PSNR (dB) and SSIM results of the denoised images for the different techniques.

## Abstract

In this chapter, a new image denoising approach is proposed. It combines two image denoising techniques. The first one is based on a wavelet transform (WT), and the second one is a two-stage image denoising by PCA (principal component analysis) with LPG (local pixel grouping). In this proposed approach, we first apply the first technique to the noisy image in order to obtain the first estimation version of the clean image. Then, we estimate the noise-level from the noisy image. This estimate is obtained by applying the third technique of noise estimation from noisy images. The third step of the proposed approach consists in using the first estimation of the clean image, the noisy image, and the estimate of the noise-level as inputs of the second image denoising system (LPG-PCA). A comparative study of the proposed technique and the two others denoising technique (one is based on WT and and the second is based on LPG-PCA), is performed. This comparative study used a number of noisy images, and the obtained results from PSNR (peak signal-to-noise ratio) and SSIM (structural similarity) computations show that the proposed approach outperforms the two other denoising approaches (the first one is based on WT and the second one is based on LPG-PCA).

### Keywords

- image denoising
- wavelet transform
- noise estimation
- LPG-PCA

## 1. Introduction

In the image acquisition process, the noise will be inevitably introduced so denoising is a necessary step for ameliorating the image quality. As a primary low-level image processing, noise suppression has been extensively studied, and numerous denoising approaches have been proposed, from the earlier frequency domain denoising approaches and smoothing filters [1] to the lately developed wavelet [2, 3, 4, 5, 6, 7, 8, 9, 10, 11], curvelet- [12] and ridgelet- [13] based approaches, sparse representation [14] and K-SVD approaches [15], shape-adaptive transform [16], bilateral filtering [17, 18], nonlocal mean-based techniques [19, 20], and nonlocal collaborative filtering [21]. With the quick development of modern digital imaging devices and their increasingly broad applications in our daily life, there are rising necessities of new denoising techniques for higher quality of image. The WT (wavelet transform) [22] proved its effectiveness in noise cancelation [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. This transform decomposes the input signal into multiple scales which represent different time-frequency components of the original signal. At each scale, some operations, such as statistical modeling [4, 5, 6] and thresholding [2, 3], can be applied for canceling noise. Noise reduction is performed by transforming back the processed wavelet coefficients into spatial domain. Late development of WT-based denoising techniques includes ridgelet- and curvelet-based techniques [12, 13] for line structure conservation. Despite WT proved its effectiveness in denoising, it uses a fixed wavelet basis (with translation and dilation) for representing image. However, for natural images, a rich amount of different local structural patterns exists and therefore cannot be well represented by using just one fixed wavelet basis. Consequently, WT-based techniques can generate many visual artifacts in the denoising output. To overcome the problem of WT, in [23], Muresan and Parks proposed a spatially adaptive principal component analysis (PCA)-based denoising technique, which computes the locally fitted basis for transforming the image. In [15, 16], Elad and Aharon proposed K-SVD-based denoising approach and sparse and redundant representation by training a highly over-complete dictionary. In [16], Foi et al. applied a shape-adaptive discrete cosine transform (DCT) to the neighborhood, which can attain very sparse representation of the image and consequently lead to efficient denoising. All these approaches proved better denoising performance than classical WT-based denoising techniques. The NLM (nonlocal means) schemes used a very different philosophy from the