METHODS AND MODELS OF GEOMETRIC RECTIFICATION OF HIGH SPATIAL RESOLUTION SATELLITE IMAGES

ABSTRACT

The article is devoted to the issues of applying methods and mathematical models for geometric rectification of satellite images with high spatial resolution for the creation of large-scale digital maps.

АННОТАЦИЯ

Статья посвящена вопросам применения методов и математических моделей геометрической коррекции космических снимков с высоким пространственным разрешением для создания крупномасштабных цифровых карт.

Keywords: geometric correction, rigorous method, geometric sensor model, RPC coefficients, parametric methods, polynomial transformation, GCP.

Ключевые слова: геометрическая коррекция, строгий метод, геометрический модел сенсора, RPC коэффициенты, параметрические методы, полиномиальное трансформирование, опорные точки.

Intrdoduction. Remote sensing data are subject to distortions due to inevitable errors, including geometric distortions that directly affect the accuracy of large-scale mapping. In order to be able to use the images for various purposes, including for the compilation of large-scale digital agricultural maps, it is necessary to perform the geometric correction.

The main reason for the appearance of geometric distortions on satellite images is the curvature of the Earth. In addition, the rotation of the Earth, the complexity of the terrain relief, the instability of platform, as well as the error of the sensor (optical distortion) also cause geometric distortions in the images [4; 7].

Ordinarily some geometric distortions are corrected in the pre-processing stage by operators of remote sensing systems. However, pre-processed images in some cases do not meet the accuracy requirements of large-scale mapping. This requires geometric correction and rectification (transformation) using ground control points (GCP) and digital elevation model (DEM).

Methods and mathematical models. Digital satellite images with high spatial resolutions for large-scale mapping are usually obtained by optical-electronic scanners. To transform these digital images to the plane, methods and mathematical models are used that provide a reliable relationship between the pixel coordinates (by row and column) of the image and the coordinates of the terrain. These methods include rigorous, parametric, and universal methods.

A rigorous method, which is the most accurate method of rectification satellite images, provides a mathematical relationship between the coordinates of the image and the terrain based on the physical parameters of the sensor. The essence of this method is to restore the spatial position of the set of rays that formed the image. To do this, it is necessary to know the trajectories, orientations (exterior orientation elements) and the geometric model (interior orientation elements) of the sensor.

The functional relationship between the coordinates of the object on the image and on the ground (GCPs), based on the well-known collinearity equations, is as follows

where  – coordinates of points in satellite image;  – are spatial coordinates of GCPs,  – are coordinates of the center of the optical system;  – focal length of the camera; …, – are directions of cosines of the main projecting ray [5].

The second most accurate rectification method is the parametric method. The essence of the parametric method is to ensure the relationship between the coordinates of the image and the terrain based on the approximation parameters of the sensor, calculated by a rigorous method. To do this, use the coefficients of rational polynomials (RPC coefficients). The relationship between the coordinates of the image and the terrain can be written in the following form

where  – normalized coordinates of points in satellite image;  – are normalized spatial coordinates of GCPs, the value of which is normalized between 0 and 1;  – are tertiary polynomials calculated by the following expressions:

where  – polynomial coefficients are provided to users by operators of remote sensing systems (RPC coefficients) [3].

The universal methods of rectification satellite images include polynomial methods of various orders. The choice of the order of the polynomial depends on the distortion properties, the number of GCPs and their geographic location. For processing satellite images, polynomials of the first, second or third order can be used. To determine the parameters of polynomials, as a rule, the method of least squares is used.

A first-order polynomial or affine transformation is a linear transformation that can be used to correct linear distortions in satellite images. It can be expressed by the following equations:

where  – coordinates of points in satellite image;  – coordinates of GCPs (map);  – polynomial coefficients [1].

Using the first-order polynomial equation, it is possible to change the position and scale of the images along the x and y axes, and to tilt or rotate them (Fig. 1).

Figure 1. Linear transformation

1) initial image; 2) scale change along the x axis; 3) scale change along the y axis; 4) deviation along the x axis; 5) deviation along the y axis; 6) rotation

Second-order polynomial transformation is typically used to reduce nonlinear distortions in satellite images. The second-order polynomial equation can be written as follows

where  – coordinates of points in satellite image;  – coordinates of GCPs;  ва  – polynomial coefficients [6].

The below figure (Fig.2) shows the modification of satellite images using nonlinear polynomial transformation.

Figure 2. Nonlinear transformation

1- initial image; 2,3,4,5 – transformed images

As a result of geometric rectification of high spatial resolution satellite images, geometric distortions in the images can be corrected and transformed to a certain cartographic projection. However, as a result of the inclination of the optical axis of the sensor too large angles relative to the nadir and the complexity of the topography, it can be lead to the displacement of points (Fig. 3). This, in turn, necessitates their orthorectification. As a result of orthorectification, the satellite image is transferred from the central projection to the orthogonal projection.

Figure 3. The displacement of points in a image into a plane

Based on the image above, the relationship between the deflection of the optical axis relative to the nadir (α), the change in topography relative to the middle horizontal plane (Δh) and the displacement of the point in the plane (ΔL) can be expressed by the following equation [4]

If the value found by formula (5) does not differ much from the spatial resolution (δl) of the satellite image used, the images can be transformed without a DEM (to the middle horizontal plane), otherwise, the satellite images will need to be orthorectification using the appropriate precision DEM.

Results and discussions. As a result of rectification (orthorectification) by the above methods and mathematical models, the image is transformed into a standard cartographic projection, which has the property of a map.

The choice of one or another method for rectification of satellite images depends on the required accuracy and the availability of the appropriate initial data.

Image rectification using a rigorous method is a complex process that requires special software and takes a long time. In addition, it is not always possible to obtain information about the physical model of the sensor, or they are not provided at all by operators of remote sensing systems, or are provided with entire scenes, which in turn lead to an increase in financial costs. Therefore, for accurate transformation of satellite images, a parametric method is used, which is an alternative to the rigorous method.

For rectification by a parametric method, it is necessary to have rational polynomial coefficients – RPС (Rational Polynomial Coefficients), which are provided by operators of remote sensing systems.

It should be noted that just one GCP can provide sufficient accuracy in the transformation of satellite images using the rational polynomial method [2].

If it is impossible to use the geometric parameters of the sensor or other information (RPC coefficients), it can be used universal methods – first or secon order polynomial transformation that provide the relationship between the coordinates of the image and the terrain based on general assumption about the geometry of the survey.

Using first-order polynomial transformation, satellite images can be transferred to a plane coordinate system or transferred from one coordinate system to another. In general, this method can be used to accurately transform images of less deformed areas, images of relatively smaller regions, to the extent that they do not take into account the curvature of the Earth. To carry out this transformation, the number of GCPs should not be less than 3-4.

Second-order polynomial transformation is typically used to convert images from a geographic coordinate system to plane coordinate system, or when transforming satellite images of large regions (if the Earth’s curvature needs to be taken into account). In this transformation, the number of GCPs should not be less than 6.

Conclusions. The analysis of scientific research shows that all of the above methods of rectification of satellite images will be able to provide the accuracy requirements for compiling large-scale maps [2]. The results of the comparative analysis show that the most accurate method is the rigorous method. But since access is not always available to the geometric models of the sensor, preference will be given to parametric and universal methods, which in turn depend on the quantity and quality of GCPs.

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