filtering
DESCRIPTION
Basis beeldverwerking (8D040) d r. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir . Marcel Breeuwer. Filtering. Contents. Sharpening Spatial Filters 1 st order derivatives 2 nd order derivatives Laplacian Gaussian derivatives - PowerPoint PPT PresentationTRANSCRIPT
Basis beeldverwerking (8D040)
dr. Andrea FusterProf.dr. Bart ter Haar Romenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer
Filtering
Contents
• Sharpening Spatial Filters• 1st order derivatives• 2nd order derivatives• Laplacian • Gaussian derivatives• Laplacian of Gaussian (LoG)• Unsharp masking
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Sharpening spatial filters
• Image derivatives (1st and 2nd order)• Define derivatives in terms of differences for the
discrete domain• How to define such differences?
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1st order derivatives
• Some requirements (1st order):• Zero in areas of constant intensity• Nonzero at beginning of intensity step or ramp• Nonzero along ramps
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1st order derivatives
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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
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2nd order derivatives
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Image Derivatives
• 1st order
• 2nd order
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1-1
1 1-2
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1st order2nd order
Zero crossing, locating edges
• 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
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2nd order
Zero crossing, locating edges
1st order
Filters related to first derivatives
• Recall: Prewitt filter, Sober filter (lecture 2 – 01/05)
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Laplacian – second derivative
• Enhances edges• Definition
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Laplacian
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Adding diagonal derivationOpposite sign for second order derivative
Laplacian
• Note: Laplacian filtering results in + and – pixel values
• Scale for image display • So: take absolute value or positive values
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Line Detector
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*
Laplacian Positive values Laplacian(figure
10.5 book)
Image sharpening - example
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4-connected Laplacian8-connected LaplacianEnhanced + Laplacian x5Enhanced + Laplacian x6Enhanced + Laplacian x8Better sharpening with 8-connected Laplacian(see figure 3.38 (d)-(e) book)
C=+1 or -1
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)
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Laplacian in the Fourier domain
• Spatial
• Fourier domain
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Blur first, take derivative later
• Smoothing is a good idea to avoid enhancement of noise. Common smoothing kernel is a Gaussian.
2 2
22x y
G e
Scale of blurring
Gaussian Derivative
• Taking the derivative after blurring gives image g
*( )g D G f
2 2
22x y
G e
Gaussian Derivative
• We can build a single kernel for both convolutions
( * )g D G f 2 2
222*
x y
xxD G e
2 2
222*
x y
yyD G e
Use the associative property of the convolution
Laplacian of Gaussian (LoG)
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LoG a.k.a. Mexican Hat
LoG applied to building
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Sharpening with LoG
25sharpening with LoG sharpening
with Laplacian
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
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• Let denote the blurred image• Obtain the mask
• Add weighted portion of mask to original image
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• If• Unsharp masking
• If• Highboost filtering
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input blurred unsharp mask u.m. result h.f. result
(see also figure 3.40 book)
Unsharp masking
• Simple and often used sharpening method• Poor result in the presence of noise – LoG performs
better in this case
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