Recent Advances in Signal Processing232 Fig. 16. Denoising results of Goldhill image corrupted by heavily correlated streak noise (top left): by NLMS denoising (top right), by BLS-GSM denoising (bottom left), by Probshrink denoising for white noise (bottom right) In a fourth denoising experiment, the Stonehenge image was used. It was treated as a color image, and used as input for a mosaicing/demosaicing experiment using the bilinear demosaicing algorithm. This results in low frequency noise structures. Then the red channel of the resulting color image was used as input for the denoising experiment. Again, it is visible that the white noise denoising algorithm Probshrink does not succeed in suppressing the noise artifacts, while the algorithms for correlated noise do. It is also visible that the BLS- GSM algorithm suffers from ringing near the top edge of the Stonehenge structure. This type of artifacts is common in wavelet-base denoising experiments and is a result from incorrectly suppressing the small coefficients that make up the edge in higher frequency scales, while keeping their respective counterparts in lower frequency scales. Fig. 17. Denoising results of Stonehenge image corrupted by simulated red channel demosaicing noise (top left): by NLMS denoising (top right), by BLS-GSM denoising (bottom left), by Probshrink denoising for white noise (bottom right) From the experiments, some conclusions can be made. White noise denoising algorithms, such as Probshrink, work well enough as long as the image is corrupted by white noise. It fails when presented with correlated noise. One reason is that the Donoho MAD estimator is often a very bad choice, leading to underestimated noise power (for low frequency noise) or severely overestimated noise power (for high frequency noise). Because of this failure of the MAD estimator, the choice was made to choose the noise variance parameter heuristically for the white noise Probshrink algorithm, in order to obtain the highest possible PSNR. It can be concluded from figures 14-17 and table 1, that for situations where image noise is correlated, a simple white noise denoising algorithm will not perform optimally and there is need for the techniques and ideas explained in this chapter. Noisy ProbShrink BLS-GSM NLMS White 22dB 29.22dB 29.76dB 29.88dB Demosaicing 27.9dB 29.8dB 32.6dB 31.4dB Thermal 24.5dB 26.0dB 31.6dB 31.5dB Streaks 16.1dB 22.8dB 25.7dB 25.9dB Table 1. PSNR table for the different denoising experiments Suppression of Correlated Noise 233 Fig. 16. Denoising results of Goldhill image corrupted by heavily correlated streak noise (top left): by NLMS denoising (top right), by BLS-GSM denoising (bottom left), by Probshrink denoising for white noise (bottom right) In a fourth denoising experiment, the Stonehenge image was used. It was treated as a color image, and used as input for a mosaicing/demosaicing experiment using the bilinear demosaicing algorithm. This results in low frequency noise structures. Then the red channel of the resulting color image was used as input for the denoising experiment. Again, it is visible that the white noise denoising algorithm Probshrink does not succeed in suppressing the noise artifacts, while the algorithms for correlated noise do. It is also visible that the BLS- GSM algorithm suffers from ringing near the top edge of the Stonehenge structure. This type of artifacts is common in wavelet-base denoising experiments and is a result from incorrectly suppressing the small coefficients that make up the edge in higher frequency scales, while keeping their respective counterparts in lower frequency scales. Fig. 17. Denoising results of Stonehenge image corrupted by simulated red channel demosaicing noise (top left): by NLMS denoising (top right), by BLS-GSM denoising (bottom left), by Probshrink denoising for white noise (bottom right) From the experiments, some conclusions can be made. White noise denoising algorithms, such as Probshrink, work well enough as long as the image is corrupted by white noise. It fails when presented with correlated noise. One reason is that the Donoho MAD estimator is often a very bad choice, leading to underestimated noise power (for low frequency noise) or severely overestimated noise power (for high frequency noise). Because of this