An Adapted Block Thresholding Method for Omnidirectional Image Denoising

The problem of image denoising is largely discussed in the literature. It is a fundamental preprocessing task, and an important step in almost all view compared to conventional perspectives images, however, the treatments are thus not appropriate for those deformed omnidirectional adaptation of an adaptation to Stein block thresholding method to omnidirectional images. We will adapt different treatments in order to take into account the nature of omnidirectional images


INTRODUCTION
Image denoising is a basic problem in image processing.It represents an important task in almost all image processing applications.It is defined as the process of removing unwanted noise in order to restore the original image.Among all image denoising techniques, wavelet based methods are known to yield the best results.This is due to their excellent localization property which became an indispensable signal and image processing tool to many image processing of application since it provides an appropriate basis for separation noisy signal from the image signal.
Over the last two decades several methods were proposed for image denoising using wavelet thresholding.These techniques can be grouped in two classes: individually thresholding (Donoho and Johnstone, 1994;Donoho, 1995;Chang Kalavathy and Suresh, 2011) and block thresholding (Efroimovich, 1986;Kerkyacharian et al 1997Kerkyacharian et al , 1999Kerkyacharian et al , 2002; Cai and Zhou, perspective images, omnidirectional images offer a large field of view.However they present a non uniform resolution and important geometric distortions.Figure 1 shows an example of omnidirectional sensor. Recently, several works have been interested in the denoising problem for omnidirectional images.Marchand (2008) used the sphere as a projection s for omnidirectional images and defined image processing tools in that space in order to perform the image denoising.Demonceaux and Vasseur (2006) used Markov Random Fields and defined an adapted system neighborhood for omnidirectional images.
, Engineering andTechnology 8(18): 1966-1972 The problem of image denoising is largely discussed in the literature.It is a fundamental preprocessing task, and an important step in almost all image processing applications.Omnidirectional images offer a large field of view compared to conventional perspectives images, however, they contain important distortions treatments are thus not appropriate for those deformed omnidirectional images.In this study adaptation of an adaptation to Stein block thresholding method to omnidirectional images.We will adapt different treatments in order to take into account the nature of omnidirectional images.thresholding, image denoising, omnidirectional image, wavelet Image denoising is a basic problem in image processing.It represents an important task in almost all image processing applications.It is defined as the process of removing unwanted noise in order to restore the original image.Among all image denoising echniques, wavelet based methods are known to yield the best results.This is due to their excellent localization property which became an indispensable signal and image processing tool to many image processing of application since it provides an te basis for separation noisy signal from the Over the last two decades several methods were proposed for image denoising using wavelet thresholding.These techniques can be grouped in two classes: individually thresholding (Donoho and tone, 1994;Donoho, 1995;Chang et al., 2000;Kalavathy andSuresh, 2011) andblock thresholding et al., 1996;Cai, and Zhou, 2009).Unlike perspective images, omnidirectional images offer a field of view.However they present a nonuniform resolution and important geometric distortions.
shows an example of omnidirectional sensor.Recently, several works have been interested in the denoising problem for omnidirectional images.Bigotused the sphere as a projection space and defined image processing tools in that space in order to perform the image denoising.Demonceaux and Vasseur (2006) used Markov Random Fields and defined an adapted neighborhood for omnidirectional images.In this study, we will adapt the Stein block thresholding algorithm to omnidirectional images.The remainder of the study is as follows: In the next section we present the Stein block thresholding approach for perspective images denoising (Chesneau The problem of image denoising is largely discussed in the literature.It is a fundamental preprocessing image processing applications.Omnidirectional images offer a large field of y contain important distortions and classical study we introduce an adaptation of an adaptation to Stein block thresholding method to omnidirectional images.We will adapt different An omnidirectional sensor mounted on mobile robot and the obtained omnidirectional image , we will adapt the Stein block thresholding algorithm to omnidirectional images.The is as follows: In the next section we present the Stein block thresholding approach for perspective images denoising (Chesneau et al., 2010).

