Sought: a generalization of the previously introduced similarity- … · 2011. 3. 7. · H....

ME-II, Kap. 4eH. Burkhardt, Institut für Informatik, Universität Freiburg 1

Affine invariant Fourier descriptors

Sought: a generalization of the previously introduced similarity-invariant Fourier descriptors


Geometrical transformations

translationscongruenciessimilarities

(preserves angles)

affine mappings

(preserves parallelisms)

central projections


Real, vectorial, parametric description of a closed contour

u

v

x(t) t=s

( )( )

( )u t

tv t⎡ ⎤

= ⎢ ⎥⎣ ⎦

x

possible parameterization:t=s (arc length)


Affine mapping of a contour

0 0

11 12 1

21 22 2

0

0 0

( ) ( ( ))

with: det( ) 0

Additonally starting point translation:( , )

In case arc length is used for parameterization: ( , ) ( )

Thus 7 degre

t t t

A A bA A b

t t

t t t t

τ

τ τ

= +

⎡ ⎤ ⎡ ⎤= = ≠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= +

x Ax b

A b A

es of freedom resultfor affine mapping!


Equivalent structures• In the equivalence class of similar maps with

equivalence relation ~ applies:

circle1 ~ circle2circle ellipseparallelogramm rectangle square

• In the equivalence class of affine maps though applies:

circle ~ ellipseparallelogramm ~ rectangle ~ square

but: circle square


Developing the contour as a periodic function into a Fourier series

2 /

2 /1

0

( )( )

( )

with the complex valued Fourier coefficient vector:

= = ( )

kj kt T

kk

Tk j kt T

k Tk t

u tt e

v t

Ut e dt

V

π

π

=+∞

=−∞

−

=

⎡ ⎤= =⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

∑

∫

x X

X x


Choosing a parameterization, which guarantees a linear (homogeneous) mapping t 0→t with the effect of the

affine map A

0 0( , ) ( )t t tμ= ⋅A A

The arc length does not meet this requirement!


Non-linear map over arc length with shear of objects

t0

t

t

t0

t

T0

verticalcompression

horizontaldilation

T0/2T0/4


Choosing an appropriate parameterization

2nd possibility: Using differential invariants of first order and additionally area centre of gravity xs (semi-differential approach). Needed are:

1st possibility: Using differential invariants of second order in form of affine length. Needed are:

[ ],x x

[ ],x x


The outer (tensor) product and its geometric meaning

The outer product of two vectors [x,y] is a (signed) real number, which corresponds to the area of the included parallelogram (or: 2 times the area of the triangle)

x

y

F

Fϕ

sin( )h ϕ= xϕ

h

y

x12/ 2 sin( )F h ϕΔ = ⋅ =y y x

[ ]

[ ]

1 11 2 2 1

2 2

outer product:

, det( , ) ( ) sin( )

, 2

x yx y x y

x y

F

ϕ

Δ

= = = − =

= ⋅

x y x y x y

x y


Results from differential geometryWe differentiate the parameterization t adequately from the arc length s.For an analytical curve (which is differentiable arbitrary number of times) applies by

means of [x,y]:

( ) ( 1)

( ) ( 1)

0

( ) ( 1) 0 0(2 1) (2 1)

0 0(2 1) (2 1)

( )

(1)( )

(2)

( ) , ,

= , n 1

( ) tangential vectorwith: e.g. is

( ) ( )

n n

n n

n nn n

n n

dt

nn

n

dt s ds ds

ds

sd xds s s

μ

κ

+

+

++ +

+ +

⎡ ⎤⎡ ⎤= =⎣ ⎦ ⎣ ⎦

⎡ ⎤ ≥⎣ ⎦

==

=

A

x x Ax Ax

A x x

x xx

x n curvature vector

0 ( )dt dtμ⇒ = ⋅A( )sx( ) tangentsx

( ) normalsn ( ) = (s) =1sx n


1st possibility: Using differential invariants of second order in form of affine length

[ ]

[ ]0

0 0

( )

0 0

, ( ) affine length

( )with: ( arc length)

it applies: , ,

and thus: ( )

t t

t ds s ds

d s sds

ds ds

t t t

μ

κ

μ

= =

=

⎡ ⎤= ⋅ ⎣ ⎦

= =

∫ ∫

∫ ∫A

x x

xx

x x A x x

A A

33

C C

3 3 3

C C

3

problem for polygons: the second derivative disappears along the line and the first derivative is non-continuous in the corners and therefore the 2nd derivative is not defined!


