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I have a requirement to calculate the ground footprint for an aerial camera. The photos are TerraPhotos. TerraPhoto user guide provide camera position and plane orientation in .IML file. Additionally, I have the camera calibration file.

In TerraPhoto guide, the yaw, pitch, and roll of the aircraft are defined as follows:

  • yaw (heading): from North clock-wise direction
  • roll: positive, if left-wing is up.
  • pitch: positive, if the nose of the aircraft is up

The camera calibration details are as follows:

[TerraPhoto calibration]
Version=20050513
Description= Nikon D800E BW 50mm
TimeOffset= 0.0000
Exposure= 0.00000
LeverArm= 0.0000 0.0000 0.0000
AntennaToCameraOffset= 0.0000 0.0000 0.0000
AttitudeCorrections(HRP)= -0.4546 0.7553 -34.7538
PlateSize= 7630.00000000 4912.00000000
ImageSize= 7630 4912
Margin= 0
FiducialRadius= 40
FiducialMarks= 0
Orientation= BOTTOM
PrincipalPoint(XoYoZo)= -77.40000000 112.80000000 -10476.54389508
LensModel=Balanced
LensK0=0.000000E+000
LensK1=0.000000E+000
LensK2=0.000000E+000
LensP1=0.000000E+000
LensP2=0.000000E+000

Here, I see that AttitudeCorrection for the camera is given. Hence, I believe it is the orientation of the aerial camera according to the local frame (i.e. aircraft).

with respect to a given aerial photo, I have the following details, which I obtained from the.IML file (please check page 344 for more info).

Image=SLR2_443_20150326_144759_C_B_3489_DUBLIN_AREA_2KM2_FL_300_2232888
Time=402972.957799
Xyz=316440.819 234424.606 312.938
Hrp=-113.33234 2.03435 -1.87426
  • Image represent the name of the image
  • XYZ (i.e. camera easting, northing, and elevation)
  • aircraft yaw, pitch, roll

With this specific information at hand, I am attempting to calculate the ground coordinates of the Image. I intend to use Horizontal FoV, and vertical FoV.

I've been attempting this for some time, but still unable to estimate the geocoordinates properly. I did attempt, pin-hole model as well. I obtain results around the area of interest, but my results do not confirm the actual geolocations.

I intend to use either pinhole model or Horizontal and Vertical field of view (FoV) to calculate my geocoordinates.

A guide in the right direction is appreciated.

Code with respect to FoV calculation is provided.

def createRollMatrix(yaw,pitch,roll):
  '''
     Uses the Eigen formatted rotation matrix
     pulled directly from Eigen base code to python
  '''
  # convert degrees to radians
  yaw = np.radians(yaw)
  pitch = np.radians(pitch)
  roll = np.radians(roll)

  su = np.sin(roll)
  cu = np.cos(roll)
  sv = np.sin(pitch)
  cv = np.cos(pitch)
  sw = np.sin(yaw)
  cw = np.cos(yaw)

  rotation_matrix = np.zeros((3,3))
  
  rotation_matrix[0][0] = cv*cw
  rotation_matrix[0][1] = su*sv*cw - cu*sw
  #rotation_matrix[0][2] = su*sw + cu - cu*sw
  rotation_matrix[0][2] = su*sw + cu*sv*cw
  
  rotation_matrix[1][0] = cv*sw
  rotation_matrix[1][1] = cu*cw + su*sv*sw
  rotation_matrix[1][2] = cu*sv*sw - su*cw

  rotation_matrix[2][0] = -sv
  rotation_matrix[2][1] = su*cv
  rotation_matrix[2][2] = cu*cv

  return rotation_matrix

#### CAMERA misalignment angles
yaw = -0.4546 #  
pitch = -34.7538  # 
roll = 0.7553 #  0 

#### aircraft's yaw pitch roll
yaw1 =  -113.33234
pitch1 =  -1.87426
roll1 = 2.03435

R = createRollMatrix(yaw,pitch,roll)
R2 = createRollMatrix(yaw1,pitch1,roll1)

