The main step of the SAR image simulation under RD imaging geometry is the multiple backscattering simulation. The ray-tracing algorithm is the key technology for multiple scattering simulations. The traditional ray-tracing architecture is mostly used for 3D scene rendering, and only the scattering amplitude information is recorded. In the process of SAR image simulation, the scattering amplitude and phase information need to be recorded at the same time, and the real-time slant range is the key element to obtain the phase information. In this paper, SAR imaging geometry is introduced into the ray-tracing algorithm, which is only suitable for computer graphics, and the SAR mode ray-tracing architecture is designed. Combined with the physical illumination model, it can provide key technical support for SAR image simulation.
2.1. Design of the SAR Mode Ray Tracing
The ray-tracing algorithm is a general architecture to study the path propagation of an EM wave. It can be used to simulate the global rendering effect of a multi-light source combined with a local illumination model. In the field of computer graphics, a large number of EM waves are transmitted from the camera position of the sensor to the pixel grids of the virtual screen. Based on Fresnel’s law of specular reflection, the specular reflection direction of an EM wave interacting with the target surface is calculated and used as a new incident direction for forward tracking; thus, the whole path of an EM wave is obtained recursively. In practical application, considering that the natural scene or man-made target surface is a complex with certain gloss and roughness, the reflected energy field has a certain width of lobes. The specular reflection direction calculated from the normal vector of the macro-facet at the intersection of an EM wave and the target surface is used as the energy principal axis direction of forward tracing. The gray value of a pixel on the generated image is the superposition of the energy of all incident rays in the case of multiple scattering. It is necessary to ensure that the position of the intersection point when scattering occurs is in view of the position of the light source and the pixel grids; otherwise, it is considered an invalid scattering energy.
As shown in
Figure 1, the 3D object can be rendered globally by combining the ray tracing of the global rendering mode with the illumination model, which can be expressed as
where
represents a pixel unit in the imaging plane;
is a ray index number that is visible to the pixel unit;
indicates the
k-th scattering of light;
represents the scattered energy and is suitable for the arbitrary illumination model. The ray-tracing architecture of the whole rendering mode only superposes the scattered energy of the incident light, which is the main difference from the SAR mode. In addition, the sensor radiation mode is also different. Commonly used global rendering light source types are divided into point light source, spotlight light source, and directional parallel light source. (1) The point light source is an ideal light source model, which assumes that the light source transmits light rays uniformly from a point in space in all directions. Its illumination range is theoretically infinite in practice; however, there is usually a limited radius of action beyond which the intensity of the light will be sharply reduced to zero. The brightness value of the light transmitted from a point light source is inversely proportional to the square of the distance from a point to the light source. (2) In addition to having a definite position point, the spotlight source has the characteristics of directivity and limiting illumination range. The light from a spotlight source does not spread evenly but is concentrated in a cone-shaped area. The inner angle of the cone identifies the central region with the highest light concentration and brightness, while the outer angle marks the boundary where the light begins to weaken noticeably. Outside the cone, the light is almost invisible or very faint. Therefore, spotlight mode can better simulate the radar beam width of 3 db. (3) Directional parallel light source is a special type of light source in a virtual environment, which is mainly used to simulate the sunlight or other extremely remote and seemingly negligible sizes of light source. This type of light source has only color and orientation attributes and no specific light source location because the directional light source assumes that the light source is infinitely far from the scene so that all the light seems to come from the same point and hit the scene in parallel and the direction of the light remains the same throughout the scene; there is no angular offset or intensity decay due to the distance of propagation. Therefore, no matter how far away the objects in the scene are from the light source, as long as they face or go back to the light source, the received light intensity is constant and only depends on the relationship between the normal of the object surface and the direction of the light angle.
