Integral and Discrete Transforms in Image Processing
Z-transform
David Svoboda
email: svoboda@fi.muni.cz
Centre for Biomedical Image Analysis
Faculty of Informatics, Masaryk University, Brno, CZ
October 24, 2022
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Outline
1 Definition
2 Properties
3 Applications
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1 Definition
2 Properties
3 Applications
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Z-Transform
Definition
(bilateral) Z-Transform
F(z) =
∞
n=−∞
f (n)z−n
(unilateral) Z-Transform
F(z) =
∞
n=0
f (n)z−n
Notice: Here z ∈ C, i.e. for z = eiω we get a special case of Z-transform,
which is DFT.
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Z-Transform
Definition
Forward transform
converts discrete series into continuous signal (Z-plane)
F(z) =
∞
n=−∞
f (n)z−n
Inverse transform
If F(z) is a polynomial, i.e. F(z) =
∞
k=−∞
ckz−k, then
f (n) = Z−1
(F(z)) =
∞
k=−∞
ckδ(n − k)
Notice: Forward transform applied to discrete signals or linear filters
always creates a polynomial with one variable z.
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Z-Transform
Some examples
Z-transform of a constant signal/averaging filter (left aligned):
f (n) = [1, 1, 1, 1, 1]
F(z) = 1 + z−1
+ z−2
+ z−3
+ z−4
Z-transform of a constant signal/averaging filter (centered):
f (n) = [1, 1, 1, 1, 1]
F(z) = z+2
+ z+1
+ 1 + z−1
+ z−2
Z-transform of Sobel filter:
f (n) = [−1, 0, 1]
F(z) = −z+1
+ z−1
Z-transform of any random filter:
f (n) = [1, 3, 1, 4, 2]
F(z) = z+1
+ 3 + z−1
+ 4z−2
+ 2z−3
Notice: With Z-transform, we always pay attention to origin!
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Z-Transform
An inverse transfom
Inverse transform for linear filters is an easy step:
Signal of length 5:
F(z) = 3z3
+ z − 3z−1
f (n) = Z−1
(F(z)) = [3, 0, 1, 0, −3]
Gaussian filter in Z-domain:
F(z) =
1
4
z +
1
2
+
1
4
z−1
f (n) = Z−1
(F(z)) =
1
4
[1, 2, 1]
Step function in Z-domain
F(z) = z−1
+ z−2
+ z−3
+ z−4
f (n) = Z−1
(F(z)) = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
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Z-Transform
Relationship to other transforms
Important notes:
green circle (z = eiφ ⇒ |z| = 1) reduces Z-transform simply to
discrete Fourier transform
DC (direct current) term is DC term from DFT
Re
Im
r=1
ω
Z−plane
DC term
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1 Definition
2 Properties
3 Applications
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Z-Transform – Properties
Delay/Shift
Z{f (n − k)} = F(z)z−k
An example:
f (n) = [1, 2, 3, 4]
g(n) = [1, 2, 3, 4]
h(n) = [1, 2, 3, 4]
F(z) = 1 + 2z−1
+ 3z−2
+ 4z−3
G(z) = z1
+ 2 + 3z−1
+ 4z−2
H(z) = z2
+ 2z + 3 + 4z−1
F(z) = G(z)z−1
F(z) = H(z)z−2
G(z) = H(z)z−1
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Z-Transform – Properties
Convolution theorem
Z{f (n) ∗ g(n)} = F(z) · G(z)
Corollary: Convolution in time domain becomes a polynomial
multiplication in Z-domain:
Z([3, 5] ∗ [−1, 1]) = 3 + 5z−1
· −1 + z−1
An example:
[3, 5] ∗ [−1, 1] = [−3, −2, 5] /Z(·)
Z([3, 5] ∗ [−1, 1]) = Z([−3, −2, 5])
Z([3, 5]) · Z([−1, 1]) = Z([−3, −2, 5])
(3 + 5z−1
) · (−1 + z−1
) = −3 − 2z−1
+ 5z−2
−3 − 2z−1
+ 5z−2
= −3 − 2z−1
+ 5z−2
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Z-Transform – Properties
Convolution theorem - cont’d
time domain Z-domain
separation of convolved signals polynomial factorization
An example (Gaussian decomposition):
[1, 2, 1] = Z−1
z + 2 + z−1
= Z−1
1 + z−1
· (z + 1)
= Z−1
1 + z−1
∗ Z−1
[z + 1]
= [1, 1] ∗ [1, 1]
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Z-Transform – Properties
Linearity
Z{af (n) + bg(n)} = aF(z) + bG(z)
An example:
f ′
(n) = f (n) ∗ [1, −1, 2]
f ′
(n) = 2f (n) − f (n + 1) + f (n + 2) /Z(·)
Z(f ′
(z)) = Z(2f (n)) − Z(f (n + 1)) + Z(f (n + 2))
F′
(z) = 2F(z) − F(z)z1
+ F(z)z2
F′
(z) = F(z)[2 − z1
+ z2
]
Notice: Keep in mind that: Z([1, −1, 2]) = z2 − z1 + 2
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Z-Transform – Properties
Symmetry
Z{f (−n)} = F
1
z
See the relationship betwenn time-revesed signals:
f (n) = [1, 2, 3, 4, 5, 6]
F(z) = 1 + 2z−1
+ 3z−2
+ 4z−3
+ 5z−4
+ 6z−5
⇓
f (−n) = [6, 5, 4, 3, 2, 1]
F(z−1
) = 1 + 2z1
+ 3z2
+ 4z3
+ 5z4
+ 6z5
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Z-Transform – Properties
Signal upsampling
Let L be a positive integer and
h(n) =
f (n/L) n/L is an integer
0 otherwise
Then
H(z) = F(zL
)
Derivation:
H(z) =
n=...,−L,0,L,...
