where He n (x) is the statistician’s Hermite polynomial of order n. The sequence of
statistician’s Hermite polynomials satisfies the recursion
He nþ1 x
ð Þ¼xHe n x
ð Þ�nHe n�1 x
ðÞ
(12)
starting from He 0 x
ð Þ¼1 and He 1 x
ð Þ¼x. This allows for efficient simultaneous
computation of all derivatives up to an order n in order to populate the n-jet space.
The n-jet components can be used to build the differential invariants up to order n.
An example is presented in Figure 6, where the five unique components of the
Gaussian jet-2 space are computed. The original dataset is present in the ImageJ
public image database.
In spite of several properties that make linear diffusion filtering useful, it also
reveals some drawbacks [17]:
1. An obvious disadvantage of Gaussian smoothing is the fact that it does not
only smooth noise but also blurs important features such as edges. Moreover, it
is uncommitted to any prior information about the image structure.
2. Linear diffusion filtering propagates edges when moving from finer to coarser
scales, which can lead to difficulties in edge identification and instabilities.
5.2 α-Scale spaces
The α-scale spaces introduce nonlinearity on the level of differentiation. Notably, the Gaussian differentiation is replaced by another convolution operation,
involving a power law. Pauwels et al. [18] and later Duits et al. [19] investigated the
use of fractional powers of the Laplacian in connection with scale invariant
smoothing and scale-space theory, respectively. This approach tries to overcome
some of the limitations of the Gaussian scale spaces identified above. The evolution
Figure 6.
Differential Gaussian 2-jet space. A microscopic image of Drosophila brain (first column) is convolved with
Gaussian derivative kernels. Different kernels are shown above the arrows. The second column shows the
components of the gradient. The third column shows the components of the Hessian. The local jet space of order k
has kkþ 1
ðÞ =2 different components.
56
Advances in Neural Signal Processing
statistician’s Hermite polynomials satisfies the recursion
He nþ1 x
ð Þ¼xHe n x
ð Þ�nHe n�1 x
ðÞ
(12)
starting from He 0 x
ð Þ¼1 and He 1 x
ð Þ¼x. This allows for efficient simultaneous
computation of all derivatives up to an order n in order to populate the n-jet space.
The n-jet components can be used to build the differential invariants up to order n.
An example is presented in Figure 6, where the five unique components of the
Gaussian jet-2 space are computed. The original dataset is present in the ImageJ
public image database.
In spite of several properties that make linear diffusion filtering useful, it also
reveals some drawbacks [17]:
1. An obvious disadvantage of Gaussian smoothing is the fact that it does not
only smooth noise but also blurs important features such as edges. Moreover, it
is uncommitted to any prior information about the image structure.
2. Linear diffusion filtering propagates edges when moving from finer to coarser
scales, which can lead to difficulties in edge identification and instabilities.
5.2 α-Scale spaces
The α-scale spaces introduce nonlinearity on the level of differentiation. Notably, the Gaussian differentiation is replaced by another convolution operation,
involving a power law. Pauwels et al. [18] and later Duits et al. [19] investigated the
use of fractional powers of the Laplacian in connection with scale invariant
smoothing and scale-space theory, respectively. This approach tries to overcome
some of the limitations of the Gaussian scale spaces identified above. The evolution
Figure 6.
Differential Gaussian 2-jet space. A microscopic image of Drosophila brain (first column) is convolved with
Gaussian derivative kernels. Different kernels are shown above the arrows. The second column shows the
components of the gradient. The third column shows the components of the Hessian. The local jet space of order k
has kkþ 1
ðÞ =2 different components.
56
Advances in Neural Signal Processing