ace
ns
mination
(same surface
endent of the
( slope of the
naging slope
ıl., 94] using
ra is observ-
> all the rays
'erpendicular
point on the
r focal plane
re parallel in
pixel, if the
face leads to
1e CCD sees
slope of the
! intensity or
maged water
ation respec-
at a certain
independent
ace ( or at
> same angle
same screen
nalyzing the
ich pixel the
r surface can
ng this color
eviously used
id spanwise)
a interaction
e. According
ve with slope
(8)
Transferfunktion
E -0.5 0 0.5 1
Wellenzahl k
ul
Transferfunktion
-1 -0.5 0 0.5 1
Wellenzahl k
Figure 9: Imaginary part of the transfer function of an optimal
Hilbert filter (a), transfer function of the approximations (b) 1,
2H and 3H (see text).
sigma denoting the surface tension and p the density of wa-
ter. The wave energy was calculated from the wave slope
images using a multi-scale Hilbert transformation technique.
A Hilbert filter converts a signal in its Hilbert transform. It
does not change the amplitude of the different spectral com-
ponents, but shifts their phase by 7/2. Therefore the magni-
tude of the transfer function is one [Jähne, 93]. Because of
the 7/2 phase shift, the transfer function is purely imaginary,
of odd symmetry, jumping from —i to ¢ at the wavenumber
k — 0, see Figure 9. The starting point for the design of a
Hilbert filter is the observation that the convolution kernels
of a first-order derivative operator (7/3 of figure 9) is odd and
shows already the main features of a Hilbert filter. For the
construction of better approximations a series of sine func-
tions with odd wave number is expanded at |k| — 2, yielding
(see [Jähne, 93]):
1 1
= -10 —1
H = 3010 -1]
2 1
= —1090 -90 —1 9
H = id (9)
MH = L302 0 150 0 —150 0 —25 0 —3]
256
The transfer functions of theses three Hilbert filters are shown
in Fig. 9. Simulations have shown, that these ordinary fil-
ters are only able to transfer a signal in its Hilbert-transform
in a limited bandwidth from 2.5 to 10 pixels. Therefore
all images were bandpass decomposed by a Laplace pyra-
mid [Burt und Adelson, 83]. On each level of the pyramid
the Hilbert transform can be computed effectively. By this
multi-grid approach structures with wavenumbers from 2.5 to
40 pixels can be phase shifted effectively. Fig. 10 shows the
energy extraction algorithm on a ring test pattern.
235
Energy [normalized]
e e o e x
o
0 6 0, : 1,0
normalized wavenumber k d
Figure 10: Test ring pattern (a), Hilbert transformation (b), En-
ergy of test pattern (c), energy profile (d).
Figure 11: Sketch of the integrated flow- and wave visualization
set-up. One camera is observing the waves from above, a second
camera is looking from the side on a light-sheet, visualizing seeding
particles.
4 SIMULTANEOUS FLOW- AND
WAVE-VISUALIZATION
Due to the presence of the color wedge of the wave visualiza-
tion the light sheet (flow visualization) cannot be generated
from below of the channel. Thus it is produced from the sides
of the wave visualization and then coupled into the channel
through a prism (see Fig. 11). An optical system consist-
ing of a spherical lens (f = 200mm) and a cylinder lens
(f = 90 mm) generates the light sheets. An immersion oil as
an optical coupling medium links the prisms with the bottom
window of the channel. As shown in Fig. 1 two illumination
systems are combined to increase intensity and homogeneity
in the image sector of 17 x 20 cm?. The oblique illumination
angle has been chosen so that most of the light is totally
reflected at the water surface. Only in rare cases - for steep
waves - the light is refracted in such a way at the water sur-
face that bright spots are observed in camera 1 for the wave
slope imaging. This residual interference was eliminated by a
blue Corion interference short wave pass filter (cut-off wave-
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
