medical image analysisgerig/cs7960-s2010/materials/perona-malik/lecture5.pdf · logo medical image...
TRANSCRIPT
![Page 1: Medical Image Analysisgerig/CS7960-S2010/materials/Perona-Malik/lecture5.pdf · logo Medical Image Analysis CS 593 / 791 Computer Science and Electrical Engineering Dept. West Virginia](https://reader030.vdocuments.us/reader030/viewer/2022040900/5e6f7792eba933686a288ecd/html5/thumbnails/1.jpg)
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Medical Image Analysis
CS 593 / 791
Computer Science and Electrical Engineering Dept.West Virginia University
23rd January 2006
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental Results
4 The rest of the paper...
5 Anisotropic Diffusion
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental Results
4 The rest of the paper...
5 Anisotropic Diffusion
![Page 4: Medical Image Analysisgerig/CS7960-S2010/materials/Perona-Malik/lecture5.pdf · logo Medical Image Analysis CS 593 / 791 Computer Science and Electrical Engineering Dept. West Virginia](https://reader030.vdocuments.us/reader030/viewer/2022040900/5e6f7792eba933686a288ecd/html5/thumbnails/4.jpg)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Edge Enhancement
Inhomogeneous diffusion may actually enhance edges, for acertain choice of g().
1D example:
Let s(x) = ∂I∂x , and φ(s) = g(s)s = g(Ix)Ix .
The 1D heat equation becomes
It =∂
∂x(g(Ix)Ix) =
∂
∂xφ(s)
= φ′(s)Ixx
∂∂t (Ix) is the rate of change of edge slope with respect to time.
∂
∂t(Ix) =
∂
∂x(It)
= φ′′(s)I2xx + φ′(s)Ixxx
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Edge Enhancement
I, Ix , Ixx , Ixxx
∂
∂t(Ix) = φ′′(s)I2
xx + φ′(s)Ixxx
For a step edge with Ix > 0 look at theinflection point, p.Observe that Ixx(p) = 0, and Ixxx(p) < 0.
∂
∂t(Ix)(p) = φ′(s)Ixxx(p)
The sign of this quantity depends only onφ′(s).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Edge Enhancement
I, Ix , Ixx , Ixxx
∂
∂t(Ix) = φ′′(s)I2
xx + φ′(s)Ixxx
For a step edge with Ix > 0 look at theinflection point, p.Observe that Ixx(p) = 0, and Ixxx(p) < 0.
∂
∂t(Ix)(p) = φ′(s)Ixxx(p)
The sign of this quantity depends only onφ′(s).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Edge Enhancement
At the inflection point:
∂
∂t(Ix)(p) = φ′(s)Ixxx(p)
If φ′(s) > 0, then ∂∂t (Ix)(p) < 0 (slope is decreasing).
If φ′(s) < 0, then ∂∂t (Ix)(p) > 0 (slope is increasing).
Since φ(s) = g(s)s, selecting the function g(s) determineswhich edges are smoothed and which are sharpened.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental Results
4 The rest of the paper...
5 Anisotropic Diffusion
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
The function φ(s) = g(s)s
φ(0) = 0
φ′(s) > 0 for s < K
φ′(s) < 0 for s > K
lims→∞ φ(s) → 0
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
The function g(s)
Perona and Malik suggest two possible functions (s = ||∇I||):
g(|∇I|) = e−( ||∇I||K )2
g(|∇I|) =1
1 + ( ||∇I||K )1+α
(α > 0)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Effect of varying K on g()
g(|∇I|) =1
1 + ( ||∇I||K )1+α
(α > 0)
Figure: K = 2, 4, 6
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Effect of varying α on g()
g(|∇I|) =1
1 + ( ||∇I||K )1+α
(α > 0)
Figure: α = 1, 3, 5, 7, 9
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Effect of varying K and α on c()
Figure: I and ||∇I||.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Effect of varying K on c()
Figure: K = 3, 5, 10, 100.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Effect of varying α on c()
Figure: α = 1, 2, 3, 5.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental ResultsDiscretized Inhomogeneous Heat EquationPerona-Malik ImplementationDiscrete Maximum PrincipleAdaptive Parameter Setting
4 The rest of the paper...
5 Anisotropic Diffusion
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discretized Inhomogeneous Heat Equation
Explicit Formulation
∂I∂t
= c(x , y , t)∇2I +∇c · ∇I
Using centered differences for the Laplacian and gradients:
I t+1x ,y − I t
x ,y
λ= cx ,y (I t
x−1,y + I tx+1,y + I t
x ,y−1 + I tx ,y+1 − 4I t
x ,y )
+ (cx+1,y − cx−1,y
2)(
Ix+1,y − Ix−1,y
2)
+ (cx ,y+1 − cx ,y−1
2)(
Ix ,y+1 − Ix ,y−1
2)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discretized Inhomogeneous Heat Equation
Implicit Formulation
Same diagonal structure as homogeneous heat equation?Yes.
