Iterative Weighted Risk Estimation for Nonlinear Image Restoration ...

3. SELECTION OF REGULARIZATION PARAMETER β3.1 Generalized Cross Validation (GCV)GCV 2 is an attractive method for selecting β, especially in the context of linear algorithms. Fora generic linear reconstruction of the form, f β (y) =F β y (F β is matrix representing some type ofinverse filtering), GCV selects β by minimizingGCV(β) = N −1 ||Af β (y) − y|| 2(1 − N −1 tr{AF β }) 2 . (6)Calculation of the trace, tr{AF β } in the denominator of (6), can be performed either analyticallyin some special cases, e.g., when F β is circulant, or stochastically using Monte-Carlo methods 17 fora general F β . GCV(β) is simple to implement and is know to yield β that asymptotically providesan optimal reconstruction for linear algorithms. 2For nonlinear algorithms (denoted by f β ), Deshpande et al. 3 proposed the following NGCVmeasure 3, 4 (GCV for nonlinear algorithms) based on the principles of cross-validation:NGCV(β) = N −1 ‖y − Af β (y)‖ 2 2(1 − N −1 tr{AJ fβ (y)}) 2 , (7)where J fβ (y) is the Jacobian matrix consisting of partial derivatives of the components {f β,n (y)} N n=1of f β (y) with respect to the components {y n } N n=1 of y: thekl-th element of J fβ (y) isgivenby[J fβ (y)] kl = ∂f β,k(z)∂z l∣∣∣∣z=y. (8)NGCV(β) is a generalization of GCV(β) for nonlinear algorithms. 3, 4 It is more involved andcomputation intensive compared to GCV(β) (6) as it requires the evaluation of J fβ (y).3.2 Stein’s Principle for Estimating MSE-type MeasuresIn image reconstruction problems, mean squared error (MSE),MSE(β) △ = N −1 ‖x − f β (y)‖ 2 2, (9)is commonly used to determine quality of a reconstructed image and is an attractive alternative to(N)GCV for tuning β. However, MSE(β) cannot be directly used in practice due to its dependenceon the unknown x. For denoising applications, i.e., A = I N in (1), one can use Stein’s principle ∗to estimate MSE(β) when noise is modeled as Gaussian. This process leads to the so-called Stein’sUnbiased Risk Estimate (SURE) 5, 9 given by SURE(β) =N −1 ‖y−f β (y)‖ 2 2−σ 2 +2σ 2 N −1 tr{J fβ (y)}.SURE(β) is unbiased, i.e., E b {MSE(β)} = E b {SURE(β)} (where E b {·} denotes the expectationoperation with respect to b), but requires the knowledge of the noise variance σ 2 unlike (N)GCV.Evaluation of J fβ (y) inSURE(β) can be performed analytically for some special nonlinear denoisingalgorithms (e.g., wavelets-based denoising 9 and nonlocal means 11 ) or numerically using the Monte-Carlo method 10 for an arbitrary linear/nonlinear, iterative/noniterative algorithm f β .∗ Application of Stein’s principle requires the hypotheses that f β is (weakly) differentiable 6, 9 and decayssufficiently rapidly 6, 9 such that lim bn→∞ p(b)f β,n (y) =0∀ n, for the Gaussian probability density functionp(b) =(2πσ 2 ) −N/2 exp(−‖b‖ 2 2/2σ 2 ).SPIE-IS&T/ Vol. 8296 82960N-4Downloaded from SPIE Digital Library on 06 Apr 2012 to 141.213.236.110. Terms of Use: http://spiedl.org/terms