20. Frame camera models

Ames Stereo Pipeline supports a generic Pinhole camera model with several lens distortion models which cover common calibration methods (Section 20.1), the somewhat more complicated panoramic (optical bar) camera model (Section 20.2), and the CSM Frame model, that has several lens distortion implementations (Section 20.3).

Bundle adjustment can refine the intrinsic and extrinsic camera parameters, including the lens distortion model (Section 12.2.1).

20.1. Pinhole models

20.1.1. Overview

The generic Pinhole model uses the following parameters:

  • fu = The focal length in horizontal pixel units.

  • fv = The focal length in vertical pixel units.

  • cu = The horizontal offset of the principal point of the camera in the image plane in pixel units, from 0,0.

  • cv = The vertical offset of the principal point of the camera in the image plane in pixel units, from 0,0.

  • pitch = The size of each pixel in the units used to specify the four parameters listed above. This will usually either be 1.0 if they are specified in pixel units or alternately the size of a pixel in millimeters or meters.

The focal length is sometimes known as the principal distance. The value \(cu\) is usually approximately half the image width in pixels times the pitch, while \(cv\) is often the image height in pixels times the pitch, though there are situations when these can be quite different.

A few sample Pinhole models are shown later in the text. The underlying mathematical model is described in Section 20.1.5.

Along with the basic Pinhole camera parameters, a lens distortion model can be added. Note that the units used in the distortion model must match the units used for the parameters listed above. For example, if the camera calibration was performed using units of millimeters the focal lengths etc. must be given in units of millimeters and the pitch must be equal to the size of each pixel in millimeters. Alternatively, units of meter can be used, and the choice of unit must be documented by the creators of the models.

The following lens distortion models are currently supported. (The formulas below may miss some small details; the implementation in LensDistortion.cc in VisionWorkbench should be the final reference.)

Note that the values below change drastically depending on whether the model creator chooses pixel units, or if measuring in millimeters or meters. In either case, all lengths must be consistent and the units documented by the model creator.

20.1.2. Lens distortion models

Here are the lens distortion models supported by ASP. Samples for each model are shown in Section 20.1.3.

20.1.2.1. Null

A placeholder model that applies no distortion.

20.1.2.2. Tsai

A common distortion model [Tsai87]. In the most recent builds (after ASP 3.3.0) this was made to agree precisely with the OpenCV radial-tangential lens distortion model. This model uses the following parameters:

K1, K2, K3 = Radial distortion parameters. The last one is optional.

P1, P2 = Tangential distortion parameters.

The lens distortion operation is computed via an explicit formula, and for undistortion a nonlinear solver is used based on Newton’s method.

This is the preferred model, unless the lens has a wide field of view, when the Fisheye model should be used (described further below).

20.1.2.3. Adjustable Tsai

A variant of the Tsai model where any number of K terms and a skew term (alpha) can be used. Can apply the AgiSoft Lens calibration parameters.

20.1.2.4. Brown-Conrady

This is an older model based on a centering angle [Bro66, Bro71]. Example usage is in Section 10.2.

This model uses the following parameters:

K1, K2, K3 = Radial distortion parameters.

P1, P2 = Tangential distortion parameters.

xp, yp = Principal point offset.

phi = Tangential distortion angle in radians.

The following equations describe the distortion. Note that this model uses non-normalized pixel units, so they can be in millimeters or meters:

\[ \begin{align}\begin{aligned}x = x_{dist} - xp\\y = y_{dist} - yp\\r^{2} = x^{2} + y^{2}\\dr = K_{1}r^{3} + K_{2}r^{5} + K_{3}r^{7}\\x_{undist} = x + x\frac{dr}{r} - (P_{1}r^{2} +P_{2}r^{4})\sin(phi)\\y_{undist} = y + y\frac{dr}{r} + (P_{1}r^{2} +P_{2}r^{4})\cos(phi)\end{aligned}\end{align} \]

The formulas start with distorted pixels that are then undistorted. This is not preferable with ASP, as then the distortion operation requires a solver, which makes bundle adjustment and mapprojection very slow. Use instead the Tsai model.

A Brown-Conrady model can be converted to a Tsai model with convert_pinhole_model (Section 16.15). The produced model can be refined with bundle adjustment (Section 12.2.1), if having several images and many interest point matches.

20.1.2.5. Photometrix

A model matching the conventions used by the Australis software from Photometrix.

K1, K2, K3 = Radial distortion parameters.

P1, P2 = Tangential distortion parameters.

xp, yp = Principal point offset.

B1, B2 = Unused parameters.

