Abstract

An important image post-processing step for optical coherence tomography (OCT) images is speckle noise reduction. Noise in OCT images is multiplicative in nature and is difficult to suppress due to the fact that in addition the noise component, OCT speckle also carries structural information about the imaged object. To address this issue, a novel speckle noise reduction algorithm was developed. The algorithm projects the imaging data into the logarithmic space and a general Bayesian least squares estimate of the noise-free data is found using a conditional posterior sampling approach. The proposed algorithm was tested on a number of rodent (rat) retina images acquired in-vivo with an ultrahigh resolution OCT system. The performance of the algorithm was compared to that of the state-of-the-art algorithms currently available for speckle denoising, such as the adaptive median, maximum a posteriori (MAP) estimation, linear least squares estimation, anisotropic diffusion and wavelet-domain filtering methods. Experimental results show that the proposed approach is capable of achieving state-of-the-art performance when compared to the other tested methods in terms of signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), edge preservation, and equivalent number of looks (ENL) measures. Visual comparisons also show that the proposed approach provides effective speckle noise suppression while preserving the sharpness and improving the visibility of morphological details, such as tiny capillaries and thin layers in the rat retina OCT images.

© 2010 Optical Society of America

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2009 (1)

2008 (1)

2007 (4)

2005 (2)

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

D. Fernandez, H. Salinas, and C. Puliafito, "Automated detection of retinal layer structures on optical coherence tomography images," Opt. Express 13, 10200-10216 (2005).
[CrossRef]

2004 (2)

2003 (3)

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

N. Iftimia, B. E. Bouma, and G. J. Tearney, "Speckle reduction in optical coherence tomography by path length encoded angular compounding," J. Biomed. Opt. 8, 260-263 (2003).
[CrossRef] [PubMed]

2002 (1)

Y. Yu and S. Acton, "Speckle reducing anisotropic diffusion," IEEE Trans. Image Process. 11(11), 1260-1270 (2002).
[CrossRef]

2000 (2)

J. Rogowska and M. E. Brezinski, "Evaluation of the adaptive speckle suppression filter for coronary optical coherence tomography imaging," IEEE Trans. Med. Imaging 19, 1261-1266 (2000).
[CrossRef]

M. Bashkansky and J. Reintjes, "Statistics and reduction of speckle in optical coherence tomography," Opt. Lett. 25, 545-547 (2000).
[CrossRef]

1999 (2)

K. Yung, S. Lee, and J. Schmitt, "Phase-Domain Processing of Optical Coherence Tomography Images," J. Biomed. Opt. 4(1), 125-136 (1999).
[CrossRef]

J. Schmitt, S. Xiang, and K. Yung, "Speckle in optical coherence tomography," J. Biomed. Opt. 4, 95-105 (1999).
[CrossRef]

1997 (1)

J. Schmitt, "Array detection for speckle reduction in optical coherence microscopy," Phys. Med. Biol. 42(7), 1427-1439 (1997).
[CrossRef] [PubMed]

1993 (1)

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

1991 (1)

M. Kobayashi, H. Hanafusa, K. Takada, and J. Noda, "Polarization-Independent Interferometric Optical-Time-Domain Reflectometer," J. Lightwave Technol. 9, 623-628 (1991).
[CrossRef]

1989 (1)

T. Loupas, W. Mcdicken, and P. Allen, "An adaptive weighted median filter for speckle suppression in medical ultrasound images," IEEE Trans. Circuits Syst. 36(1), 129-135 (1989).
[CrossRef]

1987 (1)

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

1986 (1)

J. Lee, "Speckle suppression and analysis for synthetic aperture radar," Opt. Eng. 25(5), 636-643 (1986).

1982 (1)

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

1970 (1)

W. Hastings, "Monte Carlo sampling methods using Markov chains and their applications," Biometrika 57(1), 97-109 (1970).
[CrossRef]

Acheroy, M.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

Acton, S.

Y. Yu and S. Acton, "Speckle reducing anisotropic diffusion," IEEE Trans. Image Process. 11(11), 1260-1270 (2002).
[CrossRef]

Adler, D. C.

Allen, P.

T. Loupas, W. Mcdicken, and P. Allen, "An adaptive weighted median filter for speckle suppression in medical ultrasound images," IEEE Trans. Circuits Syst. 36(1), 129-135 (1989).
[CrossRef]

Bashkansky, M.

Bilenca, A.

Bizheva, K.

Bouma, B. E.

Boyd, S.

Brezinski, M. E.

J. Rogowska and M. E. Brezinski, "Evaluation of the adaptive speckle suppression filter for coronary optical coherence tomography imaging," IEEE Trans. Med. Imaging 19, 1261-1266 (2000).
[CrossRef]

Chavel, P.