2nd possibility: using differential invariants of first order and additionally normalization by center of gravity (COG) xs (semi-differential approach)

0

0

( )det( )

dF dFdF

α= ⋅

= ⋅

AA

The vector starting from the COG to the contour are used for parameterization (outer product of pointer and tangential vector)

[ ]0

0 00

0 0

,

,

( ) ( )

dF

s

dF

s

t F ds

ds

F tμ μ

= = −

⎡ ⎤= −⎣ ⎦

= ⋅ = ⋅

∫

∫

x x x

A x x x

A A

C

C

xs0 xs

ds0dF0

dsdF

A

[ ] [ ]

due to: det( , ) det( ( , )) det(

) det( , )

, det( ) , =

⇒ =

⋅ = ⋅

⋅

Ax Ay A x y A x yAx Ay A x y


It applies: The affine transformation maps area COG to each other and areas to each other in a constant relation!

The effect of the transformation is eliminated due to the normalization to the COG

The outer product is signed! In order to avoid ambiguities the amplitude of area increment |dF| is chosen and therefore a monotonic increasing parameterization!


Affine invariant Fourier descriptors for polygons

xN=x0

x1=xN+1

x2

x3

xN-1=x4

v

u

Fi

[ ]0 1polygon: , , , iN ii

uv⎡ ⎤

= ⎢ ⎥⎣ ⎦

x x x x…

area element


Affine invariant Fourier descriptors of polygons

[ ]

[ ] [ ]

1det( , )1 1

1 1 1 1 10 01 1

3 31 1

1 10 0

0

11 1 12

area center of gravity of the whole traverse:

, ( ) ( )( )

, ,

parameter: 0

i i

i

N N

i i i i i i i i i ii i

s N N

i i i ii i

i i i i i i

F

u v u v

t

t t u v u v

+− −

+ + + + += =

− −

+ += =

+ + +

+ − += =

=

′ ′ ′ ′= + −

∑ ∑

∑ ∑

x x

x x x x x xx

x x x x

0,1, , 1 N

ss

s

i N T t

u uuv vv

= − =

′ −⎡ ⎤⎡ ⎤′ = = − = ⎢ ⎥⎢ ⎥′ −⎣ ⎦ ⎣ ⎦x x x

…


Fourier coefficients

2

10 1

0 1 1200

11

, 1 , 1(2 )0 1

1

1 , 120

,

( )( )

( ) ( )(1 ( ))( )

( ) ( ) for 0

with:

N

i i i iTi

Nk i iT

k k i k i i ikik i i

Nj

i i k i i iki

k i

Ut t

V

Ue e t t

V t t

e t t k

e

π

π

δ

δ

−

+ +=

−+

+ += +

−

+ +=

⎡ ⎤= = + −⎢ ⎥⎣ ⎦

′ ′⎡ ⎤ −= = − − −⎢ ⎥ −⎣ ⎦

′ ′+ − − ≠

=

∑

∑

∑

X x x

x xX

x x

2 /

11

1

1 if (planar increase = 0)( )

0 if first part transforms continuitiessecond part transforms discontinuities (switch

ij kt T

i ii i

i i

e

t tt t

t t

π

δ

−

++

+

=⎧− = ⎨ ≠⎩

ing through -operator)δ


Fourier coefficients of affine distorted contours

0 0

0 0 0

0

2 /

( ) ( )( ( ))

( ( ))

thus follows:

0 (eliminates translation)

k

k

kk k

j T

t tt

t

z k

z e πτ

τ

−

= + +=

=

= ≠

=

x Ax bX x

X x

X AX

F

F


A-invariants (τ=0)

