1	Announcements Project 2 more signup slots questions Picture taking at end of class
2	Image Segmentation Today’s Readings Forsyth chapter 14, 16
3	From images to objects 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
4	Image Segmentation We will consider different methods Already covered: Intelligent Scissors (contour-based) Hough transform (model-based) This week: K-means clustering (color-based) EM Mean-shift Normalized Cuts (region-based)
5	Image histograms 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
6	What do histograms look like?
7	Histogram-based segmentation 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 photoshop demo
8	Histogram-based segmentation 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 photoshop demo
9	Clustering How to choose the representative colors? This is a clustering problem!
10	Break it down into subproblems Suppose I tell you the cluster centers c_i Q: how to determine which points to associate with each c_i?
11	K-means clustering K-means clustering algorithm Randomly initialize the cluster centers, c₁, ..., c_K Given cluster centers, determine points in each cluster For each point p, find the closest c_i. Put p into cluster i Given points in each cluster, solve for c_i Set c_i to be the mean of points in cluster i If c_i 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:
12	Probabilistic clustering Basic questions what’s the probability that a point x is in cluster m? what’s the shape of each cluster? K-means doesn’t answer these questions Basic idea instead of treating the data as a bunch of points, assume that they are all generated by sampling a continuous function This function is called a generative model defined by a vector of parameters θ
13	Mixture of Gaussians One generative model is a mixture of Gaussians (MOG) K Gaussian blobs with means μ_b covariance matrices V_b, dimension d blob b defined by: blob b is selected with probability the likelihood of observing x is a weighted mixture of Gaussians where
14	Expectation maximization (EM) Goal find blob parameters θ that maximize the likelihood function: Approach: E step: given current guess of blobs, compute ownership of each point M step: given ownership probabilities, update blobs to maximize likelihood function repeat until convergence
15	EM details E-step compute probability that point x is in blob i, given current guess of θ M-step compute probability that blob b is selected mean of blob b covariance of blob b
16	EM demos http://www.cs.ucsd.edu/users/ibayrakt/java/em/ http://www.dreier.cc/index.php?topic=downloads&sub=em
17	Applications of EM Turns out this is useful for all sorts of problems any clustering problem any model estimation problem missing data problems finding outliers segmentation problems segmentation based on color segmentation based on motion foreground/background separation ...
18	Problems with EM Local minima Need to know number of segments Need to choose generative model
19	Finding Modes in a Histogram How Many Modes Are There? Easy to see, hard to compute
20	Mean Shift [Comaniciu & Meer] Iterative Mode Search Initialize random seed, and window W Calculate center of gravity (the “mean”) of W: Translate the search window to the mean Repeat Step 2 until convergence
21	Mean-Shift Approach Initialize a window around each point See where it shifts—this determines which segment it’s in Multiple points will shift to the same segment
22	Mean-shift for image segmentation Useful to take into account spatial information instead of (R, G, B), run in (R, G, B, x, y) space D. Comaniciu, P. Meer, Mean shift analysis and applications, 7th International Conference on Computer Vision, Kerkyra, Greece, September 1999, 1197-1203. http://www.caip.rutgers.edu/riul/research/papers/pdf/spatmsft.pdf
23	Region-based segmentation Color histograms don’t take into account spatial info Gestalt laws point out importance of spatial grouping proximity, similarity, continuation, closure, common fate Suggests that regions are important
24	Images as graphs Fully-connected graph node for every pixel link between every pair of pixels, p,q cost c_pq for each link c_pq measures similarity similarity is inversely proportional to difference in color and position this is different than the costs for intelligent scissors
25	Segmentation by Graph Cuts 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
26	Cuts in a graph Link Cut set of links whose removal makes a graph disconnected cost of a cut:
27	But min cut is not always the best cut...
28	Cuts in a graph
29	Interpretation as a Dynamical System Treat the links as springs and shake the system elasticity proportional to cost vibration “modes” correspond to segments
30	Interpretation as a Dynamical System Treat the links as springs and shake the system elasticity proportional to cost vibration “modes” correspond to segments
31	Color Image Segmentation
32	Normalize Cut in Matrix Form Can write normalized cut as:
33	Cleaning up the result Problem: Histogram-based segmentation can produce messy regions segments do not have to be connected may contain holes How can these be fixed?
34	Dilation operator:
35	Dilation operator Demo http://www.cs.bris.ac.uk/~majid/mengine/morph.html
36	Erosion operator:
37	Erosion operator Demo http://www.cs.bris.ac.uk/~majid/mengine/morph.html
38	Nested dilations and erosions What does this operation do?
39	Nested dilations and erosions What does this operation do?
40	Nested dilations and erosions What does this operation do?