failure of the MAD estimator, the choice was made to choose the noise variance parameter heuristically for the white noise Probshrink algorithm, in order to obtain the highest possible PSNR. It can be concluded from figures 14-17 and table 1, that for situations where image noise is correlated, a simple white noise denoising algorithm will not perform optimally and there is need for the techniques and ideas explained in this chapter. Noisy ProbShrink BLS-GSM NLMS White 22dB 29.22dB 29.76dB 29.88dB Demosaicing 27.9dB 29.8dB 32.6dB 31.4dB Thermal 24.5dB 26.0dB 31.6dB 31.5dB Streaks 16.1dB 22.8dB 25.7dB 25.9dB Table 1. PSNR table for the different denoising experiments Recent Advances in Signal Processing234 In a last experiment, we used the 3D dual tree complex wavelet denoising algorithm for MRI (Aelterman, 2008) to illustrate the denoising performance on practical MRI images. A qualitative comparison can be seen in figure 18. Fig. 18. Denoising results of noisy MRI data. (left) noisy 3D MRI sequence (middle) denoised by 2D per-slice Probshrink (right) denoised by 3D correlated noise Probshrink for MRI 7. Conclusion From the results in the previous section, it is clear that one needs to make use of specialized denoising algorithms for situations in which one encounters correlated noise in images. The short overview in section 2 shows that there are many such situations in practice. Correlated noise manifests itself as stripes, blobs or other image structures that cannot be modelled as spatially independent. Several useful noise estimation techniques were presented that can be used when creating or adapting a white noise denoising algorithm for use with correlated noise. To illustrate this, some state-of-the-art techniques were explained and compared with techniques designed for white noise. 8. References Aelterman, J.; Goossens, B.; Pizurica, A. & Philips, W. (2008) Removal of Correlated Rician Noise in Magnetic Resonance Imaging Proceedings of European Signal Processing Conference (EUSIPCO, Lausanne, 2008 Aelterman, J.; Goossens, B.; Pizurica, A. ; Philips, W. (2009) Locally Adaptive Complex Wavelet-Based Demosaicing for Color Filter Array Images Proceedings of SPIE Electronic Imaging 2009, San Jose, CA, Vol. 7248, no. 0J Bayer, B. (1976) Color Imaging Array US Patent 3,971,065 Borel, C.; Cooke, B.; Laubscher, B. (1996) Partial Removal of Correlated noise in Thermal Imagery Proceedings of SPIE, Vol. 2759, 131 Buades, A., Coll B. & Morel J. M. (2005) Image Denoising by Non-Local Averaging, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, vol. 2, pp. 25-28 Buades, A.; Coll, B & Morel, J.M (2008) Nonlocal Image and Movie Denoising Int Journal on Computer vision Vol. 76, pp. 123-139 Dabov, K.; Foi, A.; Katkovnik, V. & Egiazarian K. (2006) Image Denoising with Block- Matching and 3D Filtering, Proc. SPIE Electronic Imaging: Algorithms and Systems V, no. 6064A-30 Dabov, K.; Foi, A.; Katkovnik, V. & Egiazarian K. (2007) Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. on Im. Processing, vol. 16, no. 8 Donoho, D. & Johnstone, I. (1994) Adapting to Unknown Smoothness via Wavelet Shrinkage Journal of the American Statistics Association, Vol. 90 Donoho, D. L. (1995) De-Noising by Soft-Thresholding, IEEE Transactions on Information Theory, vol. 41, pp. 613-62. Easley, G.; Labate, D.; Lim, Wang-Q, (2006) Sparse Directional Image Representation using the Discrete Shearlet Transform Preprint submitted to Elsevier Preprint Elad, M.; Matalon, B.; & Zibulevsky, M. (2006) Image Denoising with Shrinkage and Redundant Representations Proc. IEEE Conf. on Computer Vision and Pattern Recognition vol. 2, pp. 1924-1931 Field, D. (1987) Relations between the statistics of natural images and the response properties of cortical cells J. Opt. Soc. Am. A 4, p. 2379-2394 Goossens, B.; Pizurica, A. & Philips, W. (2007) Removal of Correlated Noise by Modeling Spatial Correlations and Interscale Dependencies in the Complex Wavelet Domain Proceedings of International Conference on Image Processing (ICIP) pp. 317-320 Goossens, B.; Luong, H., Pizurica, A. Pizurica & Philips, W. (2008) An Improved Non-Local Denoising Algorithm Proceedings of international Workshop on Local and Non-Local Approximation in Image Processing, Lausanne, 2008 Goossens, B.; Pizurica, A. & Philips