2. Principal component analysis (PCA)

Let

[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 [2–11]. 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 [4–6] 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

238 Wavelet Theory and Its Applications

$$X = \begin{bmatrix} \mathbf{x}\_1^1 & \mathbf{x}\_1^2 & \dots & \mathbf{x}\_1^n \\ \mathbf{x}\_2^1 & \mathbf{x}\_2^2 & \dots & \mathbf{x}\_2^n \\ \vdots & \ddots & \vdots \\ \mathbf{x}\_n^1 & \mathbf{x}\_n^2 & & \mathbf{x}\_n^n \end{bmatrix} \tag{1}$$

$$X\_i = \begin{bmatrix} \mathbf{x}\_1^1 \ \mathbf{x}\_1^2 \dots \mathbf{x}\_1^n \end{bmatrix} \tag{2}$$

representing the sample vector of xi. The mean value of Xi is computed as follows:

$$
\mu\_i = \frac{1}{n} \sum\_{j=1}^n X\_i(j). \tag{3}
$$

And then, the sample vector is centralized as follows:

$$\overline{X}\_i = X\_i - \mu\_i = \left[ \overline{x}\_i^1 \, \overline{x}\_i^2 \dots \overline{x}\_i^n \right] \tag{4}$$

with x j <sup>i</sup> ¼ x j <sup>i</sup> � μ<sup>i</sup> . Accordingly, the centralized matrix of X is expressed as follows:

$$\overline{X} = X\_i - \mu\_i = \left[ X\_i^T X\_2^T \dots \overline{x}\_m^T \right]^T. \tag{5}$$

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

$$
\Omega = \frac{1}{n} \overline{X} \overline{X}^T.\tag{6}
$$

modeled as a vector variable and perform denoising on the vector instead of the single pixel. According to Figure 1, for an underlying pixel to be denoised, Lei Zhang et al. [25] set a K � K

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

noisy vector, xυ, they are viewed as (both noiseless and noisy) vector variables so that one can use the statistical techniques such as PCA. For canceling the noise from the noisy vector x<sup>υ</sup> by using PCA, a set of training samples of x<sup>υ</sup> is needed so that the covariance matrix of x<sup>υ</sup> and therefore the PCA transformation matrix can be computed. For this aim, Lei Zhang et al. [25] have used an L � L Lð Þ > K training block centered on x<sup>υ</sup> in order to find the training samples, as illustrated in Figure 1. The simplest manner consists in taking the pixels in each possible K � K block within the L � L training block as the samples of noisy variable xυ. In this way, for

very different blocks from the given central K � K block in the L � L training window, taking all the K � K blocks as the training samples of xυwill lead to inaccurate estimation of the matrix of covariance of xυ, which subsequently leads to inaccurate estimation of the PCA transformation matrix and finally results in much residual noise. Consequently, selecting and grouping the training samples that are similar to the central K � K block are required before image

Grouping the training samples similar to the central K � K block in the L � L training window is certainly a problem of classification, and therefore different grouping techniques such as

<sup>k</sup> <sup>¼</sup> xk <sup>þ</sup> <sup>υ</sup>k, k <sup>¼</sup> <sup>1</sup>, …, m, and <sup>x</sup><sup>υ</sup> <sup>¼</sup> <sup>x</sup><sup>υ</sup>

, m <sup>¼</sup> <sup>K</sup><sup>2</sup>

x<sup>υ</sup> ¼ x þ υ (10)

1…x<sup>υ</sup> m

<sup>k</sup> of <sup>x</sup>υ, there are in total ð Þ <sup>L</sup> � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> training samples. Though, there can be

, the vector contains all com-

Wavelets and LPG-PCA for Image Denoising http://dx.doi.org/10.5772/intechopen.74453 241

. For estimating x from the

window centered on it, and denoted by <sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>1;…; xm <sup>T</sup>

ponents within the window.

x where <sup>υ</sup> <sup>¼</sup> ½ � <sup>υ</sup>1…υ<sup>m</sup> <sup>T</sup>, <sup>x</sup><sup>υ</sup>

vector of x by [25]:

each component x<sup>υ</sup>

denoising by applying the PCA transform.

Figure 1. Illustration of the modeling of LPG-PCA-based denoising [25].

