3.3 Windowing
3.3.2 Spectral leakage and energy loss due to windowing
Due to the limited size and the non-periodicity of a typical signal a phenomenon calledspectral leakageoccurs after the DFT. All frequencies and wave numbers which do not occur in the basis of the DFT are not periodic in the image window. This results in discontinuities when the signal is periodically extended which in turn lead to the occurence of frequencies in the spectrum, which are not part of the original signal. Metaphorically speaking the “energy” of these frequencies is distributed onto the adjacent frequencies during the DFT, resulting in a leakage of energy to “wrong”
frequencies. This effect is reducing by windowing.
In general, applying a window function to a signal leads to a reduction of the total power carried by the signal. This can be corrected by a normalization factor f which is multiplied with the spectrum, resulting in a correction of the amplitudes in
3small width of the main lobe leading to better frequency selectivity
4large width of the main lobe leading to the desired small side lobe amplitudes.
Chapter 3 FOUNDATIONS IN SIGNAL PROCESSING
Fourier space. In this section, a general form of f is derived as well as the specific f for a Hann window in one and two dimensions.
According to Parseval’s theorem (seedefinition 3.4) the energy density5Eof a discrete signalS of lengthNin the spatial domain equals its energy density ˜Ein the Fourier domain
E ∶=
N−1
∑
n=0
∣S[n]∣2= 1 N
N−1
∑
k=0
∣Sˆ[k]∣2=∶E.˜ (3.6) This means that it is possible to apply a window to the signal in the spatial domain and to correct the amplitudes in the Fourier domain afterwards.
Assume a signalSwith finite extent which is (pixelwise, denoted by⋆6) multiplied by a window functionW. Let⟨○⟩denote the ensemble average and ¯○denote the time average (or the spatial average for spatial signals).
Assuming the ensemble average of the signal to be homogeneous, the ensemble average of the windowed signal is given by
⟨W⋆S⟩ =1
L
∫
Wdx⋅ ⟨S⟩ (3.7)with ¯W = 1
L ∫ Wdx.
Since the squared signal is of interest for the energy density:
⟨W⋆S2⟩ =1
L
∫
Wdx⋅ ⟨S2⟩ (3.8)Now substituteW =W˜2.
⟨W˜2⋆S2⟩ =1
L
∫
W˜2dx⋅ ⟨S2⟩ (3.9)=
⟨(W˜ ⋆S)2⟩ =1
L
∫
W˜2dx⋅ ⟨S2⟩ (3.10)For a 1D-Hann window ˜W=w(x) =0.5 (1−cos(2π⋅x
L ))the term L1 ∫ W˜2dx in
5Technically, the termsenergy,energy densityandpowerhave to be taken with care. In the language of signal processing, the signalSusually is a function of time which results in⟨S2⟩describing apowerspectrum which is a function of frequencyωand has the dimension of energytime . For application on images, the signalSis a spatial signal. Thus⟨S2⟩describes theenergy density spectrum as a function of wave number⃗k. The corresponding dimension isenergyarea .
6For two matricesA,Bof the same dimensionm×nthe Hadamard product or pixelwise product A⋆Bis the matrix of the same dimension as the operands with elements given by(A⋆B)i,j= (A)i,j⋅ (B)i,j.
34
Windowing 3.3
Equation 3.10can be evaluated as fHann1D ∶= 1
L
∫
L 0W˜2dx =1 L
∫
L 0(1 4
(1−cos(2π L ⋅x))
2
)dx
= 1 4L
∫
L 0(1−2⋅cos(2π
L x) +cos2(2π
L x))dx
= 1
4L = [x−2L
2πsin(2π L x) + x
2 + L
8π ⋅sin(2⋅2π L x)]
L 0
=3
8 (3.11)
For a 2D-Hann window ˜W =w(x)⋅w(y) =0.5 (1−cos(2πL⋅x
x ))⋅0.5 (1−cos(2πL⋅y
y )) a similar calculation results in
fHann2D ∶= 1 Lx⋅Ly
(Lx,Ly)
∫
(0,0)
W˜2dxdy= (3 8 )
2
= 9
64 (3.12)
Then the correctly normalized signal is given by⟨S2⟩ = 64
9⟨(W˜ ⋆S)2⟩ and for a 3D-Hann window and 3D signal it is⟨S2⟩ =512
27⟨(W˜ ⋆S)2⟩.
Part II.
Methods
4
The Imaging Slope Gauge (ISG) as a technique to measure water wave surface slopes
This chapter outlines the most important techniques which are used to measure water wave surface slopes. An overview of the existing techniques available for measuring the geometrical properties of water waves is given inFigure 4.1. Furthermore this chapter explains the underlying foundations as well as the limitations of the different methods. Finally, the characteristic features of the Imaging Slope Gauge are described in more detail.
Chapter 4 THE IMAGING SLOPE GAUGE (ISG) AS A TECHNIQUE TO. . .
Optical Methods
Non-optical Methods
Radar Acoustical Methods Wire Probes Accelerometer Buoys Pressure Sensors
Point-based Methods Imaging Methods
Laser Slope
Gauge Scanning Laser Slope Gauge
Statistical
Methods Imaging Slope
Gauge Color Imaging Slope Gauge
Sunglitter Method
(Cox & Munk) Reflective/Refractive Slope Gauge
Imaging Polarimeter
Resistive Wire Probes Capacitive Wire Probes Stereo Methods
Figure 4.1.: Overview of a selection of the existing techniques for measuring geometrical properties of the sea surface. Adapted fromLauer[1998].
4.1 Slope measurements vs. height measurements
When measuring geometrical properties of water waves two main approaches can be distinguished: slope measurementsandheight measurements. Both techniques are in principal equivalent because it is possible to obtain height information from slope measurements via integration at the cost of the mean surface elevation. This is feasible because water wave surface slopes are the gradient of surface elevation. A short description of this equivalence is given inJähne and Schultz[1992].
One important difference between the two approaches is due to the fact that water wave surface slope measurements require the acquisition of two slope components at the same time whereas height measurements - as surface elevation is a scalar quantity - do not. This makes water wave surface slope measurements technically more demanding. On the other hand the wave height displays a rather large variation depending on wave length whereas the variability of wave slopes is almost constant for a broad range of wave lengths (Jähne and Schultz[1992]). This is the main reason why this thesis is focused on water wave surface slope measurements.
In principle, two different types of techniques for measuring water wave surface slopes and heights are available, optical methodsand non-optical methods. Non-optical techniques include measurements of the wave amplitude using capacitive or resistive wire probes, pressure sensors or accelerometer buoys. Conventional stereophotogrammetric methods(Laas[1905,1906];Kohlschütter[1906];Laas[1921];
Schuhmacher[1939]) are an example for optical techniques used for height
measure-40
Methods for water wave surface slope measurements 4.2 ments. It has been shown that they exhibit insufficient height resolution for small
waves (seeFuß[2004]).