Basic ideas of grouping in humans: The Gestalt schoolpelillo/Didattica/Old...

Computer Vision - A Modern ApproachSet: Segmentation

Slides by D.A. Forsyth

Basic ideas of grouping in humans: The Gestalt school

Gestalt propertieselements in a collection of elements can have properties that result from relationships

• Gestaltqualitat

WertheimerWertheimer

KoehlerKoehler

KoffkaKoffka

A series of factors affect whether elements should be grouped together

• Gestalt factors

Segmentation as clustering

• Cluster together (pixels, tokens, etc.) that belong together

• Agglomerative clustering– attach closest to cluster it is

closest to– repeat

• Divisive clustering– split cluster along best

boundary– repeat

• Point-Cluster distance– single-link clustering– complete-link clustering– group-average clustering

• Dendrograms– yield a picture of output as

clustering process continues

Clustering

K-Means

• Choose a fixed number of clusters

• Choose cluster centers and point-cluster allocations to minimize error

• can’t do this by search, because there are too many possible allocations.

• Algorithm– fix cluster centers; allocate

points to closest cluster– fix allocation; compute best

cluster centers• x could be any set of features

for which we can compute a distance (careful about scaling)

x j − µ i2

j∈elements of i'th cluster∑

i∈clusters

K-Means Algorithm

1. Choose k data points to act as cluster centers2. Repeat3. Allocate each data point to cluster whose center is nearest4. Replace the cluster centers with the centroid of the elements in their clusters5. Until the cluster centers are unchanged

NOTE: In step 2), ensure that every cluster has at least one data point. Possible techniques for doing this include supplying empty clusters with a point chosen at random from points far from their cluster centers.

Image Clusters on intensity Clusters on color

K-means clustering using intensity alone and color alone

Graph-theoretic (pairwise) clustering

• Represent tokens using a weighted graph.– affinity matrix

• Cut up this graph to get subgraphs with strong interior links

Measuring AffinityIntensity

aff x, y( )= exp − 12σ i

I x( )− I y( ) 2( )

Distance

aff x, y( )= exp − 12σ d

x − y 2( )

Texture

aff x, y( )= exp − 12σ t

c x( )− c y( ) 2( )

Scale affects affinity

Eigenvector-based clustering

• This is an eigenvalue problem -choose the eigenvector of A with largest eigenvalue

• Simplest idea: we want a vector a giving the association between each element and a cluster

• We want elements within this cluster to, on the whole, have strong affinity with one another

• We could maximize

• But need the constraint aTAa

aTa = 1

Example eigenvector

points

eigenvector

matrix

More than two segments

• Two options– Recursively split each side to get a tree, continuing till the

eigenvalues are too small– Use the other eigenvectors

Clustering by eigenvectors: Algorithm

1. Construct (or take as input) the affinity matrix A2. Compute the eigenvalues and eigenvectors of A3. Repeat4. Take the eigenvector corresponding to the largest unprocessed eigenvalue5. Zero all components corresponding to elements that have already been clustered6. Threshold the remaining components to determine which elements belong to

this cluster7. If all elements have been accounted for, there are sufficient clusters8. Until there are sufficient clusters

Clustering as graph partitioning

Normalized Cut

Cut vs Association

Normalized cuts

• Current criterion evaluates within cluster similarity, but not across cluster difference

• Instead, we’d like to maximize the within cluster similarity compared to the across cluster difference

• Write graph as V, one cluster as A and the other as B

• Maximize

• i.e. construct A, B such that their within cluster similarity is high compared to their association with the rest of the graph

assoc(A,A)assoc(A,V )

assoc(B,B)assoc(B,V )

Normalized cuts algorithm

D −W( )y = λDy

Figure from “Image and video segmentation: the normalised cut framework”, by Shi and Malik, copyright IEEE, 1998

F igure from “Normalized cuts and image segmentation,” Shi and Malik, copyright IEEE, 2000

Basic ideas of grouping in humans: The Gestalt schoolpelillo/Didattica/Old...

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