W. (2009) Removal of correlated noise by modelling the signal of interest in the wavelet domain IEEE Transactions on Image Processing in press Guerrero-Colon, J. ; Simoncelli, E. & Portilla, J. (2008) Image Denoising using Mixtures of Gaussian Scale Mixtures, Proc. IEEE Int. Conf. on Image Processing (ICIP), San Diego, 2008. Hastie, Trevor; Tibshirani, Robert & Friedman, J. (2001) The Elements of Statistical Learning New York: Springer 8.5 The EM algorithm pp. 236–24 Suppression of Correlated Noise 235 In a last experiment, we used the 3D dual tree complex wavelet denoising algorithm for MRI (Aelterman, 2008) to illustrate the denoising performance on practical MRI images. A qualitative comparison can be seen in figure 18. Fig. 18. Denoising results of noisy MRI data. (left) noisy 3D MRI sequence (middle) denoised by 2D per-slice Probshrink (right) denoised by 3D correlated noise Probshrink for MRI 7. Conclusion From the results in the previous section, it is clear that one needs to make use of specialized denoising algorithms for situations in which one encounters correlated noise in images. The short overview in section 2 shows that there are many such situations in practice. Correlated noise manifests itself as stripes, blobs or other image structures that cannot be modelled as spatially independent. Several useful noise estimation techniques were presented that can be used when creating or adapting a white noise denoising algorithm for use with correlated noise. To illustrate this, some state-of-the-art techniques were explained and compared with techniques designed for white noise. 8. References Aelterman, J.; Goossens, B.; Pizurica, A. & Philips, W. (2008) Removal of Correlated Rician Noise in Magnetic Resonance Imaging Proceedings of European Signal Processing Conference (EUSIPCO, Lausanne, 2008 Aelterman, J.; Goossens, B.; Pizurica, A. ; Philips, W. (2009) Locally Adaptive Complex Wavelet-Based Demosaicing for Color Filter Array Images Proceedings of SPIE Electronic Imaging 2009, San Jose, CA, Vol. 7248, no. 0J Bayer, B. (1976) Color Imaging Array US Patent 3,971,065 Borel, C.; Cooke, B.; Laubscher, B. (1996) Partial Removal of Correlated noise in Thermal Imagery Proceedings of SPIE, Vol. 2759, 131 Buades, A., Coll B. & Morel J. M. (2005) Image Denoising by Non-Local Averaging, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, vol. 2, pp. 25-28 Buades, A.; Coll, B & Morel, J.M (2008) Nonlocal Image and Movie Denoising Int Journal on Computer vision Vol. 76, pp. 123-139 Dabov, K.; Foi, A.; Katkovnik, V. & Egiazarian K. (2006) Image Denoising with Block- Matching and 3D Filtering, Proc. SPIE Electronic Imaging: Algorithms and Systems V, no. 6064A-30 Dabov, K.; Foi, A.; Katkovnik, V. & Egiazarian K. (2007) Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. on Im. Processing, vol. 16, no. 8 Donoho, D. & Johnstone, I. (1994) Adapting to Unknown Smoothness via Wavelet Shrinkage Journal of the American Statistics Association, Vol. 90 Donoho, D. L. (1995) De-Noising by Soft-Thresholding, IEEE Transactions on Information Theory, vol. 41, pp. 613-62. Easley, G.; Labate, D.; Lim, Wang-Q, (2006) Sparse Directional Image Representation using the Discrete Shearlet Transform Preprint submitted to Elsevier Preprint Elad, M.; Matalon, B.; & Zibulevsky, M. (2006) Image Denoising with Shrinkage and Redundant Representations Proc. IEEE Conf. on Computer Vision and Pattern Recognition vol. 2, pp. 1924-1931 Field, D. (1987) Relations between the statistics of natural images and the response properties of cortical cells J. Opt. Soc. Am. A 4, p. 2379-2394 Goossens, B.; Pizurica, A. & Philips, W. (2007) Removal of Correlated Noise by Modeling Spatial Correlations and Interscale Dependencies in the Complex Wavelet Domain Proceedings of International Conference on Image Processing (ICIP) pp. 317-320 Goossens, B.; Luong, H., Pizurica, A. Pizurica & Philips, W. (2008) An Improved Non-Local Denoising Algorithm Proceedings of international Workshop on Local and Non-Local Approximation in Image Processing, Lausanne, 2008 Goossens, B.; Pizurica, A. & Philips W. (2009) Removal of correlated noise by modelling the signal of interest in the wavelet domain IEEE Transactions on Image Processing in press Guerrero-Colon, J. ; Simoncelli, E. & Portilla, J. (2008) Image Denoising using Mixtures of Gaussian Scale Mixtures, Proc. IEEE Int. Conf. on Image Processing (ICIP), San Diego, 2008. Hastie, Trevor; Tibshirani, Robert & Friedman, J. (2001) The Elements of Statistical Learning New York: Springer 8.5 The EM algorithm pp. 236–24 Recent Advances in Signal Processing236 Kingsbury, N. G. (2001) Complex Wavelets for shift Invariant analysis and Filtering of Signals, Journal of Applied and Computational Harmonic Analysis, vol. 10, no. 3, pp. 234-253 Kwon, O.; Sohn, K. & Lee, C. 