Linear filtering. Motivation: Image denoising. How can we reduce noise in a photograph?. 1. 1. 1. 1. 1. 1. 1. 1. 1. “box filter”. Moving average. Let’s replace each pixel with a weighted average of its neighborhood The weights are called the filter kernel
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“box filter”
Source: D. Lowe
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Convention: kernel is “flipped”
Source: F. Durand
full
same
valid
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Source: S. Marschner
Source: S. Marschner
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Original
Source: D. Lowe
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Original
Filtered
(no change)
Source: D. Lowe
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Original
Source: D. Lowe
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Original
Shifted left
By 1 pixel
Source: D. Lowe
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Original
Source: D. Lowe
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Original
Blur (with a
box filter)
Source: D. Lowe
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(Note that filter sums to 1)
Original
Source: D. Lowe
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Original
Source: D. Lowe
Source: D. Lowe
=
detail
smoothed (5x5)
original
Let’s add it back:
+
=
original
detail
sharpened
–
Source: D. Forsyth
“fuzzy blob”
0.003 0.013 0.022 0.013 0.003
0.013 0.059 0.097 0.059 0.013
0.022 0.097 0.159 0.097 0.022
0.013 0.059 0.097 0.059 0.013
0.003 0.013 0.022 0.013 0.003
5 x 5, = 1
Source: C. Rasmussen
σ = 2 with 30 x 30 kernel
σ = 5 with 30 x 30 kernel
Source: K. Grauman
Source: K. Grauman
Source: K. Grauman
Source: D. Lowe
*
=
=
*
2D convolution(center location only)
The filter factorsinto a product of 1Dfilters:
Perform convolutionalong rows:
Followed by convolutionalong the remaining column:
Source: K. Grauman
Source: S. Seitz
Source: M. Hebert
Smoothing with larger standard deviations suppresses noise, but also blurs the image
3x3
5x5
7x7
Source: K. Grauman
Source: K. Grauman
Median filtered
Saltandpepper noise
Source: M. Hebert
3x3
5x5
7x7
Gaussian
Median
Source: D. Lowe
=
detail
smoothed (5x5)
original
Let’s add it back:
+ α
=
original
detail
sharpened
–
unit impulse
Gaussian
Laplacian of Gaussian
image
unit impulse(identity)
blurredimage
Gaussian Filter
Laplacian Filter