1. Image Segmentation
Rob Atlas
Nick Bridle
Evan Radkoff

2. Image Segmentation
Separate an image into sets of pixels which represent some structure in the image
Main application is object and boundary detection

3. High Level Overview of Papers
Image Segmentation
Local approaches
Graph cuts
Mean Shift
Normalized Cuts
Comaniciu et al., 1999
Shi and Malik, 2000
Interactive Cuts
Rother et al., 2004
Li et al., 2004

4. Mean Shift Analysis Rutgers University, ICCV 1999
Dorin Comaniciu, Peter Meer
Rutgers University, ICCV 1999

5. { x / ∂s , y / ∂s , R / ∂r , G / ∂r , B / ∂r }
Spatial-Range Domain
Spatial domain: pixel locations
Range domain: intensity values
Spatial-Range domain: concatenation of the two
{ x, y, R, G, B }
Parameters:
∂s: Spatial domain normalization
∂r: Range domain normalization
{ x / ∂s , y / ∂s , R / ∂r , G / ∂r , B / ∂r }
5 dimensions for RGB, 3 for grayscale

6. Kernel Density Estimate
For n points {xi}i=1..n in a d-dimensional Euclidean space with kernel function K(x) and window radius h:

7. Epanechnikov Kernel
cd : volume of the unit d-dimensional sphere

8. Estimating the Density Gradient
Goal:
where Sh is a hypersphere containing nx data points

9. Mean Shift Vector
Vector with a direction towards the largest increase in density, thus leading to a local density maximum.

10. Mean Shift Procedure 1) Compute the mean shift vector Mh(x)
2) Translate the window Sh(x) by Mh(x)
3) If not converged, go to step 1.
Mathematically guaranteed to converge

11. Applying Mean Shift to Images
Spatial-Range Domain: { x / ∂s , y / ∂s , R / ∂r , G / ∂r , B / ∂r }
The authors apply the mean shift procedure to all points in the spatial-range domain.
Two applications: Filtering, Segmentation
Two parameters: spatial resolution and range resolution

12. Mean Shift Filtering xi: source pixel zi: filtered pixel for each xi
apply the mean shift procedure on xi to find the convergence point yi
spacial domain of zi = spacial domain of xi
range domain of zi = range domain of yi

13. Results

14. Results

15. Results

16. Segmentation
Instead of changing pixels ranges (colors), group them by which point their mean shift procedure converges to.

17. Advantages: Disadvantages: No oversight required
Preserves detail when appropriate
Disadvantages:
Cannot choose how many segments are made

18. Normalized Cuts and Image Segmentation
Shi and Malik, 2000
...

19. Basic Idea Big Questions:
Treat image segmentation as a graph-partitioning problem
Big Questions:
Advantage over older methods: fast and global
1. How do we define a good partitioning?
2. How do we do partition efficiently?

20. Images as a graph
G=(V,E)
Vertices: pixels
Edges: similarity

21. Graph cuts Assumes: Graph is fully connected Given: G=(V,E) Do:
Create disjoint vertex sets A, B s.t.
and

22. Similarity metric
Weight between edges in graph:
X(i): Location of pixel i
F(i): Feature vector describing pixel i
Simple case - F(i)=1 for segmenting points
Intensity, color, texture
Like mean shift, we take into account distance in space and intensity difference ,
: Parameters

23. 2-way graph cut Minimize:
Existing efficient methods for optimal 2-way cut

24. Recursive minimum cuts
Iteratively bipartition graph according to minimum cut (Wu and Leahy, 1993)
The problem: favors small clusters

25. Intuition for normalized cuts
Want to minimize cut score
But, we also want to partition out larger areas with more connections
So, what we're really interested in is the proportion of total weights that we are cutting for each segment

26. Normalized cuts
Goal: partition into disassociated groups

27. Minimizing association between groups ~ maximizing association within groups

28. Computational complexity
Minimizing normalized cut: NP-complete
Can compute approximate solution by solving a generalized eigenvalue problem
Proof - see paper

29. Eigenvalue system
D - diagonal matrix of the total weight from each node to every other
W - matrix of weights between nodes
y - eigenvectors
lambda - eigenvalues
However, this is not in standard form for solving

30. A solvable representation
First eigenvector (eigenvalue=0):

31. Second smallest eigenvalue describes solution
Remember, the smallest eigenvalue is 0

32. Recursive two-way Ncut
Third, fourth, etc eigenvalues correspond to eigenvectors that subdivide the existing graphs
However, error accumulates with each
Better to recompute partitioning on each subgraph iteratively

33. Segmentation Algorithm
1. Create weighted graph G=(V,E)
2. Solve and take the eigenvector with the second smallest eigenvalue
3. Bipartition the graph with this eigenvector
4. If Ncut is below threshold value, recursively partition each segment

34. Partitioning by an eigenvector
The partitioning eigenvector contains one continuous value per pixel
Since these are continuous, not discrete, they don't directly tell us which points are in which segment
Instead, we try different cutoffs, and select the one that produces the minimal NCut value

35. Partitioning by an eigenvector

36. Examples - segmenting point sets

37. Examples - segmenting noisy data

38. Examples - images

39. Example - image with texture metric

40. Interactive Graph Cuts
User can provide hints about where the objects are in the scene
Ideally, the less work the user has to do, the better

41. Foreground Extraction
Goal is to partition the pixels into two sets: foreground and background

42. Foreground Extraction
Given pixel intensity values
Output set of alphas
which determine how much each pixel belongs to the foreground
"Hard segmentation":
"Soft segmentation":

43. GrabCut
Rother et al.,
SIGGRAPH 2004

44. GrabCut: Handling Color Images
Previous work: histogram of gray values for foreground and background
GrabCut:
- histograms are intractable for color
- instead use Gaussian Mixture Model (GMM) to represent the distribution of color

45. GrabCut: Algorithm

46. GrabCut: Border Matting
Perform hard segmentation on image, and then relax alpha values on the border of the object

48. Lazy Snapping
Li et al., SIGGRAPH 2004

49. Lazy Snapping: Energy Function
Minimize the Gibbs energy function
Run k-means clustering on foreground and background points
E1 defined in terms of these mean colors:

50. Lazy Snapping: Speeding it up
Speed is crucial for a responsive user experience
Use the watershed algorithm to divide the image into small approximately constant patches, then do graph cut using these as the nodes

51. Lazy Snapping: Boundary Editing
Want boundary to "snap" to the user-defined polygon
Energy term E2 becomes

52. Lazy Snapping: Results

53. Thanks for listening.
Questions?