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Introduction Polarization-sensitive imaging systems have emerged as a very attractive vision technique which can reveal important information about the physical and geometrical properties of the targets. Many imaging polarimeters have been designed in the past for several fields, ranging from metrology (Ferraton et al., 2007), (Morel et al., 2006) to medical (Miura et al., 2006) and remote sensing applications (Chipman, 1993). Imaging systems that can measure the polarization state of the outgoing light across a scene are mainly based on the ability to build effective Polarization State Analyzers (PSA) in front of the camera enabling to acquire the Stokes vectors (Chipman, 1993), (Tyo et al., 2006). These Stokes polarimeters produced four images called “Stokes images” corresponding to the four Stokes parameters. Accordingly, polarization-encoded images have a multidimensional structure; i.e. multi-component information is attached to each pixel in the image. Moreover, the information content of polarization-encoded images is intricately combined in the polarization channels making awkward their proper interpretation in the presence of noise. Noise is inherent to any imaging systems and it is therefore present on Stokes images. It is of additive nature when the scene is illuminated by incoherent light and multiplicative when the illumination is coherent (Bénière et al., 2007), (Corner et al., 2003). Its presence degrades the interpretability of the data and prevents from exploring the physical potential of polarimetric information. Few works in the literature addressed the filtering of polarimetric images. We note nevertheless the use of optimization methods by (Zallat et al., 2006) to optimize imaging system parameters that condition signal to noise ratio, or the improvement of the accuracy of the degree of polarization by (Bénière et al., 2007) with the aim of reducing the noise in Stokes images. The main problem in filtering polarization-encoded images so as to remove their noise content is to respect their physical content. Indeed, mathematical operations which are performed on polarization information images while processing them alter in most cases the physical meaning of the images. The same problem has been encountered for polarization 14 Noise Estimation of Polarization-Encoded Images by Peano-Hilbert Fractal Path 237 Noise Estimation of Polarization-Encoded Images by Peano-Hilbert Fractal Path Samia Ainouz-Zemouche and Fabrice Mériaudeau This condition is known as the physical condition of Stokes formalism. An arbitrary vector that does not satisfy this condition is not a Stokes vector and doesn’t possess any physical meaning. The general scheme of Stokes images acquisition is illustrated in Figure.1 (Chipman, 1993). The device used for the acquisition is named a classical polarimeter. The wave reflected from the target, represented by a Stokes vector in S , is analyzed by a polarization-state analyzer (PSA) by measuring its projections over four linearly independent states. A PSA consists of a linear polarizer (LP) and a quarter wave (QW) rotating about four angles   41,i i   . Incoming intensities are then measured with a standard CCD camera. The complete set of 4 measurements can be written in a vectorial form as: in ASI  (3) I is a 14  intensity matrix measured by the camera. The Stokes vector in S can then easily be extracted from the raw data matrix I provided that the modulation matrix A of the PSA, is known from calibration. For the ideal case (theory), matrix A can be given as (Chipman, 1993):                      0000 0000 24 2 1 21 24 2 1 21 22 22 iii iii i sinsincos sinsincos A    (4) The angles i  are chosen such that the matrix A is invertible to easily recover the Stokes parameters from the intensity matrix. Each of the four intensity component corresponds to one image, leading to four images carrying information about the Stokes vector. Fig. 1. Stokes imaging device; classical polarimeter Figure. 