1

Lecture

Computing Optical Flow

Horn&Schunck Optical Flow

0),,(

=∂

∂+

∂

∂+

∂

∂=

t

f

dt

dx

y

f

dt

dx

x

f

dt

tyxdf

brightness constancy eq0=++ tyx fvfuf

Sequence Image ),,( tyxf

2

Horn&Schunck Optical Flow

f x y t f x y tf

xdx

f

ydy

f

tdt( , , ) ( , , )= + + +

∂∂

∂∂

∂∂

Taylor Series),,(),,( dttdyydxxftyxf +++=

brightness constancy eq0=++ tyx fvfuf

0=++ dtfdyfdxf tyx

Interpretation of optical flow eq

y

t

y

x

f

fu

f

fv --=

df

f ft

x y

=+2 2

d=normal flowp=parallel flow

Equation of st.line

0=++ tyx fvfuf

3

Horn&Schunck (contd)

variational calculus{( ) ( )}ÚÚ + + + + + +f u f v f u u v v dxdyx y t x y x y

2 2 2 2 2l

( ) ( )

( ) (( )

f u f v f f u

f u f v f f v

x y t x

x y t y

+ + + =

+ + + =

l

l

D

D

2

2

0

0

( ) ( )

( ) (( )

f u f v f f u u

f u f v f f v vx y t x av

x y t y av

+ + + - =

+ + + - =

l

l

0

0

u u fP

D

v v fP

D

av x

av y

= -

= -

P f u f v f

D f f

x av y av t

x y

= + +

= + +l 2 2

discrete version

min

yyxx uuu +=D2

Algorithm-1

• k=0

• Initialize u vK K

u u fP

D

v v fP

D

Kavk

x

avK

y

= -

= -

-

-

1

1

P f u f v f

D f f

x av y av t

x y

= + +

= + +l 2 2

• Repeat until some error measure is satisfied(converges)

4

Derivative Masks

xf

image second11

11

imagefirst 11

11

˙˚

˘ÍÎ

È

-

-

˙˚

˘ÍÎ

È

-

-

yf

image second11

11

imagefirst 11

11

˙˚

˘ÍÎ

È --

˙˚

˘ÍÎ

È --

tf

image second11

11

imagefirst 11

11

˙˚

˘ÍÎ

È

˙˚

˘ÍÎ

È

--

--

Synthetic Images

5

Results

One iteration 10 iterations

4=l