Symmetric? No.
Diagonal dominance? Data dependent.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discretized Inhomogeneous Heat Equation
Implicit Formulation
Same diagonal structure as homogeneous heat equation?Yes.
Symmetric? No.
Diagonal dominance? Data dependent.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discretized Inhomogeneous Heat Equation
Implicit Formulation
Same diagonal structure as homogeneous heat equation?Yes.
Symmetric? No.
Diagonal dominance? Data dependent.
![Page 21: Medical Image Analysisgerig/CS7960-S2010/materials/Perona-Malik/lecture5.pdf · logo Medical Image Analysis CS 593 / 791 Computer Science and Electrical Engineering Dept. West Virginia](https://reader030.vdocuments.us/reader030/viewer/2022040900/5e6f7792eba933686a288ecd/html5/thumbnails/21.jpg)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discretized Inhomogeneous Heat Equation
Implicit Formulation
Same diagonal structure as homogeneous heat equation?Yes.
Symmetric? No.
Diagonal dominance? Data dependent.
![Page 22: Medical Image Analysisgerig/CS7960-S2010/materials/Perona-Malik/lecture5.pdf · logo Medical Image Analysis CS 593 / 791 Computer Science and Electrical Engineering Dept. West Virginia](https://reader030.vdocuments.us/reader030/viewer/2022040900/5e6f7792eba933686a288ecd/html5/thumbnails/22.jpg)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
∂I∂t
= c(x , y , t)∇2I +∇c · ∇I
By splitting the Laplacian and averaging the forward and backwarddifferences in the gradient:
I t+1x ,y − I t
x ,y
λ= cx ,y [(I t
x−1,y − I tx ,y ) + (I t
x+1,y − I tx ,y )
+ (I tx ,y−1 − I t
x ,y ) + (I tx ,y+1 − I t
x ,y )]
+∂c∂x
(Ix+1,y − Ix ,y
2+
Ix ,y − Ix−1,y
2)
+∂c∂y
(Ix ,y+1 − Ix ,y
2+
Ix ,y − Ix ,y−1
2)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
∂I∂t
= c(x , y , t)∇2I +∇c · ∇I
I t+1x ,y − I t
x ,y
λ= (cx ,y −
12
∂c∂x
)(I tx−1,y − I t
x ,y )
+ (cx ,y +12
∂c∂x
)(I tx+1,y − I t
x ,y )
+ (cx ,y −12
∂c∂x
)(I tx ,y−1 − I t
x ,y )
+ (cx ,y +12
∂c∂x
)(I tx ,y+1 − I t
x ,y )
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
cx ,y +12
∂c∂x
≈ cx+ 12 ,y
cx ,y −12
∂c∂x
≈ cx− 12 ,y
cx+ 12 ,y ≈ g(
sx ,y + sx+1,y
2)
cx− 12 ,y ≈ g(
sx ,y + sx−1,y
2)
Where sx ,y = ||∇I(x , y)||.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
I t+1x ,y − I t
x ,y
λ= g(
sx ,y + sx−1,y
2)(I t
x−1,y − I tx ,y )
+ g(sx ,y + sx+1,y
2)(I t
x+1,y − I tx ,y )
+ g(sx ,y + sx ,y−1
2)(I t
x ,y−1 − I tx ,y )
+ g(sx ,y + sx ,y+1
2)(I t
x ,y+1 − I tx ,y )
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Anisotropic Implementation
Compute g() using the projection of the gradient along onedirection.For example, in g(
sx,y+sx+1,y2 ), let
sx ,y = | ∂I∂x
(x , y)|
sx+1,y = | ∂I∂x
(x + 1, y)|
Computing sx ,y using forward differences, and sx+1,y usingbackward differences
sx ,y = |Ix+1,y − Ix ,y |sx+1,y = |Ix+1,y − Ix ,y |,
so g(sx,y+sx+1,y
2 ) = g(|I(x + 1, y)− I(x , y)|).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
I t+1x ,y − I t
x ,y
λ= g(|Ix−1,y − Ix ,y |)(I t
x−1,y − I tx ,y )
+ g(|Ix+1,y − Ix ,y |)(I tx+1,y − I t
x ,y )
+ g(|Ix ,y−1 − Ix ,y |)(I tx ,y−1 − I t
x ,y )
+ g(|Ix ,y+1 − Ix ,y |)(I tx ,y+1 − I t
x ,y )
Notation:The authors use 5 to denote finite differences. This is not thegradient operator (∇).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Image neighborhoodsystem
5N Ii,j ≡ Ii−1,j − Ii,j5SIi,j ≡ Ii+1,j − Ii,j5E Ii,j ≡ Ii,j+1 − Ii,j5W Ii,j ≡ Ii,j−1 − Ii,j
cNi,j= g(| 5N Ii,j |)
cSi,j= g(| 5S Ii,j |)
cEi,j= g(| 5E Ii,j |)