2 shows an example of a set of Stokes images obtained with the Stokes imaging device. In order to test the performances of our algorithm, Stokes images were acquired in images segmentation in (Ainouz, 2006a) and (Ainouz, 2006b). Therefore, because of the duality between the polarization images filtering and their physical constraint, a trade-off is to be reached in order to minimize the effect of the noise affecting polarimetric images and to preserve their physical meaning. In this chapter, we present a technique which, estimate the additive noise (images acquired under incoherent illumination), and eliminate it such that the physical content of the polarimetric images is preserved as much as possible. Our technique combines two methods; Scatter plot (Aiazzi et al., 2002) and data masking (Corner et al., 2003) previously used in the field of multispectral imaging and to take advantage of both them. As the information content of polarization-encoded images is intricately combined in several polarization channels, Peano-Hilbert fractal path is applied on the noisy image to keep the connectivity of homogeneous areas and to minimize the impact of the outliers. The performances and the bias of our method are statistically investigated by Bootstrap method. The rest of this chapter is organized as follows: the next section deals with the principle of polarisation images acquisition, the third part details our noise estimation technique whereas part 4 presents the results obtained while filtering polarization encoded images. The chapter ends with a short conclusion. 2. Polarization images acquisition The next two subsections respectively present the principle of a Stoke’s imaging system as well as the model for the additive noise resulting from the acquisition set-up. 2.1 Stokes imaging The general polarization state of a light wave can be described by the so called Stokes vector S which fully characterizes the time-averaged polarization properties of a radiation. It is defined by the following combination of complex-valued components x E and y E of the electric field, along two orthogonal directions x and y as (Chipman, 1993):                                       * yx * yx * yy * xx * yy * xx EEIm EERe EEEE EEEE S S S S S 2 2 3 2 1 0 (1) The first parameter (S 0 ) is the total intensity of the optical field and the other three parameters (S 1 , S 2 and S 3 ) describe the polarization state (Chipman, 1993). S 1 is the tendency of the wave to look like a linear horizontal vibration (S 1 positive) or a linear vertical vibration (S 1 negative). S 2 and S 3 reflect the nature and the direction of rotation of the wave. It is straightforward to show that 2 3 2 2 2 1 2 0 SSSS  (2) Recent Advances in Signal Processing238 This condition is known as the physical condition of Stokes formalism. An arbitrary vector that does not satisfy this condition is not a Stokes vector and doesn’t possess any physical meaning. The general scheme of Stokes images acquisition is illustrated in Figure.1 (Chipman, 1993). The device used for the acquisition is named a classical polarimeter. The wave reflected from the target, represented by a Stokes vector in S , is analyzed by a polarization-state analyzer (PSA) by measuring its projections over four linearly independent states. A PSA consists of a linear polarizer (LP) and a quarter wave (QW) rotating about four angles   41,i i   . Incoming intensities are then measured with a standard CCD camera. The complete set of 4 measurements can be written in a vectorial form as: in ASI  (3) I is a 14  intensity matrix measured by the camera. The Stokes vector in S can then easily be extracted from the raw data matrix I provided that the modulation matrix A of the PSA, is known from calibration. For the ideal case (theory), matrix A can be given as (Chipman, 1993):                      0000 0000 24 2 1 21 24 2 1 21 22 22 iii iii i sinsincos sinsincos A    (4) The angles i  are chosen such that the matrix A is invertible to easily recover the Stokes parameters from the intensity matrix. Each of the four intensity component corresponds to one image, leading to four images carrying information about the Stokes vector. Fig. 1. Stokes imaging device; classical polarimeter Figure. 