Graphics Reference
In-Depth Information
for example be decomposed as
⎡
⎤
⎡
⎤
⎦
10
10
−
1
1
2
1
1
.
⎣
⎦
=
⎣
−
−
S
x
=
20
2
−
10
1
A separable filter of size 9
9 requires 18 multiplications per pixel and can be
applied in two passes. One pass performs convolution along the rows, and the
other pass performs convolution along the columns. A non-separable filter of size
9
×
9, on the other hand, requires 81 multiplications per pixel and is applied in
a single pass.
While separable filters are less computationally demanding, there are a num-
ber of image processing operations that can only be performed using non-separable
filters. The best-known non-separable filter is perhaps the Laplace (L) filter,
which can be used for edge detection. In contrast to Gaussian and Sobel filters,
it cannot be decomposed into two 1D filters. Laplace filters of size 3
×
×
3and5
×
5
can, for example, be written as
⎡
⎤
11 1 11
11 1 11
11
⎡
⎤
⎣
⎦
010
1
⎣
⎦
,
L
3
×
3
=
41
010
−
5
×
5
=
24 1 1
11 1 11
11 1 11
−
.
The quadrature filter is another popular non-separable filter, which is complex
valued in the spatial domain. The real part of the filter is a line detector and the
imaginary part is an edge detector. A 1D quadrature filter is given in Figure 5.4,
but quadrature filters of any dimension can be created. The name
quadrature
comes from electronics and describes the relation between two signals having the
same frequency and a phase difference of 90 degrees. An edge detector is an odd
function similar to a sine wave, while a line detector is an even function similar to a
cosine wave. A sine and a cosine of the same frequency always differ in phase by 90
degrees and a filter that can be described as one sine wave and one cosine wave is
therefore called a
quadrature filter
. The interested reader is referred to [Granlund
and Knutsson 95,Knutsson et al. 99] for further information about quadrature
filters and filter design. Quadrature filters can be applied for a wide range of
applications, such as image registration [Knutsson and Andersson 05, Eklund
et al. 10,Forsberg et al. 11], image segmentation [Lathen et al. 10], and image
denoising [Knutsson et al. 83, Knutsson 89, Granlund and Knutsson 95, Westin
et al. 01]. Quadrature filters are very similar to Gabor filters [Granlund 78,Jain
and Farrokhnia 91], which are also complex valued in the spatial domain.
For most algorithms using Gabor or quadrature filters, several filters are
applied along different directions. For example, estimation of a structure ten-
sor [Knutsson 89, Knutsson et al. 11] in 3D requires filtering with at least six
complex valued quadrature filters, i.e., a total of 12 filters (six line detectors
