Inpainting and Denoising Challenges

© Springer Nature Switzerland AG 2019

S. Escalera et al. (eds.)Inpainting and Denoising ChallengesThe Springer Series on Challenges in Machine Learninghttps://doi.org/10.1007/978-3-030-25614-2_1

A Brief Review of Image Denoising Algorithms and Beyond

Shuhang Gu¹   and Radu Timofte¹  

(1)

Computer Vision Laboratory, ETH Zürich, Switzerland

Shuhang Gu

Email: [email protected]

Radu Timofte (Corresponding author)

Email: [email protected]

1 Image Denoising

1.1 Problem Statement

Image denoising aims to recover a high quality image from its noisy (degraded) observation. It is one of the most classical and fundamental problems in image processing and computer vision. On one hand, the ubiquitous use of imaging systems makes image restoration very important to the system performance. On the other hand, the quality of output images plays a crucial role in the user experience and the success of the following high level vision tasks such as object detection and recognition.

A simplified general image degradation model for the denoising task largely adopted in the literature is:

(1)

where x refers to the unknown high quality image (ground truth), y is the degraded observation, and n represents the additive noise. For decades, most of the denoising research has been conducted on the additive white Gaussian noise (AWGN) case. AWGN assumes n to be the independent identically distributed Gaussian noise with zero mean and standard variance value σ. There are also some works [4, 15, 27, 33, 36, 95] which aim to solve Poisson noise removal or pepper and salt noise removal tasks. However, in this review, we focus mainly on the works and the proposed solutions for the AWGN removal task (Fig. 1).

Fig. 1

Standard Lena test image corrupted by different types of noise: Gaussian, Poisson, and pepper-and-salt

The main challenge in image denoising lies in that a significant amount of information has been lost during the degradation process, making image denoising a highly ill-posed inverse problem. In order to get a good estimation of the latent image, prior knowledge is required to provide supplementary information. Therefore, how to appropriately model the prior of high quality images is the key issue in image restoration research.

1.2 Natural Image Prior Modeling for Image Denoising

A wide range of approaches have been suggested to provide supplementary information for estimating the denoised image. According to the image information used the approaches can be divided into internal (use solely the input noisy image) [7, 26, 41] and external (use external images with or without noise) [57, 78, 97, 102] denoising methods. Some works shown that the combination or fusion of internal and external information can lead to better denoising performance [9, 38, 63, 82].

In this review, based on how the prior is exploited to generate high quality estimation, we divide the previous prior modeling methods into two categories:

1.

the implicitly modeling methods and

2.

the explicitly modeling methods.

Each category is further briefly described and reviewed in the next.

1.2.1 The Implicit Methods

The category of implicit methods adopt priors of high quality images implicitly, where the priors are embedded into specific restoration operations. Such an implicitly modeling strategy was used in most of the early years image denoising algorithms [69, 84, 89]. Based on the assumptions of high quality images, heuristic operations have been designed to generate estimations directly from the degraded images. For example, based on the smoothness assumption, filtering-based methods [20, 62, 84, 88] have been widely utilized to remove noise from noisy images. Although image priors are not modeled explicitly, the priors on high quality images are considered in designing the filters to estimate the clean images. Such implicitly modeling schemes have dominated the area of image denoising for decades. To generate the piece-wise smooth image signal, diffusion methods [69, 89] have been proposed to adaptively smooth image contents. By assuming that the wavelet coefficients of natural image are sparse, shrinkage methods have been developed to denoise images in wavelet domains [25, 30]. Based on the observation that natural image contains many repetitive local patterns, the non-local mean filtering approach has been suggested to profile from the image non-local self-similarity (NSS) prior (see Fig. 2). Although these simple heuristic operations have limited capacity in producing high-quality restoration results, these studies greatly deepen researchers’ understanding on natural image modeling. Many useful conclusions and principles are still applicable to modern image restoration algorithm design.

Fig. 2

Similar patches in an image from Set5 marked with coloured rectangles. The non-local self-similarity prior (NSS) refers to the fact that natural image usually contain many repetitive local patterns

Recently, attributed to the advances in machine learning, researchers have proposed to learn operations for image denoising. Different methods have been developed to build the complex mapping functions between noisy and its corresponding clean image [18, 23, 77, 80, 97]. Since the functions (such as neural networks) learned in these methods are often very complex, the priors embedded in these functions are very hard to analyze. As a result, the functions trained for a specific task (denoising task with different noise types) are often inapplicable to other restoration tasks. One may need to train different models for different degradation parameters. Albeit its limited generalization capacity, the highly competitive restoration results obtained by these discriminative learning methods make this category of approaches an active and attractive research topic.

1.2.2 The Explicit Methods

Besides implicitly embedding priors into restoration operations, another category of methods explicitly characterize image priors and adopt the Bayesian method to produce high quality reconstruction results. Having the degradation model p(y|x) and specific prior model p(x), different estimators can be used to estimate latent image x. One popular approach is the maximum a posterior (MAP) estimator:

(2)

with which we seek for the most likely estimate of x given the corrupted observation and prior. Compared with other estimators, the MAP estimator often leads to an easier inference algorithm, which makes it the most commonly used estimator for image restoration. However, MAP estimation still has limitations in the case of few measurements [85]. An alternative estimator is the Bayesian least square (BLS) estimator:

(3)

BLS marginalizes the posterior probability p(x|y) over all possible clean images x. Theoretically, it is the optimal estimate in terms of mean square error, and the estimator is also named as minimum mean square error (MMSE) estimator [85].