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

Desjardins, A. E.

Drexler, W.

W. Drexler, "Ultrahigh-resolution optical coherence tomography," J. Biomed. Opt. 9, 47-74 (2004).
[CrossRef] [PubMed]

Fercher, A.F.

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

Fernandez, D.

Forbes, P.

Frost, V.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

Fujimoto, J. G.

Gtzinger, E.

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

Hanafusa, H.

M. Kobayashi, H. Hanafusa, K. Takada, and J. Noda, "Polarization-Independent Interferometric Optical-Time-Domain Reflectometer," J. Lightwave Technol. 9, 623-628 (1991).
[CrossRef]

Hastings, W.

W. Hastings, "Monte Carlo sampling methods using Markov chains and their applications," Biometrika 57(1), 97-109 (1970).
[CrossRef]

Hewkoa, M.

D. Popescu, M. Hewkoa, and M. Sowa, "Speckle noise attenuation in optical coherence tomography by compounding images acquired at different positions of the sample," Opt. Commun. 1(1), 247-251 (2007).
[CrossRef]

Hitzenberger, C. K.

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

Holtzman, J.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

Iftimia, N.

N. Iftimia, B. E. Bouma, and G. J. Tearney, "Speckle reduction in optical coherence tomography by path length encoded angular compounding," J. Biomed. Opt. 8, 260-263 (2003).
[CrossRef] [PubMed]

Kim, E.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Kim, J.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Ko, T. H.

Kobayashi, M.

M. Kobayashi, H. Hanafusa, K. Takada, and J. Noda, "Polarization-Independent Interferometric Optical-Time-Domain Reflectometer," J. Lightwave Technol. 9, 623-628 (1991).
[CrossRef]

Kuan, D.

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

Laur, H.

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

Lee, J.

J. Lee, "Speckle suppression and analysis for synthetic aperture radar," Opt. Eng. 25(5), 636-643 (1986).

Lee, S.

K. Yung, S. Lee, and J. Schmitt, "Phase-Domain Processing of Optical Coherence Tomography Images," J. Biomed. Opt. 4(1), 125-136 (1999).
[CrossRef]

Leitgeb, R.

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

Lemahieu, I.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

Lopes, A.

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

Loupas, T.

T. Loupas, W. Mcdicken, and P. Allen, "An adaptive weighted median filter for speckle suppression in medical ultrasound images," IEEE Trans. Circuits Syst. 36(1), 129-135 (1989).
[CrossRef]

Malchow, D.

Mcdicken, W.

T. Loupas, W. Mcdicken, and P. Allen, "An adaptive weighted median filter for speckle suppression in medical ultrasound images," IEEE Trans. Circuits Syst. 36(1), 129-135 (1989).
[CrossRef]

Miller, D.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Milner, T.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Motaghiannezam, S. M. R.

Nezry, E.

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

Noda, J.

M. Kobayashi, H. Hanafusa, K. Takada, and J. Noda, "Polarization-Independent Interferometric Optical-Time-Domain Reflectometer," J. Lightwave Technol. 9, 623-628 (1991).
[CrossRef]

Oh, J.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Oh, S.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

Oh, W. Y.

Ozcan, A.

Philips, W.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

Pircher, M.

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
[CrossRef] [PubMed]

Pizurica, A.

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

Popescu, D.

D. Popescu, M. Hewkoa, and M. Sowa, "Speckle noise attenuation in optical coherence tomography by compounding images acquired at different positions of the sample," Opt. Commun. 1(1), 247-251 (2007).
[CrossRef]

Puliafito, C.

Puvanathasan, P.

Reintjes, J.

Ren, Z.

Rogowska, J.

J. Rogowska and M. E. Brezinski, "Evaluation of the adaptive speckle suppression filter for coronary optical coherence tomography imaging," IEEE Trans. Med. Imaging 19, 1261-1266 (2000).
[CrossRef]

Salinas, H.

Sawchuk, A.

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

Schmitt, J.

J. Schmitt, S. Xiang, and K. Yung, "Speckle in optical coherence tomography," J. Biomed. Opt. 4, 95-105 (1999).
[CrossRef]

K. Yung, S. Lee, and J. Schmitt, "Phase-Domain Processing of Optical Coherence Tomography Images," J. Biomed. Opt. 4(1), 125-136 (1999).
[CrossRef]

J. Schmitt, "Array detection for speckle reduction in optical coherence microscopy," Phys. Med. Biol. 42(7), 1427-1439 (1997).
[CrossRef] [PubMed]

Shanmugan, K.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

Sowa, M.

