Spectral Matting

A. Levin D. Lischinski and Y. Weiss. A Closed Form Solution to Natural Image Matting. IEEE Conf. on Computer Vision and

Pattern Recognition (CVPR), June 2006, New York

A. Levin, A. Rav-Acha, D. Lischinski. Spectral Matting. Best paper award runner up. IEEE Conf. on Computer Vision and Pattern

Recognition (CVPR), Minneapolis, June 2007

A. Levin1,2, A. Rav-Acha1, D. Lischinski1. Spectral Matting. IEEE Trans. Pattern Analysis and Machine Intelligence, Oct 2008.

1School of CS&Eng The Hebrew University2CSAIL MIT

1

Hard

segmentation compositing

Matte compositing

Source image

Hard segmentation and matting

2

Previous approaches to segmentation and matting

Unsupervised

Input Hard output Matte output

Spectral segmentation:Spectral segmentation: Shi and Malik 97 Yu and Shi 03 Weiss 99 Ng et al 01 Zelnik and Perona 05 Tolliver and Miller 06

3

Previous approaches to segmentation and matting

Unsupervised

Input Hard output Matte output

Supervised

0

1

July and Boykov01 Rother et al 04 Li et al 04

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Previous approaches to segmentation and matting

Unsupervised

Input Hard output Matte output

Supervised

0

1

Trimap interfaceTrimap interface: Bayesian Matting (Chuang et al 01) Poisson Matting (Sun et al 04) Random Walk (Grady et al 05)Scribbles interface:Scribbles interface: Wang&Cohen 05 Levin et al 06 Easy matting (Guan et al 06)

?

5

User guided interface

TrimapScribbles Matting result

6

Generalized compositing equation

iiiii BFI )1( 2 layers compositing

= x x+ 1 2L1L

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Generalized compositing equation

iiiii BFI )1( 2 layers compositing

= x x+ 1 2L1L

Ki

Kiii LLLI

iii ...2211

K layers compositing

= x x+

+ x x+3 4 4L3L

1 2 2L1L

Matting components

8

Generalized compositing equation

1...21 K

iii

“Sparse” layers- 0/1 for most image pixels

Matting components:

Ki

Kiii LLLI

iii ...2211

K layers compositing

= x x+

+ x x+

10 ki

1

3 4

2 2L

4L3L

1L

9

Automatically computed matting components

Input

1 2 3 4

8765

Unsupervised matting

10

Building foreground object by simple components addition

=+ +

11

Spectral segmentation

22/

),(ji CC

ejiW

WDL

j

jiWiiD ),(),(

Spectral segmentation: Analyzing smallest eigenvectors of a graph Laplacian L

E.g.: Shi and Malik 97 Yu and Shi 03 Weiss 99 Ng et al 01 Maila and shi 01 Zelnik and Perona 05 Tolliver and Miller 0612

Problem Formulation

= x x+ 1 2L1L

Assume a and b are constant in a small window

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Derivation of the cost function

14

Derivation

LJ T )(

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The matting Laplacian

LJ T )(

• semidefinite sparse matrix

• local function of the image:),( jiL

L

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The matting affinity

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The matting affinity

Color Distribution

Input

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Matting and spectral segmentation

Typical affinity function Matting affinity function

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Eigenvectors of input image

Input

Smallest eigenvectors 20

Spectral segmentationFully separated classes: class indicator vectors belong to Laplacian nullspace

General case: class indicators approximated as linear combinations of smallest eigenvectors

Null

Binary indicating

vectors

Laplacian matrix

21

Spectral segmentation

Fully separated classes: class indicator vectors belong to Laplacian nullspace

General case: class indicators approximated as linear combinations of smallest eigenvectors

Smallest eigenvectors- class indicators only up to linear transformation

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RZero eigenvectors

Binary indicating

vectors

Laplacian matrix

Smallest eigenvecto

rs

Linear transformati

on

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From eigenvectors to matting components

linear transformat

ion

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From eigenvectors to matting components

Sparsity of matting components

Minimize

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From eigenvectors to matting components

Minimize

Newton’s method

with initialization

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From eigenvectors to matting components

Smallest eigenvectors

Projection into eigs space kCTk mEE

....

K-means

..

kCmle

1) Initialization: projection of hard segments

2) Non linear optimization for sparse components26

Extracted Matting Components

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Brief Summary

LJ T )(

Construct Matting Laplacian

Smallest eigenvectors

Linear Transformation

Matting components

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Grouping Components

=+ +

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Grouping Components

Unsupervised matting User-guided matting

Complete foreground matte

=+ +

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Unsupervised matting

LJ T )(

Matting cost function

Hypothesis:Generate indicating vector b

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Unsupervised matting results

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User-guided matting Graph cut method

Energy function

Unary term Pairwise termConstrained components

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Components with the scribble interface

Components (our

approach)

Levin et al cvpr06

Wang&Cohen 05

Random Walk

Poisson 34

Components with the scribble interface

Components (our

approach)

Levin et al cvpr06

Wang&Cohen 05

Random Walk

Poisson 35

Direct component picking interface

=+ +

Building foreground object by simple components addition

36

Results

37

Quantitative evaluation

38

Spectral matting versus obtaining trimaps from a hard segmentation

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Limitations Number of eigenvectors

Ground truth matte Matte from 70 eigenvectors

Matte from 400 eigenvectors40

Limitations Number of matting components

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Conclusion Derived analogy between hard spectral

segmentation to image matting Automatically extract matting components

from eigenvectors Automate matte extraction process and

suggest new modes of user interaction

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