above approaches in noise cancelation. The NLM idea can be traced back to [24], where the similar image pixels are averaged according to their intensity distance. Similar ideas were used in the bilateral filtering schemes [17, 18], where both the spatial and intensity similarities are exploited for pixel averaging. The NLM denoising framework was well established in [19]. In the image, each pixel is estimated as the weighted average of all the pixels and the weights are determined by the similarity between the pixels. This scheme was improved in [20], where the pair-wise hypothesis testing was used in the NLM estimation. Inspired from the success of NLM schemes, Dabov et al. [21] proposed a collaborative image denoising scheme by sparse 3D transform and patch matching. They look for similar blocks in the image by using block matching and grouped these blocks into a 3D cube. Then, a sparse 3D transform was applied to that cube, and noise was canceled by Wiener filtering in the transformed domain. The so-called BM3D approach attains remarkable denoising results, yet its implementation is a little complex. Lei Zhang et al. [25] have presented an efficient PCA-based denoising approach with local pixel grouping (LPG). PCA is a classical de-correlation technique in statistical signal processing, and it is pervasively used in dimensionality reduction and pattern, etc. [26]. The original dataset is transformed into PCA domain, and only the different most significant principal components are conserved. Consequently, trivial information and noise can be eliminated. In [23], a PCA-based scheme was proposed for image denoising by using a moving window for computing the local statistics, from which the local PCA transformation matrix was estimated. This technique applies PCA directly to the noisy image without data selection, and much residual noise and visual artifacts appear in the denoised image. In the LPG-PCA-based technique, Lei Zhang et al. [25] modeled a pixel and its nearest neighbors as a vector variable. The training samples of this variable are chosen by grouping the pixels with similar local spatial structures to the underlying one in the local window. With such an LPG technique, the local statistics of the variables can be accurately calculated so that the image edge structures can be well conserved after shrinkage in the PCA domain for noise suppression. The LPG-PCA scheme proposed in [25] has two stages where the first stage yields an initial image estimation by eliminating the most of the noise and the second stage will further refine the first stage output. The two stages have the same procedures except for the noise-level parameter. Since the noise is significantly reduced in the first stage, the LPG precision will be much improved in the second stage so that the final denoised image is visually much better. When compared with WT which uses a fixed basis function for decomposing the noisy image, the proposed LPG-PCA approach is a spatially adaptive image representation so that it can better characterize the image local structures. When compared with BM3D and NLM approaches, the LPG-PCA-based technique proposed in [25] can use a relatively small local window to group the similar pixels for PCA training, yet it yields competitive results with state-of-the-art BM3D algorithm. In this paper we propose a new image denoising approach which combines the dual-tree discrete wavelet transform (DT-DWT)-based denoising approach [12] and the two-stage image denoising technique by PCA with local pixel grouping (LPG) [25]. To evaluate this proposed technique, we have compared it to the two techniques (the DT-DWT-based denoising technique [12] and LPG-PCA [25]). This comparison is based on PSNR and SSIM computation. In the rest of this paper, we first will deal with PCA. Then, we will be interested in DT-DWT [12]. After that we will deal with noise-level estimation from the noisy image proposed in [27, 28]. Then, we will present the two denoising techniques proposed in [12, 25]. After that we will detail the proposed image denoising technique, and finally we will give results and evaluation.

