5. The Hilbert transform

The Hilbert transform of a signal corresponds in Fourier plane to a filter with complex gain, �<sup>i</sup> signð Þ <sup>γ</sup> [30]. This is corresponding to an impulse response vp <sup>1</sup> πt where vp is the principal value in Cauchy sense [30]. The analytic signal is then constructed as follows:

$$\mathbf{x}(t) = \mathbf{x}(t) + iH\{\mathbf{x}(t)\} = \mathbf{x}(t) + \frac{i}{\pi}v\boldsymbol{p} \int\_{-\infty}^{+\infty} \frac{\mathbf{x}(s)}{t-s}ds\tag{18}$$

• A lower computation cost ( Two DWT),

tree (a) and tree (b), respectively [30].

Figure 6. Q-shift wavelet obtained with filters Antonini [30].

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

Wavelets and LPG-PCA for Image Denoising http://dx.doi.org/10.5772/intechopen.74453 247

Figure 5. Dual tree of real filters for the Q-shift DT-CWT, giving real and imaginary parts of complex coefficients from

• The possibility of an exact reconstruction is preserved.

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 f ; Hψa,t D E <sup>¼</sup> Hf ;ψa,t D E [30]. The latter equality is justified by the fact that the Hilbert transform is considered as a linear filter [30]. Therefore, we have the following scheme: let X nð Þ be the signal to be analyzed with real wavelet by using the Mallat algorithm in order to obtain the wavelet coefficients, d1ð Þ j; k . Then, we analyze HX nð Þ with the same wavelet, and we obtain the coefficients d2ð Þ j; k . Then, we construct the complex coefficients: dcomplex ¼ d1ð Þþ j; k i d2ð Þ j; k . As follows, the magnitude of those coefficients is named Hilbert magnitude. The drawbacks of this method are as follows: The support of the Hilbert transform of a wavelet having a compact support is infinite. There is a computing disadvantage because the cost of two wavelet transforms plus the Hilbert Transform. Theoretically speaking, it is possible to limit the drawback of the support of the Hilbert transform of the wavelet by using an approximate of the Hilbert transform. However, this approximation cannot be optimized for all scales [30]. One solution of this problem has been proposed by Kingsbury: the dual tree [30].

## 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<sup>1</sup> and g<sup>1</sup> are high-pass filters of the first and second trees, and h<sup>0</sup> and g<sup>0</sup> 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 dcomplex(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),

z tðÞ¼ x tð Þþ iH x t f g¼ ð Þ x tð Þþ <sup>i</sup>

analysis is performed with initial wavelet because we have f ; Hψa,t

the dual tree [30].

246 Wavelet Theory and Its Applications

6. Dual-tree complex wavelet transform

coefficients is named dual-tree magnitude [30].

compared to the simple Hilbert transform (Section 5) are [30]:

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

The latter equality is justified by the fact that the Hilbert transform is considered as a linear filter [30]. Therefore, we have the following scheme: let X nð Þ be the signal to be analyzed with real wavelet by using the Mallat algorithm in order to obtain the wavelet coefficients, d1ð Þ j; k . Then, we analyze HX nð Þ with the same wavelet, and we obtain the coefficients d2ð Þ j; k . Then, we construct the complex coefficients: dcomplex ¼ d1ð Þþ j; k i d2ð Þ j; k . As follows, the magnitude of those coefficients is named Hilbert magnitude. The drawbacks of this method are as follows: The support of the Hilbert transform of a wavelet having a compact support is infinite. There is a computing disadvantage because the cost of two wavelet transforms plus the Hilbert Transform. Theoretically speaking, it is possible to limit the drawback of the support of the Hilbert transform of the wavelet by using an approximate of the Hilbert transform. However, this approximation cannot be optimized for all scales [30]. One solution of this problem has been proposed by Kingsbury:

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<sup>1</sup> and g<sup>1</sup> are high-pass filters of the first and second trees, and h<sup>0</sup> and g<sup>0</sup> 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 dcomplex(j,k) = dr(j,k) + i di(j,k). The magnitude of those

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

<sup>π</sup> vp

þ ð∞

x sð Þ t � s

D E

ds (18)

¼ Hf ;ψa,t D E

[30].