3.2. Local pixel grouping (LPG)

The PCA aim consists in finding an orthonormal transformation matrix P in order to decorrelate X, i.e., Y ¼ PX, so that the matrix of covariance of Y is diagonal. Since the covariance matrix Ω is symmetrical; therefore, it can be expressed as follows:

$$
\Omega = \phi \Lambda \phi^T \tag{7}
$$

where Λ ¼ diagf g λ1; λ2;…; λ<sup>m</sup> is the diagonal eigenvalue matrix with λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ …≥ λ<sup>m</sup> and ϕ ¼ ϕ<sup>1</sup> ϕ2…ϕ<sup>m</sup> � � represents the <sup>m</sup> � <sup>m</sup> orthonormal eigenvector matrix. The terms λ1, λ2,…, λ<sup>m</sup> and ϕ1, ϕ2, …, ϕmare, respectively, the eigenvalues and the eigenvectors of Ω. By setting the matrix P as follow:

$$P = \phi^T,\tag{8}$$

<sup>X</sup> can be de-correlated, i.e., <sup>Λ</sup> <sup>¼</sup> <sup>1</sup> n Y Y <sup>T</sup> andY ¼ PX. An interesting property of PCA is that it fully de-correlates the original dataset X. In general, the signal energy will concentrate on a small subset of the PCA transformed dataset, whereas the noise energy will evenly spread over the whole dataset. Consequently, the noise and signal can be better distinguished in the domain of PCA.

#### 3. LPG-PCA denoising algorithm

#### 3.1. Modeling of spatially adaptive PCA denoising

In [25] and in previous literature, the noise υ degrading the original image I is supposed to be white and additive with standard deviation σ and zero mean, and the noisy image,Iυ, is expressed as follows:

$$I\_{\upsilon} = I + \upsilon \tag{9}$$

Both noise υ and image I are supposed to be uncorrelated. The purpose of image denoising consists in estimating the clean image I from Iυ, and the estimate is denoted by bI. The latter is expected to be as close as possible to the original image, I. Two quantities describe an image pixel. Those quantities are its intensity and the spatial location. However, the image local structure is represented as a set of neighboring pixels at different intensity levels. As most of the semantic information of an image is conveyed by its edge structures, edge conservation is highly required in denoising of this image. In [25], the pixel and its nearest neighbors were modeled as a vector variable and perform denoising on the vector instead of the single pixel. According to Figure 1, for an underlying pixel to be denoised, Lei Zhang et al. [25] set a K � K window centered on it, and denoted by <sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>1;…; xm <sup>T</sup> , m <sup>¼</sup> <sup>K</sup><sup>2</sup> , the vector contains all components within the window.

As the observed image is the original image degraded by the noise, they denote the noisy vector of x by [25]:

$$\mathbf{x}\_{\upsilon} = \mathbf{x} + \upsilon \tag{10}$$

x where <sup>υ</sup> <sup>¼</sup> ½ � <sup>υ</sup>1…υ<sup>m</sup> <sup>T</sup>, <sup>x</sup><sup>υ</sup> <sup>k</sup> <sup>¼</sup> xk <sup>þ</sup> <sup>υ</sup>k, k <sup>¼</sup> <sup>1</sup>, …, m, and <sup>x</sup><sup>υ</sup> <sup>¼</sup> <sup>x</sup><sup>υ</sup> 1…x<sup>υ</sup> m . For estimating x from the noisy vector, xυ, they are viewed as (both noiseless and noisy) vector variables so that one can use the statistical techniques such as PCA. For canceling the noise from the noisy vector x<sup>υ</sup> by using PCA, a set of training samples of x<sup>υ</sup> is needed so that the covariance matrix of x<sup>υ</sup> and therefore the PCA transformation matrix can be computed. For this aim, Lei Zhang et al. [25] have used an L � L Lð Þ > K training block centered on x<sup>υ</sup> in order to find the training samples, as illustrated in Figure 1. The simplest manner consists in taking the pixels in each possible K � K block within the L � L training block as the samples of noisy variable xυ. In this way, for each component x<sup>υ</sup> <sup>k</sup> of <sup>x</sup>υ, there are in total ð Þ <sup>L</sup> � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> training samples. Though, there can be very different blocks from the given central K � K block in the L � L training window, taking all the K � K blocks as the training samples of xυwill lead to inaccurate estimation of the matrix of covariance of xυ, which subsequently leads to inaccurate estimation of the PCA transformation matrix and finally results in much residual noise. Consequently, selecting and grouping the training samples that are similar to the central K � K block are required before image denoising by applying the PCA transform.

