60520 Presentation Image Filters

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60-520 PresentationImage Filters
School of Computer Science University 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)

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Spatial Filters

• 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

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Spatial Filters
Example 33 neighborhood about a point (x,y) in
an image
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Spatial Filters linear filters

• For linear spatial filtering, the result, R, at a
  point (x,y) is
• Rw(-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,y1) w(1,1)f(x1,y1)

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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 MN
• The filter mask is of size mn
• m2a1, n2b1

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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

Gaussian noise
Original
33 mean filter
55 mean filter
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Spatial Filters smoothing, linear

• Mean filters
• example

Salt and pepper
33 mean filter
55 mean filter
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Spatial Filters smoothing, linear

• Weighted average filters
• example
• general expression

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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

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

V.S.
Gaussian noise removed By 33 median filter
Gaussian noise removed by 33 mean filter
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Spatial Filters smoothing, nonlinear

• Median filters
• are particularly effective in salt pepper

V.S.
Salt pepper removed By 33 median filter
Salt pepper removed by 33 mean filter
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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

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Spatial Filters sharpening

• A sample
• (a) a scan line (b) image strip
• (c) first derivative (d) second derivative

(a) (b) (c) (d)
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Spatial Filters sharpening

• The Laplacian
• second derivative of a two dimensional function
  f(x,y)
• f(x1,y)f(x-1,y)f(x,y1)f(x,y-1)
  
• -4f(x,y)

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Spatial Filters sharpening

• The Laplacian
• use a convolution mask to approximate

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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

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Frequency Filters Fourier transform

• example

FT (log)
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Frequency Filters

• Basic steps for filtering in the frequency domain

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Frequency Filters

• 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

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Frequency Filters

• Lowpass filters
• attenuate high frequencies while passing low
  frequencies
• Highpass filters
• attenuate low frequencies while passing high
  frequencies

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Frequency Filters lowpass filters

• Ideal lowpass filters (ILPF)

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Frequency Filters lowpass filters

• Butterworth lowpass filters (BLPF)

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Frequency Filters lowpass filters

• Gaussian lowpass filters (GLPF)

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Frequency Filters highpass filters

• Highpass filters
• Ideal higpass filters (IHPF)
• Butterworth highpass filters (BHPF)
• Gaussian highpass filters (GHPF)

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Frequency Filters bandpass filters

• Bandpass filters
• attenuate very low frequencies and very high
  frequencies
• enhance edges while reducing the noise at the
  same time

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Frequency Filters

• Examples (lowpass filters)

ILPF with cut-off frequency of 1/3
ILPF with cut-off frequency of 1/2
BLPF with cut-off frequency of 1/3
BLPF with cut-off frequency of 1/2
Gaussian noise
Original
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Frequency Filters

• Examples (highpass filters)

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Frequency Filters

• Relationship and comparison with spatial filters
• spatial filtering
• frequency filtering

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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 ?