�∞


Figure 5. Dual tree of real filters for the Q-shift DT-CWT, giving real and imaginary parts of complex coefficients from tree (a) and tree (b), respectively [30].

Figure 6. Q-shift wavelet obtained with filters Antonini [30].

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 ψð Þ¼ x; y ψð Þ� x ψð Þy associated with the row-column implementation of the wavelet transform, where ψð Þx is a complex wavelet (approximately analytic) and is expressed as follows [31]:

$$
\psi(\mathbf{x}) = \psi\_h(\mathbf{x}) + i \,\psi\_g(\mathbf{x}).\tag{19}
$$

approximately the Hilbert transform of <sup>ψ</sup>hð Þ<sup>x</sup> (ψgð Þ<sup>x</sup> <sup>≈</sup> <sup>H</sup> <sup>ψ</sup>hð Þ<sup>x</sup> � �). Note that <sup>ψ</sup>hð Þ<sup>x</sup> <sup>ψ</sup>hð Þ<sup>y</sup> is the sub-band HH of a separable 2D real wavelet transform implemented employing the filters <sup>h</sup>0ð Þ <sup>n</sup> <sup>0</sup>;h1ð Þ <sup>n</sup> n o. The term <sup>ψ</sup>gð Þ<sup>x</sup> <sup>ψ</sup>gð Þ<sup>y</sup> is also the sub-band HH which is obtained from the application of a real separable wavelet transform. The latter is implemented by employing the

Figure 8. Idealized diagram illustrating the support of spectrum of this real wavelet, Real Part f g ψð Þ x; y [31].

2D wavelet ψ2ð Þ¼ x; y ψð Þx ψð Þy where ψð Þy is the complex conjugate of ψð Þy and, as previously mentioned, ψð Þx is approximately the analytic wavelet, ψhð Þþ x i ψgð Þx . Therefore, we

h i

¼ ψhð Þx ψhð Þþ y ψgð Þx ψgð Þþ y i ψgð Þx ψhð Þþ y ψhð Þx ψgð Þy

Real Part <sup>ψ</sup>2ð Þ <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> <sup>ψ</sup>hð Þ<sup>x</sup> <sup>ψ</sup>hð Þþ <sup>y</sup> <sup>ψ</sup>gð Þ<sup>x</sup> <sup>ψ</sup>gð Þ<sup>y</sup> : (23)

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

As above, the spectrum of the complex wavelet, ψ2ð Þ x; y , is supported in just one quadrant of the 2D frequency plane. If the real part of this complex wavelet is taken, then we

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

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 <sup>þ</sup>45<sup>∘</sup> as illustrated in the fifth panel of Figure 11. To have four more oriented real 2D wavelets, one can repeat this procedure on

Figure 9. The idealized diagram in 2D frequency plane of the spectrum of this complex wavelet [31].

h i <sup>ψ</sup>hð Þ� <sup>y</sup> <sup>i</sup>ψgð Þ<sup>y</sup>

, we consider now the complex

Wavelets and LPG-PCA for Image Denoising http://dx.doi.org/10.5772/intechopen.74453 249

h i (22)

filters <sup>g</sup>0ð Þ <sup>n</sup> <sup>0</sup>;g1ð Þ <sup>n</sup> n o. To have a real 2D wavelet oriented at <sup>þ</sup>45<sup>∘</sup>

ψ2ð Þ¼ x; y ψhð Þþ x i ψgð Þx

have

Figure 9.

have

in Figure 10.

Therefore, we obtain the following expression of ψð Þ x; y :

$$\psi(\mathbf{x}, \mathbf{y}) = \left[\psi\_h(\mathbf{x}) + i\,\psi\_g(\mathbf{x})\right] \left[\psi\_h(\mathbf{y}) + i\,\psi\_g(\mathbf{y})\right] = \psi\_h(\mathbf{x})\psi\_h(\mathbf{y}) - \psi\_g(\mathbf{x})\psi\_g(\mathbf{y}) + \dots$$

$$i\left[\psi\_g(\mathbf{x})\psi\_h(\mathbf{y}) + \psi\_h(\mathbf{x})\psi\_g(\mathbf{y})\right] \tag{20}$$

