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

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7 pages
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


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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(βσ)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(wk)p(wk) expw wkw(1) k k 2    2 1(2) (1) q(fj)p(fj) expf fjf(2) j j 2  ! X   1 2Ng q(βσ) = Γβσ; (giˆgi(f,w)), ,(3) q(f,w) 2 2 i
(1) (2) (1) (2) th ,fe par wherew,w,fj jar ameters of the distributions,gˆi(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 contextgˆi(f,w)is also a random variable, which for simplicity we denote bygˆifrom now on.Ngis 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(βσ),q(fj)for alljandq(wk)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 2E (2) w=hβσiCijkfj(4) k q(f) i j  D  E X X X X X (1) (2) =hCijkfjhwki(5) wkwkβσigiCijkhfjiq(fj)Cijkfj q(wk0) 0 0 q(f) 0 i j k6=jk j D  E X X2 (2) f=hβσiCijkwk(6) j q(w) i k  D  E X X X X X (1) (2) f f=hβσigiCijkhwkiCij kwkCijkwk.(7) j j q(wk)− hfjiq(fj0) 0 0 q(w) 0 i k j6=j k k
The details of the derivation are given next.
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