is governed by two parameters—the scale s and the order of differentiation α. The
approach leads to formulation and solving of a fractional heat problem:
u 0, x
ð
Þ¼I x
ðÞ
u s s, x
ð Þ¼� �Δ
ðÞ
α=2 u s, x
ðÞ , 1 ≤ α ≤ 2
(13)
The Riesz fractional Laplacian operator is defined in the Fourier domain by
�Δ
ðÞ
α U k
ðÞ ≔ k
jj
α U k
ðÞ
(14)
where the ∣k∣ ¼
ffiffiffiffiffiffiffiffiffi ffi
k � k
p
is the modulus of the wave vector k. In this way, the
solution can be expressed in terms of a convolution with a very general special
function—the Wright function [20]. Numerical routines for computation of the
Wright function are still not readily available; therefore the computations is easier
achieved using fast Fourier transform (FFT) and its inverse, IFFT.
5.3 Nonlinear scale spaces
Linear diffusion scale spaces are well-posed and have a solid axiomatic foundation. On the other hand, for some applications, they have the undesirable property
that they do not permit contrast enhancement and that they may blur and delocalize
structures. Nonlinear scale spaces try to overcome some of these limitations. Such
scale spaces arise in nonlinear partial differential equation framework, which will be
sketched below. The formal properties of some types of scale spaces have been
established by Alvarez et al. [4]. In particular, they established a strong link with
the related field of mathematical morphology (see Section 3). The following secondorder partial differential equation was demonstrated in particular
u t ¼ F IH u; ∇u
ðÞ ,u 0; x
ð
Þ¼fx ðÞ
(15)
where Hu are the components of the Hessian tensor, ∇u represents the components of the gradient, and f(x) is the original image. It is interesting that MM
operations can also be represented in this framework as u t ¼ �k∇uk for dilation and
erosion, respectively.
In this line of development, the Laplacian of Gaussian (LoG) operator can be
decomposed into orthogonal and tangential components ([17], Ch. 1). The representation is provided below:
Δ G ¼ G xy þ G yy ¼ Δ kG þ Δ ⊥G
G
2
x þ G
2
y
��
Δ ⊥G ¼ G
2
x
��
G xx þ 2G x G y
��
G xy þ G
2
y
��
G yy
G
2
x þ G
2
y
��
Δ kG ¼ G
2
x
��
G xx � 2G x G y
��
G xy þ G
2
y
��
G yy
(16)
The parentheses denote scalar multiplication with the component of the gradient. The orthogonal decomposition is equivalent to an effective vectorization of the
filter. The normal component is antiparallel to the gradient (i.e., in normal direction
to the isophote curve), while the tangential component is parallel to the isophote
curve passing through the point. These components can be used to segment bloblike or tubular structures. Segmentation based on the orthogonal decomposition is
illustrated in Figure 7.
The orthogonal decomposition leads naturally to anisotropic diffusion
(Figure 8). For example, if the tangential component is selected, this will lead to
57
Multiscale Segmentation of Microscopic Images
DOI: http://dx.doi.org/10.5772/intechopen.89003
approach leads to formulation and solving of a fractional heat problem:
u 0, x
ð
Þ¼I x
ðÞ
u s s, x
ð Þ¼� �Δ
ðÞ
α=2 u s, x
ðÞ , 1 ≤ α ≤ 2
(13)
The Riesz fractional Laplacian operator is defined in the Fourier domain by
�Δ
ðÞ
α U k
ðÞ ≔ k
jj
α U k
ðÞ
(14)
where the ∣k∣ ¼
ffiffiffiffiffiffiffiffiffi ffi
k � k
p
is the modulus of the wave vector k. In this way, the
solution can be expressed in terms of a convolution with a very general special
function—the Wright function [20]. Numerical routines for computation of the
Wright function are still not readily available; therefore the computations is easier
achieved using fast Fourier transform (FFT) and its inverse, IFFT.
5.3 Nonlinear scale spaces
Linear diffusion scale spaces are well-posed and have a solid axiomatic foundation. On the other hand, for some applications, they have the undesirable property
that they do not permit contrast enhancement and that they may blur and delocalize
structures. Nonlinear scale spaces try to overcome some of these limitations. Such
scale spaces arise in nonlinear partial differential equation framework, which will be
sketched below. The formal properties of some types of scale spaces have been
established by Alvarez et al. [4]. In particular, they established a strong link with
the related field of mathematical morphology (see Section 3). The following secondorder partial differential equation was demonstrated in particular
u t ¼ F IH u; ∇u
ðÞ ,u 0; x
ð
Þ¼fx ðÞ
(15)
where Hu are the components of the Hessian tensor, ∇u represents the components of the gradient, and f(x) is the original image. It is interesting that MM
operations can also be represented in this framework as u t ¼ �k∇uk for dilation and
erosion, respectively.
In this line of development, the Laplacian of Gaussian (LoG) operator can be
decomposed into orthogonal and tangential components ([17], Ch. 1). The representation is provided below:
Δ G ¼ G xy þ G yy ¼ Δ kG þ Δ ⊥G
G
2
x þ G
2
y
��
Δ ⊥G ¼ G
2
x
��
G xx þ 2G x G y
��
G xy þ G
2
y
��
G yy
G
2
x þ G
2
y
��
Δ kG ¼ G
2
x
��
G xx � 2G x G y
��
G xy þ G
2
y
��
G yy
(16)
The parentheses denote scalar multiplication with the component of the gradient. The orthogonal decomposition is equivalent to an effective vectorization of the
filter. The normal component is antiparallel to the gradient (i.e., in normal direction
to the isophote curve), while the tangential component is parallel to the isophote
curve passing through the point. These components can be used to segment bloblike or tubular structures. Segmentation based on the orthogonal decomposition is
illustrated in Figure 7.
The orthogonal decomposition leads naturally to anisotropic diffusion
(Figure 8). For example, if the tangential component is selected, this will lead to
57
Multiscale Segmentation of Microscopic Images
DOI: http://dx.doi.org/10.5772/intechopen.89003