In the field of SAR image simulation, considering the functional relationship between the EM wave propagation range and target backscattering intensity, the limitation of the 3 db radar beam width and the propagation mode of spherical wavefront, the final selection is a single spotlight radiation mode. The SAR mode ray-tracing architecture combined with EM scattering theory or a physical illumination model can be used to simulate the complex process of SAR multiple backscattering. In contrast to light rendering in the field of computer graphics, the SAR sensor, shown in
Figure 2, is an active and singular transmitter, producing an EM wave at the same location as the received echo. Therefore, it is not necessary to consider the contribution of the environment and its own radiant energy to the backscattering field; only the self-energy of the active transmission of the EM wave of the SAR sensor and the continuous decay process of its multiple scattering energies are considered. In general, the sampling time intervals along the azimuth direction are within the second level, and the sampling time intervals along the range direction are within the microsecond level [
21]; these can be defined as the slow sampling time along the azimuth direction and the fast sampling time along the range direction, respectively. In the process of SAR image simulation for local small scenes, the 2D sampling time difference can be generally met, and the “go-stop” mode can be adopted.
The main function of the SAR mode ray-tracing architecture is to determine the principal axis direction of the forward scattering lobe of the EM wave after each scattering. With the EM wave incident on the target surface, the direction of the principal axis of the EM wave forward scattering lobe can be determined by Fresnel’s law of specular reflection. As shown in
Figure 3, the normal vector of the macro-facet, the incidence direction of the EM wave and the specular reflection direction are coplanar. In practical applications, the specular reflection direction calculated by the normal vector of the macro-facet at the intersection point of the EM wave is the main axis direction of the forward scattering lobe.
where
is the principal axis direction of front scattering lobe,
is the macroscopic normal vector of facet, and
is the incident direction of the EM wave. In addition, the information recorded by the SAR mode ray-tracing architecture mainly includes the amplitude of each scattering, the real-time slant range, and the coordinates of the corresponding intersection points and the material parameters of the intersecting facets.
The phase information is obtained by using the ray-tracing algorithm to assist the real-time slant range information conversion when the electromagnetic wave records scattering, mainly introducing the real-time slant range information into the classical RD condition equation. The conversion process will be derived in detail in
Section 3.2, where the representation is simplified to show the initial architecture for obtaining the backscatter field of distributed targets. The geometric mapping between the 3D target and the 2D SAR slant-range image is represented by the key elements of the nearest slant range, the farthest slant range, the 2D sampling intervals, and the real-time slant range. Phase information is essentially the main basis for spatial location assignment of backscatter amplitude. The real-time slant range obtained by our proposed method is transformed into the slant-range coordinate matrix by using the classical RD condition equation, which is equivalent phase information. Then we combine the above designed SAR mode ray-tracing architecture with the usual physical illumination model to carry out the SAR image simulation of the 3D target:
where
is the backscattering field of a distributed target.
is the amplitude of the
-th scattering.
is the real-time equivalent phase of the
-th scattering.
is used to simplify the representation by introducing real-time slant rang
into the classical RD condition equation, which is transformed into the coordinate matrix of the slant range that is the equivalent phase information; the derivation process will be detailed in Equations (9)–(12).
is half of the sum of the propagation paths of the EM waves from the transmission to the reception. The EM waves
are transmitted by traversing the radiating area within the radar beam.
is the total number of transmitted EM waves. The transmitted EM wave is tracked by multiple scattering, and the total
times backscatter subfield is obtained. The total backscatter subfield
can be obtained by vector superposition.
Section 2.1 designs the SAR mode ray-tracing architecture and presents the method of combining it with the physical illumination model for SAR image simulation, which provides the basic conditions for the simulation of multiple backscattering of the EM wave. A distributed target of an electrically large size or larger can be considered composed of a large number of discrete scattering units, each of which can be represented by a point target or a small facet. The backscattering field of the distributed target can be calculated by vector superposition of the backscattering subfield of its independent scattering unit.
2.2. Multiple Backscattering Simulation of the Transmitted EM Wave
Based on the known coordinates of the SAR sensor, we use the SAR space traversal method to send the discrete EM wave to the corresponding radiation area of the distributed target. It is difficult to track and simulate the multiple backscattering process of the transmitted EM wave, but the traditional physical illumination model still belongs to the local scattering model; the multiple scattering mechanism between the components of distributed target is not effective. We will combine with the designed SAR mode ray-tracing architecture to simulate the multiple backscattering of the EM wave. As shown in
Figure 4, taking the transmitted discrete EM wave as an example, the amplitude and slant range of multiple scattering can be recorded in turn, which can provide key technical support for SAR image simulation. Using the map imaging method under RD imaging geometry, the amplitude of each scattering is mapped to the corresponding range gate according to the real-time slant range, which only contains the change in the incoherent scattering energy of the target. There is no definite phase relationship between inter-pulse and intra-pulse, which can be regarded as an incoherent scattering field.
