EE4328, Section 005 Introduction to Digital Image Processing Nonlinear Image Filtering Zhou Wang Dep

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EE4328, Section 005 Introduction to Digital
Image ProcessingNonlinear Image FilteringZhou
WangDept. of Electrical EngineeringThe Univ.
of Texas at ArlingtonFall 2006
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Previous Lectures

• Spatial Domain Linear Filters
• Smoothing Averaging, Gaussian
• Sharpening
• Frequency (2D-DFT) Domain Filters
• Lowpass, highpass, bandpass
• Orientation selective
• Orientation radial selective
• ..
• Linear Image Restoration Filters
• Inverse, pseudo-inverse, radially-limited inverse
• Wiener, Wiener denoising

All Linear !
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Nonlinear Filtering

• Motivation Limitation of Linear Filters
• Frequency shaping
• enhance some frequency components and suppress
  the others
• For individual frequency component, cannot
  differentiate its desirable and undesirable
  parts
• Nonlinear Filters
• Cannot be expressed as convolution
• Cannot be expressed as frequency shaping
• Nonlinear Means Everything (other than linear)
• Need to be more specific
• Often heuristic
• We will study some nice ones

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Impulsive (Salt Pepper) Noise

• Definition
• Each pixel in an image has a probability pa or pb
  of being contaminated by a white dot (salt) or a
  black dot (pepper)

X noise-free image, Y noisy image
with probability pa
noisy pixels
with probability pb
clean pixels
with probability 1 - pa - pb
add salt pepper noise
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Median Filters

• Order Statistics (OS)
• Given a set of numbers
• Denote the OS as
• such that
• Median
• Define
• Applying Median Filters
• to Images
• Use sliding windows
• (similar to spatial linear filters)
• Typical windows
• 3x3, 5x5, 7x7, other shapes

max value
min value
middle value
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Median Filters
original
noisy (pa pb 0.1)
median filtered 3x3 window
median filtered 5x5 window
From MATLAB sample images
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Iterative Median Filters

• Idea repeatedly apply median filters

1 time
2 times
3 times
From Gonzalez Woods
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Switching Median Filters

• Motivation
• Regular median filters change both bad and
  good pixels
• Idea
• Detect/classify bad and good pixels
• Filter bad pixels only

From Wang Zhang
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Switching Median Filters
original
noisy (pa pb 0.1)
regular 5x5 median filtered
switching 5x5 median filtered
From MATLAB sample images
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Order Statistics (OS) Filters

• Recall Order Statistics
• For
• OS
• such that
• OS filter General Form
• Special Cases

where
(M1)-th
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Order Statistics (OS) Filters

• Note An OS Filter is Uniquely Defined by wi
• Example 1
• Example 2

(M1)-th
M-th
(M2)-th
then
then
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Examples

• A 4x4 grayscale image is given by

impulse?
impulse?

• Filter the image with a 3x3 median filter, after
  zero-padding at the image borders

median filtering
zero-padding
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Examples

• Filter the image with a 3x3 median filter, after
  replicate-padding at the image borders

median filtering
replicate -padding
impulse cleaned!
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Examples

• Filter the image with a 3x3 OS filter, after
  replicate-padding at the image borders. The
  weighting factors of the OS filter are given by

wi i 1, , 9 0, 0, 0, ¼, ½, ¼, 0, 0, 0
OS filtering
replicate -padding
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Homomorphic Filters
f(x, y) i(x, y) r(x, y)
H(u, v)
illumination (slowly varying) (low-frequency)
reflectance (fastly varying) (high-frequency)
freq.
lnf(x, y) lni(x, y) lnr(x, y)
0
Key linear separation
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Homomorphic Filters
before
after
From Stockham