60-520 Presentation

Image Filters

Student: Xiaoliu ChenInstructor: Dr. I. Ahmad

School of Computer ScienceUniversity of Windsor

November 2003

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Outline

Introduction Spatial Filtering

– Smoothing– Sharpening

Frequency-Domain Filtering– Low pass– High pass

Summary

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Introduction

Filtering is the process of replacing a pixel with a value based on some operations or functions.

The operations/functions used on the original image are called filters.

– or masks, kernels, templates, windows…

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Introduction

In digital image processing, filters are usually used to

– suppress the high frequencies in an image• i.e., smoothing the image

– suppress the low frequencies in an image• i.e., enhancing or detecting edges in the image

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Introduction

Image filters fall into two categories:

– Spatial domain• Filters are based on direct manipulation of

pixels on an image plane.

– Frequency domain• Filters are based on modifying the Fourier

transform (FT) of an image.

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Spatial Filters

The general processes can be denoted by the expression:

– f(x,y) is the input image– g(x,y) is the processed image– T is an operator on f, defined over some

neighborhood of (x,y)

)],([),( yxfTyxg

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The principal approach in defining a neighborhood about a point (x,y)

– use a subimage area centered at (x,y)

– shapes of the neighborhood• circle• square• rectangular

Spatial Filters

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Example: 3×3 neighborhood about a point (x,y) in an image

x

(x,y)

Image f(x,y)(x+1,y+1)(x,y+1)(x-1,y+1)

(x+1,y)(x,y)(x-1,y)

(x+1,y-1)(x,y-1)(x-1,y-1)

y

Spatial Filters

Image Filters 9Pixels under mask

Image f(x,y)

f(x+1,y+1)f(x,y+1)f(x-1,y+1)

f(x+1,y)f(x,y)f(x-1,y)

f(x+1,y-1)f(x,y-1)f(x-1,y-1)

w(1,1)w(0,1)w(-1,1)

w(1,0)w(0,0)w(-1,0)

w(1,-1)w(0,-1)w(-1,-1)Mask

Mask coefficients

x

y

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Spatial Filters – linear filters

For linear spatial filtering, the result, R, at a point (x,y) is

R=w(-1,-1)f(x-1,y-1) + w(0,-1)f(x,y-1) +

…+ w(0,0)f(x,y) +…

+ w(0,1)f(x,y+1) + w(1,1)f(x+1,y+1)

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Spatial Filters – convolution

In general, linear filtering of an image is given by the expression:

– The image f is of size M×N

– The filter mask is of size m×n

m=2a+1, n=2b+1

a

as

b

bt

tysxftswyxg ),(),(),(

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Spatial Filters – smoothing

Smoothing filters are used for blurring and for noise reduction.

Smoothing, linear spatial filter

– average filters– reduce “sharp” transitions– side effect

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Spatial Filters – smoothing, linear

Mean filters– example: 111

111

111

9

1

OriginalGaussian noise 3×3 mean filter 5×5 mean filter

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Spatial Filters – smoothing, linear

Mean filters– example: 111

111

111

9

1

Salt and pepper 3×3 mean filter 5×5 mean filter

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Spatial Filters – smoothing, linear

Weighted average filters– example:

– general expression: 121

242

121

16

1

a

as

b

bt

a

as

b

bt

tsw

tysxftswyxg

),(

),(),(),(

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Spatial Filters – smoothing, nonlinear

Order-statistic filters

– nonlinear spatial filters

– order/rank the pixels contained in the image area encompassed by the filter

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Spatial Filters – smoothing, nonlinear

Median filters– replace a pixel value with the median of its

neighboring pixel values– example:

23 25 26 30 40

22 24 26 27 35

18 20 50 25 34

19 15 19 23 33

11 16 10 20 30

Neighborhood values:15, 19, 20, 23,

24, 25, 26, 27, 50

Median value: 24

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Spatial Filters – smoothing, nonlinear

Median filters– have excellent noise-reduction capabilities

Gaussian noise removedby 3×3 mean filter

Gaussian noise removed By 3×3 median filter

V.S.

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Spatial Filters – smoothing, nonlinear

Median filters– are particularly effective in salt & pepper

Salt & pepper removedby 3×3 mean filter

Salt & pepper removed By 3×3 median filter

V.S.

