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123
Transform-based features: Fourier Descriptors
- In the discrete case:
- Then:
- Fourier descriptors depend on the starting point
- For 64x64 images, it has been shown that 5 coefficients are enough to
discriminate between “2” and “Z”
∑
= − − −
∆Φ = Φ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = ∆Φ + =
j k j j j j j j j j j j
s s s x s x s y s y s s iy s x s z
1 1 1 1
) ( ) ( ) ( ) ( ) ( ) ( tan ) ( ) ( ) ( ) (
∑ ∑ ∑
= = =
∆Φ = ∆Φ − = ∆Φ − − =
m k k n m k k n m k k
L nl k n b L nl k n a k l L a
1 1 1
2 cos ) ( 1 2 sin ) ( 1 ) ( 1 π π π π π
Feature Extraction
124
Transform-based features: Fourier Descriptors
. when ) ( ) ( ˆ ), ( ) ( ˆ and length contour the is where 2 sin 2 cos ) ( ˆ 2 sin 2 cos ) ( ˆ
1 1
∞ → ≡ ≡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + =
∑ ∑
= =
N t y t y t x t x T T t n d T t n c C t y T t n b T t n a A t x
N n n n N n n n
π π π π
[ ] [ ] [ ] [ ]
pixels contour
number , , , , , / 2
sin sin 2 cos cos 2 sin sin 2 cos cos 2
1 1 2 2 1 1 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
m t t T t t y x t y y y x x x T t n t y n T d t y n T c t x n T b t x n T a
m j j m i j j i i i i i i i i i i i i m i i i n i i m i i i n i i m i i i n i i m i i i n
∑ ∑ ∑ ∑ ∑ ∑
= = − − − = − = − = − =
∆ = = ∆ = ∆ + ∆ = ∆ − = ∆ − = ∆ = − ∆ ∆ = − ∆ ∆ = − ∆ ∆ = − ∆ ∆ = π φ φ φ π φ φ π φ φ π φ φ π
Invariance to starting point
The phase shift with respecte the main axis is computed and the coefficients are rotated according to this angle
2 1 2 1 2 1 2 1 1 1 1 1 1
) ( 2 tan 2 1 d c b a d c b a − + − + =
−
θ
In the discrete case
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡
1 1 1 1 * * * *
cos sen sen cos θ θ θ θ n n n n d c b a d c b a
n n n n n n n n
Feature Extraction
Elliptic Fourier Descriptors
F.P. Kuhl, C.R. Giardina: Elliptic fourier features of a closed contour. Comput. Vis. Graphics Image Process, vol. 18, pp. 236-258, 1982