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 (ψð Þ x; y ) spectrum is supported in just one quadrant of the 2D frequency plane. That is why the complex 2D wavelet, ψð Þ x; y , is oriented. If the real part of this complex wavelet is taken, then the sum of two separable wavelets is obtained:

$$\text{Real Part}\left\{\psi(\mathbf{x},y)\right\} = \psi\_h(\mathbf{x})\psi\_h(y) - \psi\_g(\mathbf{x})\psi\_g(y). \tag{21}$$

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 �45<sup>∘</sup> . It is deserving mentioning that this construction is depending on ψð Þ¼ x ψhð Þþ x i ψgð Þx being (approximately) analytic or equivalently on ψgð Þx being

Figure 7. Idealized diagram illustrating the Fourier spectrum support of the complex wavelet, ψð Þ¼ x; y ψð Þ� x ψð Þy [31].

Figure 8. Idealized diagram illustrating the support of spectrum of this real wavelet, Real Part f g ψð Þ x; y [31].

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

To explain how the DT-CWT produces oriented wavelets, consider the 2D wavelet ψð Þ¼ x; y ψð Þ� x ψð Þy associated with the row-column implementation of the wavelet transform, where

> ψhð Þþ y i ψgð Þy h i

The following idealized diagram (Figure 7) illustrates the Fourier spectrum support of this

Since the (approximately) 1D wavelet spectrum is supported on just one side of the frequency axis, the complex 2D wavelet (ψð Þ x; y ) spectrum is supported in just one quadrant of the 2D frequency plane. That is why the complex 2D wavelet, ψð Þ x; y , is oriented. If the real part of this

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

on ψð Þ¼ x ψhð Þþ x i ψgð Þx being (approximately) analytic or equivalently on ψgð Þx being

Figure 7. Idealized diagram illustrating the Fourier spectrum support of the complex wavelet, ψð Þ¼ x; y ψð Þ� x ψð Þy [31].

Real Part f g ψð Þ x; y ¼ ψhð Þx ψhð Þ� y ψgð Þx ψgð Þy : (21)

. It is deserving mentioning that this construction is depending

complex wavelet is taken, then the sum of two separable wavelets is obtained:

i ψgð Þx ψhð Þþ y ψhð Þx ψgð Þy h i

ψð Þ¼ x ψhð Þþ x i ψgð Þx : (19)

¼ ψhð Þx ψhð Þ� y ψgð Þx ψgð Þþy

(20)

ψð Þx is a complex wavelet (approximately analytic) and is expressed as follows [31]:

Therefore, we obtain the following expression of ψð Þ x; y :

h i

ψð Þ¼ x; y ψhð Þþ x i ψgð Þx

vanishing moments).

248 Wavelet Theory and Its Applications

complex wavelet [31].

Figure 11), is oriented at �45<sup>∘</sup>

6.1. 2D DT-CWT

approximately the Hilbert transform of <sup>ψ</sup>hð Þ<sup>x</sup> (ψgð Þ<sup>x</sup> <sup>≈</sup> <sup>H</sup> <sup>ψ</sup>hð Þ<sup>x</sup> � �). Note that <sup>ψ</sup>hð Þ<sup>x</sup> <sup>ψ</sup>hð Þ<sup>y</sup> is the sub-band HH of a separable 2D real wavelet transform implemented employing the filters <sup>h</sup>0ð Þ <sup>n</sup> <sup>0</sup>;h1ð Þ <sup>n</sup> n o. The term <sup>ψ</sup>gð Þ<sup>x</sup> <sup>ψ</sup>gð Þ<sup>y</sup> is also the sub-band HH which is obtained from the application of a real separable wavelet transform. The latter is implemented by employing the filters <sup>g</sup>0ð Þ <sup>n</sup> <sup>0</sup>;g1ð Þ <sup>n</sup> n o. To have a real 2D wavelet oriented at <sup>þ</sup>45<sup>∘</sup> , we consider now the complex 2D wavelet ψ2ð Þ¼ x; y ψð Þx ψð Þy where ψð Þy is the complex conjugate of ψð Þy and, as previously mentioned, ψð Þx is approximately the analytic wavelet, ψhð Þþ x i ψgð Þx . Therefore, we have

