2 nd order derivatives

Report
Basis beeldverwerking (8D040)
dr. Andrea Fuster
Prof.dr. Bart ter Haar Romeny
dr. Anna Vilanova
Prof.dr.ir. Marcel Breeuwer
Filtering
Contents
• Sharpening Spatial Filters
•
•
•
•
•
•
1st order derivatives
2nd order derivatives
Laplacian
Gaussian derivatives
Laplacian of Gaussian (LoG)
Unsharp masking
2
Sharpening spatial filters
• Image derivatives (1st and 2nd order)
• Define derivatives in terms of differences for the
discrete domain
• How to define such differences?
3
1st order derivatives
• Some requirements (1st order):
• Zero in areas of constant intensity
• Nonzero at beginning of intensity step or ramp
• Nonzero along ramps
4
1st order derivatives
5
2nd order derivatives
• Requirements (2nd order)
• Zero in constant areas
• Nonzero at beginning and end of intensity step or ramp
• Zero along ramps of constant slope
6
2nd order derivatives
7
Image Derivatives
• 1st order
-1
1
1
-2
• 2nd order
1
8
2nd order
1st order
Zero crossing, locating edges
9
• Edges are ramp transitions in intensity
• 1st order derivative gives thick edges
• 2nd order derivative gives double thin edge with zeros in
between
• 2nd order derivatives enhance fine detail much better
10
2nd order
1st order
Zero crossing, locating edges
11
Filters related to first derivatives
• Recall: Prewitt filter, Sober filter (lecture 2 – 01/05)
12
Laplacian – second derivative
• Enhances edges
• Definition
13
Laplacian
Opposite sign for second
order derivative
Adding diagonal derivation
14
Laplacian
• Note: Laplacian filtering results in + and – pixel
values
• Scale for image display
• So: take absolute value or positive values
15
Line Detector
*
(figure 10.5
book)
Laplacian
Positive values
Laplacian
16
Image sharpening - example
C=+1 or -1
x8
8-connected
4-connected
Enhanced
Enhanced ++ Laplacian
Laplacian x5
x6
Better sharpening with 8-connected Laplacian
(see figure 3.38 (d)-(e) book)
17
Filtering in frequency domain
• Basic steps:
−
−
−
−
−
−
image f(x,y)
Fourier transform F(u,v)
filter H(u,v)
H(u,v)F(u,v)
inverse Fourier transform
filtered image g(x,y)
18
Laplacian in the Fourier domain
• Spatial
• Fourier domain
19
Blur first, take derivative later
• Smoothing is a good idea to avoid enhancement of
noise. Common smoothing kernel is a Gaussian.
Ge

x2  y 2
2 2
Scale of blurring
Gaussian Derivative
• Taking the derivative after blurring gives image g
g  D *(G  f )
Ge

x2  y 2
2 2
Gaussian Derivative
• We can build a single kernel for both convolutions
Use g
the
associative
( D * G)  fproperty of the convolution
Dx * G  
Dy * G  
x

2
y

2

e

e
x2  y 2
2 2
x2  y 2
2 2
Laplacian of Gaussian (LoG)
LoG a.k.a. Mexican Hat
23
LoG applied to building
24
Sharpening with LoG
sharpening with LoG
sharpening
with Laplacian
25
Unsharp Masking / Highboost Filtering
• Subtraction of unsharp (smoothed) version of image
from the original image.
• Blur the original image
• Subtract the blurred image from the original
(results in image called mask)
• Add the mask to the original
26
• Let
denote the blurred image
• Obtain the mask
• Add weighted portion of mask to original image
27
• If
• Unsharp masking
• If
• Highboost filtering
(see also figure 3.40 book)
input
blurred
unsharp mask
u.m. result
h.f. result
28
Unsharp masking
• Simple and often used sharpening method
• Poor result in the presence of noise – LoG performs
better in this case
29

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