#### 3.2. Local pixel grouping (LPG)

<sup>X</sup> <sup>¼</sup> Xi � <sup>μ</sup><sup>i</sup> <sup>¼</sup> XT

<sup>Ω</sup> <sup>¼</sup> <sup>1</sup> n XXT

The PCA aim consists in finding an orthonormal transformation matrix P in order to decorrelate X, i.e., Y ¼ PX, so that the matrix of covariance of Y is diagonal. Since the covariance

where Λ ¼ diagf g λ1; λ2;…; λ<sup>m</sup> is the diagonal eigenvalue matrix with λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ …≥ λ<sup>m</sup> and

de-correlates the original dataset X. In general, the signal energy will concentrate on a small subset of the PCA transformed dataset, whereas the noise energy will evenly spread over the whole dataset. Consequently, the noise and signal can be better distinguished in the domain of PCA.

In [25] and in previous literature, the noise υ degrading the original image I is supposed to be white and additive with standard deviation σ and zero mean, and the noisy image,Iυ, is

Both noise υ and image I are supposed to be uncorrelated. The purpose of image denoising consists in estimating the clean image I from Iυ, and the estimate is denoted by bI. The latter is expected to be as close as possible to the original image, I. Two quantities describe an image pixel. Those quantities are its intensity and the spatial location. However, the image local structure is represented as a set of neighboring pixels at different intensity levels. As most of the semantic information of an image is conveyed by its edge structures, edge conservation is highly required in denoising of this image. In [25], the pixel and its nearest neighbors were

� � represents the <sup>m</sup> � <sup>m</sup> orthonormal eigenvector matrix. The terms λ1, λ2,…, λ<sup>m</sup> and ϕ1, ϕ2, …, ϕmare, respectively, the eigenvalues and the eigenvectors of Ω. By

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

matrix Ω is symmetrical; therefore, it can be expressed as follows:

n Y Y

ϕ ¼ ϕ<sup>1</sup> ϕ2…ϕ<sup>m</sup>

240 Wavelet Theory and Its Applications

setting the matrix P as follow:

<sup>X</sup> can be de-correlated, i.e., <sup>Λ</sup> <sup>¼</sup> <sup>1</sup>

3. LPG-PCA denoising algorithm

expressed as follows:

3.1. Modeling of spatially adaptive PCA denoising

<sup>i</sup> X<sup>T</sup> <sup>2</sup>…x T m

h i<sup>T</sup>

: (5)

: (6)

<sup>Ω</sup> <sup>¼</sup> <sup>ϕ</sup>Λϕ<sup>T</sup> (7)

<sup>P</sup> <sup>¼</sup> <sup>ϕ</sup>T, (8)

I<sup>υ</sup> ¼ I þ υ (9)

<sup>T</sup> andY ¼ PX. An interesting property of PCA is that it fully

Grouping the training samples similar to the central K � K block in the L � L training window is certainly a problem of classification, and therefore different grouping techniques such as

Figure 1. Illustration of the modeling of LPG-PCA-based denoising [25].

correlation based matching, block matching, K-means clustering, etc. can be used based on different criteria. The block matching-based technique may be the simplest but very efficient one, and it is used in [25] for LPG. There are totally ð Þ <sup>L</sup> � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> possible training blocks of <sup>x</sup><sup>υ</sup> in the <sup>L</sup> � <sup>L</sup> training window. We will denote <sup>x</sup><sup>υ</sup> <sup>0</sup> in the column sample vector which contains the pixels in the central <sup>K</sup> � <sup>K</sup> block, and denoted by <sup>x</sup><sup>υ</sup> <sup>i</sup> , i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, Lð Þ � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> � 1, the sample vectors correspond to the other blocks. Let xi and x<sup>0</sup> be, respectively, the associated noiseless sample vectors of x<sup>υ</sup> <sup>i</sup> and x<sup>υ</sup> <sup>0</sup>. It can be simply computed that

$$\varepsilon c\_{i} = \frac{1}{m} \sum\_{k=1}^{m} \overline{\mathbf{x}}\_{0}^{\boldsymbol{\nu}}(k) - \overline{\mathbf{x}}\_{i}^{\boldsymbol{\nu}}(k)^{2} \approx \frac{1}{m} \sum\_{k=1}^{m} \overline{\mathbf{x}}\_{0}(k) - \overline{\mathbf{x}}\_{0}(k)^{2} + 2\sigma^{2} \tag{11}$$