## 2. Principal component analysis (PCA)

Let

representing the sample vector of

And then, the sample vector is centralized as follows:

with

Finally, the covariance matrix of the centralized dataset is computed as follows:

The PCA aim consists in finding an orthonormal transformation matrix

where *P* as follow:

## 3. LPG-PCA denoising algorithm

### 3.1. Modeling of spatially adaptive PCA denoising

In [25] and in previous literature, the noise

Both noise

As the observed image is the original image degraded by the noise, they denote the noisy vector of

### 3.2. Local pixel grouping (LPG)

Grouping the training samples similar to the central

In Eq. (11), the fact that noise

where

The noiseless counterpart of *X*. Once this dataset *X* is estimated, the central block and therefore we can extract the central underlying pixel. Such procedure is applied to each pixel, and then the entire image

### 3.3. Denoising refinement in the second stage

Most of the noise will be suppressed by employing the denoising procedures described in [25]. However, there is still much visually unpleasant residual noise in the denoised image. Figure 2 shows an example of image denoising where (a) is the original image Cameraman, (b) the noisy version of it with

According to this figure, we remark that the visual quality is much ameliorated after the second round of refinement. As shown in Figure 3, in the second round of LPG-PCA denoising technique [25], the noise-level should be updated.

Denote by

where

We have

In practice,

where

## 4. The proposed image denoising technique

As previously mentioned, in this chapter, a new image denoising technique is proposed. It combines two denoising approaches. The first one is a dual-tree discrete wavelet (DT-DWT)-based denoising method [12], and the second one is a two-stage image denoising by PCA with LPG [25]. This proposed technique consists at the first step in applying the first denoising approach [12] to the noisy image in order to obtain the first estimation of the clean image (the cleaned image). Then, we estimate the level of noise corrupting the clean image. The cleaned image, the noisy image, and the noise-level are used for applying the second approach which is two-stage image denoising by PCA with LPG [25]. Figure 4 illustrates the block diagram of the proposed technique.

According to this figure, the first step of the proposed image denoising technique consists in applying the first denoising approach based on DT-DWT [12] to the noisy image,

## 5. The Hilbert transform

The Hilbert transform of a signal corresponds in Fourier plane to a filter with complex gain,

This analytic signal has only positive frequencies. The Hilbert transform of a real signal is also real. Instead of considering the Hilbert transform of the wavelet (which is defined through the associated filters), we can consider the Hilbert transform of the signal, and the analysis is performed with initial wavelet because we have

## 6. Dual-tree complex wavelet transform

The dual tree complex wavelet (DT-CWT) permits to make signal analysis by using two different trees of DWT, with filters selected in such manner to obtain approximately a signal decomposition using analytic wavelet [30]. Figure 5 shows a tree of DT-CWT, using two different filter banks: *h*_{1} and *g*_{1} are high-pass filters of the first and second trees, and *h*_{0} and *g*_{0} are low-pass filters of the same two trees [30]. The first tree gives the coefficients of the real part, *dr*(*j,k*), and the second tree gives those of the imaginary part, *di*(*j,k*). After that, we construct the complex coefficients *d*_{complex}(*j,k*) = *dr*(*j,k*) + *i di*(*j,k*). The magnitude of those coefficients is named dual-tree magnitude [30].

This Q-shift dual-tree complex wavelet transform (Figure 5) is in 1D. Synthesis of the filters adapted to this structure has been performed by many research works. Particularly, Kingsbury [30] proposed some filters named Q-shift. In [30], some filters are employed, and their utilization is equivalent to the signal analysis by wavelets illustrated in Figure 6.

We can see in this figure that the wavelet corresponding to the imaginary part tree is very near to the Hilbert transform of the wavelet corresponding to the real part tree [30]. Finally, the utilization of this structure requires an operation of prefiltering; it means that the filters used in the first step are not the same as those used in the next step. The advantages of this method compared to the simple Hilbert transform (Section 5) are [30]:

A lower computation cost ( Two DWT),

An approximate of the Hilbert transform, is optimized for each scale,

The possibility of an exact reconstruction is preserved.

The principal drawback of the DT-CWT is the non-possibility of the use of the well-known wavelets of the DWT (Daubechies wavelet, Spline, etc.) and therefore the non-possibility to choose the number of vanishing moments (all the Q-shift filter gives wavelets with two vanishing moments).

### 6.1. 2D DT-CWT

To explain how the DT-CWT produces oriented wavelets, consider the 2D wavelet

Therefore, we obtain the following expression of **:**

The following idealized diagram (Figure 7) illustrates the Fourier spectrum support of this complex wavelet [31].

Since the (approximately) 1D wavelet spectrum is supported on just one side of the frequency axis, the complex 2D wavelet (

Since the real function spectrum should be symmetric with respect to the origin, then the spectrum of this real wavelet is supported in two quadrants of the 2D frequency plane (Figure 8).

Unlike the real separable wavelet, the support of the spectrum of this real wavelet has not the checkerboard artefact and consequently this real wavelet (illustrated in the second panel of Figure 11), is oriented at

The support in the 2D frequency plane of this complex wavelet spectrum is illustrated in Figure 9.

As above, the spectrum of the complex wavelet,

The spectrum of which is supported in two quadrants of the 2D frequency plane as illustrated in Figure 10.

