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Anisotropic Diffusion’s Extension to
Constrained Line Processes

Anisotropic Diffusion’s Application in
3D Confocal Microscopy Image Processing
Cédric Dufour
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Contents
 Anisotropic diffusion’s basics
 Extension to constrained line processes
 Anisotropic diffusion vs. constrained line processes
 3D microscopy image processing
 Conclusions
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Anisotropic diffusion’s basics (1)
 Underlying principle: standard heat diffusion
Grayscale intensity value
Diffusion coefficient
I x, y, z 
 div c  I x, y, z 
t
Time (iteration) variable
Equivalent to gaussian local meaning
(the variance being related unequivocally to the diffusion coefficient)
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Anisotropic diffusion’s basics (2)
 Problem: diffusion occurs in all direction, regardless
of edges
Blurring
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Anisotropic diffusion’s basics (3)
5
 Solution: bind the diffusion coefficient to the gradient
of the intensity
Anisotropic diffusion coefficient
(“edge stopping” function)
I x, y, z 
 div g  I x, y, z   I x, y, z 
t
Care must be taken in the choice of the edge
stopping function for the problem to be well-posed
More info: edge stopping function
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Anisotropic diffusion’s basics (4)
 Results: diffusion is inhibited when the gradient gets
more important (edges)
Piecewise smooth image
A.D.
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Extension to constrained line processes (1)
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 Anisotropic diffusion can be derived from the
minimization of a smoothness functional:
Smoothness functional
Smoothness norm

E  I      I s , p
I s I p s
 x   0, x  0
 x   0, x  0
 ( x)  0, x  0

min
I
I
 div g  I  I 
t
  x 
g x  
x
More info: 3D neighborhood for anisotropic diffusion
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Extension to constrained line processes (2)
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 Expressing this minimization problem according to
the line process formulation, we have:
Line process
 
E I , l      I s , p  ls , p   ls , p
2
I s I ps
Fitting constant
Line process penalty function
 Adding explicit spatial constraints:
Spatial constraints
 
E I , l      I s , p  ls , p   ls , p  EC ls , p 
2
I s I ps
More info: line process and penalty function characteristics
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Extension to constrained line processes (3)
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 The line process formulation is related to the
standard anisotropic formulation through:
I  x and ls,p  z


 x     x 2
      
1   x  1
z x   
  g x 
z      
2
x
2
2
 z    x     z  x
 z       z      
x  g  2  z 
def
def
1
1
z 
More info: starting relating axiom between the standard anisotropic formulation and the line process formulation
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Extension to constrained line processes (4)
 Computational results:
Image
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
Gradient
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Extension to constrained line processes (5)
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 Adding spatial constraints...
EC ls , p   Chyst  1  ls , p  1  lu ,v   Csupp  1  ls , p  1  l p ,w 
… we obtain the following iterative formula:
I
l
t 1
s
t 1
s, p

I 
s
t
s


 l
p
s, p
s
1

1  Chyst  1  l
Hysteresis term
 I s , p
t
u ,v
 C
supp

 1 l
t
p,w
 2
Non-maximum suppression term
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
1
2
 I s , p
2
More info: spatial constraints clique
Extension to constrained line processes (6)
 Results: the diffusion is inhibited by the spatial
constraints
Sharper details and smoother contours
C.L.P.
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Anisotropic diffusion vs. constrained line processes (7)
 Comparative MSE and variance:
A.D.
C.L.P.
MSE
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
C.L.P.
A.D.
Variance
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Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
3D microscopy image processing (1)
 Goal: obtain correlation statistics in multi-channel 3D
confocal microscopy images
CH.1
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
CH.2
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3D microscopy image processing (2)
 Step 1: de-noising (using anisotropic diffusion)
Smooth image
CH.1
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
CH.2
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3D microscopy image processing (3)
 Step 2: thresholding
Proteins mask
CH.1
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
CH.2
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3D microscopy image processing (4)
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 Step 3: skeleton and labeling
Disjointed protein labeled skeleton
CH.1
CH.2
More info: disjointed clusters
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
3D microscopy image processing (5)
 Step 4: geodesic growth
Disjointed protein labeled mask
CH.1
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
CH.2
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3D microscopy image processing (6)
 Step 5: compute distance table
Distance between proteins
CH.1  CH.2
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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3D microscopy image processing (7)
 Step 6: clustering
Group proteins according to the separating distance
D = 5 x ‘mean size’
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
D = 7.5 x ‘mean size’
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3D microscopy image processing (8)
 Step 7: compute the statistics
Proper correlation statistics and interpretation
To do!
(IBCM’s biologists task)
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Conclusions
 Anisotropic diffusion is a powerful tool for de-noising
 Spatial constraints (added through the line process
formulation) allow to obtain better quality denoising
 Application of the anisotropic diffusion along with
other morphological and clustering tools allowed
efficient segmentation and classification of proteins
appearing in 3D confocal microscopy images.
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne
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Thank you for your attention !
Signal Processing Laboratory
Swiss Federal Institute of Technology, Lausanne