Non-blind and Blind Deconvolution Under Poisson Noise Using Fractional-Order Total Variation
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Non-blind and Blind Deconvolution Under Poisson Noise Using Fractional-Order Total Variation Mujibur Rahman Chowdhury1 · Jing Qin2 · Yifei Lou1 Received: 7 April 2020 / Accepted: 7 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In a wide range of applications such as astronomy, biology, and medical imaging, acquired data are usually corrupted by Poisson noise and blurring artifacts. Poisson noise often occurs when photon counting is involved in such imaging modalities as X-ray, positron emission tomography, and fluorescence microscopy. Meanwhile, blurring is also inevitable due to the physical mechanism of an imaging system, which can be modeled as a convolution of the image with a point spread function. In this paper, we consider both non-blind and blind image deblurring models that deal with Poisson noise. In the pursuit of high-order smoothness of a restored image, we propose a fractional-order total variation regularization to remove the blur and Poisson noise simultaneously. We develop two efficient algorithms based on the alternating direction method of multipliers, while an expectation-maximization algorithm is adopted only in the blind case. A variety of numerical experiments have demonstrated that the proposed algorithms can efficiently reconstruct piecewise smooth images degraded by Poisson noise and various types of blurring, including Gaussian and motion blurs. Specifically for blind image deblurring, we obtain significant improvements over the state of the art. Keywords Blind deconvolution · Poisson noise · Expectation-maximization · Fractional-order total variation Mathematics Subject Classification 65F22 · 68U10 · 52A41 · 49N45
1 Introduction Data acquired by any imaging sensor usually undergo many types of degradations, in which two prominent ones are noise and blur. Specifically in photon-counting systems, such as X-ray, positron emission tomography (PET), single-photon emission computerized tomography (SPECT), fluorescence microscopy, and telescope, Poisson distribution is more Qin is supported by the NSF DMS-1941197. Lou acknowledges the NSF CAREER award DMS-1846690.
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Yifei Lou [email protected] Mujibur Rahman Chowdhury [email protected] Jing Qin [email protected]
1
Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA
2
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
appropriate to describe the noise statistics than the standard Gaussian distribution. Moreover, the recorded images are typically blurry due to the physical mechanism of an imaging device. The blurring process can be characterized by a point spread function (PSF), which is an impulse response of the imaging system to a point source. For a linear shift-invariant (LSI) system, blurring can be mathematically described by a convolution of a clean image with a PSF, where the PSF serves as a blurring kernel (also known as convolution kernel). Therefore, image deblurring is often referred to as image d
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