Again, neither the wavelet nor the spectrum of this real wavelet has the spectrum of the checkerboard artifact. This real 2D wavelet is oriented at

By taking the real part of each of these wavelets, one can obtain four real oriented 2D wavelets. Moreover, the two already obtained in Eqs. (21) and (23). Precisely, we have six wavelets expressed as follows:

For

The normalization factor

## 7. The technique of Noise-level estimation

In many image processing applications, the noise-level is an important parameter. For example, the performance of an image denoising technique can be much degraded due to the poor noise-level estimation. The most available denoising techniques simply supposed that the noise-level is known that largely prevents them from practical employment. Furthermore, even with the given true noise-level, those denoising techniques still cannot achieve the best performance, precisely for scenes with rich texture. Xinhao Liu et al. [28, 29] have proposed a technique of patch-based noise-level estimation, and they suggested that the noise-level parameter should be tuned according to the complexity of the scene. Their approach [28, 29] includes the process of selecting low-rank patches without high-frequency components from a single noisy image. Then, the noise-level was estimated from the selected patches employing principal component analysis. Because the exact noise-level does not always provide the best performance for non-blind denoising. Experiments prove that both the stability and precision are superior to the state-of-the-art noise-level estimation technique for different noise-levels and scenes.

## 8. Evaluation criteria

In this section, we will evaluate the three techniques which are the proposed image denoising techniques: the first image denoising approach based on DT-CWT [12] and the second denoising approach and the two-stage image denoising by principal component analysis with local pixel grouping [25]. This evaluation is based on the computation of PSNR and SSIM which are detailed in [32].

## 9. Results and discussion

In this work, we have applied the proposed image denoising technique, the first image denoising technique based on DT-CWT [12] and the second denoising technique and the two-stage image denoising by principal component analysis with local pixel grouping [25], on a number of digital images such as “House,” “Lena,” and “Cameraman.” These images are degraded by additive white noise with different values of noise-level,

Technique | The first image denoising technique based on DT-DWT [12] | Two-stage image denoising by principal component analysis with local pixel grouping [25]: first stage | Two-stage image denoising by principal component analysis with local pixel grouping [25]: second stage | The proposed technique |
---|---|---|---|---|

House (σ = 10) | 34.7138 (0.8778) | 35.4 (0.9003) | 35.6 (0.9012) | 36.1223 (0.9130) |

House (σ = 20) | 31.6671 (0.8253) | 31.8 (0.8084) | 32.5 (0.8471) | 33.0828 (0.8677) |

House (σ = 30) | 29.8494 (0.7877) | 29.3 (0.7225) | 30.4 (0.8185) | 31.2095 (0.8393) |

House (σ = 40) | 28.5744 (0.8084) | 27.3(0.6243) | 28.9 (0.7902) | 29.7344 (0.8084) |

Lena (σ = 10) | 33.6767(0.9170) | 33.6 (0.9218) | 33.7 (0.9243) | 34.0765 (0.9271) |

Lena (σ = 20) | 30.0002 (0.8539) | 29.5 (0.8346) | 29.7 (0.8605) | 30.5415 (0.8765) |

Lena (σ = 30) | 27.9859 (0.8016) | 27.1 (0.7441) | 27.6 (0.8066) | 28.3595 (0.8292) |

Lena (σ = 40) | 26.6364 (0.7585) | 25.4 (0.6597) | 26.0 (0.7578) | 26.8566 (0.7882) |

Cameraman (σ = 10) | 32.7481 (0.8989) | 33.9 (0.9261) | 34.1 (0.9356) | 33.6141 (0.9241) |

Cameraman (σ = 20) | 28.9990 (0.8175) | 29.8 (0.8320) | 30.1 (0.8902) | 29.7184 (0.8575) |

Cameraman (σ = 30) | 27.1022 (0.7641) | 27.3 (0.7395) | 27.8(0.8558) | 27.8174 (0.8151) |

Cameraman (σ = 40) | 25.7866 (0.7241) | 25.5 (0.6393) | 26.2 (0.8211) | 26.4954 (0.7826) |

