Image Processing Algorithms with Structure Transferring Propertie on the Basis of Gamma-normal Model

1. Image
processing
algorithms
with
structure
transferring
proper4es
on
the
basis
of
gamma-­‐normal
model
Gracheva
Inessa,
gia1509@mail.ru,
Kopylov
Andrey,
and.kopylov@gmail.com,
Russia,
Tula,
Tula
State
University,
AIST
Conference,
Yekaterinburg,
2016

2. Related
Work
( , )tY y t T= ∈ ( , )tX x t T= ∈
Analyzed Image Processing Result
1 2 1 1 2 2{ ( , ): 1,..., , 1,..., }T t t t t N t N= = = =
We will consider an analyzed image and processing result
as, respectively, the observed and hidden components of the two-component random
field (X, Y ).
( , )tY y t T= ∈ ( , )tX x t T= ∈

3. Probabilis4c
Data
Model
Joint conditional probability density:
where is the variance of the observation noise.
A priori joint distribution:
where is factors of the unknown local variability
of the sought for processing result; V is the neighborhood
graphs of image elements having the form of a lattice.
1 2 1 2
2
( )/2 ( )/2
1 1
( | , ) exp( ( ) )
(2 ) 2
t tN N N N
t T
Y X y xδ
δ π δ ∈
Φ = − −∑
1 2
2
1/2
,( )/2
1 1 1
( | , ) exp ( )
2
(2 )
t t
t t V tN N
t
t T
X x xδ
δλ
δλ π
ʹ′ ʹ′ʹ′
ʹ′ ʹ′ʹ′∈
∈
⎛ ⎞
Ψ Λ ∝ × − −⎜ ⎟
⎛ ⎞ ⎝ ⎠
⎜ ⎟
⎝ ⎠
∑
∏
δ=)( 2
teE
tλ
1 N1t1
1
N2
t2

4. Gamma-­‐Normal
Model
2 1
2
(1/ | , , ) (1/ ) exp (1/ )
2
t t t
µ
δµ λ
γ λ δ λ µ λ λ
δµ
+
⎛ ⎞
∝ −⎜ ⎟
⎝ ⎠
2
1)1(
2)/1(,
1)1(
)/1(
λ
µδ
δµλ
λ
µδ
λ
++
=
++
= tt VarE
1 1 1
( | , , ) exp ln
2
t
t T t
G δ λ µ λ λ
δµ λ λ∈
⎡ ⎤⎛ ⎞
Λ = − +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
∑
Gamma-distribution of the inverse factors :tλ/1
with mathematical expectations and variances:
A priori distribution density:
Joint prior normal gamma-distribution: ( , | , , ) ( | , ) ( | , , )H X X Gδ λ µ δ δ λ µΛ = Ψ Λ Λ
,
2 2
' ''
', ''
( , | , ) argmin ( , | , , ),
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
λ µ λ
λ µ µ
Λ
∈ ∈
⎧ Λ = Λ
⎪⎪
⎨ ⎧ ⎫⎡ ⎤
⎪ Λ = − + − + + +⎨ ⎬⎢ ⎥
⎪ ⎣ ⎦⎩ ⎭⎩
∑ ∑
) )
Bayesian estimate of :( , )X Λ
Gracheva I., Kopylov A., Krasotkina O.: Fast global image denoising algorithm on the
basis of nonstationary gamma-normal statistical model. Communications in Computer and
Information Science. Springer. 542, 71-83 (2015).

5. New
Formula4on
of
the
Problem
He K., Sun J., Tang X.: Guided Image Filtering. IEEE Trans. on Pattern Analysis and
Machine Intel. 35(6), 1397-1409 (2013).
Analyzed Image
Guided Image
( , )tY y t T= ∈
( , )g g
tX x t T= ∈
( , )tX x t T= ∈
( , , , ) arg min ( | , , , )
X
X Y J X Yµ µΛ = Λλ λ
( , , ) arg min ( | , , )g g
X J Xµ µ
Λ
Λ = Λλ λ
Processing Result

6. 1 2 1 1 2 2{ ( , ): 1,..., , 1,..., }T t t t t N t N= = = =
Image haze removal
problem:
Analyzed Image Processing Result Guided Image
( , )tY y t T= ∈ ( , )tX x t T= ∈ ( , )g g
tX x t T= ∈
HDR image
compression
problem:
Edge refinement of
an image:
Examples
of
Problems

7. Structure-­‐Transferring
Proper4es
2
' ''
'
(1/ )( ) 1/
( , , )
1 1/
g g
g t t
t
x x
X
λ µ
λ λ µ λ
µ
− +
=
+
)
( , )t t TλΛ = ∈
))
( , )g g
tX x t T= ∈
( , )tX x t T= ∈ ( , )tY y t T= ∈
The estimates of factors represent the edges of objects in the guide
image . This makes it possible to transfer structure from the guide image
to the output image , even if the analyzed image
is smooth.
( , )g g
tX x t T= ∈
2 2
' ''
', '' '
( , ) argmin ( , | , , )
1
argmin ( ) ( ) .
X
t t t t
X t T t t V t
X x t T J X Y
y x x x
λ µ
λ∈ ∈
= ∈ = Λ =
⎧ ⎫
= − + −⎨ ⎬
⎩ ⎭
∑ ∑
) )

8. Edge
Refinement
of
an
image
( , )g g
tX x t T= ∈
( , )tY y t T= ∈
( , )t t TλΛ = ∈
))
( , )tX x t T= ∈

9. Image
Haze
Removal
3
Ra ∈
g
t
t
t
x a
x a
x
−
= +%
( , )t t TλΛ = ∈
))
( , )g g
tX x t T= ∈
( , )tY y t T= ∈
( , )tX x t T= ∈% %
( , )tX x t T= ∈

10. Compression
HDR
( ( ))t t tx y x mean xα= − +
) )
%
α is contrast sensitivity and will compress contrasts for values<1,0.
( , )tY y t T= ∈ ( , )g g
tX x t T= ∈( , )tX x t T= ∈

11. Experimental
Result
Comparison experimental results on structure-transferring filtering.
a) Original image; b) Binary mask; c) Our algorithm; d) Guided filter;
e) Fast Guided filter; f) In the zoom-in patches, our algorithm
compare with the Guided filter and the fast Guided filter.

12. Experimental
Result

13. Experimental
Result

14. Computa4on
Time
Computation time of Guided filter, fast Guided filter (with s = 2), fast
Guided filter (with s = 4) and our algorithm for processing of images of
different sizes.

15.
THANK
YOU
FOR
ATTENTION!