![Robert Collins
CSE486
Recall
• Cascaded Gaussians
– Repeated convolution by a smaller Gaussian
to simulate effects of a larger one.
• G*(G*f) = (G*G)*f [associativity]](https://image.slidesharecdn.com/lecture10-110524023512-phpapp02/85/Lecture10-2-320.jpg)
![Robert Collins
CSE486
Example: Cascaded Convolutions
[1 1] * [1 1] --> [1 2 1]
[1 1] * [1 2 1] --> [1 3 3 1]
[1 1] * [1 3 3 1] --> [1 4 6 4 1]
..and so on…
1 1
1 2 1 Pascal’s
1 3 3 1 Triangle
1 4 6 4 1
1 5 10 10 5 1
and so on…](https://image.slidesharecdn.com/lecture10-110524023512-phpapp02/85/Lecture10-3-320.jpg)
![Robert Collins
CSE486
Aside: Binomial Approximation
Look at odd-length rows of Pascal’s triangle:
1 1 [1 2 1]/4 - approximates
1 2 1 Gaussian with sigma=1/sqrt(2)
1 3 3 1
1 4 6 4 1
[1 4 6 4 1]/16 - approximates
1 5 10 10 5 1
Gaussian with sigma=1
and so on…
An easy way to generate integer-coefficient
Gaussian approximations.](https://image.slidesharecdn.com/lecture10-110524023512-phpapp02/85/Lecture10-5-320.jpg)
![Robert Collins
CSE486
Specific Example
From Crowley et.al., “Fast Computation of Characteristic Scale
using a Half-Octave Pyramid.” Proc International Workshop
on Cognitive Vision (CogVis), Zurich, Switzerland, 2002.
General idea: cascaded filtering using [1 4 6 4 1] kernel to generate
a pyramid with two images per octave (power of 2 change in
resolution). When we reach a full octave, downsample the image.
blur blur
downsample
blur blur](https://image.slidesharecdn.com/lecture10-110524023512-phpapp02/85/Lecture10-24-320.jpg)
Lecture10
- 1. Robert Collins CSE486 Lecture 10: Pyramids and Scale Space
- 2. Robert Collins CSE486 Recall • Cascaded Gaussians – Repeated convolution by a smaller Gaussian to simulate effects of a larger one. • G*(G*f) = (G*G)*f [associativity]
- 3. Robert Collins CSE486 Example: Cascaded Convolutions [1 1] * [1 1] --> [1 2 1] [1 1] * [1 2 1] --> [1 3 3 1] [1 1] * [1 3 3 1] --> [1 4 6 4 1] ..and so on… 1 1 1 2 1 Pascal’s 1 3 3 1 Triangle 1 4 6 4 1 1 5 10 10 5 1 and so on…
- 4. Robert Collins CSE486 Aside: Binomial Approximation 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 and so on… Pascal’s Binomial Triangle Coefficients
- 5. Robert Collins CSE486 Aside: Binomial Approximation Look at odd-length rows of Pascal’s triangle: 1 1 [1 2 1]/4 - approximates 1 2 1 Gaussian with sigma=1/sqrt(2) 1 3 3 1 1 4 6 4 1 [1 4 6 4 1]/16 - approximates 1 5 10 10 5 1 Gaussian with sigma=1 and so on… An easy way to generate integer-coefficient Gaussian approximations.
- 6. Robert Collins CSE486 From Homework 2
- 7. Robert Collins CSE486 More about Cascaded Convolutions (for the mathematically inclined) Fun facts: The distribution of the sum of two random variables X + Y is the convolution of their two distributions Given N i.i.d. random variables, X1 … XN, the distribution of their sum approaches a Gaussian distribution (aka the central limit theorem) Therefore: The repeated convolution of a (nonnegative) filter with itself takes on a Gaussian shape.
- 8. Robert Collins CSE486 Gaussian Smoothing at Different Scales original sigma = 1
- 9. Robert Collins CSE486 Gaussian Smoothing at Different Scales original sigma = 3
- 10. Robert Collins CSE486 Gaussian Smoothing at Different Scales original sigma = 10
- 11. Robert Collins CSE486 Idea for Today: Form a Multi-Resolution Representation original sigma = 1 sigma = 3 sigma = 10
- 12. Robert Collins CSE486 Pyramid Representations Because a large amount of smoothing limits the frequency of features in the image, we do not need to keep all the pixels around! Strategy: progressively reduce the number of pixels as we smooth more and more. Leads to a “pyramid” representation if we subsample at each level.
- 13. Robert Collins CSE486 Gaussian Pyramid • Synthesis: Smooth image with a Gaussian and downsample. Repeat. • Gaussian is used because it is self-reproducing (enables incremental smoothing). • Top levels come “for free”. Processing cost typically dominated by two lowest levels (highest resolution).
- 14. Robert Collins CSE486 Gaussian Pyramid Low res High res
- 15. Robert Collins CSE486 Emphasis: Smaller Images have Lower Resolution
- 16. Robert Collins CSE486 Generating a Gaussian Pyramid Basic Functions: Blur (convolve with Gaussian to smooth image) DownSample (reduce image size by half) Upsample (double image size)
- 17. Robert Collins CSE486 Generating a Gaussian Pyramid Basic Functions: Blur (convolve with Gaussian to smooth image) DownSample (reduce image size by half) Upsample (double image size)
- 18. Robert Collins CSE486 Downsample By the way: Subsampling is a bad idea unless you have previously blurred/smoothed the image! (because it leads to aliasing)
- 19. Robert Collins CSE486 To Elaborate: Thumbnails 131x97 65x48 32x24 original image downsampled (left) 262x195 vs. smoothed then downsampled (right)
- 20. Robert Collins CSE486 To Elaborate: Thumbnails original image downsampled (left) 262x195 vs. smoothed then downsampled (right) 131x97
- 21. Robert Collins CSE486 To Elaborate: Thumbnails original image downsampled (left) 262x195 vs. smoothed then downsampled (right) 65x48
- 22. Robert Collins CSE486 To Elaborate: Thumbnails original image downsampled (left) 262x195 vs. smoothed then downsampled (right) 32x24
- 23. Robert Collins CSE486 Upsample How to fill in the empty values? Interpolation: • initially set empty pixels to zero • convolve upsampled image with Gaussian filter! e.g. 5x5 kernel with sigma = 1. • Must also multiply by 4. Explain why.
- 24. Robert Collins CSE486 Specific Example From Crowley et.al., “Fast Computation of Characteristic Scale using a Half-Octave Pyramid.” Proc International Workshop on Cognitive Vision (CogVis), Zurich, Switzerland, 2002. General idea: cascaded filtering using [1 4 6 4 1] kernel to generate a pyramid with two images per octave (power of 2 change in resolution). When we reach a full octave, downsample the image. blur blur downsample blur blur
- 25. Robert Collins CSE486 Effective Sigma at Each Level Crowley etal.
- 26. Robert Collins CSE486 Effective Sigma at Each Level CAN YOU EXPLAIN HOW THESE VALUES ARISE?
- 27. Robert Collins CSE486 Concept: Scale Space Basic idea: different scales are appropriate for describing different objects in the image, and we may not know the correct scale/size ahead of time.
- 28. Robert Collins CSE486 Example: Detecting “Blobs” at Different Scales. But first, we have to talk about detecting blobs at one scale...