Monarch (σ = 10) | 32.9907 (0.9369) | 34.0 (0.9522) | 34.2 (0.9594) | 34.0698 (0.9553) |

Monarch (σ = 20) | 29.1114 (0.8811) | 29.6 (0.8859) | 30.0 (0.9202) | 30.0384 (0.9145) |

Monarch (σ = 30) | 27.0058 (0.8346) | 27.0 (0.8071) | 27.4 (0.8769) | 27.7209 (0.8735) |

Monarch (σ = 40) | 25.5973 (0.7950) | 25.2 (0.7267) | 25.9 (0.8378) | 26.0832 (0.8293) |

Peppers (σ = 10) | 33.4942 (0.9056) | 33.4 (0.8909) | 33.3 (0.8943) | 33.7904 (0.9189) |

Peppers (σ = 20) | 29.8124 (0.8424) | 29.9 (0.8177) | 30.1 (0.8413) | 30.5252 (0.8743) |

Peppers (σ = 30) | 27.7810 (0.7924) | 27.5 (0.7332) | 27.9 (0.7973) | 28.4765 (0.8356) |

Peppers (σ = 40) | 26.4045 (0.7507) | 25.9 (0.6447) | 26.7(0.7648) | 26.9883 (0.8013) |

Paint (σ = 10) | 32.5488 (0.9165) | 33.5 (0.9280) | 33.6 (0.9311) | 33.3567 (0.9276) |

Paint (σ = 20) | 28.5980 (0.8416) | 26.8 (0.7467) | 29.5 (0.8683) | 29.4699 (0.8648) |

Paint (σ = 30) | 26.6067 (0.7817) | 26.8 (0.7467) | 27.2 (0.8088) | 27.2540 (0.8077) |

Paint (σ = 40) | 25.2968 (0.7330) | 25.0 (0.6590) | 25.6 (0.7569) | 25.6389 (0.7560) |

These obtained results (Table 1) show clearly that the proposed technique outperforms the denoising technique based on DT-CWT proposed in [12] and the denoising approach based on LPG-PCA [25]. Figures 12–15 show four examples of image denoising using the proposed technique.

These figures show that the noise corrupting the original images is sufficiently suppressed. Moreover, the proposed technique permits to obtain denoised images with good perceptual quality. In each of these figures, the image (c) is obtained after the first denoising stage in the proposed technique. In this image (c), some noise is still existing, whereas it is considerably reduced into the image (d) obtained after the second denoising step. In the following subsection, we will give the results obtained by applying the proposed technique, the LPG-PCA-based denoising technique [25, 27] and the DT-DWT-based denoising one to a number of grayscale images. Those results are in terms of SNR and MSE and are listed in Table 2.

Technique | The first image denoising technique based on DT-DWT [12] | Two-stage image denoising by principal component analysis with local pixel grouping [25] | The proposed technique |
---|---|---|---|

House (σ = 10) | SNR = 78.00, MSE = 21.96 | SNR = 79.41, MSE = 15.88 | SNR = 79.44, MSE = 15.75 |

House (σ = 20) | SNR = 74.95, MSE = 44.29 | SNR = 76.37, MSE = 31.97 | SNR = 76.38, MSE = 31.92 |

House (σ = 30) | SNR = 73.14, MSE = 67.31 | SNR = 74.50, MSE = 49.21 | SNR = 74.50, MSE = 49.21 |

House (σ = 40) | SNR = 71.86, MSE = 90.28 | SNR = 73.02, MSE = 69.16 | SNR = 73.02, MSE = 69.14 |

