Filtering

Basis beeldverwerking (8D040)

dr. Andrea Fusterdr. 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-like 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 – 14/05/13)

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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 - eqs. (2.6-10, 2.6-11)• Or: take absolute value or positive values

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Line Detector

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*

scaled 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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