BAE 790I BMME 231 Fundamentals of Image Processing Class 11

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BAE 790I / BMME 231Fundamentals of Image
ProcessingClass 11

• Fourier Domain Filtering
• Low-pass filters Ideal, Hamming, Hanning,
  Butterworth, Gaussian
• Sharpening Filters Derivative operators,
  Laplacians, Unsharp masking

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Fourier-domain Filters

• Filters can be designed in the Fourier domain as
  well as the spatial domain

FFT
FFT-1
H(wx,wy)
Objective Determine a suitable form of this
function
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Ideal Low-pass Filter

• Consider the ideal low-pass filter in 1D

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Ideal Low-pass Filter
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Ideal low-pass filter
SNR 21.4 dB

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Commonly-used Fourier Filters

• All are given as isotropic filters

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Commonly-used Fourier Filters

• Hanning filter

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Commonly-used Fourier Filters

• Butterworth filter

The order N determines rate of cutoff
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Commonly-used Fourier Filters

• Gaussian filter

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Filter Comparison
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Butterworth filterorder 8, cutoff .2 cycles/pix
SNR 21.8 dB

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Sharpening Filters

• Consider taking the directional first derivative
  of an image
• This is the same as applying a filter with a
  transfer function of jwx

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Gradient Operators

• Simple derivatives can be applied as convolution
  masks
• These take a local slope of intensity.
• Results are the x- and y-components of the
  intensity gradient.

0 0 0 -1 1 0 0 0 0
0 -1 0 0 1 0 0 0 0
x-gradient
y-gradient
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Gradient Operators

• Derivatives in other directions can be determined
  from x- and y- derivatives.
• Are these LSI operations?
• Results of gradient operators are sensitive to
  regions of high Dintensity/Dspace, which we
  perceive as what?

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Gradient Operators

0 0 0 -1 1 0 0 0 0
0 -1 0 0 1 0 0 0 0
X gradient
Y gradient
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Second Derivative Operators

• Consider second derivatives
• In terms of convolution masks,

0 0 0 -1 2 -1 0 0 0
0 -1 0 0 2 0 0 -1 0
1 -1 0 -1 1 0 0 0 0
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Second Derivative Operators

• The Laplacian
• Rotationally invariant
• Sensitive to regions of high curvature

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Laplacian Operator

0 0 0 0 0 0 0 -1 0 0 0
-1 4 -1 0 0 0 -1 0 0 0 0
0 0 0
Mask
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Unsharp Masking

• Old photographic sharpening technique
• To sharpen an image, take a blurred version and
  subtract it from the original.
• image low-pass(image)
• Equivalent to taking a high-pass filtered version
  and adding it.
• image high-pass(image)

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Unsharp Masking

0 0 0 0 0 0 0 -1 0 0 0
-1 5 -1 0 0 0 -1 0 0 0 0
0 0 0
Mask
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Profiles from Unsharp Masking

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Fourier-domain Root Filtering

a .6
a 1.4