1
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- vote for Project 3 artifacts
- Project 4 (due next Wed night)
- Questions?
- Late day policy: everything must
be turned in by next Friday
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2
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- Today’s Readings
- Shapiro, pp. 279-289
- http://www.dai.ed.ac.uk/HIPR2/morops.htm
- Dilation, erosion, opening, closing
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3
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- What Defines an Object?
- Subjective problem, but has been well-studied
- Gestalt Laws seek to formalize this
- proximity, similarity, continuation, closure, common fate
- see notes by Steve Joordens, U. Toronto
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4
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- We will consider different methods
- Already covered:
- Intelligent Scissors (contour-based)
- Hough transform (model-based)
- This week:
- K-means clustering (color-based)
- Normalized Cuts (region-based)
- Forsyth, chapter 16.5 (supplementary)
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5
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- How many “orange” pixels are in this image?
- This type of question answered by looking at the histogram
- A histogram counts the number of occurrences of each color
- Given an image
- The histogram is defined to be
- What is the dimension of the histogram of an RGB image?
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6
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7
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- Goal
- Break the image into K regions (segments)
- Solve this by reducing the number of colors to K and mapping each pixel
to the closest color
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8
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- Goal
- Break the image into K regions (segments)
- Solve this by reducing the number of colors to K and mapping each pixel
to the closest color
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9
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- How to choose the representative colors?
- This is a clustering problem!
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10
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- Suppose I tell you the cluster centers ci
- Q: how to determine which points
to associate with each ci?
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11
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- K-means clustering algorithm
- Randomly initialize the cluster centers, c1, ..., cK
- Given cluster centers, determine points in each cluster
- For each point p, find the closest ci. Put p into cluster i
- Given points in each cluster, solve for ci
- Set ci to be the mean of points in cluster i
- If ci have changed, repeat Step 2
- Java demo: http://www.cs.mcgill.ca/~bonnef/project.html
- Properties
- Will always converge to some solution
- Can be a “local minimum”
- does not always find the global minimum of objective function:
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12
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- Problem:
- Histogram-based segmentation can produce messy regions
- segments do not have to be connected
- may contain holes
- How can these be fixed?
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13
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14
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- Demo
- http://www.cs.bris.ac.uk/~majid/mengine/morph.html
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15
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16
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- Demo
- http://www.cs.bris.ac.uk/~majid/mengine/morph.html
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17
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- What does this operation do?
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18
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- What does this operation do?
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19
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- What does this operation do?
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20
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21
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- Fully-connected graph
- node for every pixel
- link between every pair of pixels, p,q
- cost cpq for each link
- cpq measures similarity
- similarity is inversely proportional to difference in color and
position
- this is different than the costs for intelligent scissors
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22
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- Break Graph into Segments
- Delete links that cross between segments
- Easiest to break links that have high cost
- similar pixels should be in the same segments
- dissimilar pixels should be in different segments
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23
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- Link Cut
- set of links whose removal makes a graph disconnected
- cost of a cut:
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24
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25
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26
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- Treat the links as springs and shake the system
- elasticity proportional to cost
- vibration “modes” correspond to segments
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27
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- Treat the links as springs and shake the system
- elasticity proportional to cost
- vibration “modes” correspond to segments
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28
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29
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- Can write normalized cut as:
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30
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- Things to take away from this lecture
- Image histogram
- K-means clustering
- Morphological operations
- dilation, erosion, closing, opening
- Normalized cuts
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