With the explosion in the number of digital images taken every day, the demand for more accurate and visually pleasing images is increasing. However, the images captured by modern cameras are inevitably degraded by noise, which leads to deteriorated visual image quality. Therefore, work is required to reduce noise without losing image features (edges, corners, and other sharp structures). So far, researchers have already proposed various methods for decreasing noise. Each method has its own advantages and disadvantages. In this paper, we summarize some important research in the field of image denoising. First, we give the formulation of the image denoising problem, and then we present several image denoising techniques. In addition, we discuss the characteristics of these techniques. Finally, we provide several promising directions for future research.
where y is the observed noisy image, x is the unknown clean image, and n represents additive white Gaussian noise (AWGN) with standard deviation σn, which can be estimated in practical applications by various methods, such as median absolute deviation [5], block-based estimation [6], and principle component analysis (PCA)-based methods [7]. The purpose of noise reduction is to decrease the noise in natural images while minimizing the loss of original features and improving the signal-to-noise ratio (SNR). The major challenges for image denoising are as follows:
VB6 Image Resizing and more with FREEIMAGE
Owing to solve the clean image x from the Eq. (1) is an ill-posed problem, we cannot get the unique solution from the image model with noise. To obtain a good estimation image \( \hatx \), image denoising has been well-studied in the field of image processing over the past several years. Generally, image denoising methods can be roughly classified as [3]: spatial domain methods, transform domain methods, which are introduced in more detail in the next couple of sections.
Originally, linear filters were adopted to remove noise in the spatial domain, but they fail to preserve image textures. Mean filtering [14] has been adopted for Gaussian noise reduction, however, it can over-smooth images with high noise [15]. To overcome this disadvantage, Wiener filtering [16, 17] has further been employed, but it also can easily blur sharp edges. By using non-linear filters, such as median filtering [14, 18] and weighted median filtering [19], noise can be suppressed without any identification. As a non-linear, edge-preserving, and noise-reducing smoothing filter, Bilateral filtering [10] is widely used for image denoising. The intensity value of each pixel is replaced with a weighted average of intensity values from nearby pixels. One issue concerning the bilateral filter is its efficiency. The brute-force implementation takes O(Nr2) time, which is prohibitively high when the kernel radius r is large.
Spatial filters make use of low pass filtering on pixel groups with the statement that the noise occupies a higher region of the frequency spectrum. Normally, spatial filters eliminate noise to a reasonable extent but at the cost of image blurring, which in turn loses sharp edges.
Starting with Tikhonov regularization [20, 21], the advantages of non-quadratic regularizations have been explored for a long time. Although the Tikohonov method [20, 21] is the simplest one in which R(x) is minimized with the L2 norm, it over-smooths image details [22, 23]. To solve this problem, anisotropic diffusion-based [24, 25] methods have been used to preserve image details, nevertheless, the edges are still blurred [26, 27].
At present, most research on image denoising has shifted from local methods to non-local methods [50,51,52,53,54,55]. For instance, extensions of non-local methods to TV regularization have been proposed in refs. [37, 56]. Considering the respective merits of the TV and NLM methods, an adaptive regularization of NLM (R-NL) [56] has been proposed to combine NLM with TV regularization. The results showed that the combination of these two models was successful in removing noise. Nevertheless, structural information is not well preserved by these methods, which degrades the visual image quality. Moreover, further prominent extensions and improvements of NSS methods are based on learning the likelihood of image patches [57] and exploiting the low-rank property using weighted nuclear norm minimization (WNNM) [58, 59].
Sparse representation-based methods encode an image over an over-complete dictionary D with L1-norm sparsity regularization on the coding vector, i.e., \( \underset\boldsymbol\upalpha\min \left\Vert \boldsymbol\upalpha \right\Vert_1\ s.t.x=\mathbfD\boldsymbol\upalpha \), resulting in a general model:
As a dictionary learning method, the sparse representation model can be learned from a dataset, as well as from the image itself with the K-singular value decomposition (K-SVD) algorithm [61, 62]. The basic idea behind K-SVD denoising is to learn the dictionary D from a noisy image y by solving the following joint optimization problem:
Coupled with the NSS prior [37], the sparsity from self-similarity properties of natural images, which has received significant attention in the image processing community, is widely applied for image denoising [64,65,66]. One representative work is the non-local centralized sparse representation (NCSR) model [66].
