September 19, 2013 Computer Vision Lecture 6: Image Filtering 1 Image Filtering • Many basic image processing techniques are based on convolution. • In a convolution, a convolution filter W is applied to every pixel of an image I to create a filtered image I*. • The filter W itself is a 2D matrix of real values. • To simplify the mathematics, we could consider W to have a center [0, 0] and extend from –m to m vertically and –n to n horizontally. • This means that W is of size (2m +
Transcript
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Image Filtering
• Many basic image processing techniques are based on convolution.
• In a convolution, a convolution filter W is applied to every pixel of an image I to create a filtered image I*.
• The filter W itself is a 2D matrix of real values.• To simplify the mathematics, we could consider W
to have a center [0, 0] and extend from –m to m vertically and –n to n horizontally.
• This means that W is of size (2m + 1)×(2n + 1).
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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j
i
ConvolutionExample: Averaging filter:
1/9 1/9 1/9
1/9 1/9 1/9
1/9 1/9 1/9
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Convolution
Grayscale Image:
1 6 3 2 9
2 11 3 10 0
5 10 6 9 7
3 1 0 2 8
4 4 2 9 10
1/9 1/9 1/9
1/9 1/9 1/9
1/9 1/9 1/9
Averaging Filter:
September 19, 2013 Computer Vision Lecture 6: Image Filtering
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Convolution
Original Image:
1 6 3 2 9
2 11 3 10 0
5 10 6 9 7
3 1 0 2 8
4 4 2 9 10
Filtered Image:
0 0 0 0 0
0 0
0 0
0 0
0 0 0 0 0
5 7 5
5
5
5 6
64
Now you can see the averaging (smoothing) effect of the 33 filter that we applied.
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Convolution
• It needs to be noted that for convolution the filter needs to be rotated by 180 before starting the computations (otherwise it’s a correlation).
• An intuitive explanation is that we would like convolution to be like a “local multiplication of patterns.”
• For example, if our image contains a few 1s and otherwise 0s, we would expect the convolution result to contain a copy of the filter pattern centered at each 1.
• Let us look at an image with one 1-pixel:
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Convolution
Image:
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
1 2 3
4 5 6
7 8 9
Filter: Result:
0 0 0 0 0
0 9 8 7 0
0 6 5 4 0
0 3 2 1 0
0 0 0 0 0
Oops! The copy of the filter is rotated by 180 !
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Convolution
• If we rotate the filter by 180 beforehand, we get the desired result.
• This leads to the following definition of convolution for image I, filter W, and result I*:
m
mk
n
nl
ljkilkjijiji ],[],[],[*],[],[* IWIWI
• This formula needs to be applied to all coordinates [i, j] in I in order to create the complete image I*.
• Convolution is commutative, i.e., W and I are exchangeable.
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Image FilteringMore common: Gaussian Filters
2
22
222
1),(),(
yx
eyxGyxW
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14741
Discrete version: 1/273
• implement decreasing influence by more distant pixels
Continuous version:
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Image Filtering
original 33 99 1515
Effect of Gaussian smoothing:
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Properties of Gaussian Filters
The application of Gaussian convolution filters can be made more efficient.
This is important, for example, if we want to apply different Gaussian filters to a large number of big input images.
The basic idea is to separate the convolution with the 2D Gaussian filter into two successive convolutions with 1D Gaussian filters.
One of these filters is vertical, the other one horizontal.
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Properties of Gaussian FiltersThe general form of the Gaussian filter, without a normalizing factor, is given by:
2
22
2],[ ji
eji
G
The convolution of an image F with a Gaussian filter G of size (2m + 1)(2n + 1) is given by:
This formula needs to be applied to all coordinates [i, j] in F in order to create the convoluted image.
m
mk
n
nl
ljkilkjiji ],[],[],[*],[ FGFG
September 19, 2013 Computer Vision Lecture 6: Image Filtering
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Properties of Gaussian FiltersThen we have:
m
mk
n
nl
ljkilkjiji ],[],[],[*],[ FGFG
m
mk
n
nl
lk
ljkie ],[2
22
2 F
m
mk
n
nl
lk
ljkiee ],[2
2
2
2
22 F
m
mk
n
nl
lk
ljkiee ],[2
2
2
2
22 F
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Properties of Gaussian Filters
The formula in the curly braces describes the convolution of F[i, j] with a horizontal one-dimensional Gaussian filter.
The remainder of the formula takes the result of this first convolution and performs a convolution with a vertical one-dimensional Gaussian filter on it.
So instead of applying an (2m + 1)(2n + 1) Gaussian convolution filter, we can successively apply a 1(2n + 1) filter and an (2m + 1)1 filter.
This increases the efficiency of the computation.
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Different Types of Filters
• Smoothing can reduce noise in the image.• This can be useful, for example, if you want to find
regions of similar color or texture in an image.
• However, there are different types of noise.• For so-called “salt-and-pepper” noise, for example,
a median filter can be more effective.• Note that it is not a convolution filter, but it works
similarly.
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Median Filter
• Use, for example, a 33 filter and move it across the image like we did before.
• For each position, compute the median of the brightness values of the nine pixels in question.– To compute the median, sort the nine values in
ascending order.– The value in the center of the list (here, the fifth
value) is the median.• Use the median as the new value for the center
pixel.
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Median Filter
• Advantage of the median filter: Capable of eliminating outliers such as the extreme brightness values in salt-and-pepper noise.
• Disadvantage: The median filter may change the contours of objects in the image.
September 19, 2013 Computer Vision Lecture 6: Image Filtering