2 shows an example of a set of Stokes images obtained with the Stokes imaging device. In order to test the performances of our algorithm, Stokes images were acquired in Noise Estimation of Polarization-Encoded Images by Peano-Hilbert Fractal Path 239                 jiSjiS jinAjiIA jinjiIAjiS a a ,, ,, ,,, ˆ 11 1       (8) The estimated Stokes vector S ˆ is an independent sum of the theoretical Stokes vector (noise free) S and the term S  due to the additive noise effects. The estimation of the noise distribution is then needed to reconstruct the noisy image S  and minimize its effects on the estimated Stokes image S ˆ . However, as explained above, in the introduction, direct filtering of polarimetric measurements can induce a non physical meaning of the filtered Stokes image SS ˆ   . This means that for an important number of pixels, the vector SS ˆ   does not satisfy the physical constraint stated in equation (2) and therefore cannot be considered as a Stokes vector which fully fulfils physical constraints. In such conditions, these images have no interest and additional steps have be taken to obtain a trade-off between filtering and physical meaning for as many pixels as possible. The following section presents three methods for noise estimation in the case of additive Gaussian noise. 3. Parametric noise estimation Inasmuch as the estimated noised Stokes image is an independent sum of the noise and the noise free image, the estimation of the S  distribution is sufficient to have information about the additive noise. Two multi-spectral filtering methods: Scatter plot method (SP) (Aiazzi et al., 2002) and data masking method (DM) (Corner et al., 2003) were used to process the images. The proposed filtering algorithm takes advantages of both methods. In order to eliminate the impact of non relevant data, the image is first transformed to a Peano-Hilbert fractal path. This method is applied onto gray level images and the results are compared to the results obtained with SP and DM methods. 3.1 Scatter plot and Data masking methods reminder 3.1.1 Scatter plot method (SP) In the SP method, the standard deviation of the noisy observed image can be evaluated in homogeneous areas (Aiazzi et al., 2002). Under the assumption that the noise is Gaussian with zero mean, local means  and local standard deviations  are calculated in a sliding small window within the whole image. The scatter plot plane of local standard deviations () versus local means () is plotted and then partitioned into rectangular blocks of size LL  ( 100100  for example). After sorting the blocks by decreasing number of points, denser blocks are considered as the homogeneous areas of the image. The estimated standard deviation of the noise,  ˆ , is found as the intersection between the linear regression of the data set corresponding to homogenous areas with the ordinate ( y ) axis. strong noisy conditions. In order to have these strong noisy conditions, addition to the natural noise, a hair dryer is turned between the PSA and the camera. The scene is made of 4 small elements of different composition glued on a cardboard. Objects A and D are transparents whereas objects B and C are darks. Fig. 2. Stokes image of four small objects glued on a cardboard The following subsection describes the noise model which was used through out this study. 2.1 Noise in Stokes images It has been established that under incoherent illumination, the noise affecting the images can be modelled as additive and independent (Corner et al., 2003). This type of noise can be modelled by a zero mean random Gaussian distribution which probability density function (PDF) is expressed as follows (Aiazzi et al., 2002):            2 2 σ2 π2σ 1 n n x x expxf (5) Where 2 n  is the noise variance. The effect of an additive noise a n on a digital image g at the pixel position   j,i is expressed as the sum of the noise free image I and the noise in the form : a nIg  (6) In perfect acquisition conditions, the Stokes vector is recovered from equation (3) such that:     j,iIAj,iS 1  (7) In the presence of noise, equation (7) becomes: Recent Advances in