Image Analysis

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Image Analysis

• Image Restoration

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Image Restoration
Image enhancement tries to improve subjective
image quality. Image restoration tries to recover
the original image.
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Noise Models

• Noise may arise during image due to sensors,
  digitization, transmission, etc.
• Most of the time, it is assumed that noise is
  independent of spatial coordinates, and that
  there is no correlation between noise component
  and pixel value.
• Noise may be considered as a random variable,
  its statistical behavior is characterized by a
  probability density function (PDF).

Gaussian noise
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Noise Models
Uniform noise
Impulse (salt-and-pepper) noise
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Noise Models
Original
Noisy images and their histograms
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Noise Models

• How to estimate noise parameters?
• If imaging device is available
• Take a picture of a flat surface.
• See the shape of the histogram decide on the
  noise model.
• Estimate the parameters. (e.g., find mean and
  standard deviation.)
• When only images already generated are available
• Get a small patch of image with constant gray
  level
• Inspect histogram
• Estimate the parameters

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Restoration When There is Only Noise

• Low-Pass Filters
• Smoothes local variations in an image.
• Noise is reduced as a result of blurring.
• For example, Arithmetic Mean Filter is

? Convolve with a uniform filter of size m-by-n.
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Restoration When There is Only Noise

• Adaptive, local noise reduction filter
• Let be the noise variance at
  (x,y).
• Let be the local variance of
  pixels around (x,y).
• Let be the local mean of pixels
  around (x,y).

• We want a filter such that
• If noise variance is zero, it should return
  g(x,y).
• If local variance is high relative to noise
  variance, the filter should return a value close
  to g(x,y). (Therefore, edges are preserved!)
• If two variances are equal, the filter should
  return the average of the pixels within the
  neighborhood.

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Restoration When There is Only Noise
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Restoration When There is Only Noise

• Median Filter
• Replaces the value of a pixel by the median of
  intensities in the neighborhood of that pixel.
• Is very effective against the salt-and-pepper
  noise.

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Restoration When There is Only Noise

• Adaptive Median Filter The basic idea is to
  avoid extreme values
• Let
• z_min minimum gray level value in a neigborhood
  of a pixel at (x,y).
• z_max maximum gray level value
• z_med median
• z(x,y) gray level at (x,y).
• Is z_medz_min or z_medz_max? (That is, is
  z_med an extreme value?)
• No
• Is z(x,y) an extreme value? (Is z(x,y)z_min or
  z(x,y)z_max?)
• No Output is z(x,y)
• Yes Output is z_med.
• Yes Increase window size (to find a non-extreme
  z_med) and go to the first step. (When a maximum
  allowed window size is reached, stop and output
  z(x,y).)

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Restoration When There is Only Noise
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Restoration When There is Only Noise
Removing Periodic Noise with Band-Reject Filters
Spikes are due to noise
Periodic Noise
Band-reject filter
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Restoration When There is Only Noise
Finding Periodic Noise from the Spectrum and
Using Notch Filters
Filter out these spikes
Noise due to interference
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Image Restoration
Spatial domain
Frequency domain
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Image Restoration
Inverse Filtering
This could dominate signal.
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Image Restoration
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Image Restoration
Cut off the inverse filter for large frequencies.
(Signal-to-noise ratio is typically low for large
frequencies.)
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Image Restoration

• Minimum Mean Square Error (Wiener) Filtering
• Find such that the expected
  value of error is minimized

• Solution is

Investigate this equation for different
signal-noise ratios.
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Original image
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Image Restoration
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Least Squares Filtering
Find F(u,v) that minimizes the following cost
function
The solution is
(Unconstrained solution)
(See the derivations in the classroom)
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Least Squares Filtering
Find F(u,v) that minimizes the following cost
function
Choose a P(u,v) to have a smooth solution. (A
high-pass filter would do the trick.)
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Least Squares Filtering
The frequency domain solution to this
optimization problem is
(Constrained solution)
where P(u,v) is the Fourier Transform of p(x,y),
which is typically chosen as a high-pass filter.
Example
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Least Squares Filtering