A wide range of models, such as Independent Component Analysis (ICA) [5], variational models [75], dictionary learning approaches [3] and Markov Random Field (MRF) [35, 72], have been utilized to characterize priors of natural image. Early studies tend to analyze image signals with analytical mathematical tools and manually designed functional forms to describe natural image prior. Later methods tend to take advantage of training data and learn parameters to better model high quality image priors. Compared with implicit prior modeling methods, these explicit priors are limited in generating highly competitive denoising results [54, 55], but them often have a stronger generalization capacity and can be applied to different image restoration applications.

Both of the two categories of prior modeling approaches have delivered several classical denoising algorithms. In the very beginning of image denoising research, implicit approaches used to dominate the field. Different filtering and diffusion approaches have been designed for image denoising. Since two decades ago, due to the hardware development, the availability of large amount computational resources and the development of optimization algorithms, sparse and low-rank models have been suggested to provide priors for denoising in an explicit way. Most recently, state-of-the-art denoising results are achieved by deep neural networks (DNN)-based denoisers, which directly learn the mapping function between noisy and clean image. Both the degradation model as well as implicit image prior are embedded in the networks.

In this paper, we review previous algorithms in chronological order. As some classical filtering, diffusion, and wavelet based algorithms have been thoroughly reviewed in previous papers [8] and [64], we focus more on recently proposed algorithms. Concretely, we start from the sparse-based models and then introduce the low-rank methods and the DNN-based approaches. In Fig. 3, we provide the timeline of some representative denoising approaches.

Fig. 3

Timeline with a selection of representative denoising approaches

2 Sparse Models for Image Denoising

The idea of using sparse prior for denoising has been investigate from a very early stage of image denoising research. Researches on image statistics have shown that the marginal distributions of bandpass filter responses to natural images exhibit clearly non-Gaussianity and heavy tails [34]. Based on the above observation, shrinkage or optimization approaches have been suggested to get sparse coefficients in the transformation domain. During the last several decades, many attempts have been made to find more appropriate transformation domains as well as sparsity measurements for image denoising. According to different mechanisms to achieve the representation coefficients, Elad et al. [32] divided the sparse representation models into analysis-based and synthesis-based methods. In this section, we review both the two categories of works. In addition, as there are some state-of-the-art algorithms exploiting both the sparse and the non-local self-similarity prior, we also introduce these methods and show how sparse and NSS prior can be combine to achieve good denoising performance.

2.1 Analysis Sparse Representation Models for Image Denoising

The analysis representation approaches represent a signal in terms of its product with a linear operator:

(4)

where x is the signal vector and α a is its analysis representation coefficients. Linear operator P is often referred to as the analysis dictionary [74].

Some early works directly adopt orthogonal wavelet basis as dictionary, and conduct shrinkage operation to sparsify coefficients. Then, inverse transform is applied on the sparse coefficients to reconstruct denoised estimation. A wide range of wavelet basis and shrinkage operations have been investigated to get better denoising performance. A good review of wavelet-based approach can be found in [64].

Although sophisticated shrinkage operations have been designed from different points of view, such a single step shrinkage operation can not achieve very good denoising performance. Iteration algorithms have been suggested to get better results. Under the MAP framework, most of analysis sparse representation models share a similar form:

(5)

where Υ(x, y) is the data fidelity term which depends on the degradation model, and Ψ(Px) is the regularization term which imposes sparsity prior on the filter responses Px.

The analysis dictionary P and the penalty function Ψ(⋅) play a very important role in the analysis sparse representation model. Early studies utilize signal processing and statistical tools to analytically design dictionaries and penalty functions. One of the most notable analysis-based methods is the Total-Variation (TV) approach [75], which uses a Laplacian distribution to model image gradients, resulting in an ℓ 1 norm penalty on the gradients of estimated image. In addition to TV and its extensions [14, 16, 17], researchers have also proposed wavelet filters [10, 28, 58] for analysis sparse representation. In these methods, the gradient operator in TV methods is replaced by wavelet filters to model image local structures. Besides dictionaries, the penalty functions have also been well investigated. Different statistical models have been introduced to model the heavy-tailed distributions of coefficients in natural, which leads to a variety of robust penalty functions, such as ℓ p norm [103], normalized sparsity measure [51], etc.

Although these analytic methods have greatly deepened our understanding on image modeling, they are considered to be over-simplistic to model the complex natural phenomena. With the advance of computing power, machine learning methods have been introduced to learn better priors. From a probabilistic image modeling point of view, Zhu et al. [101] proposed the filters, random fields and maximum entropy (FRAME) framework, which characterizes the distribution of the filter responses over latent image to model image textures. For image denoising, field-of-expert (FoE) [72] is one of the representative works which learn filters (analysis dictionary) for predefined potential (penalty) functions. Inspired by FoE [72], a lot of methods have been proposed to learn better filters for image denoising from a conditional random field (CRF) [81] point of view. All the potential functions adopted in these methods have been selected to lead to a sparse coefficients. Besides the probabilistic point of view, there are still other works proposed to learn analysis dictionary from different frameworks. Ravishankar et al. [71] proposed the transform learning framework, which aims to learn better analytical sparsifying transforms for image restoration. Rubinstein et al. [74] proposed an analysis-KSVD algorithm, which borrows idea from the K-SVD algorithm [3] and learns analysis dictionary from image patches. All the above approaches learn image priors in a generative way, only high quality images are involved in the training phase. Recently, discriminative learning methods have also been utilized to train priors for specific tasks [22, 44]. By using the image pairs as training data, these discriminative learning methods are able to deliver highly competitive restoration results. However, the learning is often achieved by solving a bi-level optimization problem, which is time-consuming.

2.2 Synthesis Sparse Representation Models for Image Denoising

Different from the analysis representation models, the synthesis sparse representation models represent a signal x as the linear combination of dictionary atoms:

(6)

where α s is the