Combined with
Figure 2 and
Figure 4, it can be seen that when the sensor moves in turn according to the preset sampling interval along the azimuth direction and sends EM waves, it receives the target echo signal at the same position. The scattering amplitude and the instantaneous slant range when a discrete EM wave has its
k-th scattering are two key parameters for calculating the multiple backscattering field.
(1) We follow Nyquist’s sampling law to set the sampling interval along the azimuth and range directions, respectively; the EM wave transmitting density is equivalent to the 2D grid density divided by the 2D sampling interval, and the size of the sampling interval is about half of the theoretical resolution. The simulated EM wave is transmitted into the radar beam radiation area by SAR spatial traverse mode in combination with the instantaneous position of the sensor, can be expressed as:
where
is the number of azimuth sampling points, and
is the number of slant-range gates.
and
are the azimuth coordinates of the scene;
and
are the near range and far range, respectively;
is the slant-range sampling space;
is the slant range within the zero-doppler plane;
is the incidence angle; and
is the incidence direction.
is the sensor position;
is the sensor azimuthal coordinate;
is the azimuthal sampling space;
is the incidence angle;
is the simulated incident EM wave;
is the coordinates of any point on the target surface;
is the platform coordinates, which can be determined by the discrete time axis along the azimuth direction, the platform velocity, and the scene range.
(2) According to Fresnel’s law of reflection, the forward propagation direction of the EM wave is calculated at the point of intersection of the EM wave and the target surface. In the plane in which the reflected EM wave, the incident EM wave, and the normal vector of the intersection point lie, the reflection angle is equal to the incident angle and can be expressed as:
where
denotes the specular reflection direction of the
k-th scattering of the EM wave, and the EM wave continues to propagate forward along the specular reflection direction of the energy principal axis;
is the incident direction of the
k-th scattering of the EM wave;
is the instantaneous slant range of the
k-th scattering;
is the intersection coordinate of the
k-th scattering. The process of simulating multiple scattering of the EM wave is essentially the process of taking the specular reflection direction of the principal axis of EM wave as the incident direction of EM wave.
(3) The intersection of the EM wave and the target model panel is taken as the starting point of EM wave forward propagation, and the specular reflections are taken as the new incident direction of the EM wave. The recursive depth of multiple scattering is determined by the actual propagation path and energy decay condition of the EM wave. In the process of multiple scattering, the EM wave follows the conservation of energy, which is manifested in the fact that the outgoing energy is not higher than the incident energy, and the propagating energy will decrease continuously with the occurrence of multiple scattering. Usually, the surface material of the target has a certain glossiness and roughness at the same time, and it is easy to occur in the directional scattering dominated by the specular reflection mechanism. The macro-specular reflection direction is the main energy axis, and some diffuse energy including subsurface scattering continues to propagate along the main energy axis. In order to calculate the forward propagation energy of the EM wave after each scattering, the macroscopic energy decay coefficient is set for each component facet of the distributed target to quantify the energy loss value of the EM wave after each scattering; the energy remaining value is used as the main energy of the forward propagation. The EM wave intersects with different parts of the target surface, and the material properties of each part are different, which has a different influence on the propagation energy decay. In the experiment, we set the energy threshold to prevent the EM wave from entering the loop, and we can judge and track the complex multiple scattering path of each discrete EM wave.
(4) Due to the complexity of the geometry structure of distributed targets, the multiple scattering intersection point and the instantaneous sensor position cannot always be intervisible. It is necessary to add the intervisibility condition, that is, a ray is transmitted from the intersection point to the instantaneous sensor position to see whether there is an obstruction. If there is, the
k-th scattering intersection point and the sensor are not intervisible, and the backscattering energy and phase of the intersection point cannot be recorded. The specific process can be expressed in the following Equation (6):
where
is the vector direction of the instantaneous sensor
transmitting rays towards the current intersection point
;
is the instantaneous distance from
to
;
is the actual intersection point position. If
and
are located at the same position, record the current backscatter amplitude and real-time slant range; Otherwise, this situation will be considered an occlusion state, and the contribution of the current backscatter information will be directly ignored.
