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Figure 5.11. Illustration of shock filtering. (a) A smooth step edge. (b) First derivative
of the edge. (c) Second derivative of the edge. (d) A shock filter applies a dilation
where the second derivative is positive, and an erosion where it is negative, resulting in
For each iteration, first- and second-order derivatives, as well as the smoothed
structure tensor, have to be calculated. Another limitation is that Weickert's
shock filter introduces shocks in almost smooth regions, resulting in maze-like
artifacts. The next section discusses an alternative approach.
Gradient-Directed Shock Filter
The idea in obtaining a fast shock filter implementation is to approximate the
general working principle discussed earlier and illustrated in Figure 5.11. First,
whether a pixel is in the neighborhood of a minimum or maximum is detected.
Then, correspondingly, either an erosion or dilation is performed. Both operations
are guided by the structure tensor.
Derivative operators are highly sensitive to noise, and sensitivity increases
with order. Therefore, the second=derivative operator for the sign computation
must be regularized to avoid artifacts. In addition, the regularization allows for
artistic control over the resulting line thickness. Two strategies are at hand. First,
the image can be isotropically smoothed prior to derivative computation, using
a Gaussian filter with standard deviation σ i . This helps to remove noise and
allows for aggressive image simplification. Secondly, the smoothing and deriva-
tive operators can be consolidated into a single operator,since convolution and
differentiation commute. Inspired by the flow-based difference of Gaussians filter,
and its separable implementation [Kyprianidis and Dollner 09], the second-order
derivative in direction of the major eigenvector is implemented by convolving the
image locally with a one-dimensional (scale-normalized) second-order Gaussian
x 2
σ g
2 πσ g
d 2 G σ g ( t )
d t 2
t 2
2 σ g
σ g G σ g ( t )= σ g
in the direction of the minor eigenvector. This operation will be referred to as
flow-guided Laplacian of Gaussian (FLoG). More specifically, let L be the input
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