Mean-Shift and Normalized Cuts for Image Segmentation: A Comprehensive Study - Prof. Marsh, Study notes of Computer Science

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Lecture 11: Mean-shift and

Normalized Cuts

Segmentation

CAP 5415

Fall 2009

Mean-Shift

• Like EM, this algorithm is built on

probabilistic intuitions.

• To understand EM we had to understand

mixture models

• To understand mean-shift, we need to

understand kernel density estimation

(Take Pattern Recognition!)

Using a Parametric Model

• Could fit a parametric model (like a Gaussian)
• Why:
  • Can express distribution with a few number of parameters (like mean and variance)
• Why not:
  • Limited in flexibility

Non-Parametric Methods

• We’ll focus on kernel-density estimates
• Basic Idea: Use the data to define the

distribution

• Intuition:
  • If I were to draw more samples from the same probability distribution, then those points would probably be close to the points that I have already drawn
  • Build distribution by putting a little mass of probability around each data-point

Formally

• Most Common Kernel: Gaussian or Normal Kernel
• Another way to think about it:
  • Make an image, put 1(or more) wherever you have a sample
  • Convolve with a Gaussian Kernel

Gradient Ascent?

• Actually, no.
• A set of iterative steps can be taken that

will monotonically converge to a mode

• No worries about step sizes
• This is an adaptive gradient ascent (x = y j )

Results

Normalized Cuts

• Clustering approach based on graphs
• First some background

Graphs

• A graph G(V,E) is a triple consisting of a

vertex set V(G) an edge set E(G) and a

relation that associates with each edge

two vertices called its end points.

(From Slides by Khurram Shafique)

Can represent a graph with a

matrix

a e d c b

[

]

Adjacency Matrix: W One Row Per Node (Based on Slides by Khurram Shafique)

Can add weights to edges

[ 0 1 3 ∞ ∞ 1 0 4 ∞ 2 3 4 0 6 7 ∞ ∞ 6 0 1 ∞ 2 7 1 0 ] Weight Matrix: W (Based on Slides by Khurram Shafique)

Minimum Cut

• There can be more than one minimum cut in a given graph
• All minimum cuts of a graph can be found in polynomial time 1 . 1 H. Nagamochi, K. Nishimura and T. Ibaraki, “Computing all small cuts in an undirected network. SIAM J. Discrete Math. 10 (1997) 469-481. (Based on Slides by Khurram Shafique)

How does this relate to image

segmentation?

• When we compute the cut, we've divided

the graph into two clusters

• To get a good segmentation, the weight on

the edges should represent pixels affinity

for being in the same group

(Images from Khurram Shafique)