1

Efficient, Interpretable Deep Blind ImageDeblurring Via Unrolling

Supplementary Document

Yuelong Li, Student Member, IEEE, Mohammad Tofighi, Student Member, IEEE,Junyi Geng, Student Member, IEEE, Vishal Monga, Senior Member, IEEE, and Yonina C. Eldar, Fellow, IEEE

Let L be the cost function defined in (12). We derive its gradients w.r.t. its variables using the chain rule as follows:

∇wliL = ∇wl

iyli∇yl

iL = Rwl

iFdiag

(yl+1i

)F∗∇yl

iL,

∇ζliL = ∇ζlizl+1i ∇zl+1

iL

=

kl∗�(kl � gli − yli

)(∣∣∣kl∣∣∣2 + ζli

)2

T

F∗(I{|Pgg

l+1i |>bli} � ∇zl+1

iL),

∇bliL = ∇blizl+1i ∇zl+1

iL =

(I{gl+1

i <−bli} − I{gl+1i >bli}

)T∇zl+1

iL,

where Rwli

is the operator that extracts the components lying in the support of wli. Again using the chain rule,

∂L∂kl

=∂L∂zl+1

i

∂zl+1i

∂kl,

∂L∂zli

=∂Lzl+1i

∂zl+1i

∂zli+∂L∂kl

∂kl

∂zli,

∂L∂yli

=∂Lzl+1i

∂zl+1i

∂yli+

∂L∂kl+1

∂kl+1

∂yli+

∂L∂yl−1i

∂yl−1i

∂yli. (1)

We next derive each individual term in (1) as follows:

∂zl+1i

∂gl+1i

=∂zl+1

i

gl+1i

∂gl+1i

∂gl+1i

= diag(I{|Pgg

l+1i |>bli}

)F∗,

∂zl+1i

∂zli=∂zl+1

i

∂gl+1i

∂gl+1i

∂zli

∂zli∂zli

= diag(I{|Pgg

l+1i |>bli}

)F∗diag

ζli∣∣∣kl∣∣∣2 + ζli

F, (2)

∂zl+1i

∂yli=∂zl+1

i

∂gl+1i

∂gl+1i

∂yli

∂yli∂yli

= diag(I{|Pgg

l+1i |>bli}

)F∗diag

kl∗∣∣∣kl∣∣∣2 + ζli

F, (3)

and

∂zl+1i

∂kl=∂zl+1

i

∂gl+1i

(∂gl+1

i

∂kl

∂kli∂kli

+∂gl+1

i

∂kl∗∂kli∗

∂kli

)(4)

= diag(I{|Pgg

l+1i |>bli}

)F∗

diag ζli y

li(∣∣∣kl∣∣∣2 + ζli

)2

F∗ − diag

(kl∗)2� yli(∣∣∣kl∣∣∣2 + ζli

)2

F

,

∂kl+1

∂kl+13

=∂kl+1

∂kl+23

∂kl+23

∂kl+13

∂kl+13

∂kl+13

=I(1Tkl+

23

)− kl+

231T(

1Tkl+23

)2 diag

(I{

Pkkl+1

3>0})F∗,

2

∂kl+1

∂yli=∂kl+1

∂kl+13

kl+13

yli

yliyli

=I(1Tkl+

23

)− kl+

231T(

1Tkl+23

)2 · diag(I{

Pkkl+1

3>0})F∗diag

∑Ci=1 z

l+1i

∗

∑Ci=1

∣∣zl+1i

∣∣2 + ε

F, (5)

∂kl+1

∂zl+1i

=∂kl+1

∂kl+13

∂kl+ 13

∂zl+1i

∂zl+1i

∂zl+1i

+∂kl+

13

∂∂zl+1i

∗∂zl+1

i

∗

∂zl+1i

=I(1Tkl+

23

)− kl+

231T(

1Tkl+23

)2 diag

(I{

Pkkl+1

3>0})F∗· (6)

