while for grayscale discrete images
I ⊕ E ¼ max
y ∈ B
Ixþ y
ð
ÞþBy ðÞ
ðÞ
(20)
In the continuous approximation case, the minimum and maximum should be
replaced by infimum and supremum, respectively. Consider the set X. The infimum
inf, or the greatest lower bound, is the greatest number, which is not necessarily in X
but is smaller (or equal) than all members of X. For a finite set, the infimum
coincides with the minimum. The supremum sup, or the least upper bound, is the
smallest number, which is not necessarily in X but is greater (or equal) than all
members of X. For a finite set, the supremum coincides with the maximum.
A.2 Some useful Fourier transforms
The concept of frequency and the decomposition of waveforms into elementary
“harmonic” wave motions first arose in the context of music and sound. The Fourier
transform and its inverse in the continuous domain are defined as
U k
ð Þ¼Fu x
ð Þ¼
ðð
�∞
e
�2πik�x ux ðÞ dx
2 ⇔u x
ð Þ¼F
�1 U k
ð Þ¼
ðð
�∞
e
2πik�x Uk ðÞ dk
2 (21)
The reader is directed to the book of [27] for an introduction on the topic.
The Fourier transform of the Gaussian is given by
~
Gf ð Þ¼e
�2 π
2 sf
2
(22)
and of its derivatives by
~
G n f
ð Þ¼ i2πfs
ðÞ
n e
�2π
2 sf
2
(23)
In the Fourier domain, the fractional heat kernel is expressed as
~
G s, ω, η
ð
Þ¼e
� 2π
ðÞ
α k
jj
2α s
(24)
Integer-order powers. Integer powers of the Laplacian operators are successive
compositions of the Laplacian operator [28]:
2
n ~
L n ω, η
ð
Þ¼ �1
ðÞ
n 2π s
ðÞ
2n e
�2π
2 sk
2 k
2n
(25)
By substitution with the radial wavenumber k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ω 2 þ η 2
p
, it can be demonstrated that the kernel is radially symmetric about the vector k and gets sharper
with increasing the order n:
~
L n k
ð Þ¼ �1
ðÞ
n 2
n π
2 n k
2 n e
�2 π
2 k
2 s
(26)
Fractional-order powers. In the fractional domain the operator can be
expressed as a direct generalization:
~
L n, α k
ð Þ¼ �1
ðÞ
n 2
n π
2 n k
2 α e
�2π
2 k
jj
2α s
(27)
Therefore, the kernel bandwidth can be controlled by the fractional power α.
61
Multiscale Segmentation of Microscopic Images
DOI: http://dx.doi.org/10.5772/intechopen.89003
I ⊕ E ¼ max
y ∈ B
Ixþ y
ð
ÞþBy ðÞ
ðÞ
(20)
In the continuous approximation case, the minimum and maximum should be
replaced by infimum and supremum, respectively. Consider the set X. The infimum
inf, or the greatest lower bound, is the greatest number, which is not necessarily in X
but is smaller (or equal) than all members of X. For a finite set, the infimum
coincides with the minimum. The supremum sup, or the least upper bound, is the
smallest number, which is not necessarily in X but is greater (or equal) than all
members of X. For a finite set, the supremum coincides with the maximum.
A.2 Some useful Fourier transforms
The concept of frequency and the decomposition of waveforms into elementary
“harmonic” wave motions first arose in the context of music and sound. The Fourier
transform and its inverse in the continuous domain are defined as
U k
ð Þ¼Fu x
ð Þ¼
ðð
�∞
e
�2πik�x ux ðÞ dx
2 ⇔u x
ð Þ¼F
�1 U k
ð Þ¼
ðð
�∞
e
2πik�x Uk ðÞ dk
2 (21)
The reader is directed to the book of [27] for an introduction on the topic.
The Fourier transform of the Gaussian is given by
~
Gf ð Þ¼e
�2 π
2 sf
2
(22)
and of its derivatives by
~
G n f
ð Þ¼ i2πfs
ðÞ
n e
�2π
2 sf
2
(23)
In the Fourier domain, the fractional heat kernel is expressed as
~
G s, ω, η
ð
Þ¼e
� 2π
ðÞ
α k
jj
2α s
(24)
Integer-order powers. Integer powers of the Laplacian operators are successive
compositions of the Laplacian operator [28]:
2
n ~
L n ω, η
ð
Þ¼ �1
ðÞ
n 2π s
ðÞ
2n e
�2π
2 sk
2 k
2n
(25)
By substitution with the radial wavenumber k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ω 2 þ η 2
p
, it can be demonstrated that the kernel is radially symmetric about the vector k and gets sharper
with increasing the order n:
~
L n k
ð Þ¼ �1
ðÞ
n 2
n π
2 n k
2 n e
�2 π
2 k
2 s
(26)
Fractional-order powers. In the fractional domain the operator can be
expressed as a direct generalization:
~
L n, α k
ð Þ¼ �1
ðÞ
n 2
n π
2 n k
2 α e
�2π
2 k
jj
2α s
(27)
Therefore, the kernel bandwidth can be controlled by the fractional power α.
61
Multiscale Segmentation of Microscopic Images
DOI: http://dx.doi.org/10.5772/intechopen.89003