Fullwave

Fullwave processing

HELIOS++ supports simulation of full waveform data by simulating a laser beam cone of finite divergence via sampled subrays.

Subray quantification

The quality of the simulation can be adjusted by the beamSampleQuality attribute of the <FWFSettings [...]/> tag in the survey XML. If no information is given there, the scanner definition may have a set of FWFSettings instead. If this is also not present, hardcoded default values are used, as shown in the following table:

Attribute	Default value	Comment
beamSampleQuality	3	3 concentric circles of subrays, 19 subrays total, discretization in space
binSize_ns	0.25	discretization in time
maxFullwaveRange_ns	0	maximum time to record for a single pulse. 0 means no limit. (wait for last pulse)
winSize_ns	pulseLength_ns/4	pulseLength as defined in the scanner

The beamSampleQuality $bSQ$ is the number of concentric circles where subrays are sampled from. The actual (angular) distance of circle $i$ depends on the beam divergence $\beta$ and is calculated as $i \beta / bSQ$, so the outermost circle lies at an angular distance of twice the beam divergence (assumed to be defined at the $1/e^2$ points of the maximum energy).

On one of these circles, $k$ subrays are sampled, where $k=\lfloor2 \pi n\rfloor$. The subrays are distributed evenly on the circle. The number of subrays for a certain $bSQ$ can be calculated as:

$$\Large n_{Rays} = 1+ \sum_{i=1}^{bSQ} \lfloor 2 \pi i \rfloor$$

For example, the subrays for a beamSampleQuality of three look like this (color represents relative amplitude, see next section):

Amplitude evaluation

For every subray, the returned waveform is calculated by intersecting it with the scene. If the subray returns a hit, it does not continue onwards.

The received amplitude is derived from the LiDAR equation, considering the following items:

1. Transmitted energy: The energy at a certain offset from the center ray is calculated as follows:

$w_0$ ... beam waist radius (see explanation on the Scanner page),

$\lambda$ ... wavelength,

$r$ ... radius (offset from center beam),

$R$ ... target range,

$R_0$ ... minimum range (range of beam waist),

$I_0$ ... average power,

$\Omega = \frac{\lambda R}{\pi w_0^2}$

$\Omega_0 = 1-\frac{R}{R_0}$

$w = w_0 \sqrt{\Omega_0^2 + \Omega^2}$

$I = I_0 \exp{-2r^2 / w^2}$

where $I$ is the power of the subray (Carlsson et al. 2001).

2. Material - see different options and material definitions on the Scene page, using Phong's Bi-Directional Reflection Distribution Function (BDRF)(Phong, 1975)
3. Target cross section, using the area illuminated by the subray and the local incidence angle
4. Atmospheric and system efficiency.

In the time domain, the beam is modelled with the following function, where $t_\tau = t / \tau$ and $\tau=pulseLength / 1.75$ (Carlsson et al. 2001):

$P(t) = t_\tau^2 \exp{-t_\tau}$

which gives the following shape:

Calculating full waveform

After the subrays have been cast and intersected with the scene, they are aligned according to their range, where the maximum position (the peak in the previous figure) corresponds to the position of the return, and the waveform shape is not altered e.g. with respect to the incidence angle. Although the central pulse and each subray are cast at the same time, they inevitably make contact with different parts of the scene, leading to a temporal offset in the returning pulses. The waveform is the sum of these returning pulses and the maxima of this waveform are used to extract points for the output array. The output array is created with bins of size binSize_ns and a maximum length of maxFullwaveRange_ns (if not set to 0), which starts at the first return (pulseLength_ns before the hit). Subsequently, the recorded waveforms, scaled with their respective amplitudes are added into the array. As the bins are not necessarily aligned, they are cast to the next lowest bin.

The following two graphs illustrate the calculation and analysis of one such full waveform based on the return of one laser beam with a central pulse and six subrays which, for the sake of a simplified example, is aimed at an imaginary target at roughly 3 metres distance. For the output, maxima are detected using a window approach, meaning that a maximum is counted as valid if a bin has the highest value in a neighborhood of winSize_ns. The local maxima of the waveform are indicated in the graphs, along with windows for the validation of the maxima for the final return. The examples show how using full waveform can lead to more detailed and/or precise results.

In the case of the first graph, the subrays are returned relatively close temporally to the central pulse. The resulting waveform has three local maxima, of which only one can be validated using the given window size (default=pulselength/4). In consequence, this full waveform pulse would result in this one point being added to the output array, in this case at a distance of c*10.16 ns (3.05 m) to the scanner. In comparison, the result without full waveform and evaluation of only the central pulse would have led to a less precise peak at around c*9 ns (2.7 m).

In the case of the second graph, the subrays are further spaced around the central pulse. This leads to the development of four local maxima, with three of them being validated and included in the output array. In this example the increase in the number of points leads to an evident increase in detail as well as an increase in precision.

Echo width determination

If the command line parameter --calcEchoWidth was provided, a Gaussian curve is fitted to the waveform. This allows the determination of an echo width as the standard deviation of this Gaussian. As shown in the following figure, the maximum of the Gaussian does not align with the maximum of the pulse shape. For the distance measurement, the position of the maximum, as detected with the window approach, is used. Note that due to the binning in the full waveform calculation, this distance may be off by a maximum of binSize_ns, which is 0.25 by default, corresponding to 7.5 cm. This offset only concerns non-first returns, though, and can be decreased by choosing a lower binSize_ns.

Output

The point position, amplitude (as detected by the maximum search), echo width (if calculated) as well as information about the return number are written to the point output file (either LAS or ASCII). The signal strength levels of the individual bins are stored - along with the beam origin and the unit beam vector - in the fullwave output file. For details on these files, see the page on Output.

References

Carlsson, T., Steinvall, O. & Letalick, D. (2001): Signature simulation and signal analysis for 3-D laser radar: scientific report. Linköping: Sensor Technology, Swedish Defence Research Agency (Sensorteknik, Totalförsvarets forskningsinstitut (FOI). Online access.

Phong, B.T. (1975): Illumination for computer generated pictures. In: Communications of the ACM 18.6, pp. 311-317. DOI: 10.1145/360825.360839.