The document proposes a method for edge-preserving image decomposition using L1 fidelity with L0 gradient. It decomposes an input image into a base layer and detail layer by minimizing the L1 difference between the base layer and input, along with the L0 gradient of the base layer. This encourages sparsity of gradients in the base layer to preserve edges. The method estimates the base layer, detail layer, and other parameters using alternating minimization. It effectively removes noise and artifacts while preserving edges compared to other decomposition methods.

1. Chih-Tsung Shen Feng-Ju Chang Yi-Ping Hung Soo-Chang Pei
NTU, Academia Sinica Academia Sinica NTU, Academia Sinica NTU
Edge-Preserving Image Decomposition
using L1 Fidelity with L0 Gradient

2. Image Decomposition
• Image decomposition : a tool for image editing.
• The intensity can be decomposed into base
layer and detail layer.
• The basic example is un-sharp masking.
Input Base layer Output
( , ) ( , ) ( , )L x y B x y D x y  ( , )B x y ( , ) ( , ) ( , )ENL x y B x y k D x y  

3. Image Decomposition
• Pyramid: [Burt & Adelson 83]
• Retinex: [Jobson et al. 97]
• EMD: [Huang 98]
• Bilateral Filtering: [Tomasi & Manduchi 98]
• Fast Bilateral Filtering: [Durand & Dorsey 02]
• Weighted Least Square: [Farbman et al.08]
• Local Extrema: [Subr et al. 09]
• Guided Filtering: [He et al. 10]
• L0 Gradient Minimization: [Xu et al. 11]

4. Edge-Preserving
Image Decomposition
• Weighted Least Square: [Farbman et al.08]
• L0 Gradient Minimization: [Xu et al. 11]
• We decompose the signal using L1 fidelity with
L0 gradient. (sparsity measurement)
2 2 2
( , ) ( , )
( , ) ( , )
| ( , ) ( , ) | ( ( , ) ( ) ( , ) ( ) )min x y
B x y x y
B x y B x y
B x y L x y b x y b x y
x y

 
     
 

2
( , ) ( , )
| ( , ) ( , ) | ( , )minB x y x y
B x y L x y C x y  
( , ) ( , )
( , ) # | | | | 0
B x y B x y
C x y
x y
   
   
   
( , ) ( , )
( , ) # | | | | 0
B x y B x y
C x y
x y
   
   
   ( , ) ( , )
| ( , ) ( , ) | ( , )minB x y x y
B x y L x y C x y  

5. Image Smoothing
• The input signal can be
decomposed into base
layer and detail layer:
• using minimization:
• Approximate L1 fidelity
by using :
( , ) ( , )
| ( , ) ( , ) | ( , )minB x y x y
B x y L x y C x y  
2
, ( , ) ( , )
| ( , ) ( , ) ( , ) | | ( , ) | ( , )minB w x y x y
B x y L x y w x y w x y C x y       
( , ) ( , ) ( , )L x y B x y D x y 
( , )w x y
Error between B(x,y)-L(x,y) and w(x,y) Approximation

6. Image Smoothing
• Approximate L0 gradient using
where the L0 norm of gradient (L0 gradient):
2
, , , ( , ) ( , )
2 2
| ( , ) ( , ) ( , ) | | ( , ) | ( ( , ), ( , ))
( , ) ( , )
(| ( , ) | | ( , ) | )
minx y
x y
B w a a x y x y
x y
B x y L x y w x y w x y C a x y a x y
B x y B x y
a x y a x y
x y
 

     
 
    
 
 
( , ) ( , )
( ( , ), ( , )) # ( , ) | | | | 0x y
B x y B x y
C a x y a x y x y
x y
   
   
   
( ( , ), ( , ))x yC a x y a x y
(Equation* )

7. • Since is independent to , and
,we can solve the equation* by using
alternating minimization:
2
( , )
2 2
| ( , ) ( , ) ( , ) |
( , ) ( , )
| ( , ) | | ( , ) |
minB x y
x y
B x y L x y w x y
B x y B x y
a x y a x y
x y


  
 
   
 

* *
1
( ) ( ) ( ) ( ) ( ) ( )
( , )
*( ) ( ) *( ) ( )
x yF L F w F x F a F y F a
B x y F
F x F x F y F y
   
  

      
  
       
Solver: Computing B(x,y)
( , )B x y ( , )w x y ( , )xa x y
( , )ya x y
• Using FFT to speed up the whole process:

8. • Moreover, is also independent to
and , we can solve first:
2
( , ) ( , )
| ( , ) | | ( , ) ( , ) ( , ) |minw x y x y
w x y B x y L x y w x y   
Solver: Computing w(x,y), ax(x,y) & ay(x,y)
1 ( , ) ( , )
( , ) max | ( , ) ( , ) | ,0
2 | ( , ) ( , ) |
B x y L x y
w x y B x y L x y
B x y L x y
 
   
 
( , )w x y ( , )xa x y
( , )ya x y ( , )w x y

9. • Using a binary map , we can
obtain the L0 norm of gradient :
Solver: Computing w(x,y), ax(x,y) & ay(x,y)
2 2
,
( , ) ( , )
( ( , ), ( , )) | ( , ) | | ( , ) |minx y
x y x y
a a
B x y B x y
C a x y a x y a x y a x y
x y


 
    
 
1,| ( , ) | | ( , ) | 0
( ( , ), ( , )) (| ( , ) | | ( , ) |)
0,| ( , ) | | ( , ) | 0
x y
x y x y
x y
a x y a x y
C a x y a x y H a x y a x y
a x y a x y
 
   
 
(| ( , ) | | ( , ) |)x yH a x y a x y
( ( , ), ( , ))x yC a x y a x y

10. • Since we obtain the final base layer , we
can also obtain our corresponding detail
layer :
• The general form for image enhancement:
is the scale parameter, is the white value,
is the parameter for gamma correction, and
is the gain for detail boosting.
General Form for Enhancement
ˆ ˆ( , ) ( , ) ( , )D x y L x y B x y 
ˆ( , ) ˆ( , ) ( ) ( , )EN
B x y
L x y s W G D x y
W

    
ˆ( , )B x y
ˆ( , )D x y
s

G
W

11. Image Smoothing
Input Ours

12. Noise Removal
Noisy Input WLS,
L0 Smooting, Ours,
Using the same
balancing
parameter
2 2
|...| (...) 
2
0|...| ||...||  0| ...| || ...|| 
0.1 

13. Noise Removal
Noisy Input Ours

14. Artifact Removal
Ours
Non-Degraded BMP
Mean Shift
Degraded JPG

15. Artifact Removal
Degraded JPG

16. Artifact Removal
Ours

17. Edge Detection
Input Ours

18. Edge Detection
Input Ours

19. Stylization
OursInput

20. Stylization
Input Ours

21. Image Enhancement
Input Ours

22. Image Enhancement
Input Ours

23. Low Backlight
Original, 100% Original, 40% Our, 40%
Backlight:
Scale down to 40%
Image:
Compensate the brightness
Boost the detail parts
Backlight:
Scale down to 40%