Corrected_R = (R2.dot(R))

yaw = math.atan(Corrected_R[1][0]/ Corrected_R[0][0])
yaw

roll =  math.atan(Corrected_R[2][1]/ Corrected_R[2][2])
roll

pitch = math.atan(-Corrected_R[2][0]/ math.sqrt( (math.pow(Corrected_R[2][1], 2) + math.pow(Corrected_R[2][2], 2))))
pitch

Subsequently, I use the following code to calculate the geocoordinates.

import math
import numpy as np 

# pip install vector3d
from vector3d.vector import Vector


class CameraCalculator:
    """Porting of CameraCalculator.java
    This code is a 1to1 python porting of the java code:
        https://github.com/zelenmi6/thesis/blob/master/src/geometry/CameraCalculator.java
    referred in:
        https://stackoverflow.com/questions/38099915/calculating-coordinates-of-an-oblique-aerial-image
    The only part not ported are that explicetly abandoned or not used at all by the main
    call to getBoundingPolygon method.
    by: milan zelenka
    https://github.com/zelenmi6
    https://stackoverflow.com/users/6528363/milan-zelenka
    example:
        c=CameraCalculator()
        bbox=c.getBoundingPolygon(
            math.radians(62),SLR2_443_20150326_144759_C_B_3489_DUBLIN_AREA_2KM2_FL_300_2233046
            math.radians(84),
            117.1, 
            math.radians(0),
            math.radians(33.6),
            math.radians(39.1))
        for i, p in enumerate(bbox):
            print("point:", i, '-', p.x, p.y, p.z)
    """

    def __init__(self):
        pass

    def __del__(delf):
        pass

    @staticmethod
    def getBoundingPolygon(FOVh, FOVv, altitude, roll, pitch, heading):
        '''Get corners of the polygon captured by the camera on the ground. 
        The calculations are performed in the axes origin (0, 0, altitude)
        and the points are not yet translated to camera's X-Y coordinates.
        Parameters:
            FOVh (float): Horizontal field of view in radians
            FOVv (float): Vertical field of view in radians
            altitude (float): Altitude of the camera in meters
            heading (float): Heading of the camera (z axis) in radians
            roll (float): Roll of the camera (x axis) in radians
            pitch (float): Pitch of the camera (y axis) in radians
        Returns:
            vector3d.vector.Vector: Array with 4 points defining a polygon
        '''
        # import ipdb; ipdb.set_trace()
        ray11 = CameraCalculator.ray1(FOVh, FOVv)
        ray22 = CameraCalculator.ray2(FOVh, FOVv)
        ray33 = CameraCalculator.ray3(FOVh, FOVv)
        ray44 = CameraCalculator.ray4(FOVh, FOVv)

        rotatedVectors = CameraCalculator.rotateRays(
                ray11, ray22, ray33, ray44, roll, pitch, heading)
        
        #origin = Vector(0, 0, altitude) # 
        #origin = Vector(0, 0, altitude) # 

   
   ###   FW ---- SLR1
    

        #  origin = Vector(316645.779, 234643.179, altitude)

        '''
        BW ===== SLR2 
        '''
        origin = Vector(316440.819, 234424.606, altitude)
        #origin = Vector(316316, 234314, altitude)
        intersections = CameraCalculator.getRayGroundIntersections(rotatedVectors, origin)

        return intersections


    # Ray-vectors defining the the camera's field of view. FOVh and FOVv are interchangeable
    # depending on the camera's orientation
    @staticmethod
    def ray1(FOVh, FOVv):
        '''
        tasto
        Parameters:
            FOVh (float): Horizontal field of view in radians
            FOVv (float): Vertical field of view in radians
        Returns:
            vector3d.vector.Vector: normalised vector
        '''
        pass
        ray = Vector(math.tan(FOVv / 2), math.tan(FOVh/2), -1)
        return ray.normalize()

