Fast total least squares vectorization

Abstract

This paper proposes a novel algorithm for the vectorization of ordered sets of points, named Fast Total Least Squares (FTLS) vectorization. The emphasis was put on low computational complexity, which allows it to be run online on a mobile device at a speed comparable to the fastest algorithms, such as the Douglas–Peucker (DP) algorithm, while maintaining a much higher quality of the approximation. Our approach is based on the total least squares method, therefore all the points from the cloud contribute to its approximation. This leads to better utilization of the information contained in the point cloud, compared to those algorithms based on point elimination, such as DP. Several experiments and performance comparisons are presented to demonstrate the most important attributes of the FTLS algorithm.

Appendix

In this appendix, we are going to show the rearrangement from Eq.  (8) to (9). By substituting (7) into (8), we get:

$$\sum \limits _{i=1}^n \frac{\partial \left( ax_i + \sqrt{1 - a^2} y_i + c \right) ^2}{\partial a} = 0,$$

(17)

$$\sum \limits _{i=1}^n \frac{\partial \left( ax_i + \sqrt{1 - a^2} y_i + c \right) ^2}{\partial c} = 0.$$

(18)

After differentiation, the first equation is as follows:

$$2 \sum \limits _{i=1}^n \left( a x_i^2 - a y_i^2 + \frac{1 - 2 a^2}{\sqrt{1 - a^2}} x_i y_i + cx_i - c\frac{a}{\sqrt{1 - a^2}} y_i \right) = 0.$$

(19)

The Eq. (18) gives a simple result:

$$2 \sum \limits _{i=1}^n \left( a x_i + \sqrt{1 - a^2} y_i + c \right) = 0,$$

(20)

which can be rewritten into the form:

$$c = \frac{-1}{n} \left( a \sum \limits _{i=1}^n x_i + \sqrt{1 - a^2} \sum \limits _{i=1}^n y_i\right) .$$

(21)

Similar rearrangement of (19) and substitution of (21) yields the equation:

$$\begin{aligned} a \sum \limits _{i=1}^n x_i^2 - a \sum \limits _{i=1}^n y_i^2 + \frac{1 - 2 a^2}{\sqrt{1 - a^2}} \sum \limits _{i=1}^n x_i y_i - \\ - \frac{a}{n} \left( \sum \limits _{i=1}^n x_i \right) ^2 - \frac{\sqrt{1 - a^2}}{n} \sum \limits _{i=1}^n x_i \sum \limits _{i=1}^n y_i + \\ + \frac{a^2}{n \sqrt{1 - a^2}} \sum \limits _{i=1}^n x_i \sum \limits _{i=1}^n y_i + \frac{a}{n} \left( \sum \limits _{i=1}^n y_i \right) ^2 = 0, \end{aligned}$$

(22)

which gives the following identity:

$$\begin{aligned} a \left( \sum \limits _{i=1}^n x_i^2 - \sum \limits _{i=1}^n y_i^2 - \frac{1}{n} \left( \sum \limits _{i=1}^n x_i \right) ^2 + \frac{1}{n} \left( \sum \limits _{i=1}^n y_i \right) ^2 \right) \\ = \frac{1 - 2 a^2}{\sqrt{1 - a^2}} \left( \frac{1}{n} \sum \limits _{i=1}^n x_i \sum \limits _{i=1}^n y_i - \sum \limits _{i=1}^n x_i y_i \right) . \end{aligned}$$

(23)

Simple rearrangement gives the Eq. (9), which is further used in Sect. 2.2:

$$\begin{aligned} \frac{a \sqrt{1-a^2}}{1-2a^2} = \frac{\sum \nolimits _{i=1}^n x_i \sum \nolimits _{i=1}^n y_i - n \sum \nolimits _{i=1}^n x_i y_i}{n \sum \nolimits _{i=1}^n x_i^2 - n \sum \nolimits _{i=1}^n y_i^2 - \left( \sum \nolimits _{i=1}^n x_i \right) ^2 + \left( \sum \nolimits _{i=1}^n y_i \right) ^2}. \end{aligned}$$

(24)