0 0 0

with:

det , det( ) det , det( )

from that result complete and minimal invariants:

det , det( )det ,

kp k p k p kp

k pkpk

pp p p

Q

∗ ∗

∗

∗

⎡ ⎤ ⎡ ⎤Δ = = ⋅ = ⋅Δ⎣ ⎦ ⎣ ⎦

⎡ ⎤Δ ⎣ ⎦= = =Δ ⎡ ⎤⎣ ⎦

X X A X X A

X X AX X det( )A

0 0 0* 0 00

0 0 0 0 0

0 0

0 1, 2, 3,

for 0 results though:

kp k p k pk

pp p p p p

k p kk k kp

U V V UQ

U V V U

p const k

Q Q z Q z

τ

∗

∗ ∗

−

Δ −= =

Δ −

= ≠ = ± ± ±

≠

= ⋅ = ⋅

…

has to be eliminated


Additional starting point invariance(τ≠0)

(special solution of second order)( ) ( )

arg( )

with:

( , ) are integral solutions ofthe following linear diophantic equation:

( ) ( ) 1 0a solution exists for:gcd( , ) 1

(solution with extended Euclid

k

k p k pk k q r

j Qk k k k

I Q

Q Q Q e

q p r p

q p r p

λ η

λ η

λ η

− −= Φ Φ

= Φ =

− + − + =

− − =ean algorithm)

These invariants are also complete and minimal!The approach realizes also a compensation of phases, which are unknown mod 2π.


1 16 7

for example:7, 6, 15

gcd(5,6) 16

5 6 1 0

holds for: 1, 1k k

k k

r q pq p

p

I

r

Q

λ η

λ η− −

= = =

− = ⎫=⎬− = ⎭

⇒ ⋅ + ⋅ + =

= = −

⇒ = Φ Φ

Also in this case one representative from the equivalence class results from the invariants, i.e. a contour in a certain location and view!Also a linear complexity results for a constant number of Fourier descriptors :

O(N)


Properties of Fourier seriesSince the parametrical description of contours still contains

discontinuities (polygon section in radial direction with planarincrease 0), the magnitude of the FC is proportional to 1/n and thus tend to 0, which is slower than for continuous functions.

|cn|

n

1/ n∼


a b c

0 2 2 5 5 2 2 7 7(5) 0 3 2 mit: 0,5

0 0 5 5 7 7 9 9 11 11 1 3⎡ ⎤ ⎡ ⎤′ = ⋅ = =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

F A F F A

Affine invariant Fourier descriptors

n a b c-5 0.107 0.004 0.146 -0.001 -0.086 0.284 0.075 0.094 0.075 -4 -0.006 -0.034 -0.047 -0.058 -0.086 0.311 0.057 0.084 0.057 -3 -0.036 -0.055 -0.001 -0.074 0.126 0.481 0.029 0.014 0.029 -2 0.283 -0.477 0.227 -0.490 -1.560 0.392 0.315 0.290 0.315 -1 -0.263 -0.779 -0.178 -0.733 -5.370 0.661 0.000 0.000 0.000 0 --- --- --- --- --- --- --- --- --- 1 -1,120 -1,730 -1,090 -1,650 0,743 -7,330 1,000 1,000 1,0002 -0.024 -0.375 -0.064 -0.467 0.927 -0.751 0.229 0.252 0.229 3 -0.169 -0.104 -0.191 -0.096 0.702 0.030 0.104 0.111 0.104 4 -0.081 0.182 -0.063 0.175 0.476 -0.385 0.119 0.126 0.119 5 0.066 -0.020 0.057 0.014 0.046 -0.201 0.061 0.059 0.061

Invariantena b c

Fourierkoeffizientena b c


Power spectra of the difference of the invariants of both objects

difference for real affine map considering the quantization error

difference for real structure changes

Affine invariant Fourier descriptorsGeometrical transformationsReal, vectorial, parametric description of a closed contourAffine mapping of a contourEquivalent structuresDeveloping the contour as a periodic function into a Fourier seriesChoosing a parameterization, which guarantees a linear (homogeneous) mapping t 0t with the effect of the affine map A Non-linear map over arc length with shear of objectsChoosing an appropriate parameterizationThe outer (tensor) product and its geometric meaningResults from differential geometryAffine invariant Fourier descriptors for polygonsAffine invariant Fourier descriptors of polygonsFourier coefficients Fourier coefficients of affine distorted contoursA-invariants (=0)Additional starting point invariance�(0)�(special solution of second order)Properties of Fourier seriesPower spectra of the difference of the invariants of both objects

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Sought: a generalization of the previously introduced similarity- … · 2011. 3. 7. · H....

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