$$\begin{split} \psi\_{2}(\mathbf{x}, \mathbf{y}) &= \left[ \psi\_{h}(\mathbf{x}) + i \,\psi\_{\mathcal{g}}(\mathbf{x}) \right] \left[ \psi\_{h}(\mathbf{y}) - i \psi\_{\mathcal{g}}(\mathbf{y}) \right] \\ &= \psi\_{h}(\mathbf{x}) \psi\_{h}(\mathbf{y}) + \psi\_{\mathcal{g}}(\mathbf{x}) \psi\_{\mathcal{g}}(\mathbf{y}) + i \left[ \psi\_{\mathcal{g}}(\mathbf{x}) \psi\_{h}(\mathbf{y}) + \psi\_{h}(\mathbf{x}) \psi\_{\mathcal{g}}(\mathbf{y}) \right] \end{split} \tag{22}$$

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, ψ2ð Þ x; y , is supported in just one quadrant of the 2D frequency plane. If the real part of this complex wavelet is taken, then we have

$$\text{Real } Part\left\{\psi\_2(\mathbf{x},y)\right\} = \psi\_h(\mathbf{x})\psi\_h(y) + \psi\_g(\mathbf{x})\psi\_g(y). \tag{23}$$

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 <sup>þ</sup>45<sup>∘</sup> as illustrated in the fifth panel of Figure 11. To have four more oriented real 2D wavelets, one can repeat this procedure on

Figure 9. The idealized diagram in 2D frequency plane of the spectrum of this complex wavelet [31].

Figure 10. Idealized diagram in 2D frequency plane of the spectrum Real Part <sup>ψ</sup>2ð Þ <sup>x</sup>; <sup>y</sup> � � [31].

Figure 11. Typical wavelets associated with the real oriented 2D dual-tree wavelet transform. Top row illustrates the wavelets in the space domain: bottom row illustrates the (idealized) support of the Fourier spectrum of each wavelet in 2D frequency plane. The absence of the checkerboard phenomenon is observed in both frequency and spatial domains.

the complex wavelets expressed as follows: ϕð Þx ψð Þy , ψð Þx ϕð Þy , ϕð Þx ψ ⇀ ð Þy , and ψð Þx ϕð Þy where we have

$$
\psi(\mathbf{x}) = \psi\_h(\mathbf{x}) + i \,\psi\_g(\mathbf{x}) \tag{24}
$$

<sup>ψ</sup>1,1ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>hð Þ<sup>x</sup> <sup>ψ</sup>hð Þ<sup>y</sup> <sup>h</sup>

<sup>ψ</sup>1,2ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ψ</sup>hð Þ<sup>x</sup> <sup>ϕ</sup>hð Þ<sup>y</sup> <sup>h</sup>

<sup>ψ</sup>1,3ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ψ</sup>hð Þ<sup>x</sup> <sup>ψ</sup>hð Þ<sup>y</sup> <sup>h</sup>

2

7. The technique of Noise-level estimation

The normalization factor 1= ffiffiffi

frame [31].

and scenes.

8. Evaluation criteria

which are detailed in [32].

,ψ2, <sup>1</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ϕ</sup>gð Þ<sup>x</sup> <sup>ψ</sup>gð Þ<sup>y</sup> <sup>g</sup>

,ψ2, <sup>2</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ψ</sup>gð Þ<sup>x</sup> <sup>ϕ</sup>gð Þ<sup>y</sup> <sup>g</sup>

,ψ2, <sup>3</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>ψ</sup>gð Þ<sup>x</sup> <sup>ψ</sup>gð Þ<sup>y</sup> <sup>g</sup>

an orthonormal operation. In Figure 11 the six real oriented wavelets derived from a pair of typical wavelets satisfying <sup>ψ</sup>gð Þ<sup>x</sup> <sup>≈</sup> <sup>H</sup> <sup>ψ</sup>hð Þ<sup>x</sup> � � are illustrated. Compared to separable wavelets, these six non-separable wavelets succeed in isolating different orientations. Each of these six wavelets are aligned with a specific direction. Moreover, no checkerboard effect appears. In addition, they cover more distinct orientations than the separable wavelets obtained from the application of DWT. Moreover, since the sum/difference operation is orthonormal, the wavelet set is obtained from integer translates and dyadic dilations from a

