non-linear filtering

Non-linear filtering

• Example: Median filter

• Replaces pixel value by median value over neighborhood

• Generates no new gray levels

I=(1 2 3 2 3 2 1)2 22 3 2

Median filters

Advantage (?): the “odd-man-out” effect

e.g.1,1,1,7,1,1,1,1

?,1,1,1.1,1,1,?

Median filters: example

filter width = 5

Linear (averaging)

Input

Median


filter width = 5

Linear (averaging)

Input

MedianDiscards

spike

Preservesjump exactly

at correctlocation


filter width = 5

Linear (averaging)

Input

Median

Smooths spike

Blursjump

Median filter

3 x 3 median filter

Sharpens edges Destroys edge cusps and protrusions

Original

Keeps edges sharp Odd Man Out can destroy details

Median filter vs Gaussian

Gaussian

Eye has more detail.Blurrier edges

3 x 3 median filter

Median filter: summary

• Purpose– Smooth image without blurring boundaries

Input Median Filter

Median filter: summary

• Purpose– Smooth image without blurring boundaries

• Problem– Loses small details

Input Median Filter

Another approach: bilateral filter

Input Gaussian Smoothed



[

OK to average these brightness values



[

Don’t average these(brightness too different,

probably from different objects)



• Bilateral smoothing: basic idea

– Ordinary smoothing…– But only sum over pixels with similar brightness





Sum over all 3 pixels





Sum over all 3 pixels (similar brightness)





Don’t sum over 3rd pixel (too dark)





Don’t sum over 3rd pixel (too bright)





Sum over all 3 pixels (similar brightness))


Input Bilateral Smoothing







Smoothed





Not smoothed



• Bilateral smoothing: implementation

– Ordinary smoothing…– But only sum over pixels with similar brightness– Pixels with similar brightness weighted more

Bilateral filter• Weight pixel contributions by nearness in distance and

nearness in intensity

– Old: Gaussian smoothing

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– New: bilateral smoothing

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Bilateral filter

• Weight pixel contributions by nearness in distance and nearness in intensity


– New: bilateral smoothing

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Suppresses contributionfrom pixels with different brightness

Bilateral filter

Pattern detection with filters

• Biggest response to brightness patterns resembling them!

• Example: 1D correlation with mask

W I 1 0 1( )W w w w



• Example: 1D correlation with mask Compute at position 3

W I 1 0 1( )W w w w

3W I




I1 I2 I3 I4 I5 I6Image (1D)

mask weights 1 0 1w w w

W I 1 0 1( )W w w w

3W I

…





mask weights 1 0 1w w w

1 2 0 3 1 4w I w I w I 3W I

W I 1 0 1( )W w w w

3W I

…





mask weights

Vector i

Vector w

2 3 4( )I I I

1 0 1( )w w w1 0 1w w w

3W I

W I 1 0 1( )W w w w

3W I

…

1 2 0 3 1 4w I w I w I





mask weights

Vector i

Vector w

dot product

2 3 4( )I I I

1 0 1( )w w w1 0 1w w w

1 2 0 3 1 4w I w I w I w i 3W I

W I 1 0 1( )W w w w

3W I

…


• Gaussian filter– Detects blobs

• “Derivative” filter e.g., [-1/2 0 1/2]– Detects brightness jumps

• eye detector?


• Summary: detect pattern (object) by correlating with similar mask

• Problems when lighting changes – Overall brightness can change between images

– Contrast may change

– Or both…

new oldI I c

new oldI k I

new oldI k I c

To improve detection: Compensate for changes in brightness and contrast

• Want to detect pattern Old

– Correlate image with

– Pattern “detected” wherever is large

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P ' '''

ij kl k li k kj l l

P I P I

| |P I


• Want to detect pattern Old

– Correlate image with

– Pattern “detected” wherever is large

New: at each pixel (i,j)– Subtract average brightness (to eliminate problems from additive shifts)

Do weighted sum

P

P

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1'kl kl kl k lk l

P P P PN

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1'kl kl kl k lk l

I I I IN

' '''

ij kl k li k kj l l

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| |P I

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C P I


New: at each pixel (i,j)

– Subtract average brightness

Do weighted sum

– Normalize (avoids problems from contrast changes)

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P P P PN

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Normalized Correlation

,

1/ 2 1/ 22 2

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( , ) , ,,

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i jNC

i j i j

P i j P I x i y j I x yI x y

P i j P I x i y j I x y

Important: for each (x,y) (i.e., each pixel in result )

you must compute the average for window centered over (x,y) ,I x y

NCI

Normalized Correlation

• MATLAB demo (eye detection)

Detection with Normalized Correlation

• ProblemResult of normalized correlation

Suppose you want 3 best candidates for an eye…


• ProblemResult of normalized correlation

Suppose you want 3 best candidates for an eye…

The three highest NCvalues are all here(at best eye position plus2 nearby pixels)


• SolutionResult of normalized correlation

Only consider pixels with higher NC value than at all other “nearby” pixels



Only consider pixels with higher NC value than at all other “nearby” pixels (Non maximum suppression)

(Non Ma



Result of NC and NonMax Suppression

non-linear filtering

Documents

similar brightness sum

similar brightness dont

pixelsanother approach

darkanother approach

brightanother approach

similar brightnesspixels

median value

median filtersadvantage