Gracheva Inessa - Fast Global Image Denoising Algorithm on the Basis of Nonstationary Gamma-Normal Statistical Model
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The document proposes a fast global image denoising algorithm based on a nonstationary gamma-normal statistical model. The algorithm effectively removes Gaussian and Poisson noise while satisfying constraints on computational cost to process large datasets with minimal user input. It develops a probabilistic data model and defines the joint prior distribution, leading to a Bayesian estimate of the hidden image field. The algorithm uses a Gauss-Seidel procedure on a trellis of neighborhood graphs to iteratively find optimal hidden variable values. Experimental results show the algorithm achieves similar denoising performance to other techniques but with significantly less computation time.
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Gracheva Inessa - Fast Global Image Denoising Algorithm on the Basis of Nonstationary Gamma-Normal Statistical Model
- 1. Fast Global Image Denoising Algorithm on the Basis of Nonstationary Gamma- Normal Statistical Model Gracheva Inessa, gia1509@mail.ru, Kopylov Andrey, and.kopylov@gmail.com, Krasotkina Olga, O.V.Krasotkina@yandex.ru Tula State University, Tula, Russia AIST Conference, April 2015, Ekaterinburg
- 2. Formulation of the problem What do we want? the method should effectively remove Gaussian noise as well as Poissonian noise; the method should satisfy strict constraints in terms of computational cost, so as to be able to process large data sets; the algorithm required as less user input as possible in order to facilitate its application and to enhance the reproducibility of its results. To achieve these purpose a nonstationary gamma-normal noise model has been proposed in the framework of Bayesian approach to the problem of image processing. This model allows us to develop a fast global algorithm on the basis of Gauss-Seudel procedure and Kalman filter-interpolator.
- 3. Image denoising ( , )tY y t T= ∈ ( , )tX x t T= ∈ 1 2 1 1 2 2{ ( , ): 1,..., , 1,..., }T t t t t N t N= = = = x X∈y Y∈ The analyzed image The processing result
- 4. Probabilistic data model The joint conditional probability density: where is the variance of the observation noise. The a priori joint distribution: where is the proportionality coefficients to the variance of the observation noise ; V is the neighborhood graphs of image elements having the form of a lattice. 2 /2 /2 1 1 ( | , ) exp( ( ) ) (2 ) 2 t tN N t T Y X y xδ δ π δ ∈ Φ = − −∑ 2 1/2 ,( 1)/2 1 1 1 ( | , ) exp ( ) 2 (2 ) t t t t V tn N t t T X x xδ δλ δλ π ′ ′′ ′ ′′∈− ∈ Ψ Λ µ × − − ÷ ÷ ∑ ∏ δ=)( 2 teE tλ t tr λδ= 1 N1t1 1 N2 t2
- 5. Gamma - distribution ))/1(exp()/1(),|/1( 1 ttt λϑλϑαλγ α −∝ − 2 1 2 (1/ | , , ) (1/ ) exp (1/ ) 2 t t t µ δµ λ γ λ δ λ µ λ λ δµ + µ − ÷ 2 1)1( 2)/1(, 1)1( )/1( λ µδ δµλ λ µδ λ ++ = ++ = tt VarE2 (1/ ) , (1/ )t tE Var α α λ λ ϑ ϑ = = 1 1 ( | , ) ( | , ) 1 1 exp exp ( 1) ln t t T t T t t t T t T t Tt t G α α ϑ γ λ α ϑ λ ϑ α λ ϑ λ λ − ∈ ∈ ∈ ∈ ∈ Λ = µ × ÷ × − = − − − ÷ ∏ ∏ ∑ ∑ ∑ ( | , , ) 1 1 1 exp ln , 2 t t T t G δ λ µ λ λ δµ λ λ∈ Λ = = − + ÷ ∑ where 1 1 1 1 1 , 2 2 λ α ϑ δ µ δµ = + + = ÷ Gamma-distributed of the inverse factors :tλ/1 with mathematical expectations and variances: We come to the a priori distribution density:
- 6. Normal gamma-distribution So, have completely defined the joint prior normal gamma-distribution of both hidden fields and Coupled with the conditional density of the observable field, it makes basis for Bayesian estimation of the field . The joint a posteriori distribution of hidden elements, namely, those of field and its instantaneous factors ( , )tX x t T= ∈ ( , )t t TλΛ = ∈ ( , | , , ) ( | , ) ( | , , )H X X Gδ λ µ δ δ λ µΛ = Ψ Λ Λ ( , )tX x t T= ∈ ( | , ) ( | , ) ( | , ) ( , | , , , ) ( '| ', ) ( '| , ) ( | ', ) ' ' X G Y X P X Y X G Y X dX d δ α ϑ δ δ α ϑ δ α ϑ δ Ψ Λ Λ Φ Λ = Ψ Λ Λ Φ Λ∫∫ ( , )tX x t T= ∈ ( , )t t TλΛ = ∈