Signal Processing240                 jiSjiS jinAjiIA jinjiIAjiS a a ,, ,, ,,, ˆ 11 1       (8) The estimated Stokes vector S ˆ is an independent sum of the theoretical Stokes vector (noise free) S and the term S  due to the additive noise effects. The estimation of the noise distribution is then needed to reconstruct the noisy image S  and minimize its effects on the estimated Stokes image S ˆ . However, as explained above, in the introduction, direct filtering of polarimetric measurements can induce a non physical meaning of the filtered Stokes image SS ˆ   . This means that for an important number of pixels, the vector SS ˆ   does not satisfy the physical constraint stated in equation (2) and therefore cannot be considered as a Stokes vector which fully fulfils physical constraints. In such conditions, these images have no interest and additional steps have be taken to obtain a trade-off between filtering and physical meaning for as many pixels as possible. The following section presents three methods for noise estimation in the case of additive Gaussian noise. 3. Parametric noise estimation Inasmuch as the estimated noised Stokes image is an independent sum of the noise and the noise free image, the estimation of the S  distribution is sufficient to have information about the additive noise. Two multi-spectral filtering methods: Scatter plot method (SP) (Aiazzi et al., 2002) and data masking method (DM) (Corner et al., 2003) were used to process the images. The proposed filtering algorithm takes advantages of both methods. In order to eliminate the impact of non relevant data, the image is first transformed to a Peano-Hilbert fractal path. This method is applied onto gray level images and the results are compared to the results obtained with SP and DM methods. 3.1 Scatter plot and Data masking methods reminder 3.1.1 Scatter plot method (SP) In the SP method, the standard deviation of the noisy observed image can be evaluated in homogeneous areas (Aiazzi et al., 2002). Under the assumption that the noise is Gaussian with zero mean, local means  and local standard deviations  are calculated in a sliding small window within the whole image. The scatter plot plane of local standard deviations () versus local means () is plotted and then partitioned into rectangular blocks of size LL  ( 100100  for example). After sorting the blocks by decreasing number of points, denser blocks are considered as the homogeneous areas of the image. The estimated standard deviation of the noise,  ˆ , is found as the intersection between the linear regression of the data set corresponding to homogenous areas with the ordinate ( y ) axis. strong noisy conditions. In order to have these strong noisy conditions, addition to the natural noise, a hair dryer is turned between the PSA and the camera. The scene is made of 4 small elements of different composition glued on a cardboard. Objects A and D are transparents whereas objects B and C are darks. Fig. 2. Stokes image of four small objects glued on a cardboard The following subsection describes the noise model which was used through out this study. 2.1 Noise in Stokes images It has been established that under incoherent illumination, the noise affecting the images can be modelled as additive and independent (Corner et al., 2003). This type of noise can be modelled by a zero mean random Gaussian distribution which probability density function (PDF) is expressed as follows (Aiazzi et al., 2002):            2 2 σ2 π2σ 1 n n x x expxf (5) Where 2 n  is the noise variance. The effect of an additive noise a n on a digital image g at the pixel position   j,i is expressed as the sum of the noise free image I and the noise in the form : a nIg   (6) In perfect acquisition conditions, the Stokes vector is recovered from equation (3) such that:     j,iIAj,iS 1  (7) In the presence of noise, equation (7) becomes: Noise Estimation of Polarization-Encoded Images by Peano-Hilbert Fractal Path 241 [...]