Stein block thresholding for perspective images
Let's consider the nonparametric regression  The observed sequence of coefficients is defined by: Figure 2 shows a representation of the Wavelet Transform.The subbands HH j , HL j and the details.The subband LL j is the low resolution residual.
The block thresholding methods was proposed in Hall et al. (1999) and developed, generalized to any dimension and applied to image denoising in Chesneau et al. (2010).The main of this method is to increase the quality of estimation by using the neighborhood information of the wavelet coefficients.The procedure first divides the wavelet coefficients at each resolution level into non-overlapping blocks and then keeps all the coefficients within a block if and only if, the magnitude of the sum of the squared empirical coefficients within that block is greater than a fixed threshold (Chesneau et al., 2010).
Let A j = {1,…, 2 j L -1 } be the set indexing the blocks at scale j where L is the block length.For each block index The observed sequence of coefficients is defined by: (2) Figure 2 shows a representation of the Wavelet and LH j are called is the low resolution The block thresholding methods was proposed in developed, generalized to any denoising in Chesneau (2010).The main of this method is to increase the quality of estimation by using the neighborhood information of the wavelet coefficients.The procedure divides the wavelet coefficients at each resolution overlapping blocks and then keeps all the and only if, the magnitude of the sum of the squared empirical coefficients within d threshold (Chesneau be the set indexing the blocks at scale j where L is the block length.For each be the set indexing the block: B = (x, y); (K -1)L +1 x KL and 1 y j, K The rule of shrinkage of James Eq. ( 3):

ADAPTED METHOD FOR OMNIDIRECTIONAL IMAGES
Omnidirectional images offer a large field of view, nevertheless, they contains significant radial distortions and present non-uniform resolution due to the non linear projection.Consequently, denoising such images in the same way as a perspective image will lead to mistaken results.In the literature there are two ways to treat omnidirectional images.One is treating them such as perspective images by adapting their characteristic The other, is to use the projection on the sphere and perform all treatments in this domain. (3) James-stein at block

ADAPTED METHOD FOR OMNIDIRECTIONAL IMAGES
Omnidirectional images offer a large field of view, nevertheless, they contains significant radial distortions uniform resolution due to the non-Consequently, denoising such images in the same way as a perspective image will lead to mistaken results.In the literature there are two ways to treat omnidirectional images.One is treating them such as perspective images by adapting their characteristic.The other, is to use the projection on the sphere and perform all treatments in this domain.projection to a virtual sphere followed by a projection from the sphere to the retina.This second projection depends on the shape of the mirror.
shows the equivalence between the catadioptric projection and the two step mapping sphere.
App. Sci. Eng. Technol., 8(18): 1966-1972, 20141968 Equivalence between the catadioptric projection and the two step mapping via the sphere The subband of DWT (left) subband projection to hemi-sphere (right) to a virtual sphere followed by a projection from the sphere to the retina.This second projection depends on the shape of the mirror.Figure 3 and 4 uivalence between the catadioptric mapping via the The parameter ξ defines the shape of mirror.In our case, we consider parabolic mirror where However, the method can easily be adapted to the general case, let ( , ) (X , Y , Z ) s s s s s P P θ ϕ = the sphere.The Cartesian coordinates of this point are given by: defines the shape of mirror.In our case, we consider parabolic mirror where ξ = 1.However, the method can easily be adapted to the  The stereographic projection of P s on the image plane yields point P i (x, y) given by: By combining Eq. ( 5) and ( 6) we obtain the spherical coordinates of point P i : cot( ) cos 2 y = cot( ) sin 2 Adapted spherical block estimator: The block as defined in the classic stein block thresholding method has the shape of a rectangle.In our case we need to define the block in the sphere in order to take into account the radial distortions present in omnidirectional images.Each subband is mapped on the sphere (Fig. 5) and a spherical block is defined according to the spherical coordinates θ and as shown in Fig. 6.
The set indexing the position of coefficients within the K th block in the sphere is defined by: where L θ is the block length, where U p is the spherical block as defined in Eq. ( 8) for omnidirectional images.

APPLICATION AND RESULTS
To show the improvement given by our proposed adapted method, we have applied it on synthetic omnidirectional images of 512*512 pixels obtained using the ray tracing program POV-Ray.We created a scene where a parabolic camera, initially at the origin of the three dimensional Cartesian coordinate system and looking in the y-axis.This camera observes two planes   c) and (d) denoising using respectively the soft (Donoho, 1995), the classical Stein block thresholding method result of denoising methods (Chesneau et al., 2010) and our proposed method We have compared our results with the classical method proposed in Chesneau et al.

