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Alternating Direction Method for Image Restoration

Jivnesh Dongre16EC65R12

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IntroductionImage restoration: Operation of taking

corrupt image and estimating clean, original image.

Image restoration is performed by reversing the process that blurred the image

Objective of image restoration: Reduce noise and recover resolution loss

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IntroductionImage restoration applications:Science and engineering such as medical and astronomical imaging, film restoration, image and video codingOriginal image corrupted by:Invariant blur, build in nonlinearities and additive Gaussian white noise

Objective function

Nonlinear least square (NLS)data fitting term

Total variation(TV) regularization term

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Nonlinear image degraded model: where =observed image, =true image, =blurring matrix, =noise vectorNonlinear least square problem:

Zervakis and venetsanopoulos used steepend descent method.Zervakis and venetsanopoulos further considered Gauss-Newton(GN) algorithm for NLS problem.

(1)

(2)

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TV based nonlinear least square problem :

+where = regularization parameter=discrete total variation of , =discrete gradient of at pixel

Main idea:Original optimization

problem

Easier subproblems under ADM

split

(3)

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Alternating Direction Method Of Multipliers(ADM):

Subject to , , , , , and and are indicator functions given by,

=

=

2

21 22 2

1

1argmin ( ) ( ) ( )2

m

ii

s z g p k u k v

1( )k u 2 ( )k v

1( )k u

2 ( )k v

(4)

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Algorithm

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Experimental ResultsPeak signal to noise ratio(PSNR):PSNR=

Structural similarity (SSIM) index

PSNR should be high

(5)

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Nonlinear Image Restoration

PSNR:b)26.95db, c)28.68dbSSIM:b)0.8036,c)0.8848

a)True image, b)observed image, c)NLS model,d)TVNLS model

Natural logarithm nonlinearity

Fig. 1[1]

Fig. 2[1]

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Power Nonlinearity

PSNR:b)27.80db, c)30.27dbSSIM:b)0.8401, c)0.9124

a)True imageb)Observed imagec)NLS modeld)TVNLS model

Fig. 3[1]

Fig. 4[1]

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Restoration results with different blurs and noise level

NLS TVNLSNONLINEARTITY

BLUR STANDARD DEVIATION

PSNR SSIM PSNR

SSIM

Logarithm

Gaussian

0.001 26.95 0.8036 28.68

0.8848

0.01 23.75 0.6012 25.76

0.8120

Moffat 0.001 28.52 0.7987 30.81

0.9039

0.01 24.26 0.6201 26.44

0.8225

Power Gaussian

0.001 27.80 0.8401 30.27

0.9124

0.01 25.07 0.7451 27.30

0.8619

Moffat 0.001 29.82 0.8599 33.49

0.9304

0.01 26.12 0.7609 28.88

0.8793

Table [1]

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High Dynamic Range ImagingThe nonlinear response is formulated as:

where is true HDR radiance, is observed LDR image, is the camera response

Idea of majorize-minimize(MM) method: use reweighted least squares technique to tackle the non smooth TV term and linearized technique to tackle the non linear least square data fitting term.

(6)

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a)Tone mapped LDR image from true HDRb)Noisy observed LDR imagec)Tone mapped LDR image from recovered HDR image by MM methodd) Tone mapped LDR image from recovered HDR image by ADM method

Fig. 5[1] Fig. 6[1]

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a)Tone mapped LDR image from true HDRb)Noisy observed LDR imagec)Tone mapped LDR image from recoverd HDR image by MM methodd) Tone mapped LDR image from recoverd HDR image by ADM method

Fig. 7[1] Fig. 8[1]

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ConclusionTV based variation model to tackle

nonlinear image restoration problem.An efficient alternating direction

method of multipliers to solve the model.

Numerical examples including nonlinear image restoration and HDR imaging are shown by author to illustrate the effectiveness and efficiency of numerical scheme.

REFERENCES[1] C. Chen, M. K. Ng and X. L. Zhao, "Alternating Direction

Method of Multipliers for Nonlinear Image Restoration Problems," in IEEE Transactions on Image Processing, vol. 24, no. 1, pp. 33-43, Jan. 2015.

[2]B. K. Gunturk and X. Li, Image Restoration: Fundamentals and Advances. Boca Raton, FL, USA: CRC Press, 2012.

[3] Zhou Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, "Image quality assessment: from error visibility to structural similarity," in IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600-612, April 2004.

[4] S. Kim, Y. W. Tai, S. J. Kim, M. S. Brown and Y. Matsushita, "Nonlinear camera response functions and image deblurring," Computer Vision and Pattern Recognition (CVPR), IEEE Conference on, Providence, RI, 2012, pp. 25-32.

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Questions?