# Normalized Cuts and Image Segmentation 1 Normalized Cuts and Image Segmentation Jianbo Shi, Upenn...

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Normalized Cuts and Image Segmentation

Jianbo Shi, Upenn Jitendra Malik, Berkeley

10/7/2003 2

Perceptual Grouping and Organization (Wertheimer)

Similarity Proximity Common fate Good continuation Past experience

= Visual grouping

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History

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How to Partition?

There is no single answer Prior World Knowledge:

Low Level Coherence of Brightness Color Texture Motion

Mid or High Level Symmetries of Objects

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How to Partition? (Cont’d)

inherently hierarchical TREE STRUCTURE instead of flat partitioning OBJECTIVE:

low level cues for hierarchical partitions. High level knowledge for further partitioning.

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Tools

G=(V,E) An edge is formed between every pair (Complete graph) w(i,j) function of similarity. Partition V into disjoint sets V1, V2,…, Vm

Similarity within sets maximum. Similarity between sets minimum.

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Graph Vs. Image

What is the best criterion i.e. the weighting function How to compute efficiently

The graph is huge

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Definitions

Optimal bipartition is minimum cut Minimum cut

computed efficiently partitions out small chunks

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Example of a Bad Partition

Similarity decreases as the distance increases

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New Definition: Normalized Cut

Normalize the cut. Still a disassociation (dissimilarity) measure.

Smaller the better.

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New Definition: Normalized Association

Total association (similarity) within groups. The bigger the better.

assoc(a,a): total connectivity within A.

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Normalized Cut and Normalized Association Are Related

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Formulation for Normal Cut

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System Linearization

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Setting up the matrices

d1 0 0 0 0 0 …… 0 0 d2 0 0 0 0 …… 0 D-W=

D=0 0 d3 0 0 0 …… 0 ………………………. 0 0 0 0 0 0 …… dN

0 w12 w13 …. w1N W= w21 0 w23 …. w2N

…………………….. wN1 wN2 ….. …. 0

dN…-wN3-wN2-wN1

……………

-w3N…d3-w32-w31

-w2N…-w23d2-w21

-w1N…-w13-w12d1

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Rayleigh Quotient

Generalized eigenvalue system Minimized if y can be real.

Convert generalized eigensystem to a standard eigensystem:

Where

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Results

D-W is positive semidefinite eigenvectors are perpendicular.

Second smallest eigenvector minimizes the normalized cut.

Rayleigh quotient

Constraints (to be justified later): y can take on either 1 or –b

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Problem

First constraint not satisfied The solution gives real valued eigenvectors.

Which pixels are above threshold, which ones are not?

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Algorithm

Set up G=(V,E) All the points in the image as nodes Complete graph.

Weights proportional to the similarity between two pixels.

Similarity measure is yet to be decided.

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Algorithm (cont’d)

Solve (D-W)x=λDx for smallest eigenvectors Method is yet to be decided.

Second smallest eigenvector bipartitions the graph.

Must decide the break point. Decide if current partition is good, if not recursively subdivide.

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Example: Brightness Image

Define the weighting function To build matrices D and W:

Sigma: 10 to 20 percent of the total range of the distance function.

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Example (Cont’d)

Solve for Transform into a standard eigenvalue problem

Normally takes O(n^3) operations Use Lanczos method to reduce computation.

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Lanczos Method

Graphs are locally connected, thus sparse Eigensystems are therefore sparse

Only few smallest eigenvectors needed Precision requirement for eigenvectors is loose.

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Dividing Point for Eigenvector

Ideal case: second smallest eigenvector is integer. But in reality it takes real values

Use the median, or Use 0, or Use the value that makes Ncut(A,B) smallest.

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Stability Criterion

Sometimes an eigenvector takes on continues values. Not suitable for partitioning purpose

Hard to find a cut point Similar Ncut values.

Compute the histogram ratio between the minimum and maximum values in the bins.

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Drawbacks of 2-Way Cut

Stability criterion is good but it gets rid of subsequent eigenvectors

Recursive 2-way cut is inefficient uses only the second smallest eigenvector

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Simultaneous K-Way Cut with Multiple Eigenvectors

k-means is used for oversegmentation. Second step can vary

Greedy Pruning Global Recursive Cut

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Greedy Pruning

Iteratively merge two segments until k segments are left. Each iteration choose the pair when merged minimizes the k-way Ncut criterion:

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Global Recursive Cut

Build a condensed graph Each node of the graph corresponds to one initial segment Weights Defined as the total connection from one initial partition to another Use exhaustive search or generalized eigenvalue system for solution.

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Experiments

Brightness Color Texture Motion Following equation always holds; it is F(i) that changes according to the application

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Similarity * Spatial Proximity

F(i)= 1, when segmenting point sets I(i), the intensity value for segmenting scalar images [v,v.s.sin(h), v.s.cos(h)](i) for color images [|I*f1|, … , |I*fn|](i), for texture.

The weight is always zero if proximity criterion is not met, i.e. The pixels are more than r pixels apart.

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MOTION

Spatiotemporal neighborhood

Dm: motion distance

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Experimental Results

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Experimental Results

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Experimental Results

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Motion

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Linearize the System

Xi > 0 if node i belongs to A, Xi

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Linearize the System (2)

The equation linearizes to:

D is a diagonal NxN matrix with d on its diagonal, W is the NxN weight matrix where w(i,j) is the weight of the edge between node i and j. 1 is a vector of all ones.

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Linearize the System (3)

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Linearize the System (4)

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Linearize the System (5)

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