- 7. The Bayesian estimate of the hidden random field The Bayesian estimate of is the maximum point of the numerator The conditionally optimal factors are defined independently of each other: The zero conditions for the derivatives, excluding the trivial solutions lead to the equalities ( , )X Λ , 2 2 ' '' ', '' ( , | , ) arg min ( , | , , ), ( , | , , ) ( ) 1 ( ) / (1 1/ )ln . X t t t T t t t t t V t X J X Y J X Y y x x x λ µ λ µ λ µ λ µ µ λ λ Λ ∈ ∈ Λ = Λ Λ = − + + − + + + ∑ ∑ ) ) , 2 1 2 ' '' ', '' ( , | , ) arg min ( , | , , ), ( , | , , ) ( ) (1/ )( ) 1/ (1 1/ )ln . 1 1/ X N t t t t t t t V X J X Y J X Y y x x x λ µ λ µ λ µ λ µ µ µ Λ = ∈ Λ = Λ Λ = − + − + + + ∑ ∑ ) ) ( , , ) [ ( , , ), ]tX X t Tλ µ λ λ µΛ = ∈ )) 2 ' '' 1 ( , , ) arg min ( | , , ) : [( ) / ] (1 1/ )ln 0t t t t t X J X x xλ µ λ µ λ µ µ λ λ λΛ ∂ Λ = Λ − + + + = ∂ ) ,∞→tλ 2 ' ''(1/ )( ) 1/ ( , , ) 1 1/ t t t x x X λ µ λ λ µ λ µ − + = + )
- 8. “The saturation effect” “The saturation effect” of the unsmoothness penalty for sufficiently large values of the parameter µ and fixed value λ =1.
- 9. The Gauss - Seudel procedure Minimize of the objective function gives the approximation to the estimate of the mutually agreed field starting with the initial values , and new value of the hidden component gives the solution of the conditional optimization problem ( , )k k tX x t T= ∈ ) ) 2 2 ' '' ', '' ' ( , 1,..., ) arg min ( , | , , ) 1 arg min ( ) ( ) . k k k X t t t t X t T t t V t X x t N J X Y y x x x λ µ λ∈ ∈ = = = Λ = − + − ∑ ∑ ) ) 1 2 2 ' '' ', '' ' arg min ( , | , , ) 1 arg min ( ) ( ) . k k k k t t t t t T t t V t J X Y y x x x λ µ λ + Λ Λ ∈ ∈ Λ = Λ = = − + − ∑ ∑ ) ) ) ) ) 0 0 ( , )t t Tλ λΛ = = ∈ ))
- 10. Approximation trellis neighborhood graph sequence trees For finding the values of the hidden field at each vertical row of the picture, we use a separate pixel neighborhood tree which is defined, on the whole pixel set and has the same horizontal branches as the others. The resulting image processing procedure is aimed at finding optimal values only for the hidden variables at the stem nodes in each tree. The simplicity of such a succession of elementary trees allows for a significant simplification of the optimization procedure. 1
- 11. Comparison of Poissonian and Gaussian noise removal algorithms Image Peak intensity Input PSNR Peppers 256x256 120 60 30 23.92 20.92 17.91 Haar-Fisz 29.27 27.30 24.54 Platelet 29.07 27.44 25.73 PURE-LET 30.28 28.51 26.72 Our algorithm 29.53 28.65 25.77 Image Peak intensity Input PSNR Einstein 512x512 120 60 30 8.34 14.35 19.23 FBF 24.86 26.21 29.63 SURE-LET 24.56 26.30 29.43 BM3D 25.03 26.59 29.99 Our algorithm 24.76 26.44 29.88 Gaussian noisePoissonian noise
- 12. Relative computation time of various denoising techniques (sec) Methods Image size 256x256 512x512 Platelet 856 1112 PURE-LET 4.6 8.2 BM3D 4.1 7.2 Haar-Fisz 1.3 3.2 FBF 0.5 1.8 SURE-LET 0.4 1.6 Our algorithm 0.16 0.53 A plot of the time of the algorithm (s) on the image size.
- 13. Experimental results a) b) c) d) a)The original Einstein image. b)Noisy version with simulated Gaussian noise. c)Denoised with our algorithm. d)Denoised with BM3D.
- 14. Experimental results a) b) c) a)Part of the original MRI slice from the IGBMC Imaging Center data sets. b)Denoised with our algorithm. c)Denoised with PURE-LET
- 15. THANKS FOR YOUR ATTENTION! Acknowledgements This research is funded by RFBR grant,13-07-00529.