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

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

<sup>p</sup> is used only so that the sum/difference operation constitutes

, (28)

251

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

, (29)

, (30)

$$\phi(\mathbf{x}) = \phi\_h(\mathbf{x}) + i \,\phi\_g(\mathbf{x}) \tag{25}$$

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:

$$
\psi\_i(\mathbf{x}, y) = \frac{1}{\sqrt{2}} \left( \psi\_{1,i}(\mathbf{x}, y) - \psi\_{2,i}(\mathbf{x}, y) \right) \tag{26}
$$

$$
\psi\_{i+3}(\mathbf{x}, y) = \frac{1}{\sqrt{2}} \left( \psi\_{1,i}(\mathbf{x}, y) + \psi\_{2,i}(\mathbf{x}, y) \right) \tag{27}
$$

For i ¼ 1, 2, 3, the two separable 2-D wavelet bases are expressed as follow:

$$
\psi\_{1,1}(\mathbf{x}, \mathbf{y}) = \phi\_h(\mathbf{x})\psi\_h(\mathbf{y})^h,\\
\psi\_{2,1}(\mathbf{x}, \mathbf{y}) = \phi\_g(\mathbf{x})\psi\_g(\mathbf{y})^g,\tag{28}
$$

$$
\psi\_{1,2}(\mathbf{x}, \mathbf{y}) = \psi\_h(\mathbf{x})\phi\_h(\mathbf{y})^\hbar , \psi\_{2,2}(\mathbf{x}, \mathbf{y}) = \psi\_g(\mathbf{x})\phi\_g(\mathbf{y})^\hbar , \tag{29}
$$

$$
\psi\_{1,3}(\mathbf{x}, \mathbf{y}) = \psi\_h(\mathbf{x})\psi\_h(\mathbf{y})^\hbar \, \vert \, \psi\_{2,3}(\mathbf{x}, \mathbf{y}) = \psi\_g(\mathbf{x})\psi\_g(\mathbf{y})^\mathbf{y} \,. \tag{30}
$$

The normalization factor 1= ffiffiffi 2 <sup>p</sup> is used only so that the sum/difference operation constitutes an orthonormal operation. In Figure 11 the six real oriented wavelets derived from a pair of typical wavelets satisfying <sup>ψ</sup>gð Þ<sup>x</sup> <sup>≈</sup> <sup>H</sup> <sup>ψ</sup>hð Þ<sup>x</sup> � � are illustrated. Compared to separable wavelets, these six non-separable wavelets succeed in isolating different orientations. Each of these six wavelets are aligned with a specific direction. Moreover, no checkerboard effect appears. In addition, they cover more distinct orientations than the separable wavelets obtained from the application of DWT. Moreover, since the sum/difference operation is orthonormal, the wavelet set is obtained from integer translates and dyadic dilations from a frame [31].

#### 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

the complex wavelets expressed as follows: ϕð Þx ψð Þy , ψð Þx ϕð Þy , ϕð Þx ψ

Figure 10. Idealized diagram in 2D frequency plane of the spectrum Real Part <sup>ψ</sup>2ð Þ <sup>x</sup>; <sup>y</sup> � � [31].

ψi

ð Þ¼ x; y

<sup>ψ</sup><sup>i</sup>þ<sup>3</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup>

For i ¼ 1, 2, 3, the two separable 2-D wavelet bases are expressed as follow:

1 ffiffiffi 2 p ψ1,i

> 1 ffiffiffi 2 p ψ1,i

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

Figure 11. Typical wavelets associated with the real oriented 2D dual-tree wavelet transform. Top row illustrates the wavelets in the space domain: bottom row illustrates the (idealized) support of the Fourier spectrum of each wavelet in 2D frequency plane. The absence of the checkerboard phenomenon is observed in both frequency and spatial domains.

ð Þ� x; y ψ2,i

� �

ð Þþ x; y ψ2,i

� �

where we have

250 Wavelet Theory and Its Applications

expressed as follows:

⇀

ψð Þ¼ x ψhð Þþ x i ψgð Þx (24)

ϕð Þ¼ x ϕhð Þþ x i ϕgð Þx (25)

ð Þ x; y

ð Þ x; y

ð Þy , and ψð Þx ϕð Þy

(26)

(27)

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].
