Least Squares – Best Fit

All measurements contain errors. These errors can be mitigated by applying least square adjustments which is also known as best fit. With global positioning systems, total station instruments, laser scanners and satellite imaging systems now generating millions of points per second, adjustment for errors is crucial to accurate interpretation. Laser scanning tools bring the outside as built world into the office so now the evaluation of this data can be accomplished in a safe and comfortable environment.

LiDAR scanning provides two categories of data: the primary data or coordinates obtained by differences in time of the received laser signals, and the intensity value data, an indicator of the differences within the reflected laser beam of dissimilar materials present in the scene. The combination of coordinates and intensity provide valuable information during the evaluation and feature extraction process.

LiDAR data contains considerable information about the objects being scanned. Within the intensity data there are errors from random noise and signal anomalies caused by sensor scanning patterns, also called range noise. The point data that a laser scanner collects has inherent random noise, some more than others. Typically longer range scanners have more range noise than short range scanners mostly due to the lasers spot size. The lasers spot size increases the further it travels from the scanner. Range noise can be reduced by using the power of averaging or Least Squares.

An old survey saying is two points make a line, three points make a triangle and four or more points make a mess. Take the example of a three legged table, it is always stable and never rocks. Add a fourth leg and if they are not all exactly the same length the table will rock. Least squares essentially takes the points that cause the table rock and creates the best fit plane.

There are a number of tools available to assist in analyzing laser data. Classification separates the data into useful levels such as Ground, Low, Medium and High vegetation or even more specifically into features such as light poles, street signs, buildings etc. Laser data can also be viewed by intensity values which is useful in finding features with like properties. These techniques are very valuable when analyzing data for accuracy and or feature extraction.

When two variables are highly correlated how do you find the line or plane that best fits the points? Any line or plane you choose will involve some compromise: moving the line closer to some points will increase or decrease the distance from others.

Applying least squares adjustment allows the technician to find the line or plane that minimizes the "average distance" from the line or plane to all the points. This is a powerful tool that can greatly reduce the random error value by averaging the random error in some cases over thousands of points. That line is called the regression line. The use of regression lines, or what is also referred to as averaging through the point data is illustrated below:

According to 3D Laser Mapping Executive Chairman, Dr. Graham Hunter, and the developer of the StreetMapper mobile LiDAR system, The average distance is random, meaning that the regression line is the correct measurement of the line in the middle of a noisy set of (random) laser measurements.

An example of using least squares or averaging is scanning a control target that has an existing known coordinate value. The range noise tends to make the target area look thick. As stated above the thickness depends on the scan system used, however this range noise, large or small can be averaged out to create a coordinate value that is far more accurate by averaging the many points than using a single scan point.

Fig. 1 Average line through the points

Fig. 2 Residual measurements from the points to the line

Once all the control points are extracted they can then be evaluated as a network comparing the scan points to the known coordinate values. This is where the least square adjustment analysis is paramount to the final positional accuracies of the laser data. The positional accuracy to the known coordinates after adjustment is a measure of how well the scan data fits the known coordinates. The scan data relies on the accuracy of the known coordinates for a final positional accuracy. Using Terrestrial Mobile LiDAR Scanning we have found errors in the known coordinate values. Every measurement has an error budget and is also subject to blunders so careful examination is prudent states Michael Frecks.

There are many papers written on this subject and one only needs to type in Least Squares on the internet for a full days reading, but it will be time well spent .

About the Author

Cyn René Whitfield

Cyn René Whitfield. Cyn has a master's degree in managerial business communications and is a nationally published journalist who has been involved in marketing for land surveying and laser scanning companies since 1983. She is currently the Business Development Coordinator for TREKK Design Group a D/WBE full-service civil engineering design and consulting firm headquartered in Kansas City, Mo.
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