In Eq. (11), the fact that noise υ is white and uncorrelated with signal is used. With Eq. (11), if we have the following condition

$$
\sigma\_i < T + 2\sigma^2 \tag{12}
$$

where T designates a preset threshold, then we select x<sup>υ</sup> <sup>i</sup> as a sample vector of xυ. Assume that n sample vectors of x<sup>υ</sup> are selected including the central vector x<sup>υ</sup> <sup>0</sup>. For the expression convenience, these sample vectors are denoted as x<sup>υ</sup> <sup>0</sup>, x<sup>υ</sup> <sup>1</sup>, …, x<sup>υ</sup> <sup>n</sup>�<sup>1</sup>. The noiseless counterparts of those vectors are denoted as x0, x1,…, xn�1, accordingly. Then, the training dataset for x<sup>υ</sup> is constituted by.

$$X\_{\upsilon} = \begin{bmatrix} \overline{\mathbf{x}}\_{0}^{\upsilon}, \overline{\mathbf{x}}\_{1}^{\upsilon}, \dots, \overline{\mathbf{x}}\_{n-1}^{\upsilon} \end{bmatrix} \tag{13}$$

estimation bias of the PCA transformation matrix and therefore deteriorates the denoising performance; second, the strong noise in the original dataset will also lead to LPG errors, which therefore results in estimation bias of the covariance matrix Ω<sup>x</sup><sup>υ</sup> or Ωx. Consequently, it is essential to further process the denoising output for a better image denoising. As the noise has been much canceled in the first round of LPG-PCA denoising, the LPG correctness and the estimation of Ω<sup>x</sup><sup>υ</sup> or Ωx can be much ameliorated with the denoised image. Consequently, the LPG-PCA denoising procedure for the second round for enhancing the denoising results.

Figure 2. (a) Original image Cameraman, (b) corresponding noisy image (PSNR ¼ 22:1 dB), (c) denoised image after the first round of the technique proposed in [25] (PSNR ¼ 29:8 dB), and (d) denoised image after the second round of the

Wavelets and LPG-PCA for Image Denoising http://dx.doi.org/10.5772/intechopen.74453 243

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.

Figure 3. Flowchart of the two-stage LPG-PCA denoising technique proposed in [25].

proposed technique (PSNR ¼ 30:1 dB) [25].

The noiseless counterpart of X<sup>υ</sup> is designated as X ¼ ½ � x0; x1; …; xn�<sup>1</sup> . To insure the existence of enough samples in calculating the PCA transformation matrix, ncould not be too small. Practically speaking, it will be used in denoising at least c � m training samples of xυ, with c ¼ 8˜10. That is to say that in case of n < c � m, we will use the best c � m-matched samples in PCA training. Often, the best c � m-matched samples are robust for estimating the local statistics of image, and this operation makes the algorithm more stable for computing the PCA transformation matrix. The problem now is how to estimate from the noisy data Xυ, the noiseless dataset 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 I<sup>υ</sup> can be denoised. The LPG-PCA-based denoising is detailed in [25], and the denoising refinement in the second stage will be detailed in the next part of this paper.

#### 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 PSNR ¼ 22:1 dB and σ ¼ 20, and ð Þa is the denoised image with PSNR ¼ 29:8 dB by employing the LPG-PCA technique proposed in [25]. Despite the remarkable improvement of PSNR, one can still see much residual noise in the denoising output. There are mainly two reasons for the residual noise. First, because of the strong noise in the original dataset Xυ, the covariance matrix Ω<sup>x</sup><sup>υ</sup> is much noise degraded, which leads to

correlation based matching, block matching, K-means clustering, etc. can be used based on different criteria. The block matching-based technique may be the simplest but very efficient one, and it is used in [25] for LPG. There are totally ð Þ <sup>L</sup> � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> possible training blocks of <sup>x</sup><sup>υ</sup>

ple vectors correspond to the other blocks. Let xi and x<sup>0</sup> be, respectively, the associated

In Eq. (11), the fact that noise υ is white and uncorrelated with signal is used. With Eq. (11), if

<sup>i</sup> ð Þ<sup>k</sup> <sup>2</sup> <sup>≈</sup> <sup>1</sup> m Xm k¼1

<sup>0</sup>. It can be simply computed that

<sup>0</sup> in the column sample vector which contains

<sup>i</sup> , i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, Lð Þ � <sup>K</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> � 1, the sam-