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Spatial Filters – smoothing, nonlinear

Max filters– maximum of neighboring pixel values– useful for finding the brightest points in an

image

Min filters– minimum of neighboring pixel values– useful for finding the darkest points in an

image

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Spatial Filters – sharpening

Principal objective– highlight fine detail in an image– enhance detail that has been blurred

Sharpening can be accomplished by spatial differentiation

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Spatial Filters – sharpening

For one dimensional function f(x)– first order derivative

– second order derivative

)()1( xfxfx

f

)(2)1()1(2

2

xfxfxfx

f

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Spatial Filters – sharpening

– A sample

(a) a scan line (b) image strip

(c) first derivative (d) second derivative

7777000013100006000123455

0007000-1-221000-6600-1-1-1-1-1

00-770011-411006-126010000-1

(a)

(b)

(c)

(d)

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Spatial Filters – sharpening

The Laplacian– second derivative of a two dimensional

function f(x,y)

= [f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)]

-4f(x,y)

2

2

2

22

y

f

x

ff

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Spatial Filters – sharpening

The Laplacian– use a convolution mask to approximate

010

1-41

010

111

1-81

111

-12-1

2-42

-12-1

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Spatial Filters – sharpening

The Laplacian– example:

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Spatial Filters – sharpening

The Laplacian– example:

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Frequency Filters – Fourier transform

Fourier transform (FT)– decompose an image into its sine and

cosine components– transform real space images into Fourier or

frequency space images– In a frequency space image, each point

represents a particular frequency contained in the real domain image.

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Frequency Filters – Fourier transform

Discrete Fourier transform (DFT)

Inverse DFT

1

0

1

0

)//(2),(1

),(M

x

N

y

NvyMuxjeyxfMN

vuF

1

0

1

0

)//(2),(),(M

u

N

v

NvyMuxjevuFyxf

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Frequency Filters – Fourier transform

– example:

FT

(log)

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Basic steps for filtering in the frequency domain

Frequency Filters

Filterfunction

FilterfunctionDFTDFT Inverse

DFT

InverseDFT

f(x,y)Input image

g(x,y)Processed image

F(u,v) H(u,v)F(u,v)

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Frequencies in an image correspond to the rate of change in pixel values

– High frequencies• rapid changes of gray level values

– Low frequencies• slow changes of gray level values

Frequency Filters

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Lowpass filters– attenuate high frequencies while “passing”

low frequencies

Highpass filters– attenuate low frequencies while “passing”

high frequencies

Frequency Filters

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Ideal lowpass filters (ILPF)

Frequency Filters – lowpass filters

0

0

),( if0

),( if1),(

DvuD

DvuDvuH

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Butterworth lowpass filters (BLPF)

Frequency Filters – lowpass filters

nDvuDvuH

20 ]/),([1

1),(

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Gaussian lowpass filters (GLPF)

Frequency Filters – lowpass filters

22 2/),(),( vuDevuH

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Highpass filters

– Ideal higpass filters (IHPF)

– Butterworth highpass filters (BHPF)

– Gaussian highpass filters (GHPF)

Frequency Filters – highpass filters

),(1),( vuHvuH lphp

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Bandpass filters– attenuate very low frequencies and very

high frequencies

–

– enhance edges while reducing the noise at the same time

Frequency Filters – bandpass filters

),(),( vuHvuHH lphpbp

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Frequency Filters

Original Gaussian noise ILPF withcut-off frequency of 1/3

ILPF withcut-off frequency of 1/2

BLPF withcut-off frequency of 1/3

BLPF withcut-off frequency of 1/2

Examples: (lowpass filters)

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Examples: (highpass filters)

Frequency Filters

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Frequency Filters

Relationship and comparison with spatial filters– spatial filtering

– frequency filtering

–

),(),(),( yxfyxhyxg

),(),( vuHyxh

),(),(),( vuFvuHvuG

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Frequency Filters

Comparison with spatial filters

– more computational efficient– more intuitive

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Summary

Filtering is the operation of applying a transform on an image in order to enhance it.

Filtering techniques can be subdivided into two types– Spatial domain filtering– Frequency domain filtering

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Summary

Filtering techniques are very useful in image analysis and processing– Noise removal– Edge detection

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The end

Thank you

&

Questions ?