D algorithm, recently proposed for additive white Gaussian noise denoising, but modifies its major processing steps in order to take into account the peculiarities of SAR images. A probabilistic similarity measure is used for the block-matching step, while the wavelet shrinkage is developed using an additive signal-dependent noise model and looking for the optimum local linear minimum-mean-square-error estimator in the wavelet domain. The proposed technique compares favorably w.r.t. several state-of-the-art reference techniques, with better results both in terms of signal-to-noise ratio (on simulated speckled images) and of perceived image quality.

A Nonlocal SAR Image Denoising Algorithm Based on LLMMSE Wavelet Shrinkage

The last decade has seen an astronomical shift from imaging with DSLR and point-and-shoot cameras to imaging with smartphone cameras. Due to the small aperture and sensor size, smartphone images have notably more noise than their DSLR counterparts. While denoising for smartphone images is an active research area, the research community currently lacks a denoising image dataset representative of real noisy images from smartphone cameras with high-quality ground truth. We address this issue in this paper with the following contributions. We propose a systematic procedure for estimating ground truth for noisy images that can be used to benchmark denoising performance for smartphone cameras. Using this procedure, we have captured a dataset - the Smartphone Image Denoising Dataset (SIDD) - of ~30,000 noisy images from 10 scenes under different lighting conditions using five representative smartphone cameras and generated their ground truth images. We used this dataset to benchmark a number of denoising algorithms. We show that CNN-based methods perform better when trained on our high-quality dataset than when trained using alternative strategies, such as low-ISO images used as a proxy for ground truth data.

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A High-Quality Denoising Dataset for Smartphone Cameras

The removal of Poisson noise is often performed through the following three-step procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest. The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied. We introduce optimal inverses for the Anscombe transformation, in particular the exact unbiased inverse, a maximum likelihood (ML) inverse, and a more sophisticated minimum mean square error (MMSE) inverse. We then present an experimental analysis using a few state-of-the-art denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the low-count regime. This results in a very efficient filtering solution that is competitive with some of the best existing methods for Poisson image denoising.

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Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

Many digital imaging devices operate by successive photon-to-electron, electron-to-voltage, and voltage-to-digit conversions. These processes are subject to various signal-dependent errors, which are typically modeled as Poisson-Gaussian noise. The removal of such noise can be effected indirectly by applying a variance-stabilizing transformation (VST) to the noisy data, denoising the stabilized data with a Gaussian denoising algorithm, and finally applying an inverse VST to the denoised data. The generalized Anscombe transformation (GAT) is often used for variance stabilization, but its unbiased inverse transformation has not been rigorously studied in the past. We introduce the exact unbiased inverse of the GAT and show that it plays an integral part in ensuring accurate denoising results. We demonstrate that this exact inverse leads to state-of-the-art results without any notable increase in the computational complexity compared to the other inverses. We also show that this inverse is optimal in the sense that it can be interpreted as a maximum likelihood inverse. Moreover, we thoroughly analyze the behavior of the proposed inverse, which also enables us to derive a closed-form approximation for it. This paper generalizes our work on the exact unbiased inverse of the Anscombe transformation, which we have presented earlier for the removal of pure Poisson noise.

Optimal Inversion of the Generalized Anscombe Transformation for Poisson-Gaussian Noise

We presented an exact unbiased inverse of the Anscombe variance-stabilizing transformation in [M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process., vol. 20, no. 1, pp. 99-109, Jan. 2011.] and showed that when applied to Poisson image denoising, the combination of variance stabilization and state-of-the-art Gaussian denoising algorithms is competitive with some of the best Poisson denoising algorithms. We also provided a MATLAB implementation of our method, where the exact unbiased inverse transformation appears in nonanalytical form. Here, we propose a closed-form approximation of the exact unbiased inverse in order to facilitate the use of this inverse. The proposed approximation produces results equivalent to those obtained with the accurate (nonanalytical) exact unbiased inverse, and thus, notably better than one would get with the asymptotically unbiased inverse transformation that is commonly used in applications.

A Closed-Form Approximation of the Exact Unbiased Inverse of the Anscombe Variance-Stabilizing Transformation

In digital imaging, there is often a need to produce estimates of the parameters that define the chosen noise model. We investigate how the mismatch between the estimated and true parameter values affects the stabilization of variance of signal-dependent noise. As a practical application of the general theoretical considerations, we devise a novel approach for estimating Poisson–Gaussian noise parameters from a single image, combining variance-stabilization and noise estimation for additive Gaussian noise. Furthermore, we construct a simple algorithm implementing the devised approach. We observe that when combined with optimized rational variance-stabilizing transformations, the algorithm produces results that are competitive with those of a state-of-the-art Poisson–Gaussian estimator.

Noise parameter mismatch in variance stabilization, with an application to Poisson-Gaussian noise estimation.

Synthetic-aperture radar (SAR) imaging has become an efficient tool for obtaining and retrieving useful information about surfaces of Earth and other planets. However, the formed images suffer from speckle noise, especially if single-look observation mode is used. Then, filtering is often applied to improve image quality and provide better estimation of radar cross-section and other parameters of sensed scenes. Recently, a novel class of image filters has proved to be very successful in the removal of additive white Gaussian noise from natural images; these filters are based on nonlocal image modeling, i.e. they exploit the mutual self-similarity of image patches at different locations in the image. These filters have been shown in several benchmarks to significantly outperform all previous techniques. In this paper, we evaluate the performance of nonlocal filters applied to the denoising of single-look SAR images corrupted by speckle with a Rayleigh distribution, taking advantage of exact forward and inverse variance-stabilizing transformations. Numerical simulations demonstrate the success of this approach against several known despeckling methods.

Markku J. Mäkitalo

Papers

Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

Optimal Inversion of the Generalized Anscombe Transformation for Poisson-Gaussian Noise

A Closed-Form Approximation of the Exact Unbiased Inverse of the Anscombe Variance-Stabilizing Transformation

Noise parameter mismatch in variance stabilization, with an application to Poisson-Gaussian noise estimation.

Denoising of single-look SAR images based on variance stabilization and nonlocal filters