<sup>x</sup>0ð Þ� <sup>k</sup> <sup>x</sup>0ð Þ<sup>k</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>σ<sup>2</sup> (11)

<sup>i</sup> as a sample vector of xυ. Assume that n

<sup>n</sup>�<sup>1</sup>. The noiseless counterparts of those vectors

� � (13)

<sup>0</sup>. For the expression convenience,

ei <sup>&</sup>lt; <sup>T</sup> <sup>þ</sup> <sup>2</sup>σ<sup>2</sup> (12)

in the <sup>L</sup> � <sup>L</sup> training window. We will denote <sup>x</sup><sup>υ</sup>

ei <sup>¼</sup> <sup>1</sup> m Xm k¼1 xυ <sup>0</sup>ð Þ� <sup>k</sup> <sup>x</sup><sup>υ</sup>

where T designates a preset threshold, then we select x<sup>υ</sup>

sample vectors of x<sup>υ</sup> are selected including the central vector x<sup>υ</sup>

<sup>0</sup>, x<sup>υ</sup> <sup>1</sup>, …, x<sup>υ</sup>

<sup>X</sup><sup>υ</sup> <sup>¼</sup> <sup>x</sup><sup>υ</sup>

are denoted as x0, x1,…, xn�1, accordingly. Then, the training dataset for x<sup>υ</sup> is constituted by.

<sup>0</sup>; x<sup>υ</sup> <sup>1</sup>; …; x<sup>υ</sup> n�1

The noiseless counterpart of X<sup>υ</sup> is designated as X ¼ ½ � x0; x1; …; xn�<sup>1</sup> . To insure the existence of enough samples in calculating the PCA transformation matrix, ncould not be too small. Practically speaking, it will be used in denoising at least c � m training samples of xυ, with c ¼ 8˜10. That is to say that in case of n < c � m, we will use the best c � m-matched samples in PCA training. Often, the best c � m-matched samples are robust for estimating the local statistics of image, and this operation makes the algorithm more stable for computing the PCA transformation matrix. The problem now is how to estimate from the noisy data Xυ, the noiseless dataset 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 I<sup>υ</sup> can be denoised. The LPG-PCA-based denoising is detailed in [25], and the

denoising refinement in the second stage will be detailed in the next part of this paper.

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 PSNR ¼ 22:1 dB and σ ¼ 20, and ð Þa is the denoised image with PSNR ¼ 29:8 dB by employing the LPG-PCA technique proposed in [25]. Despite the remarkable improvement of PSNR, one can still see much residual noise in the denoising output. There are mainly two reasons for the residual noise. First, because of the strong noise in the original dataset Xυ, the covariance matrix Ω<sup>x</sup><sup>υ</sup> is much noise degraded, which leads to

noiseless sample vectors of x<sup>υ</sup>

242 Wavelet Theory and Its Applications

we have the following condition

these sample vectors are denoted as x<sup>υ</sup>

3.3. Denoising refinement in the second stage

the pixels in the central <sup>K</sup> � <sup>K</sup> block, and denoted by <sup>x</sup><sup>υ</sup>

<sup>i</sup> and x<sup>υ</sup>

Figure 2. (a) Original image Cameraman, (b) corresponding noisy image (PSNR ¼ 22:1 dB), (c) denoised image after the first round of the technique proposed in [25] (PSNR ¼ 29:8 dB), and (d) denoised image after the second round of the proposed technique (PSNR ¼ 30:1 dB) [25].

estimation bias of the PCA transformation matrix and therefore deteriorates the denoising performance; second, the strong noise in the original dataset will also lead to LPG errors, which therefore results in estimation bias of the covariance matrix Ω<sup>x</sup><sup>υ</sup> or Ωx. Consequently, it is essential to further process the denoising output for a better image denoising. As the noise has been much canceled in the first round of LPG-PCA denoising, the LPG correctness and the estimation of Ω<sup>x</sup><sup>υ</sup> or Ωx can be much ameliorated with the denoised image. Consequently, the LPG-PCA denoising procedure for the second round for enhancing the denoising results.

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.

Figure 3. Flowchart of the two-stage LPG-PCA denoising technique proposed in [25].

