Transcript Retinex theoretical study

Approaches for Retinex and Their
Relations
Yu Du
March 14, 2002
Presentation Outline
Introductions to retinex
Approaches for retinex
The variational framework
Relation of these approaches
Conclusions
2
What Is Retinex
Lightness and retinex theory

E. H. Land 1971
Visual system of human

Retina: the sensory membrane lining the eye that receives the
image formed by the lens (Webster)

Reflectance and illumination

Edges and independent color senstion
3
Model of retinex (1)
S ( x, y )  R ( x, y )  L ( x, y )
The given image
The illumination part
The reflectance part
4
Model of retinex (2)
s ( x, y )  r ( x, y )  l ( x, y )
S
Input Image
Log
s

Estimate the
+

r̂
Exp
lˆ
Illumination
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R̂
Three Types of Previous Approaches
Random walk algorithms

E. H. Land (1971)
Homomorphic filtering

E. H. Land (1986), D. J. Jobson (1997)
Solving Poisson equation

B. K. P. Horn (1974)
6
Random Walk Algorithms (1)
First retinex algorithm
A series of random paths

Starting pixel

Randomly select a neighbor pixel as next pixel on path
x1
Accumulator and counter
A( xi )  A( xi )  log( f ( xi ))  log( f ( x1 ))
N ( xi )  N ( xi )  1
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Random Walk Algorithms (2)
Adequate number of random paths

Cover the whole image

Small variance
Length of paths

>200 for 10x10 image (D. H. Brainard)
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Special Smoothness of Random Walk
The value in the accumulator
A( x) 
 log( f ( x))  log( f ( x ))
i
paths that
passed
pixel x
The illumination part
l ( x)  log( G( x))
G( x)  N f ( x1 )  f ( xN )
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Homomorphic Filtering
Assume illumination part to be smooth
Apply low pass filter
H (u, v)  ( H   L )(1  e
D 2 ( u ,v )
c
D02
)  L
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Poisson Equation Solution (1)
Derivative of illumination part close to zero
Reflectance part to be piece-wise constant
Get the illumination part

Take the derivative of the image

Clip out the high derivative peaks
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Poisson Equation Solution (2)
s
s  T
 (s)  
 0 other wise
Solve Poisson equation
lˆ   (s)
Iterative method
Apply low-pass filter (invert Laplacian operator)
12
Comments on Above Approaches
Random walk algorithm

Too slow
Homomorphic filtering

Low-pass filtering first or log first?
More work needed to be done on Poisson equation
solving
13
Variational Framework
Presented by R. Kimmel etc.
From assumptions to penalty function
From penalty function to algorithm
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Assumptions On Illumination Image
Spatial smoothness of illumination
Reflectance is not pure white
Illumination close to intensity image
Spatial smoothness of reflectance
Continues smoothly beyond boundaries
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Penalty Function and Restrictions
Goal to minimize:
2
F (l )   l   (l  s)   (l  s) )dxdy
2
2

Subject to:
And
ls

l , n  0
on

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Solve the Penalty Function (1)
Euler-Lagrange equations
F (l )
 0  l   (l  s )  (l  s )
l
And
ls
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Solve the Penalty Function (2)
Projected normalized steepest descent (PNSD)
Iteratively to get illumination part
l j  min{ l j 1   NSD  G, s}
G  l j 1  (  )(l j 1  s)
 NSD 
 ( G
G
2
2
 (1   ) G )
2
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Multi-resolution
Make PNSD algorithm converges faster
Illumination part is smooth
Coarse resolution image first
Upscale coarse illumination as initial of finer resolution
layer
Not multi-scale technique
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Relationship of Different Approaches (1)
Random walk and Homomorphic filtering
R. Kimmel’s words on Homomorphic filtering
  0
and remove constraint
ls
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Relationship of Different Approaches (2)
Apply appropriate scaling on images,
Homomorphic filtering satisfies constrain
ls
and
 0
Poisson equation approach:
 ( x, y )   (s)
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Conclusions
Retinex is trying to simulate human vision process
Different approaches are from same assumptions
Implementation details are important for results
22
Thank You
March 14, 2002