Lena (σ = 10) | SNR = 74.67, MSE = 27.88 | SNR = 75.28, MSE = 24.17 | SNR = 75.28, MSE = 24.19 |

Lena (σ = 20) | SNR = 70.99, MSE = 65.02 | SNR = 71.53, MSE = 57.40 | SNR = 71.55, MSE = 57.19 |

Lena (σ = 30) | SNR = 68.97, MSE = 103.39 | SNR = 69.35, MSE = 94.87 | SNR = 69.37, MSE = 94.36 |

Lena (σ = 40) | SNR = 67.62, MSE = 141.07 | SNR = 67.85, MSE = 134.09 | SNR = 67.87, MSE = 133.35 |

Cameraman (σ = 10) | SNR = 75.33, MSE = 34.53 | SNR = 76.19, MSE = 28.29 | SNR = 76.23, MSE = 28.06 |

Cameraman (σ = 20) | SNR = 71.58, MSE = 81.88 | SNR = 72.30, MSE = 69.38 | SNR = 72.33, MSE = 68.80 |

Cameraman (σ = 30) | SNR = 69.68, MSE = 126.72 | SNR = 70.39, MSE = 107.48 | SNR = 70.45, MSE = 106.00 |

Cameraman (σ = 40) | SNR = 68.36, MSE = 171.56 | SNR = 69.07, MSE = 145.72 | SNR = 69.14, MSE = 143.51 |

Monarch (σ = 10) | SNR = 74.94, MSE = 32.65 | SNR = 76.02, MSE = 25.47 | SNR = 76.01, MSE = 25.55 |

Monarch (σ = 20) | SNR = 71.06, MSE = 79.78 | SNR = 71.99, MSE = 64.45 | SNR = 71.98, MSE = 64.53 |

Monarch (σ = 30) | SNR = 68.96, MSE = 129.56 | SNR = 69.67, MSE = 109.89 | SNR = 69.68, MSE = 109.62 |

Monarch (σ = 40) | SNR = 67.55, MSE = 179.20 | SNR = 68.01, MSE = 161.05 | SNR = 68.03, MSE = 160.25 |

Peppers (σ = 10) | SNR = 76.07, MSE = 29.08 | SNR = 76.65, MSE = 25.43 | SNR = 76.63, MSE = 25.56 |

Peppers (σ = 20) | SNR = 72.39, MSE = 67.89 | SNR = 73.10, MSE = 57.61 | SNR = 73.12, MSE = 57.43 |

Peppers (σ = 30) | SNR = 70.36, MSE = 108.38 | SNR = 71.05, MSE = 92.34 | SNR = 71.07, MSE = 92.02 |

Peppers (σ = 40) | SNR = 68.98, MSE = 148.80 | SNR = 69.57, MSE = 130.09 | SNR = 69.58, MSE = 129.58 |

Those results show that the proposed technique outperforms the two other techniques (the LPG-PCA-based denoising technique [25, 27] and the DT-DWT-based denoising one [12]). In fact the proposed techniques are the highest values of SNR and lowest values of MSE.

## 10. Conclusion

In this chapter, a new image denoising technique is proposed. It combines two denoising approaches. The first one is a dual-tree discrete wavelet transform (DT-DWT)-based denoising technique, and the second one is a two-stage image denoising by principal component analysis with local pixel grouping (LPG-PCA). The first step of this proposed technique consists in applying the first approach to the noisy image in order to obtain a first estimate of the clean image. Then, we estimate the level of noise corrupting the original image. This estimation is performed by using a method of noise estimation from noisy images. The third step of the proposed technique consists in using this first clean image estimation, the noisy image, and this noise-level estimate as inputs of the second image denoising system (LPG-PCA-based image denoising) in order to obtain the final estimation of the clean image. A comparative study is performed between the proposed image denoising technique and two others denoising approaches where the first is based on DT-DWT and the second is based on LPG-PCA. This study is based on PSNR and SSIM computations, and the obtained results show that the proposed technique outperforms the two other denoising approaches. We also computed SNR (Signal to Noise Ratio) and MSE (Mean Square Error) and the obtained results also show that the proposed technique outperforms the others techniques.

## Acknowledgments

We would like to thank all the people who contributed in some way to this work which was supported by the CRTEn (Center of Research and Technology of Energy) of Borj Cedria, Tunisia, and the Ministry of Higher Education and Scientific Research.