Denote bybI, the denoised version of the noisy image in the first stage. ThebI can be expressed as

$$
\widehat{I} = I + \nu\_s \tag{14}
$$

where υ<sup>s</sup> is the residual noise in the denoised image. The level estimation ofυ<sup>s</sup> is denoted by σ<sup>s</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffi E υ<sup>2</sup> s <sup>q</sup> � � and inputs it to the second round of LPG-PCA denoising algorithm. In [25], <sup>σ</sup><sup>s</sup> is estimated based on the difference between I<sup>υ</sup> and bI. Let

$$
\hat{I} = I\_\upsilon - \hat{I} = \upsilon - \upsilon\_s \tag{15}
$$

We have E ~I <sup>2</sup> h i <sup>¼</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup> � � <sup>þ</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup> s � � � <sup>2</sup><sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ �¼ <sup>σ</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>s</sup> � 2E υ � υ<sup>s</sup> ½ �. The υ<sup>s</sup> can be seen as the smoothed version of noise υ, and it mainly contains the low-frequency component of υ. Let υ~ ¼ υ � υ<sup>s</sup> be their difference, and υ~ mainly contains the high-frequency component of υ. There is <sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ �¼ <sup>E</sup> <sup>υ</sup><sup>~</sup> � <sup>υ</sup>s� þ <sup>E</sup> <sup>υ</sup><sup>2</sup> s � � �. Generally, compared to E υ<sup>2</sup> s � �, <sup>E</sup> <sup>υ</sup><sup>~</sup> � <sup>υ</sup><sup>s</sup> <sup>½</sup> � is much smaller, and we can obtain the following approximation: <sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ � <sup>≈</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup> s � � <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>s</sup> . Thus, from <sup>E</sup> <sup>~</sup><sup>I</sup> <sup>2</sup> h i <sup>¼</sup> <sup>σ</sup><sup>2</sup><sup>þ</sup> σ2 <sup>s</sup> � 2Eð Þ υ � υ<sup>s</sup> , we obtain

$$
\sigma\_s^2 \approx \sigma^2 - E\left[\tilde{I}^2\right] \tag{16}
$$

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

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, Ib, in order to obtain a first estimate of the clean image, Id, and then estimates the noise-level, υ, from Ib. The noisy images Ib, Id, and υ constitute the inputs of the second image denoising system proposed in [25, 27]. The output of this system and the overall proposed one are the final denoised image, Id1. In the image denoising system (LPG-PCA denoising) proposed in [25, 27], Lei Zhang et al. have used the clean image, I, and the noise-level, υ, as the inputs of this system [27]. However, only the noisy image, Ib, is available, and for this raison, we have used in our proposed technique the denoising approach based on DT-DWT [12] in order to obtain a cleaned image, Id, which is then used as a clean image, I. This clean image is one important input of the denoising system proposed by Lei Zhang et al. [27]. In the following two subsections, we will be interested in the first image denoising approach based on DT-DWT [12] and the technique of noise-level estimation

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

principal value in Cauchy sense [30]. The analytic signal is then constructed as follows:

πt

Wavelets and LPG-PCA for Image Denoising http://dx.doi.org/10.5772/intechopen.74453 245

where vp is the

gain, �<sup>i</sup> signð Þ <sup>γ</sup> [30]. This is corresponding to an impulse response vp <sup>1</sup>

the proposed technique.

proposed in [28, 29], from the noisy image, Ib.

Figure 4. The block diagram of the proposed image denoising technique.

5. The Hilbert transform

In practice, υ<sup>s</sup> will include not only the residual noise but also the estimation error of noiseless image I. Consequently, in the implementation [25], of Lei Zhang et al. let

$$
\sigma\_s = \mathbb{C}\_s \sqrt{\sigma^2 - E\left[\tilde{I}^2\right]} \tag{17}
$$

where Cs is a constant satisfying Cs < 1. In [25], Lei Zhang et al. found experimentally that setting Cs around 0:35 can lead to satisfying denoising results for most of the testing images. Figure 2d shows the denoised image (PSNR ¼ 30:1 dB) after the second round of the LPG-PCA denoising technique [25]. Although the PSNR is not too much ameliorated on this image, we can remark clearly that the visual quality is much ameliorated by efficiently eliminating the residual noise obtained from the first round of denoising.

#### 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

Figure 4. The block diagram of the proposed image denoising technique.

