Tracking Image dengan Metode feature...

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Tracking Image Tracking Image dengandengan MetodeMetodefeature Lucasfeature Lucas--KanadeKanade

SumberSumber::

--Forsyth & Ponce Chap. 19Forsyth & Ponce Chap. 19

--TomashiTomashi, Lucas & , Lucas & KanadeKanade: : Good Feature to TrackGood Feature to Track

--StandfordStandford Vision & ModelingVision & Modeling

• Ulasan metode Lucas-Kanade + Implementasi dengan Matlab• Analisa metode Lucas-Kanade

• Support Maps / Layers:- Robust Norm- Layered Motion- Background Subtraction- Color Layers

AgendaAgenda

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-

Lucas-Kanade: Minimisasi fungsi: Image 1D

Intensitas

x

u ?F G

∑ −≈x

tx xFuxF 2))()((

∑ −+=x

xGuxFuE 2))()(()(

Linierisasi:

Spatial Gradient Temporal Gradient

∑∈

−++=ROIyx

xGvyuxFvuE,

2))(),((),(

∑∈

−+≈ROIyx

tyx yxFvyxFuyxF,

2)),(),(),((

Spatial Gradient Temporal Gradient

ROI

ROI

(u,v)

F G


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Minimisasi fungsi E(u,v):

0

0

=∂∂

=∂∂

v

Eu

E

=

∑∑

∑∑∑∑

yt

xt

yyx

yxx

FF

FF

v

u

FFF

FFF2

2

=>

C

D

=

v

u

=

v

uC

-1

D


• Step 0: Inisialisasi (dengan manual)

• Step 1: hitung: C dan D dan cari penyelesaian (u,v):

- Hitung image derivatives Fx,Fy,Ft

• Step 2: re-warp image G:

- Sub-pixel image interpolation

• Step 3: Loop:

- Ukur error / terminate

Impelementasi Lucas-Kanade

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A) Fx, Fy: Filter dengan Gaussian Derivative Kernel:




B) Ft: Finite Difference of Blurred F and G:


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• Step 1: hitung: C dan D dan cari penyelesaian

(u,v):


- Hitung dengan Gaussian kernel

(menggunakan coarse-to-fine strategy denganpengurangan sigma)


•Step 2: re-warp image G:

- Sub-pixel image interpolation


Operasi Warping gunakan fungsiinterp2 dari Matlab

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• Step 3: Loop:

- Ukur error / terminate


2|||| tFperhatikan:

AnalisaAnalisa GrafisGrafis metodemetode LucasLucas--KanadeKanade

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Minimisasi fungsi E(u,v):

0

0

=∂∂

=∂∂

v

Eu

E

=

∑∑

∑∑∑∑

yt

xt

yyx

yxx

FF

FF

v

u

FFF

FFF2

2

=>

C

D

=

v

u

=

v

uC

-1

D

Lucas-Kanade: problem singulariti

=

∑∑

∑∑∑∑

yt

xt

yyx

yxx

FF

FF

v

u

FFF

FFF2

2


=

00

00C

Fx=0, Fy=0

8

=

∑∑

∑∑∑∑

yt

xt

yyx

yxx

FF

FF

v

u

FFF

FFF2

2


=

00

00C

Fx=0, Fy=0

=

00

0aC

Fy=0

=

∑∑

∑∑∑∑

yt

xt

yyx

yxx

FF

FF

v

u

FFF

FFF2

2


=

00

00C

Fx=0, Fy=0

=

00

0aC

Fy=0

)0,1()( eCeig =

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Lucas-Kanade: Aperture Problem

=

00

00C

Fx=0, Fy=0

)0,1()( eCeig =

Lucas-Kanade: Aperture Problem

Bergen et al.

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∑∑∑∑

2

2

yyx

yxx

FFF

FFF

Aperture Problem: Bisa diatasi ???

)2,1()( eeCeig =

- Hindari sebisa mungkin !

- Gunakan nilai Eigenvalues untuk inisialisasi “Good Features”

(lihat paper “Good Features to track” Shi-Tomasi)

- Lokasi Good Feature berada pada: min(eig1,eig2) > a


“Good Features” (Shi-Tomasi)

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∑∑∑∑

2

2

yyx

yxx

FFF

FFF


)2,1()( eeCeig =

- coba di Hack !

- regularisasi C:

+=

λλ0

0CCreg

∑∑∑∑

2

2

yyx

yxx

FFF

FFF


)2,1()( eeCeig =

-

- Simoncelli et al 1991:

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∑∑∑∑

2

2

yyx

yxx

FFF

FFF


)2,1()( eeCeig =

- Tambah Aperture (window feature) !

- Coarse-to-fine Pyramids (Bergen et al, Simoncelli)

∑∑∑∑

2

2

yyx

yxx

FFF

FFF


)2,1()( eeCeig =

- Tambah Aperture (window feature) !

- akibat: integrasi ROI lebih besar -> motion model jadi lebih komplex

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Pengembangan Lucas-Kanade secaraAffine

+

=

++

6

5

43

21

a

a

y

x

aa

aa

vy

ux

Affine Motion Model:

- 2D Translation- 2D Rotation- Scale in X / Y- Shear

Matlab demo ->

Pengembangan Lucas-Kanade secaraAffine

Affine Motion Model -> digunakan pada Lucas-Kanade:

Matlab demo ->

∑∈

−++=ROIyx

yxGvyuxFvuE,

2)),(),((),(

321 ayaxa ++ 654 ayaxa ++

D

a

a

C =

6

1

...

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• Support Maps / Layers:- Robust Norm- Layered Motion- Background Subtraction- Color Based Tracking

Support Maps / LayersSupport Maps / Layers

- L2 Norm vs Robust Norm

- Bahaya dari fitting secara least square:

Akibat adanya Outliers (gangguan pixel luar) menyebabkan square error menjadi sangat besar

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L2

D




L2 robust

D D

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- Robust Norm -- baik untuk menangani outliers

- nonlinear optimization

robust

D


- Black-Jepson-95

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- Layered Motion (Jepson/Black, Weiss/Adelson, …)


- Kasus spesial dari Layered Motion:

- Background substraction

- Outlier rejection (== robust norm)

- Kasus sederhana: Tiap Layer punya warna seragam

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- Color Layers:

P(skin | F(x,y))

Tracking Image dengan Metode feature...

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