... complex LPC speech analysis methods have already been proposed for an analytic signal [10][11][12] An analytic signal is a complex signal having an observed signal in real part and a Hilbert transformed signal for the observed signal in imaginary part Since the analytic signal provides the spectrum only on positive frequencies, the signals can be decimated by a factor of 2 with no degradation As a result,... additive noise in not the only noise affecting polarimetric measurement, other sources of noise are currently being investigated especially multiplicative noise 250 (a) Fig 8 (a) Noisy Stokes image, (b) Filtered Stokes image Recent Advances in Signal Processing (b) 6 References B Aiazzi, L Alparone, A Barducci, S Baronti & I Pippi (2002) Estimating noise and information of multispectral imagery, In Opt Eng,... follows (16) 256 Recent Advances in Signal Processing Pss ( ) Pss ( )  Pww ( ) From Eq.(12) and (15), the enhanced speech is estimated in the frequency domain by H ( )  S ( )  (17) Pss ( ) X ( ) Pss ( )  Pww ( ) ( 18) and then inverse FFT is operated to obtain enhanced speech as: s (t )  IFFT S ( )  (19) Finally, the OLA (OverLap Add) procedure is carried out in the time domain between adjacent... [13] (3) Babble noise [13] (4) Car internal noise [13] 20,10,5,0,-5[dB] LPC Cepstral Distance 20[msec]/ 20[msec] 16 for 8KHz, 32 for 16KHz 16 for 8KHz, 32 for 16KHz 262 Recent Advances in Signal Processing (1) CDs for additive white Gauss noise (8KHz) (2) CDs for additive Pink noise (8KHz) Fig 7 CDs for 8KHz speech Speech Enhancement based on Iterative Wiener Filter using Complex LPC Speech Analysis (3)... noise (8KHz) (4) CDs for additive Car Internal noise (8KHz) Fig 7 CDs for 8KHz speech 263 264 Recent Advances in Signal Processing (1) CDs for additive white Gauss noise (16KHz) (2) CDs for additive Pink noise (16KHz) Fig 8 CDs for 16KHz speech Speech Enhancement based on Iterative Wiener Filter using Complex LPC Speech Analysis (3) CDs for additive Babble noise (16KHz) (4) CDs for additive Car Internal...242 Recent Advances in Signal Processing 3.1.2 Data masking method (DM) DM method deals by first filtering the image to remove the image structure, leaving only the noise (Corner et al., 2003) The Laplacian Kernel presented in equation.9 is used for that purpose The image obtained after the convolution with the Laplacian kernel (Laplacian image) mainly contains the noise as well as... estimated by applying the same instructions on the plotted plane  ,   The advantage of the fractal path is double: it eases the task of calculating the local statistics within the image with keeping at most the neighbourhood of image pixels Moreover, the vectorization of the image disperses so much the isolated points of the plane  ,  244 Recent Advances in Signal Processing preventing the regression... of complex speech analysis will be explained in Section 4 We will explain the experiments evaluating the performance for additive white Gaussian, pink, babble, or car internal noise in Section 5 2 TV-CAR Speech Analysis 2 1 Analytic speech signal Target signal of the time-varying complex AR (TV-CAR) method is an analytic signal that is complex-valued signal defined by y c (t )  y (2t )  j  yH (2t... (2003) Noise estimation in remote sensing imagery using data masking, In Int J Remote sensing, vol.4, pp 689 –702 M Ferraton, C Stolz & F Meriaudeau (2007) 3D reconstruction of transparent objects using image polarization, EOS topical meeting, pp 110-111, Lille, France N Kazakova, M Margala & N G Durdle (2004) Sobel edge detection processor for a realtime volume rendering system, in ISCAS’04, vol.2, pp... four intensity channels corresponding to S 0 , S1 , S 2 and S 3 are respectively 0. 18, 0.19, 0.194 and 0. 187 2 48 Recent Advances in Signal Processing Fig 5 Noisy Stokes channels Gaussian zero-mean noise and of variance 0.2 is added to the correspondent intensity channels These values are very close to the simulated variance The results also show that the noise affecting the four polarimetric measurements . modelling the signal of interest in the wavelet domain IEEE Transactions on Image Processing in press Guerrero-Colon, J. ; Simoncelli, E. & Portilla, J. (20 08) Image Denoising using Mixtures. modelling the signal of interest in the wavelet domain IEEE Transactions on Image Processing in press Guerrero-Colon, J. ; Simoncelli, E. & Portilla, J. (20 08) Image Denoising using Mixtures. S. Ainouz & M-P. Stoll. (2006). Optimal configuration for imaging polarimeters : impact of image noise and systematic errors, in J. Opt. vol .8, pp. 80 7 81 4. Recent Advances in Signal Processing2 50 Speech