D. Popescu, M. Hewkoa, and M. Sowa, "Speckle noise attenuation in optical coherence tomography by compounding images acquired at different positions of the sample," Opt. Commun. 1(1), 247-251 (2007).
[CrossRef]

Stiles, J.

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

Strand, T.

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

Takada, K.

M. Kobayashi, H. Hanafusa, K. Takada, and J. Noda, "Polarization-Independent Interferometric Optical-Time-Domain Reflectometer," J. Lightwave Technol. 9, 623-628 (1991).
[CrossRef]

Tearney, G. J.

Touzi, R.

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

Vakoc, B. J.

Xiang, S.

J. Schmitt, S. Xiang, and K. Yung, "Speckle in optical coherence tomography," J. Biomed. Opt. 4, 95-105 (1999).
[CrossRef]

Yu, Y.

Y. Yu and S. Acton, "Speckle reducing anisotropic diffusion," IEEE Trans. Image Process. 11(11), 1260-1270 (2002).
[CrossRef]

Yung, K.

K. Yung, S. Lee, and J. Schmitt, "Phase-Domain Processing of Optical Coherence Tomography Images," J. Biomed. Opt. 4(1), 125-136 (1999).
[CrossRef]

J. Schmitt, S. Xiang, and K. Yung, "Speckle in optical coherence tomography," J. Biomed. Opt. 4, 95-105 (1999).
[CrossRef]

Biometrika (1)

W. Hastings, "Monte Carlo sampling methods using Markov chains and their applications," Biometrika 57(1), 97-109 (1970).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

D. Kuan, A. Sawchuk, T. Strand, and P. Chavel, "Adaptive restoration of images with speckle," IEEE Trans. Acoust. Speech Signal Process. 35(3), 373-383 (1987).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

T. Loupas, W. Mcdicken, and P. Allen, "An adaptive weighted median filter for speckle suppression in medical ultrasound images," IEEE Trans. Circuits Syst. 36(1), 129-135 (1989).
[CrossRef]

IEEE Trans. Image Process. (1)

Y. Yu and S. Acton, "Speckle reducing anisotropic diffusion," IEEE Trans. Image Process. 11(11), 1260-1270 (2002).
[CrossRef]

IEEE Trans. Med. Imag. (1)

A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, "A versatile wavelet domain noise filtration technique for medical imaging," IEEE Trans. Med. Imag. 22(3), 323-331 (2003).
[CrossRef]

IEEE Trans. Med. Imaging (1)

J. Rogowska and M. E. Brezinski, "Evaluation of the adaptive speckle suppression filter for coronary optical coherence tomography imaging," IEEE Trans. Med. Imaging 19, 1261-1266 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

V. Frost, J. Stiles, K. Shanmugan, and J. Holtzman, "A model for radar images and its application to adaptive digital filtering for multiplicative noise," IEEE Trans. Pattern Anal. Machine Intell. 4(2), 157-166 (1982).
[CrossRef]

Int. J. Remote Sens. (1)

A. Lopes, E. Nezry, R. Touzi, and H. Laur, "Structure detection and adaptive speckle filtering in SAR images," Int. J. Remote Sens. 14(9), 1735-1758 (1993).
[CrossRef]

J. Biomed. Opt. (6)

K. Yung, S. Lee, and J. Schmitt, "Phase-Domain Processing of Optical Coherence Tomography Images," J. Biomed. Opt. 4(1), 125-136 (1999).
[CrossRef]

W. Drexler, "Ultrahigh-resolution optical coherence tomography," J. Biomed. Opt. 9, 47-74 (2004).
[CrossRef] [PubMed]

J. Schmitt, S. Xiang, and K. Yung, "Speckle in optical coherence tomography," J. Biomed. Opt. 4, 95-105 (1999).
[CrossRef]

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. Milner, "Optical coherence tomography speckle reduction by a partially spatially coherent source," J. Biomed. Opt. 10, 640349 (2005).

M. Pircher, E. Gtzinger, R. Leitgeb, A.F. Fercher, and C. K. Hitzenberger, "Speckle reduction in optical coherence tomography by frequency compounding," J. Biomed. Opt. 8, 565-569 (2003).
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Figures (7)

Fig. 1.
Fig. 1.

Step-by-step flowchart of the proposed despeckling algorithm.

Fig. 2.
Fig. 2.