    @staticmethod
    def ray2(FOVh, FOVv):
        '''
        Parameters:
            FOVh (float): Horizontal field of view in radians
            FOVv (float): Vertical field of view in radians
        Returns:
            vector3d.vector.Vector: normalised vector
        '''
        ray = Vector(math.tan(FOVv/2), -math.tan(FOVh/2), -1)
        return ray.normalize()

    @staticmethod
    def ray3(FOVh, FOVv):
        '''
        Parameters:
            FOVh (float): Horizontal field of view in radians
            FOVv (float): Vertical field of view in radians
        Returns:
            vector3d.vector.Vector: normalised vector
        '''
        ray = Vector(-math.tan(FOVv/2), -math.tan(FOVh/2), -1)
        return ray.normalize()

    @staticmethod
    def ray4(FOVh, FOVv):
        '''
        Parameters:
            FOVh (float): Horizontal field of view in radians
            FOVv (float): Vertical field of view in radians
        Returns:
            vector3d.vector.Vector: normalised vector
        '''
        ray = Vector(-math.tan(FOVv/2), math.tan(FOVh/2), -1)
        return ray.normalize()

    @staticmethod
    def rotateRays(ray1, ray2, ray3, ray4, roll, pitch, yaw):
        """Rotates the four ray-vectors around all 3 axes
        Parameters:
            ray1 (vector3d.vector.Vector): First ray-vector
            ray2 (vector3d.vector.Vector): Second ray-vector
            ray3 (vector3d.vector.Vector): Third ray-vector
            ray4 (vector3d.vector.Vector): Fourth ray-vector
            roll float: Roll rotation
            pitch float: Pitch rotation
            yaw float: Yaw rotation
        Returns:
            Returns new rotated ray-vectors
        """
        sinAlpha = math.sin(yaw) #sw OK
        sinBeta = math.sin(pitch) #sv OK
        sinGamma = math.sin(roll) #su OK
        cosAlpha = math.cos(yaw) #cw OK
        cosBeta = math.cos(pitch) #cv OK
        cosGamma = math.cos(roll) #cu OK
        m00 = cosBeta * cosAlpha # cosAlpha * cosBeta  #cw*cv 
        m01 = sinGamma * sinBeta * cosAlpha - cosGamma * sinAlpha # cosAlpha * sinBeta * sinGamma - sinAlpha * cosGamma     #cw*sv#cu
        m02 = sinGamma * sinAlpha +  cosGamma * cosAlpha * sinBeta#cosAlpha * sinBeta * cosGamma + sinAlpha * sinGamma
        m10 = sinAlpha * cosBeta
        m11 = sinAlpha * sinBeta * sinGamma + cosAlpha * cosGamma
        m12 = sinAlpha * sinBeta * cosGamma - cosAlpha * sinGamma
        m20 = -sinBeta
        m21 = cosBeta * sinGamma
        m22 = cosBeta * cosGamma
        
        # Matrix rotationMatrix = new Matrix(new double[][]{{m00, m01, m02}, {m10, m11, m12}, {m20, m21, m22}})
        rotationMatrix = np.array([[m00, m01, m02], [m10, m11, m12], [m20, m21, m22]])

        # Matrix ray1Matrix = new Matrix(new double[][]{{ray1.x}, {ray1.y}, {ray1.z}})
        # Matrix ray2Matrix = new Matrix(new double[][]{{ray2.x}, {ray2.y}, {ray2.z}})
        # Matrix ray3Matrix = new Matrix(new double[][]{{ray3.x}, {ray3.y}, {ray3.z}})
        # Matrix ray4Matrix = new Matrix(new double[][]{{ray4.x}, {ray4.y}, {ray4.z}})
        ray1Matrix = np.array([[ray1.x], [ray1.y], [ray1.z]])
        ray2Matrix = np.array([[ray2.x], [ray2.y], [ray2.z]])
        ray3Matrix = np.array([[ray3.x], [ray3.y], [ray3.z]])
        ray4Matrix = np.array([[ray4.x], [ray4.y], [ray4.z]])
        