f (n/L)z−n
=
m=...,−1,0,1,...
f (m)z−mL
=
m=...,−1,0,1,...
f (m) zL
−m
= F(zL
)
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Z-Transform – Properties
Signal decomposition with downsampling
F(z) = Feven(z2
) + z−1
Fodd (z2
)
An example:
f (n) = [a, b, c, d, e, f ]
feven(n) = [a, c, e]
fodd (n) = [b, d, f ]
F(z) = a + bz−1
+ cz−2
+ dz−3
+ ez−4
+ fz−5
Feven(z) = a + cz−1
+ ez−2
Fodd (z) = b + dz−1
+ fz−2
Feven(z2
) = a + cz−2
+ ez−4
Fodd (z2
) = b + dz−2
+ fz−4
Feven(z2
) + z−1
Fodd (z2
) = a + bz−1
+ cz−2
+ dz−3
+ ez−4
+ fz−5
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Z-Transform – Properties
Signal decomposition without downsampling
F(z) =
1
2
[F(z) + F(−z)] +
1
2
[F(z) − F(−z)]
where
Feven(z2
) =
1
2
[F(z) + F(−z)] . . . even component
Fodd (z2
) =
1
2
[F(z) − F(−z)] . . . odd component
An example:
f = [a, b, c, d, e, f ] /Z(·)
F(z) = a + bz−1
+ cz−2
+ dz−3
+ ez−4
+ fz−5
EVEN =
1
2
[F(z) + F(−z)] = a + cz−2
+ ez−4
= Z−1
[a, 0, c, 0, e, 0]
ODD =
1
2
[F(z) − F(−z)] = bz−1
+ dz−3
+ fz−5
= Z−1
[0, b, 0, d, 0, f ]
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Z-Transform
Transfer function
Definition
Let f and h be input and output signals, respectively, and let g be a linear
filter such that h = f ∗ g, then Z(g) is called a transfer function.
An example:
h = f ∗ [1, 2, 1] /Z(·)
H(z) = F(z) · z + 2 + z−1
H(z)
F(z)
= z + 2 + z−1
G(z) =
H(z)
F(z)
= z + 2 + z−1
Notice: Transfer function expresses the frequency response of selected linear filter.
In optics, G(z), when assigning z = eiφ
, is called a optical transfer function
(OTF) and its time-domain counter part is a point spread function (PSF).
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1 Definition
2 Properties
3 Applications
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Z-Transform
Applications
The areas that rely on Z-transform
Linear recursive filters
Optimization of wavelet transform
Understanding and frequency analysis of linear filters
. . .
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Bibliography
Strang G., Nguyen T. Wavelets and Filter Banks,
Wellesley-Cambridge Press, 1997, ISBN 0-9614088-7-1
Steven W. Smith. The scientist and engineer’s guide to digital signal
processing. California Technical Publishing, USA. 1997
https://www.youtube.com/watch?v=B4IyRw1zvvA
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You should know the answers . . .
Describe the relationship between Fourier transform and Z-transform.
What is the Z-transform of signal [1, 2, 4, 3, 20, −5, 23]?
What is the Z-transform of Gaussian filter?
How is the time delay expressed with Z-transform?
Is it possible to perform inverse Z-transform for a linear filter?
How is the signal upsampling expressed with Z-transform?
What is a transfer function?
What does the polynomial z − 1
z mean?
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