−diag(∑C

j=1 zl+1j

∗� ylj

)� zl+1

i

∗

(∑Cj=1

∣∣∣zl+1j

∣∣∣2 + ε

)2

F + diag

yli �

(∑Cj=1

∣∣∣zl+1j

∣∣∣2 + ε

)−(∑C

j=1 zl+1j

∗� ylj

)� zl+1

i(∑Cj=1

∣∣∣zl+1j

∣∣∣2 + ε

)2

F∗

,

∂yl−1i

∂yli=∂yl−1i

∂yl−1i

∂yl−1i

∂yli

∂yli∂yli

= F∗diag

(wl−1i

)F. (7)

Plugging (2) (3) (4) (5) (6) (7) into (1), we obtain

∇klL =

F∗diag ζli y

li(∣∣∣kl∣∣∣2 + ζli

)2

− Fdiag

(kl∗)2� yli(∣∣∣kl∣∣∣2 + ζli

)2

F∗

(I{|Pgg

l+1i |>bli} � ∇zl+1

iL)

∇gliL = Fdiag

ζli∣∣∣kl∣∣∣2 + ζli

F∗(I{|Pgg

l+1i |>bli} � ∇zl+1

iL)+

−Fdiag(∑L

j=1 glj

∗� yl−1j

)� gli

∗

(∑Lj=1

∣∣∣glj∣∣∣2 + ε

)2

+ F∗diag

yl−1i �

(∑Lj=1

∣∣∣glj∣∣∣2 + ε

)−(∑L

j=1 glj

∗� yl−1j

)� gli(∑L

j=1

∣∣∣glj∣∣∣2 + ε

)2

F∗

1

1Tkl−13

I{Pkk

l− 23>0

} �∇klC −I{

Pkkl− 2

3>0}kl− 1

3T

(1Tkl−

13

)2 ∇klL

∇yliL = Fdiag

kl∗∣∣∣kl∣∣∣2 + ζli

F∗(I{|Pgg

l+1i |>bli} � ∇zl+1

iL)+ Fdiag

∑Li=1 z

l+1i

∗

∑Li=1

∣∣zl+1i

∣∣2 + ε

F∗

1

1Tkl+23

I{Pkk

l+13>0

} �∇kl+1C −I{

Pkkl+1

3>0}kl+ 2

3T

(1Tkl+

23

)2 ∇kl+1C

+ Fdiag

(wl−1i

)F∗∇yl−1

iL

3

I. ADDITIONAL EXPERIMENTAL RESULTS

(a) Groundtruth (b) Perrone et al. [1] (c) Nah et al. [2] (d) Chakrabarti [3] (e) Xu et al. [4] (f) Kupyn et al. [5] (g) DUBLID

Fig. 1. Qualitative comparisons on the dataset from [6]. The blur kernels are placed at the right bottom corner.

REFERENCES

[1] D. Perrone and P. Favaro, “A Clearer Picture of Total Variation Blind Deconvolution,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 38, no. 6, pp.1041–1055, Jun. 2016.

[2] S. Nah, T. H. Kim, and K. M. Lee, “Deep multi-scale convolutional neural network for dynamic scene deblurring,” in Proc. IEEE Conf. CVPR, vol. 1,2017, p. 3.

[3] A. Chakrabarti, “A Neural Approach to Blind Motion Deblurring,” in Proc. ECCV, Oct. 2016.[4] X. Xu, J. Pan, Y. J. Zhang, and M. H. Yang, “Motion Blur Kernel Estimation via Deep Learning,” IEEE Trans. Image Process., vol. 27, no. 1, pp.

194–205, Jan. 2018.[5] O. Kupyn, V. Budzan, M. Mykhailych, D. Mishkin, and J. Matas, “Deblurgan: Blind motion deblurring using conditional adversarial networks,” in Proc.

IEEE Conf. CVPR, Jun. 2018.[6] L. Sun, S. Cho, J. Wang, and J. Hays, “Edge-based blur kernel estimation using patch priors,” in Proc. IEEE ICCP, Apr. 2013.