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Basic Image Processing
February 1
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Homework
Second homework set will be online Friday (2/2).
First programming assignment will be online Monday (2/5).
Slides will be posted online before the end of next Monday.
Today, we will finish Chapter 6. We will work on Chapter 7 and 8 next Tuesday.
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Recall that we introduced a linear noise model
Optimal filtering: Find a filter that maximally suppresses the noise.
An application of the math we have been studied so far!
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Changing the notation slightly (following the textbook)
o is the output signal, with h the (unknown) filter kernel (point-spread function) that we want to figure out.
Find h that minimizes this error functional.
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The expression for E become complicated (only in appearance !)
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Auto-correlation and cross-correlationGiven two functions, a, b, their cross-correlation is defined as
The auto-correlation of a function is the cross-correlation between itself:
Recall that cross-correlation and auto-correlation are functions, not just a number.
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Some important properties of correlations.
(0, 0) is a global maximum of auto-correlation.
If then
The Fourier transform of autocorrelation is called power spectrum.
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The cross-correlation of the two images have maximum at (100, 200).
The two auto-correlations are the same (invariant under translation).
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Back to optimal filter design
and
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Remember, we want to determine h. This is a calculus of variation problem!
That is,
Want to find h such that
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How to get h?
That is, we only need to know the power spectra.
From
Assume that the noise and signal are not correlated.
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We have
is the signal-to-noise ratio (for each frequency).
SNR is high, the gain is almost unity.
When SNR is low (mostly noise), the gain is small.
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Similar applications
Suppose now we have
Examples:
Image Blurring:
Defocusing:
Motion Smear: image points smeared into a line.
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Assume that the noise and signal are not correlated.
Need to figure out
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SNR is high
In parts where SNR is low
The gain is roughly