1	Linear Filtering About modifying pixels based on neighborhood. Local methods simplest. Linear means linear combination of neighbors. Linear methods simplest. Useful to: Integrate information over constant regions. Scale. Detect changes. Fourier analysis. Many nice slides taken from Bill Freeman.
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4	Correlation
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6	Boundaries Zeros Repeat values Cycle Produce shorter result Examples
7	Correlation
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9	Some Examples
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24	Filtering to reduce noise Noise is what we’re not interested in. We’ll discuss simple, low-level noise today: Light fluctuations; Sensor noise; Quantization effects; Finite precision Not complex: shadows; extraneous objects. A pixel’s neighborhood contains information about its intensity. Averaging noise reduces its effect.
25	Additive noise I = S + N. Noise doesn’t depend on signal. We’ll consider:
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27	Averaging Filter and noise reduction Example: try executing: k=1; figure(1); hist(sum((1/k)*rand(k,1000))) for different values of k. The average of noise is smaller than one example. This is intuitive Can be proven in many cases (some technical conditions: noise must be independent, many samples….) Actually true for many real examples: Gaussian noise, flipping a coin many times
28	Filtering reduces noise if signal stable Suppose I(i) = I+n(i), I(i+1) = I+n(i+1) I(i+2) = I+n(i+2). Average of I(i), I(i+1), I(i+2) = I + average of n(i), n(i+1), n(i+2). When there is no noise, averaging smooths the signal. So in real life, averaging does both.
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31	Gaussian Averaging Rotationally symmetric. Weights nearby pixels more than distant ones. This makes sense as probabalistic inference.
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36	Box Filter
37	Gaussian Filter
38	Smoothing as Inference About the Signal: Non-linear Filters.
39	Filtering to reduce noise: Lessons Noise reduction is probabilistic inference. Depends on knowledge of signal and noise. In practice, simplicity and efficiency important.
40	Filtering and Signal Smoothing also smooths signal. Matlab Removes detail Matlab This is good and bad: - Bad: can’t remove noise w/out blurring shape. - Good: captures large scale structure; allows subsampling.
41	Subsampling