Denote bybI, the denoised version of the noisy image in the first stage. ThebI can be expressed as

where υ<sup>s</sup> is the residual noise in the denoised image. The level estimation ofυ<sup>s</sup> is denoted by

smoothed version of noise υ, and it mainly contains the low-frequency component of υ. Let υ~ ¼ υ � υ<sup>s</sup> be their difference, and υ~ mainly contains the high-frequency component of υ.

<sup>s</sup> <sup>≈</sup> <sup>σ</sup><sup>2</sup> � <sup>E</sup> <sup>~</sup><sup>I</sup>

In practice, υ<sup>s</sup> will include not only the residual noise but also the estimation error of noiseless

where Cs is a constant satisfying Cs < 1. In [25], Lei Zhang et al. found experimentally that setting Cs around 0:35 can lead to satisfying denoising results for most of the testing images. Figure 2d shows the denoised image (PSNR ¼ 30:1 dB) after the second round of the LPG-PCA denoising technique [25]. Although the PSNR is not too much ameliorated on this image, we can remark clearly that the visual quality is much ameliorated by efficiently eliminating the

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

<sup>2</sup> h i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>σ</sup><sup>2</sup> � <sup>E</sup> <sup>~</sup><sup>I</sup> 2

r h i

� � �. Generally, compared to E υ<sup>2</sup>

<sup>q</sup> � � and inputs it to the second round of LPG-PCA denoising algorithm. In [25], <sup>σ</sup><sup>s</sup> is

σ<sup>s</sup> ¼

σ2

ffiffiffiffiffiffiffiffiffiffiffiffi E υ<sup>2</sup> s

244 Wavelet Theory and Its Applications

We have E ~I

<sup>2</sup> h i

<sup>s</sup> � 2Eð Þ υ � υ<sup>s</sup> , we obtain

There is <sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ �¼ <sup>E</sup> <sup>υ</sup><sup>~</sup> � <sup>υ</sup>s� þ <sup>E</sup> <sup>υ</sup><sup>2</sup>

estimated based on the difference between I<sup>υ</sup> and bI. Let

s

s

and we can obtain the following approximation: <sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ � <sup>≈</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup>

residual noise obtained from the first round of denoising.

4. The proposed image denoising technique

� � � <sup>2</sup><sup>E</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> ½ �¼ <sup>σ</sup><sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

σ2

σ<sup>s</sup> ¼ Cs

image I. Consequently, in the implementation [25], of Lei Zhang et al. let

<sup>¼</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup> � � <sup>þ</sup> <sup>E</sup> <sup>υ</sup><sup>2</sup>

bI ¼ I þ υ<sup>s</sup> (14)

<sup>~</sup><sup>I</sup> <sup>¼</sup> <sup>I</sup><sup>υ</sup> �b<sup>I</sup> <sup>¼</sup> <sup>υ</sup> � <sup>υ</sup><sup>s</sup> (15)

s

s � � <sup>¼</sup> <sup>σ</sup><sup>2</sup>

<sup>s</sup> � 2E υ � υ<sup>s</sup> ½ �. The υ<sup>s</sup> can be seen as the

� �, <sup>E</sup> <sup>υ</sup><sup>~</sup> � <sup>υ</sup><sup>s</sup> <sup>½</sup> � is much smaller,

<sup>2</sup> h i

<sup>¼</sup> <sup>σ</sup><sup>2</sup><sup>þ</sup>

(16)

(17)

<sup>s</sup> . Thus, from <sup>E</sup> <sup>~</sup><sup>I</sup>

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, Ib, in order to obtain a first estimate of the clean image, Id, and then estimates the noise-level, υ, from Ib. The noisy images Ib, Id, and υ constitute the inputs of the second image denoising system proposed in [25, 27]. The output of this system and the overall proposed one are the final denoised image, Id1. In the image denoising system (LPG-PCA denoising) proposed in [25, 27], Lei Zhang et al. have used the clean image, I, and the noise-level, υ, as the inputs of this system [27]. However, only the noisy image, Ib, is available, and for this raison, we have used in our proposed technique the denoising approach based on DT-DWT [12] in order to obtain a cleaned image, Id, which is then used as a clean image, I. This clean image is one important input of the denoising system proposed by Lei Zhang et al. [27]. In the following two subsections, we will be interested in the first image denoising approach based on DT-DWT [12] and the technique of noise-level estimation proposed in [28, 29], from the noisy image, Ib.
