Modeling of Retinal Optical Coherence Tomography Based on Stochastic Differential Equations: Application to Denoising

In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increm...

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Vydáno v:	IEEE transactions on medical imaging Ročník 40; číslo 8; s. 2129 - 2141
Hlavní autoři:	Tajmirriahi, Mahnoosh, Amini, Zahra, Hamidi, Arsham, Zam, Azhar, Rabbani, Hossein
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	United States IEEE 01.08.2021 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Algorithms Alternating direction method of multipliers~(ADMM) algorithm Differential equations Formulas (mathematics) Fractals fractional Laplacian Goodness of fit Histograms Image classification Image segmentation Laplace equations Laplace transforms Mathematical model Mathematical models Noise Noise reduction Operators (mathematics) Optical Coherence Tomography optical coherence tomography~(OCT) image Optimization Pixels Regularization Retina Statistical analysis Statistical models Statistical tests stochastic differential equation (SDE) Stochastic processes Stochasticity Technological innovation Tomography
ISSN:	0278-0062, 1558-254X, 1558-254X
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Abstract	In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula>) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> distribution. We applied fractional Laplacian operator to image and fitted s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities.
AbstractList	In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s [Formula Omitted]) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s [Formula Omitted] distribution. We applied fractional Laplacian operator to image and fitted s [Formula Omitted] to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s [Formula Omitted] distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities. In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula>) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> distribution. We applied fractional Laplacian operator to image and fitted s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s <inline-formula> <tex-math notation="LaTeX">\boldsymbol {\alpha }\text{s} </tex-math></inline-formula> distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities. In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s [Formula: see text]) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s [Formula: see text] distribution. We applied fractional Laplacian operator to image and fitted s [Formula: see text] to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s [Formula: see text] distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities.In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s [Formula: see text]) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s [Formula: see text] distribution. We applied fractional Laplacian operator to image and fitted s [Formula: see text] to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s [Formula: see text] distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities. In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (sαs) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as sαs distribution. We applied fractional Laplacian operator to image and fitted sαs to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled sαs distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities.
Author	Amini, Zahra Tajmirriahi, Mahnoosh Hamidi, Arsham Zam, Azhar Rabbani, Hossein
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Snippet	In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In...
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SubjectTerms	Algorithms Alternating direction method of multipliers~(ADMM) algorithm Differential equations Formulas (mathematics) Fractals fractional Laplacian Goodness of fit Histograms Image classification Image segmentation Laplace equations Laplace transforms Mathematical model Mathematical models Noise Noise reduction Operators (mathematics) Optical Coherence Tomography optical coherence tomography~(OCT) image Optimization Pixels Regularization Retina Statistical analysis Statistical models Statistical tests stochastic differential equation (SDE) Stochastic processes Stochasticity Technological innovation Tomography
Title	Modeling of Retinal Optical Coherence Tomography Based on Stochastic Differential Equations: Application to Denoising
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Volume	40
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