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New Geomatics Technologies and Applications

changing the deflection angles, consecutive lines are simplified in such a way that they do not go beyond the error band [18].
The proposed model of this research is based on least squares regression and is presented to solve the above problem. As a result,
in order to fit the line, it considers all of the points, and as a result, the fit of the summarized line is affected by all of the points
of each line of the desired feature. The aim of this research is to present a method for summarizing the geometry of multiple
lines and to compare it to the most commonly used algorithms in the field [19,20,21].

2. Least Squares

The least squares (LS) method is one of the most popular regression analysis techniques. It is used to solve equations in
which the number of equations of observations is greater than the number of unknowns. Since fitting curves to data is one of the
most important applications of the least squares process, it was used in this study to fit a line to each of the line components. If

the line equation is assumed to be in accordance with Equation (1), where x and y are the values of the coordinates of the points
of each piece of the line (observations), v is the value of the corrections, a and b are the coefficients of the line (unknowns) that
must be Be calculated and the line equation is formed [22, 23].

y  ax  b; v  ax  b  y (1)

The number of equations equals the number of points observed. The coefficients of unknowns (a and b) and observations
(y) of the equations of relation (1) must then be determined in order to solve these equations using the least squares method. The

^
unknown coefficients matrix is denoted by A, the observation coefficients by L, and the value of unknowns ( X ) is determined
using Equation (2) [23].

^
X  ( A T pA) 1  A T pL (2)
In Equation (2), p is the point weight matrix, which is used if different points have different accuracy or importance, and

the identity matrix is used otherwise [23]. The model minimizes the value of the square of the y error or the correction (v) and
has least squares in both linear and nonlinear forms. The LS model, which is a global model based on the assumption of constant
relation in the sampling space, is the most suitable nonlinear linear estimator. The points on the line are used to estimate the
unknowns in this model. Equation (3) illustrates how to use observed or independent parameters (X i) to measure an unknown or
dependent parameter (y) [24].

i 
y  B ( u  v  P B ( u  v ) X  E (3)
)
i
i
i
i
i
0
i1
Where ui and vi are the coordinates of the i-th point, the intersection of the i-th position, the local coefficient of the
independent variable i-th and E is also a random error assuming the data is normal [24].

3