The following equations describe the undistortion. Note that this model uses non-normalized pixel units, so they are in mm.

\[ \begin{align}\begin{aligned}x = x_{dist} - xp\\y = y_{dist} - yp\\r^{2} = x^{2} + y^{2}\\dr = K_{1}r^{3} + K_{2}r^{5} + K_{3}r^{7}\\x_{undist} = x + x\frac{dr}{r} + P_{1}(r^{2} +2x^{2}) + 2P_{2}xy\\y_{undist} = y + y\frac{dr}{r} + P_{2}(r^{2} +2y^{2}) + 2P_{1}xy\end{aligned}\end{align} \]

These formulas also start with distorted pixels and undistort them, just as the Brown-Conrady model. This is not preferred. Use instead the Tsai model.

20.1.2.6. Fisheye

A four-parameter model for wide field-of-view lenses, with the same implementation as OpenCV and rig_calibrator (Section 16.58).

The parameters are named k1, k2, k3, k4.

To apply the lens distortion with this model, the undistorted pixels are first shifted relative to the optical center, divided by the focal length, producing pixel (x, y), and then the following equations are applied:

\[ \begin{align}\begin{aligned}r = \sqrt{x^2 + y^2}\\\theta = \arctan(r)\\\theta_d = \theta (1 + k_1 \theta^2 + k_2 \theta^4 + k_3 \theta^6 + k_4 \theta^8)\\s = \frac{\theta_d}{r}\\x_{dist} = s \cdot x\\y_{dist} = s \cdot y\end{aligned}\end{align} \]

These values are then multiplied by the focal length, and the optical center is added back in.

The undistortion operation goes in the opposite direction. It requires inverting a nonlinear function, which is done with Newton’s method.

Care is needed around the origin to avoid division of small numbers.

20.1.2.7. FOV

A field-of-view model with a single parameter, for wide-angle lenses.

This is in agreement with rig_calibrator (Section 16.58).

The implementation is as follows. Let k1 by the distortion parameter. Given an undistorted pixel, shift it relative to the optical center, divide by the focal length, producing pixel (x, y). Then, the following equations are applied:

\[ \begin{align}\begin{aligned}p_1 = 1 / k_1\\p_2 = 2 \tan(k_1 / 2)\\r_u = \sqrt{x^2 + y^2}\\r_d = p_1 \arctan(r_u p_2)\\s = r_d / r_u\\x_{dist} = s \cdot x\\y_{dist} = s \cdot y\end{aligned}\end{align} \]

These values are then multiplied by the focal length, and the optical center is added back in.

The undistortion operation goes in the opposite direction, and an explicit formula exists for that.

Care is needed around the origin to avoid division of small numbers.

20.1.2.8. RPC

The distortion is based on a rational polynomial coefficient (RPC) model. This is different than going from ground to image coordinates via RPC (Section 8.19).

In this model, one goes from undistorted normalized pixels \((x, y)\) to distorted normalized pixels via the formulas

\[ \begin{align}\begin{aligned}x_{dist} = \frac{P_1(x, y)}{Q_1(x, y)}\\y_{dist} = \frac{P_2(x, y)}{Q_2(x, y)}\end{aligned}\end{align} \]

The functions in the numerator and denominator are polynomials in \(x\) and \(y\) with certain coefficients. The degree of polynomials can be any positive integer. A degree of 3 or 4 is usually more than sufficient.

The inputs and output pixels are normalized, that is, shifted relative to the optical center, and (in the most latest builds) are also divided by the focal length. Such normalizations are applied before distortion / undistortion operations, and then undone after them. This is consistent with the radial-tangential and fisheye models.

RPC distortion models can be generated as approximations to other pre-existing models with the tool convert_pinhole_model (Section 16.15).

Also in the latest builds, the RPC undistortion is computed via a solver based on Newton’s method, as for the fisheye lens distortion model.

20.1.3. File formats

ASP Pinhole model files are written in an easy to work with plain text format using the extension .tsai. A sample file is shown below.

VERSION_4
PINHOLE
fu = 28.429
fv = 28.429
cu = 17.9712
cv = 11.9808
u_direction = 1  0  0
v_direction = 0  1  0
w_direction = 0  0  1
C = 266.943 -105.583 -2.14189
R = 0.0825447 0.996303 -0.0238243 -0.996008 0.0832884 0.0321213 0.0339869 0.0210777 0.9992
pitch = 0.0064
Photometrix
xp = 0.004
yp = -0.191
k1 = 1.31024e-04
k2 = -2.05354e-07
k3 = -5.28558e-011
p1 = 7.2359e-006
p2 = 2.2656e-006
b1 = 0.0
b2 = 0.0

The first half of the file is the same for all Pinhole models:

  • VERSION_X = A header line used to track the format of the file.

  • PINHOLE = The type of camera model, so that other types can be stored with the .tsai extension.

  • fu, fv, cu, cv = The first four intrinsic parameters described in the previous section.

  • u, v, and w_direction = These lines allow an additional permutation of the axes of the camera coordinates. By default, the positive column direction aligns with x, the positive row direction aligns with y, and downward into the image aligns with z.

