Non uniform Deblurring for Shaken Images: Derivation of parameter update equations for blind de blurring

Oliver Whyte

Description

Non-uniform Deblurring for Shaken Images: Derivation of parameter update equations for blind de-blurring Oliver Whyte Abstract This note outlines the derivation of the parameter update formulas for the variational non-uniform blind deblurring algorithm described in Whyte et al. [4]. First, using the calculus of variations, we find the optimal forms of the factorized approximating distributions and arrive at the same formulas as in the uniform blind deblurring of Miskin & MacKay [3] and Fergus et al. [2]. Next, we derive the parameter update equations, which differ significantly from [3, 2]. 1. Summary We derive here the optimal forms and parameters of the approximating distributions q(f), q(w) and q(??) used in the variational inference of the blind deblurring algorithms of Miskin & MacKay [3] and Fergus et al. [2]. Using the calculus of variations, we find that the optimal distributions for the latent variables are (the same as Equations (42, 43, 17) of [3]): q(wk) ? p(wk) exp ( ? 1 2 w(2)k ( wk ? w (1) k )2 ) (1) q(fj) ? p(fj) exp ( ? 1 2 f (2)j ( fj ? f (1) j )2 ) (2) q(??) = ? ( ??

• th blurry


• optimal forms


• fj ?


• blurry pixel


• uniform deblurring


• blind deblurring


• dfj? dw


• ???? ∑


• variational method


• deblurring algorithm

Sujets

• Variational method

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

Non-uniform Deblurring for Shaken Images: Derivation of parameter update equations for blind de-blurring

Oliver Whyte

Abstract

This note outlines the derivation of the parameter update formulas for the variational non-uniform algorithm described in Whyteet al. [4]. First, using the calculus of variations, we find the optimal forms approximating distributions and arrive at the same formulas as in the uniform blind deblurring of Miskin & Ferguset al. [2]. Next, we derive the parameter update equations, which differ significantly from [3,2].

blind deblurring of the factorized MacKay [3] and

1. Summary We derive here the optimal forms and parameters of the approximating distributionsq(f),q(w)andq(β σ )used in the variational inference of the blind deblurring algorithms of Miskin & MacKay [3] and Ferguset al. [2]. Using the calculus of variations, we find that the optimal distributions for the latent variables are (the same as Equations (42, 43, 17) of [3]):     2 1 (2) (1) q(w k )∝p(w k ) exp−w w k −w(1) k k 2     2 1 (2) (1) q(f j )∝p(f j ) exp−f f j −f(2) j j 2  ! X   1 2 N g q(β σ ) = Γβ σ ; (g i −ˆg i (f,w)), ,(3) q(f,w) 2 2 i

(1) (2) (1) (2) th ,fe par wherew,w,f j j ar ameters of the distributions,gˆ i (f,w)is the “reconstruction” of theiblurry pixel using k k the model in Equation (7) of Whyteet al. [4that]. Note fandware random variables, so in this contextgˆ i (f,w)is also a random variable, which for simplicity we denote bygˆ i from now on.N g is the number of observed blurry pixels, andh∙i q represents the expectation with respect to the the distributionqeach latent variable, the parameters of its distribution. For (1) (2) depend on the distributions of all the other latent variables,e.g.wandwdepend onq(β σ ),q(f j )for alljandq(w k )for 0 k k 0 allk6=k. For our non-uniform blur model, we find the following optimal values for the parameters (c.f. Equations (46–49) of [3]): D X X  2 E (2) w=hβ σ iC ijk f j (4) k q(f) i j  D  E  X X X X X (1) (2) =hC ijk f j hw k i(5) w k w k β σ ig i C ijk hf j i q(f j ) −C ijk f j q (w k 0 ) 0 0 q(f) 0 i j k6= jk j D  E X X 2 (2) f=hβ σ iC ijk w k (6) j q(w) i k  D  E  X X X X X (1) (2) f f=hβ σ ig i C ijk hw k iC ij k w k C ijk w k .(7) j j q(w k ) − hf j i q(f j 0 ) 0 0 q(w) 0 i k j6=j k k

The details of the derivation are given next.

1