UHROCT images of the rat retina acquired near the optic disc (A) and away from the optic disc (B). NFL nerve fiber layer; GCL ganglion cell layer; IPL inner plexiform layer; INL inner nuclear layer; OPL outer plexiform layer; ONL outer nuclear layer; ELM external limiting membrane; IS inner segment; OS outer segment of the photoreceptor layer; RPE retinal pigmented epithelium; C- choroid and S sclera. The red arrows mark tiny capillaries imbedded in the retinal OPL, while the yellow arrows mark large blood vessels in the choroid. The red line boxes mark sections of the retinal image that were enlarged for more direct visual comparison of the performance of the speckle denoising algorithms (see Figs. 3,4 and 5).

Fig. 3.
Fig. 3.

An OCT image of the rat retina acquired away from the optic disc and processed with the following filters: Original image, Adaptive median filter, Linear least square estimation, Anisotropic Diffusion, MAP estimation, general Wavelet, Type II Fuzzy AD, and the new proposed algorithm. The blue and red-line boxes in the original image mark regions in the image used for quantitative comparison of the performance of all image processing algorithms applied to the original image.

Fig. 4.
Fig. 4.

3X enlargement of the region in the original OCT image marked with the red box #1 in Fig. 1(b) and processed with the following filters: AMF (adaptive median filter); LLSQ (linear least square estimation); AD (adaptive diffusion); general Wavelet; MAP (maximum a posteriori estimation); Type II fuzzy AD (fuzzy rules controlled adaptive diffusion). Features in the original image such as the large blood vessel in the NFL and the three capillaries positioned at the boundary between the IPL and INL appear very sharp and distinct on the image processed with the proposed algorithm. In contrast, the same image features appear with low contrast and blurred boundaries on the images processed with the rest of the algorithms.

Fig. 5.
Fig. 5.

3X enlargement of the region in the original OCT image marked with the red box #2 in Fig. 1(b) and processed with the following filters: AMF (adaptive median filter); LLSQ (linear least square estimation); AD (adaptive diffusion); general Wavelet; MAP (maximum a posteriori estimation); Type II fuzzy AD (fuzzy rules controlled adaptive diffusion). The RPE, IS and OS of the photoreceptor layers of the retina appear very sharp and distinct on the image processed with the proposed novel algorithm. In contrast, the same image features appear with low contrast and blurred boundaries on the images processed with the rest of the algorithms.

Fig. 6.
Fig. 6.

Magnified view of a choroidal blood vessel, corresponding to the area marked with red box #3 in the original image [Fig. 1(b)] and processed with the different speckle reduction algorithms. AMF adaptive median filter; LLSQ linear least square estimation; AD anisotropic diffusion; general Wavelet; MAP maximum a posteriori estimation; Type II fuzzy AD (fuzzy rules controlled adaptive diffusion). The proposed method results in very clear delineation of the blood vessel walls as compared to the significant blur and / or image artefacts introduced by the other speckle denoising methods.

Fig. 7.
Fig. 7.

An OCT image of the rat retina acquired at the optic disc and processed with the following filters: Original image, Adaptive median filter, Linear least square estimation, Anisotropic Diffusion, MAP estimation, general Wavelet, Type II Fuzzy AD, and the new proposed algorithm. The blue and red boxes in the original image mark regions in the image used for quantitative comparison of the performance of all image processing algorithms applied to the original image.

Tables (1)

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Table 1. Image quality metrics evaluated for the rat retina image [Fig. 2(b)]. Values are relative to the original image.

Equations (12)

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m ( s ) = a ( s ) · n ( s ) .
log m ( s ) = m l ( s ) = log [ a ( s ) . n ( s ) ] = log { a ( s ) } + log { n ( s ) } = a l ( s ) + n l ( s ) .
a ̂ l ( s ) = arg min a l ( s ) { E [ ( a l ( s ) a ̂ l ( s ) ) 2 m l ( s ) ] } .
a ̂ l ( s ) = p ( a l ( s ) m l ( s ) ) a l ( s ) d a l ( s ) .
Q ( s | s ) = 1 2 π σ spatial e ( s ' s 2 2 σ spatial 2 ) ,
μ ( s ) μ ( s ) < 2 σ ,
w ( s ' i | s ) = exp [ | μ ( s ' i ) μ ( s ) 2 σ 2 ] .
p ̂ ( a l ( s ) | m l ( s ) ) = k = Ω w ( s k | s ) δ ( a l m l ( s k ) ) Z ,
SNR = 10 log 10 [ max ( A 2 ) / σ 2 ] ,
ENL = 1 H [ h 1 H μ h 2 σ h 2 ] ,
CNR = 1 R [ r = 1 R ( μ r μ b ) σ r 2 + σ b 2 ] ,
η = ( 2 M 2 M ¯ ) ( 2 A 2 A ¯ ) ( 2 M 2 M ¯ ) 2 . ( 2 A 2 A ¯ ) 2 ,

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