1. Filtering Images Work in the spatial domain
Convert the filter into a matrix, the filter mask
eg 3x3 box filter performs averaging of a neighborhood:

2. 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

3. Box Filter Box filters smooth by averaging neighbors
In frequency domain, clearly a low-pass filter

4. Bartlett Filter Triangle shaped filter in spatial domain
In frequency domain, product of two box filters, so attenuates high frequencies more than a box

5. Constructing Masks: 2D 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

6. Constructing Masks: 3D
Multiply 2 2D masks together using outer product
M is 3D mask, m is 2D 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

7. Guassian Filter Attenuates high frequencies even further
In 2d, rotationally symmetric, so fewer artifacts

8. 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

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 2D Reduction Example (Bartlett)

15. 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

16. 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

17. Enlargement

18. Ideal Enlargement Reconstruct original function
Resample at higher frequency
Original function was band-limited, so resampling does not add any extra frequency information

19. Image Morphing Process to turn one image into another
Define parameterized path from each point in the original image to its destination in the output image
Animate points along paths, while interpolating colors from source to destination

20. 2D Morphing Example
Final resample

21. Morphing Issues
User interface to specify pixel mapping and paths is important
Different interpolation schemes are possible to get different effects
Simple morphs, like cross dissolve, are key to video effects:
Cross dissolve simply interpolates colors between the two images. Pixels move on straight paths