Filtering Images

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
0
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 dont 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) xo,
  shifts an image

13
Reducing Image Size

• Warp function f(x)kx, k lt 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 gt 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