        res1 = rotationMatrix.dot(ray1Matrix)
        res2 = rotationMatrix.dot(ray2Matrix)
        res3 = rotationMatrix.dot(ray3Matrix)
        res4 = rotationMatrix.dot(ray4Matrix)
        
        rotatedRay1 = Vector(res1[0, 0], res1[1, 0], res1[2, 0])
        rotatedRay2 = Vector(res2[0, 0], res2[1, 0], res2[2, 0])
        rotatedRay3 = Vector(res3[0, 0], res3[1, 0], res3[2, 0])
        rotatedRay4 = Vector(res4[0, 0], res4[1, 0], res4[2, 0])
        rayArray = [rotatedRay1, rotatedRay2, rotatedRay3, rotatedRay4]
        
        return rayArray

    @staticmethod
    def getRayGroundIntersections(rays, origin):
        """
        Finds the intersections of the camera's ray-vectors 
        and the ground approximated by a horizontal plane
        Parameters:
            rays (vector3d.vector.Vector[]): Array of 4 ray-vectors
            origin (vector3d.vector.Vector): Position of the camera. The computation were developed 
                                            assuming the camera was at the axes origin (0, 0, altitude) and the python
                                            results translated by the camera's real position afterwards.
        Returns:
            vector3d.vector.Vector
        """
        # Vector3d [] intersections = new Vector3d[rays.length];
        # for (int i = 0; i < rays.length; i ++) {
        #     intersections[i] = CameraCalculator.findRayGroundIntersection(rays[i], origin);
        # }
        # return intersections

        # 1to1 translation without python syntax optimisation
        intersections = []
        for i in range(len(rays)):
            intersections.append( CameraCalculator.findRayGroundIntersection(rays[i], origin) )
        return intersections

    @staticmethod
    def findRayGroundIntersection(ray, origin):
        """
        Finds a ray-vector's intersection with the ground approximated by a planeç
        Parameters:
            ray (vector3d.vector.Vector): Ray-vector
            origin (vector3d.vector.Vector): Camera's position
        Returns:
            vector3d.vector.Vector
        """
        # Parametric form of an equation
        # P = origin + vector * t
        x = Vector(origin.x,ray.x)
        y = Vector(origin.y,ray.y)
        z = Vector(origin.z,ray.z)
        
        # Equation of the horizontal plane (ground)
        # -z = 0
        
        # Calculate t by substituting z
        t = - (z.x / z.y)
        
        # Substitute t in the original parametric equations to get points of intersection
        return Vector(x.x + x.y * t, y.x + y.y * t, z.x + z.y * t)
8
  • @scottbb, no, my question inquires about the relationship of Field of View and the principal point.
    – CoolCK
    May 4 at 5:22
  • How does the principal point deviate from image center? Is the lens decentered, shifted, or tilted?
    – scottbb
    May 4 at 5:44
  • The camera is mounted to an aircraft with 34 degrees of pitch. Also, it has a dimension of 7360x4196 pixels. The principal point is given as follows" "PrincipalPoint(XoYoZo)= -77.40000000 112.80000000 -10476.54389508"
    – CoolCK
    May 4 at 5:58
  • 1
    @slingeraap it was my mistake, you are correct, I updated the description.
    – CoolCK
    May 4 at 17:05
2

Your assumption is correct, the formula

a = 2 arctan(d/2f)

assumes the principal point in the image center. With r=d/2 being the "radius" from the principal point to the sensor edge, you can read it as

a = arctan(r/f) + arctan(r/f)

still assuming that the radii to e.g. the left and right edge are equal. If the aren't, simply use:

a = arctan(r1/f) + arctan(r2/f)

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