Graphics Reference
In-Depth Information
Figure 5.4.
A 1D quadrature filter; the real part (green) is a line detector (a cosine
modulated with a Gaussian window) and the imaginary part (blue) is an edge detector
(a sine modulated with a Gaussian window).
and six edge detectors). The Morphon [Knutsson and Andersson 05] is an image
registration algorithm that uses quadrature filters to estimate the displacement
between two images or volumes. To improve the registration, estimation of a
structure tensor is performed in each iteration, thus requiring an ecient imple-
mentation of filtering. Monomial filters [Knutsson et al. 11,Eklund et al. 11] are
a good example of filters appropriate for non-separable filtering in 4D.
5.3 Convolution vs. FFT
Images and filters can be viewed directly in the image domain (also called the
spatial domain) or in the frequency domain (also denoted Fourier space) after the
application of a Fourier transform. Filtering can be performed as a convolution
in the spatial domain or as a multiplication in the frequency domain, according
to the convolution theorem
F
[
s ∗ f
]=
F
[
s
]
· F
[
f
]
,
where
F
[ ] denotes the Fourier transform,
s
denotes the signal (image),
f
denotes
the filter,
denotes pointwise multiplication. For large
non-separable filters, filtering performed as a multiplication in the frequency do-
main can often be faster than convolution in the spatial domain. The transfor-
mation to the frequency domain is normally performed using the fast Fourier
transform (FFT), for which very optimized implementations exist. However,
FFT-based approaches for 4D data require huge amounts of memory and current
GPUs have only 1-6 GB of global memory. Spatial approaches can therefore be
advantageous for large datasets. Bilateral filtering [Tomasi and Manduchi 98],
which is a method for image denoising, requires that a range function is evaluated
for each filter coecient during the convolution. Such an operation is hard to do
∗
denotes convolution, and
·