cWi,j= g(| 5W Ii,j |)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Perona-Malik Implementation
Explicit Formulation
The previous explicit formulation
I t+1x ,y − I t
x ,y
λ= g(|Ix−1,y − Ix ,y |)(I t
x−1,y − I tx ,y )
+ g(|Ix+1,y − Ix ,y |)(I tx+1,y − I t
x ,y )
+ g(|Ix ,y−1 − Ix ,y |)(I tx ,y−1 − I t
x ,y )
+ g(|Ix ,y+1 − Ix ,y |)(I tx ,y+1 − I t
x ,y )
can be rewritten as
I t+1x ,y = I t
x ,y + λ(cNi,j5N Ii,j + cSi,j
5S Ii,j
+ cEi,j5E Ii,j + cWi,j
5W Ii,j)t
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discrete Maximum Principle
Let IMij= max(I, IN , IS, IE , IW )
I t+1x ,y = I t
x ,y + λ(cNi,j5N Ii,j + cSi,j
5S Ii,j+ cEi,j
5E Ii,j + cWi,j5W Ii,j)
I t+1x ,y = I t
x ,y (1− λ(cNi,j+ cSi,j
+ cEi,j+ cWi,j
))
+ λ(cNi,jINi,j
+ cSi,jISi,j
+ cEi,jIEi,j
+ cWi,jIWi,j
)
Since all c’s and λ are positive and between 0,1:
I t+1x ,y ≤ I t
Mij(1− λ(cNi,j
+ cSi,j+ cEi,j
+ cWi,j))
+ I tMij
(λ(cNi,j+ cSi,j
+ cEi,j+ cWi,j
))
= I tMij
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Discrete Maximum Principle
Let IMij= max(I, IN , IS, IE , IW )
I t+1x ,y = I t
x ,y + λ(cNi,j5N Ii,j + cSi,j
5S Ii,j+ cEi,j
5E Ii,j + cWi,j5W Ii,j)
I t+1x ,y = I t
x ,y (1− λ(cNi,j+ cSi,j
+ cEi,j+ cWi,j
))
+ λ(cNi,jINi,j
+ cSi,jISi,j
+ cEi,jIEi,j
+ cWi,jIWi,j
)
Since all c’s and λ are positive and between 0,1:
I t+1x ,y ≤ I t
Mij(1− λ(cNi,j
+ cSi,j+ cEi,j
+ cWi,j))
+ I tMij
(λ(cNi,j+ cSi,j
+ cEi,j+ cWi,j
))
= I tMij
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Adaptive Parameter Setting
Set K every iteration
Compute a histogram, fi , of ||∇I||
Find K such that 90% of the pixels have gradient magnitude < K .(If
∑bi=1 fi ≥ 0.9n2 then bin b corresponds to gradient magnitude K ).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental Results
4 The rest of the paper...
5 Anisotropic Diffusion
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Edge Detection
Due to
Edge enhancement
Edge localization
edge detection algorithms will benefit from using this nonlineardiffusion process rather than using linear diffusion (Gaussianconvolution).
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Energy Minimization and Markov Random FieldModels
The ’anisotropic’ diffusion algorithm minimizes ’someenergy function.’
Smoothness → conditional dependence on nearestneighbors (Markovian property)
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Outline
1 Recap
2 Edge Enhancement
3 Experimental Results
4 The rest of the paper...
5 Anisotropic DiffusionIntroduction
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Introduction
The problem with inhomogeneous diffusion
Noise on edges will not be reduced.
Even with the anisotropic implementation of Perona-Malik,noise on diagonal edges will not be handled properly.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Introduction
The problem with inhomogeneous diffusion
The inhomogeneous heat equation
∂I∂t
= div(c(x , y , t)∇I)
only controls the magnitude of intensity (or heat) flow.The anisotropic heat equation
∂I∂t
= div(D(x , y , t)∇I),
where D(x , y , t) is a matrix-valued function, can also control thedirection of intensity (or heat) flow.
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Recap Edge Enhancement Experimental Results The rest of the paper... Anisotropic Diffusion
Introduction
Next:
Wednesday:
Susan Lemieux will continue the MRI acquisition lecture.
Friday:
Weickert ”A review of nonlinear diffusion filtering”.
The physical laws of heat flow and diffusion.
Anisotropic diffusion filtering.