Different from the sparse representation model, this low-rank-based model formats similar patches as a matrix. Each column of this matrix is a stretched patch vector. By exploiting the low-rank prior of the matrix, this model can effectively reduce the noise in an image [67, 68]. The low-rank method first appeared in the field of matrix filling, and it has made great progress under the drive of Cand \( \overset`\mathrme \) s and Ma [69]. In recent years, the low-rank model has achieved good denoising results, resulting in low-rank denoising methods being studied more often.
From ref. [58], we know that WNNM achieves advanced denoising performance and is more robust to noise strength than other NNMs. Besides, the low-rank theory has been widely used in artificial intelligence, image processing, pattern recognition, computer vision, and other fields [63]. Although most low-rank minimization methods (especially the WNNM method) outperform previous denoising methods, the computational cost of the iterative boosting step is relatively high.
In contrast with spatial domain filtering methods, transform domain filtering methods first transform the given noisy image to another domain, and then they apply a denoising procedure on the transformed image according to the different characteristics of the image and its noise (larger coefficients denote the high frequency part, i.e., the details or edges of the image, smaller coefficients denote the noise). The transform domain filtering methods can be subdivided according to the chosen basis transform functions, which may be data adaptive or non-data adaptive [84].
As an effective and powerful extension of the NLM approach, BM3D, which was proposed by Dabov et al. [55], is the most popular denoising method. BM3D is a two-stage non-locally collaborative filtering method in the transform domain. In this method, similar patches are stacked into 3D groups by block matching, and the 3D groups are transformed into the wavelet domain. Then, hard thresholding or Wiener filtering with coefficients is employed in the wavelet domain. Finally, after an inverse transform of coefficients, all estimated patches are aggregated to reconstruct the whole image. However, when the noise increases gradually, the denoising performance of BM3D decreases greatly and artifacts are introduced, especially in flat areas.
Recently, CNN-based methods have been developed rapidly and have performed well in many low-level computer vision tasks [101, 102]. The use of a CNN for image denoising can be tracked back to [103], where a five-layer network was developed. In recent years, many CNN-based denoising methods have been proposed [99, 104,105,106,107,108]. Compared to that of ref. [103], the performance of these methods has been greatly improved. Furthermore, CNN-based denoising methods can be divided into two categories: multi-layer perception (MLP) models and deep learning methods.
Zhang et al. [106] introduced residual learning and batch standardization into image denoising for the first time; they also proposed feed-forward denoising CNNs (DnCNNs). The aim of the DnCNN model is to learn a function \( \hatx=F\left(y;\Theta_\sigma\right) \) that maps between y and \( \hatx \). The parameters Θσ are trained for noisy images under a fixed variance σ. There are two main characteristics of DnCNNs: the model applies a residual learning formulation to learn a mapping function, and it combines it with batch normalization to accelerate the training procedure while improving the denoising results. Specifically, it turns out that residual learning and batch normalization can benefit each other, and their integration is effective in speeding up the training and boosting denoising performance. Although a trained DnCNN can also handle compression and interpolation errors, the trained model under σ is not suitable for other noise variances.
Here, we compare the denoising results of the CNN-based methods (DnCNN [106] and FFDNet [107]) with those of several current effective image denoising methods, including BM3D [55] and WNNM [58]. To the best of our knowledge, BM3D has been the most popular denoising method over recent years, and WNNM is a successful scheme that has been proposed recently.
In Fig. 4, we can see that the details of the antennas and contour areas are difficult to recover. BM3D [55] and WNNM [58] blur the fine textures, whereas the other two methods restore more textures. This is because Monarch has many repetitive structures, which can be effectively exploited by NSS. Moreover, the contour edges of these regions are much sharper and look more natural. Overall, FFDNet [107] produces the best perceptual quality of denoised images. 2ff7e9595c
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