soft-thresholding method proposed in Donoho
We have applied the orthogonal wavelet transform to get the wavelet coefficients using the Symmlet wavelet with 6 order vanishing moments.
In order to measure methods calculate the Peak Signal to Noise Ratio (PSNR), the Mean Square Error (MSE) and the Signal to Noise Ratio (SNR) given by: where f and f are respectively the reference and the denoised image.Figure 7 and 8 respectively show the evolution of the PSNR and the SNR against the noise level.Overall, both the PSNR and the SNR values of our proposed approach remain higher than those for the other two methods.The evolution of the MSE level is displayed in Fig. 9 and 10 shows a visual 4.5 -thresholding method (Donoho, 1995), the classical Stein block result of denoising methods , 2010) and our proposed method We have compared our results with the classical (2010) and with the thresholding method proposed in Donoho (1995).We have applied the orthogonal wavelet transform to get the wavelet coefficients using the Symmlet wavelet In order to measure methods performances we calculate the Peak Signal to Noise Ratio (PSNR), the Mean Square Error (MSE) and the Signal to Noise   (Donoho, 1995) Classical blockJS (Chesneau et al., 2010) Proposed method Table 2: SNR evolution against the noise level for different image denoising methods on real image Noise Soft-thresholding (Donoho, 1995) Classical blockJS (Chesneau et al., 2010) Proposed method Table 3: MSE evolution against the noise level for different image denoising methods on real image Noise Soft-thresholding (Donoho, 1995) Classical blockJS (Chesneau et al., 2010) Proposed method (a) Fig. 11: Visual comparison of denoising methods on real image the soft-thresholding method (Donoho, 1995), the classical (Chesneau et al., 2010) and our proposed method comparison of denoising results obtained by different methods using the same noise level σ seen that our method achieves the smallest values of MSE compared with the other two methods.We used also real omnidirectional images.They are captured using a catadioptric camera embedded on a mobile robot as shown in Fig. 1.Table 1 to respectively, the evolution of the PSNR, the SNR and the MSE against the noise level.Figure 11 shows a visual comparison of denoising results obtained by different methods using the same noise level These results confirm the previous positive results obtained on synthetic images and show that the classical approaches (Donoho, 1995;Chesneau 2010), even if these methods work well images, are not appropriate to omnidirectional images.

CONCLUSION
Omnidirectional images are rich in information since they depict almost the whole scene.Unfortunately, they include severe distortions.That is why classical image denoising techniques th App.Sci. Eng. Technol., 8(18): 1966-1972 denoising methods on real image (a) noisy 30 σ = , (b), (c) and (d) denoising using respectively thresholding method (Donoho, 1995), the classical stein block thresholding method result of denoising methods , 2010) and our proposed method comparison of denoising results obtained by different 30 σ = .It can be seen that our method achieves the smallest values of MSE compared with the other two methods.
We used also real omnidirectional images.They using a catadioptric camera embedded on a Table 1 to 3 shows, respectively, the evolution of the PSNR, the SNR and the MSE against the noise level.Figure 11 shows a visual comparison of denoising results obtained by ent methods using the same noise level 30 σ = .These results confirm the previous positive results obtained on synthetic images and show that the classical approaches (Donoho, 1995;Chesneau et al., well for perspectives images, are not appropriate to omnidirectional images.
Omnidirectional images are rich in information since they depict almost the whole scene.Unfortunately, they include severe distortions.That is why classical image denoising techniques that work for perspectives images need to be adapted for omnidirectional ones.
In this study we have proposed an adaptation to Stein block thresholding (Chesneau applied our approach in synthetic and real images and we compared it to the classical methods (Donoho, 1995;Chesneau et al., 2010).The comparison shows that our adapted method has the best overall results over any other method.perspectives images need to be adapted for we have proposed an adaptation to Stein block thresholding (Chesneau et al., 2010).We applied our approach in synthetic and real images and assical methods (Donoho, 2010).The comparison shows that our adapted method has the best overall results

Fig. 1 :
Fig. 1: An omnidirectional sensor mounted on mobile robot and the obtained omnidirectional image

Fig. 2 :
Fig. 2: Subbands of 2-D orthogonal wavelet transform be the set indexing the position of coefficients within the K th block:App.Sci.Eng.Technol., 8(18): 1966-1972, 2014   1967    D orthogonal wavelet transform of Y, the matrix of Unknown coefficients and a sequence of noise random variables, where is the dimensional dyadic orthogonal wavelet transform is the scale parameter, is the threshold.For each j = {0,…, J} if the mean energy within the block , Fig.3: Representation of a block as defined by Fig. 4: Equivalence between the catadioptric projection and the two step mapping via the sphere be the point on the sphere.The Cartesian coordinates of this point are

Fig. 6 :
Fig. 6: Representation of the spherical block as defined in Eq. (8) the blocks in the sphere at scale j.The suitable estimator for omnidirectional images is given using the Stereographic Projection:

Fig. 7 :
Fig. 7: Comparison of PSNR evolution against the noise level for different image denoising methods on synthetic image

Figure 7
Figure 7 and 8 respectively show the evolution of the PSNR and the SNR against the noise level.Overall, both the PSNR and the SNR values of our proposed er than those for the other two MSE against the noise and 10 shows a visual ), (c) and (d) denoising using respectively result of denoising methods

Table 1 :
PSNR evolution against the noise level for different image denoising methods on real image