(5) The coordinates of the intersection point of the
k-th scattering of the EM wave and the target surface are obtained by steps (1) to (4), and the instantaneous slant range of the
k-th scattering is calculated by the following equation.
where
denotes the slant of the
k-th scattering equal to half of the sum of the distances from the sensor’s instantaneous position
to the point of intersection
of the
k-th scattering and then to
.
denotes the coordinates of the intersection of each scattering.
is the instantaneous position of the sensor.
In the process of multiple scattering, the EM wave always intersects the surface facet with different material properties. The normal vector of the facets is divided into the macro-normal vector and the micro-normal vector. The macro-normal vector of the facet is an important variable for recursively determining the forward tracking direction of the EM wave, and it is also a key element for calculating the backscattering field of the facet.
The normal vector of the micro-facet is the reasonable correction of backscattered field calculation value from the perspective of the micro-facet. The index for quantifying the glossiness and roughness of the target surface material is set according to the regularity degree of the normal vector distribution of the micro-facet. The specular reflectivity coefficient, diffuse reflectivity coefficient, specular index, energy decay coefficient, and permittivity are set according to the type of material parameters in the physical illumination model. The normal vector and index number of the intersecting panel are recorded when calculating the coordinates of the intersection point of the k-th scattering of the discrete EM wave and the target surface. The index number is used to determine the permittivity, diffuse reflection coefficient, specular reflection coefficient, specular index, energy decay coefficient of materials, and facet normal vectors.
According to the physical illumination model of SAR mode, the backscatter energy of the target mainly includes diffuse reflection energy and specular reflection energy. The ray-tracing algorithm is used to track the propagation path of each discrete EM wave. When the geometry structure of the selected distributed target is relatively regular, the termination condition of multiple scattering is that there is no collision point to end recursion. Assuming that the recursive depth of path tracing for discrete EM waves is
k, the
k-th backscattering field can be expressed as
where
indicates the
-th backward sub-scattering.
is the backscattering energy,
is the incident energy,
is the initial incident energy of the EM wave, and
is the residual energy value at the
-th scattering.
is the backscatter coefficient.
is the normal vector of the surface facet;
indicates the diffuse scattering mechanism to the surrounding area.
is the specular reflection coefficient,
is the energy decay coefficient, and
is the specular index.
is the horizontal polarimetric scatter coefficient, and
is the vertical polarimetric scatter coefficient.
is the relative permittivity.
is the direction of EM wave incidence, and
is the direction of EM wave specular reflection.
is the direction of energy reception, and
is the energy decay threshold, which can be used to terminate the tracking of the complex multiple scattering path of each discrete EM wave.
The value of is defined adaptively by the recursive energy decay under the constraint of the law of conservation of energy. In fact, the iterative depth value of multiple scattering of electromagnetic waves is determined by the complexity of the geometric structure of distributed targets to a greater extent. For example, the geometric structure of the local scene is relatively flat, and the path tracking will stop after only one scattering. If the geometric structure of the local scene is complex, the incident electromagnetic wave will scatter many times. Every time the incident electromagnetic wave scatters on the target surface, its energy decays and then stops tracking when the energy decays below the energy threshold .
It is concluded that the simulation of multiple backscattering based on SAR mode ray tracing is an important step in SAR image simulation, which directly affects the quality of SAR images. Firstly, the distributed targets within the SAR beam radiation area are transformed into discrete EM waves by using the SAR beam scanning method. Combined with SAR-mode ray-tracing architecture and the physical illumination model, the amplitude and real-time slant range of each EM wave when multiple scattering occurs are recorded, which are the necessary parameters for the calculation of the target multiple backscattering field. Then, the amplitude and slant range of multiple scattering are brought into the range condition equation, and the amplitude is directly mapped to the 2D SAR slant-range image to generate a SAR simulation image.