1. (c) 2002 University of Wisconsin, CS 559
Last Time
Signal Processing
Filtering basics
Homework 2
Project 1
09/24/02
(c) 2002 University of Wisconsin, CS 559

2. (c) 2002 University of Wisconsin, CS 559
Today
More Filters
Sampling and Reconstruct
Image warping
09/24/02
(c) 2002 University of Wisconsin, CS 559

3. (c) 2002 University of Wisconsin, CS 559
Box Filter
Box filters smooth by averaging neighbors
In frequency domain, keeps low frequencies and attenuates (reduces) high frequencies, so clearly a low-pass filter
Spatial: Box
Frequency: sinc
09/24/02
(c) 2002 University of Wisconsin, CS 559

4. (c) 2002 University of Wisconsin, CS 559
Box Filter
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5. (c) 2002 University of Wisconsin, CS 559
Filtering Algorithm
If Iinput is the input image, and Ioutput is the output image, M is the filter mask and k is the mask size:
Care must taken at the boundary
Make the output image smaller
Extend the input image in some way
09/24/02
(c) 2002 University of Wisconsin, CS 559

6. (c) 2002 University of Wisconsin, CS 559
Bartlett Filter
Triangle shaped filter in spatial domain
In frequency domain, product of two box filters, so attenuates high frequencies more than a box
Spatial: Triangle (BoxBox)
Frequency: sinc2
09/24/02
(c) 2002 University of Wisconsin, CS 559

7. (c) 2002 University of Wisconsin, CS 559
Constructing Masks: 1D
Sample the filter function at matrix “pixels”
eg 2D Bartlett
Can go to edge of pixel or middle of next: results are slightly different 1 1 3 1 5 1 2 1 1 2 1 4
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8. (c) 2002 University of Wisconsin, CS 559
Constructing Masks: 2D
Multiply 2 1D masks together using outer product
M is 2D mask, m is 1D mask
0.2
0.6
0.2
0.2
0.04
0.12
0.04
0.6
0.12
0.36
0.12
0.2
0.04
0.12
0.04
09/24/02
(c) 2002 University of Wisconsin, CS 559

9. (c) 2002 University of Wisconsin, CS 559
Bartlett Filter
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10. (c) 2002 University of Wisconsin, CS 559
Guassian Filter
Attenuates high frequencies even further
In 2d, rotationally symmetric, so fewer artifacts
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11. (c) 2002 University of Wisconsin, CS 559
Gaussian Filter
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12. Constructing Gaussian Mask
Use the binomial coefficients
Central Limit Theorem (probability) says that with more samples, binomial converges to Gaussian 1 1 2 1 4 1 1 4 6 4 1 16 1 1 6 15 20 15 6 1 64
09/24/02
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13. (c) 2002 University of Wisconsin, CS 559
High-Pass Filters
A high-pass filter can be obtained from a low-pass filter
If we subtract the smoothed image from the original, we must be subtracting out the low frequencies
What remains must contain only the high frequencies
High-pass masks come from matrix subtraction:
eg: 3x3 Bartlett
09/24/02
(c) 2002 University of Wisconsin, CS 559

14. (c) 2002 University of Wisconsin, CS 559
High-Pass Filter
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15. (c) 2002 University of Wisconsin, CS 559
Edge Enhancement
High-pass filters give high values at edges, low values in constant regions
Adding high frequencies back into the image enhances edges
One approach:
Image = Image + [Image – smooth(Image)]
Low-pass
High-pass
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(c) 2002 University of Wisconsin, CS 559

16. (c) 2002 University of Wisconsin, CS 559
Edge-Enhance Filter
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17. (c) 2002 University of Wisconsin, CS 559
Edge Enhancement
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(c) 2002 University of Wisconsin, CS 559

18. Fixing Negative Values
The negative values in high-pass filters can lead to negative image values
Most image formats don’t support this
Solutions:
Truncate: Chop off values below min or above max
Offset: Add a constant to move the min value to 0
Re-scale: Rescale the image values to fill the range (0,max)
09/24/02
(c) 2002 University of Wisconsin, CS 559

19. (c) 2002 University of Wisconsin, CS 559
Image Warping
An image warp is a mapping from the points in one image to points in another
f tells us where in the new image to put the data from x in the old image
Simple example: Translating warp, f(x) = x+o, shifts an image
09/24/02
(c) 2002 University of Wisconsin, CS 559

20. (c) 2002 University of Wisconsin, CS 559
Reducing Image Size
Warp function: f(x)=kx, k > 1
Problem: More than one input pixel maps to each output pixel
Solution: Filter down to smaller size
Apply the filter, but not at every pixel, only at desired output locations
eg: To get half image size, only apply filter at every second pixel
09/24/02
(c) 2002 University of Wisconsin, CS 559

21. 2D Reduction Example (Bartlett)
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22. Ideal Image Size Reduction
Reconstruct original function using reconstruction filter
Resample at new resolution (lower frequency)
Clearly demonstrates that shrinking removes detail
Expensive, and not possible to do perfectly in the spatial domain…
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23. (c) 2002 University of Wisconsin, CS 559
Enlarging Images
Warp function: f(x)=kx, k < 1
Problem: Have to create pixel data
More pixels in output than in input
Solution: Filter up to larger size
Apply the filter at intermediate pixel locations
eg: To get double image size, apply filter at every pixel and every half pixel
New pixels are interpolated from old ones
Filter encodes interpolation function
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24. (c) 2002 University of Wisconsin, CS 559
Enlargement
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25. (c) 2002 University of Wisconsin, CS 559
Ideal Enlargement
Reconstruct original function
Resample at higher frequency
Original function was band-limited, so re-sampling does not add any extra frequency information
09/24/02
(c) 2002 University of Wisconsin, CS 559