Download Formula collection for Digital Image Processing

Formula collection for
Digital Image Processing
Maria Magnusson, [email protected]
May 2013
1
1-D Continuous Fourier Transforms
Definition:
Signal domain, t ∈ R
∞
G(f )ej2πf t df
g(t) =
Frequency domain,f ∈ R
∞
G(f ) =
g(t)e−j2πf t dt
Real signal:
Linearity:
g(t) real
ag1 (t) + bg2 (t)
G(−f ) = G∗ (f )
aG1 (f ) + bG2 (f )
−∞
e−j2πf a G(f )
Translation, time: g(t − a)
Translation, freq:
Scaling:
Convolution:
Correlation:
Multiplication:
Derivative:
Dirac impulse:
Impulse train:
ej2πat g(t)
g(at)
(g ∗ h)(t)
(g2h)(t)
g(t) · h(t)
d
g(t)
dt
δ(t)
δ(t − nΔ)
sΔ (t) =
unit step:
u(t) =
Rectangle puls:
Π(t) =
Sinc function :
sinc(t)
Triangle puls :
Λ(t) =
n
1, t ≥ 0
0, t < 0
1, |t| ≤ 0.5
0, |t| > 0.5
1 − |t|, |t| ≤ 1
0,
|t| > 1
Cosinine wave:
cos 2πf0 t
2 Dirac impulses: (δ(t − Δ) + δ(t + Δ)) /2
Sine wave:
sin 2πf0 t
2 Dirac impulses: (−δ(t − Δ) + δ(t + Δ)) /2
Constant:
1
Gauss:
Miscellaneous:
−∞
2
G(f − a)
(1/|a|) · G(f /a)
G(f ) · H(f )
G∗ (f ) · H(f )
(G ∗ H)(f )
j2πf G(f )
1
1
1 k
δ f−
s1/Δ (f ) =
Δ
Δ k
Δ
1
1
+ δ(f )
j2πf
2
sin(πf )
sinc(f ) =
πf
Π(f )
sinc2 (f )
(δ(f + f0 ) + δ(f − f0 )) /2
cos(2πΔf )
j (δ(f + f0 ) − δ(f − f0 )) /2
j sin(2πΔf )
δ(f )
2
e−πt
e−at u(t)
te−at u(t)
e−πf
1/(a + j2πf )
1/(a + j2πf )2
e−a|t|
2a/(a2 + (2πf )2 )
2πf0
(a + j2πf )2 + (2πf0 )2
a + j2πf
(a + j2πf )2 + (2πf0 )2
e−at sin(2πf0 t)u(t)
e−at cos(2πf0 t)u(t)
2
2-D Continuous Fourier Transforms
Definition:
Spatial domain, x, y ∈ R
Frequency domain, u, v ∈ R
f (x, y) =
F (u, v) =
∞
∞
j2π(xu+yv)
F (u, v)e
dudv
f (x, y)e−j2π(xu+yv) dxdy
Real signal:
Linearity:
f (x, y) real
af1 (x, y) + bf2 (x, y)
F (−u, −v) = F ∗ (u, v)
aF1 (u, v) + bF2 (u, v)
Translation, time:
f (x − a, y − b)
e−j2π(au+bv) F (u, v)
Translation, freq:
Scaling:
Convolution:
Correlation:
Multiplication:
ej2π(ax+by) f (x, y)
f (ax, by)
(f ∗ g)(x, y)
(f 2g)(x, y)
f (x, y) · g(x, y)
∂
f (x, y)
∂x
∂
f (x, y)
∂y
∇2 f (x, y) =
2
∂
∂2
f (x, y)
+
∂x2 ∂y 2
x
f (Ax), x =
y
x
f (Rx), x =
y
F (u − a, v − b)
(1/|ab|) · F (u/a, v/b)
F (u, v) · G(u, v)
F ∗ (u, v) · G(u, v)
(F ∗ G)(u, v)
−∞
Derivative in x:
Derivative in y:
Laplace:
General scaling:
Rotation 1:
−∞
j2πu · F (u, v)
j2πv · F (u, v)
− 4π 2 (u2 + v 2 ) · F (u, v)
1
F ((A−1 )T u), u =
| det A|
u
F (Ru), u =
v
f (r, θ + θ0 )
x = r cos θ, y = r sin θ
Separable function: f (x, y) = g(x) · h(y)
Dirac impulse:
δ(x, y) = δ(x) · δ(y)
Box:
Π(x, y) = Π(x) · Π(y)
Bend pyramid:
Λ(x, y) = Λ(x) · Λ(y)
F (ω, ϕ + θ0 )
u = ω cos ϕ, v = ω sin ϕ
F (u, v) = G(u) · H(v)
1
sinc(u) · sinc(v)
sinc2 (u) · sinc2 (v)
Rotation 2:
Gauss:
e−π(x
2 +y 2 )
2
= e−πx · e−πy
2
e−π(u
2 +v 2 )
2
= e−πu · e−πv
2
u
v
3
Definitions, Properties and Relations
DFT and IDFT, 1D and 2D:
F [k] =
N
−1
f [n] · e
−j 2π
nk
N
,
n=0
F [k, l] =
f [n, m] =
−1
N
−1 M
f [n, m] · e−j2π(nk/N +ml/M ) ,
n=0 m=0
−1
N
−1 M
1
NM
N −1
2π
1 f [n] =
F [k] · ej N nk
N k=0
F [k, l] · ej2π(nk/N +ml/M )
k=0 l=0
Symmetric DFT and IDFT, 1D and 2D:
M/2−1
F [k] =
f [m] · e
−j 2π
km
M
,
m=−M/2
M/2−1
F [k, l] =
1
f [m] =
M
M/2−1
2π
F [k] · ej M km
k=−M/2
N/2−1
f [m, n] · e−j2π(km/M +ln/N ) ,
m=−M/2 n=−N/2
1
f [m, n] =
MN
M/2−1
N/2−1
F [k, l] · ej2π(km/M +ln/N )
k=−M/2 l=−N/2
Sifting property, 1D and 2D:
∞
x(t )δ(t − t0 ) dt = x(t0 ),
−∞
∞ ∞
f (x , y )δ(x − x0 , y − y0 ) dx dy = f (x0 , y0 )
−∞
−∞
Convolution with a shifted dirac impulse, 1D and 2D:
∞
x(t − t )δ(t − t0 ) dt = x(t − t0 )
x(t) ∗ δ(t − t0 ) =
−∞
∞ ∞
f (x, y) ∗ δ(x − x0 , y − y0 ) =
f (x − x , y − y )δ(x − x0 , y − y0 ) dx dy −∞
−∞
= f (x − x0 , y − y0 )
Parseval’s formula, 1D and 2D:
∞
∞
∗
x(t)y (t) dt =
X(f )Y ∗ (f ) df
−∞
∞ ∞
−∞
∞ ∞
∗
f (x, y)g (x, y) dxdy =
F (u, v)G∗ (u, v) dudv
−∞
−∞
−∞
−∞