  • C = The location of the camera center, usually in the geocentric coordinate system (GCC/ECEF).

  • R = The rotation matrix describing the camera’s absolute pose in the world coordinate system (camera-to-world rotation, Section 20.1.5).

  • pitch = The pitch intrinsic parameter described in the previous section.

The second half of the file describes the lens distortion model being used. The name of the distortion model appears first, followed by a list of the parameters for that model. The number of parameters may be different for each distortion type.

Partial samples of each format are shown below. The part up to and including the line having the pitch is the same for all models and not shown in the examples.

  • Null

    NULL
    
  • Tsai

    TSAI
    k1 = 1.31024e-04
    k2 = -2.05354e-07
    p1 = 0.5
    p2 = 0.4
    k3 = 1e-3
    

The k3 parameter is optional in the Tsai model. It is stored last, as done in OpenCV.

  • Adjustable Tsai

    AdjustableTSAI
    Radial Coeff: Vector3(1.31024e-04, 1.31024e-07, 1.31024e-08)
    Tangential Coeff: Vector2(-2.05354e-07, 1.05354e-07)
    Alpha: 0.4
    
  • Brown-Conrady

    BrownConrady
    xp = 0.5
    yp = 0.4
    k1 = 1.31024e-04
    k2 = -2.05354e-07
    k3 = 1.31024e-08
    p1 = 0.5
    p2 = 0.4
    phi = 0.001
    
  • Photometrix

    Photometrix
    xp = 0.004
    yp = -0.191
    k1 = 1.31024e-04
    k2 = -2.05354e-07
    k3 = -5.28558e-011
    p1 = 7.2359e-006
    p2 = 2.2656e-006
    b1 = 0.0
    b2 = 0.0
    
  • Fisheye

    FISHEYE
    k1 = -0.036031089735101024
    k2 = 0.038013929764216248
    k3 = -0.058893197165394658
    k4 = 0.02915171342570104
    
  • RPC

    RPC
    rpc_degree = 1
    image_size = 5760 3840
    distortion_num_x   = 0 1 0
    distortion_den_x   = 1 0 0
    distortion_num_y   = 0 0 1
    distortion_den_y   = 1 0 0
    

This sample RPC lens distortion model represents the case of no distortion, when the degree of the polynomials is 1, and both the distortion and undistortion formula leave the pixels unchanged, that is, the distortion transform is

\[(x, y) \to (x, y) = \left(\frac{ 0 + 1\cdot x + 0\cdot y}{1 + 0\cdot x + 0\cdot y}, \frac{0 + 0\cdot x + 1\cdot y}{1 + 0\cdot x + 0\cdot y}\right).\]

In general, if the degree of the polynomials is \(n\), there are \(2(n+1)(n+2)\) coefficients. The zero-th degree coefficients in the denominator are always set to 1.

20.1.4. Notes

For several years Ames Stereo Pipeline generated Pinhole files in the binary .pinhole format. That format is no longer supported.

Also in the past Ames Stereo Pipeline has generated a shorter version of the current file format, also with the extension .tsai, which only supported the TSAI lens distortion model. Existing files in that format can still be used by ASP.

Note that the orbitviz tool can be useful for checking the formatting of .tsai files you create and to estimate the position and orientation. To inspect the orientation use the optional .dae model file input option and observe the rotation of the 3D model.

20.1.5. How the pinhole model is applied

As mentioned in Section 20.1.3, the ASP Pinhole models store the focal length as \(fu\) and \(fv\), the optical center \((cu, cv)\) (which is the pixel location at which the ray coming from the center of the camera is perpendicular to the image plane, in units of the pixel pitch), the vector \(C\) which is the camera center in world coordinates system, and the matrix \(R\) that is the transform from camera to world coordinates.

To go in more detail, a point \(Q\) in the camera coordinate system gets transformed to a point \(P\) in the world coordinate system via:

\[P = RQ + C\]

Hence, to go from world to camera coordinates one does:

\[Q = R^{-1} P - R^{-1} C\]

From here the undistorted pixel location is computed as:

\[\frac{1}{p} \left(fu \frac{Q_1}{Q_3} + cu, fv \frac{Q_2}{Q_3} + cv\right)\]

where \(p\) is the pixel pitch. Next, a distortion model may be applied, as discussed earlier.

20.2. Panoramic Camera Model

ASP also supports a simple panoramic/optical bar camera model for use with images such as the declassified Corona KH4 and Keyhole KH9 images. It implements the model from [SCS03] with the motion compensation from [SKY04].

Such a model looks as follows:

VERSION_4
OPTICAL_BAR
image_size = 110507 7904
image_center = 55253.5 3952
pitch = 7.0e-06
f = 0.61000001430511475
scan_time = 0.5
forward_tilt = -0.261799
iC = -1047140.9611702315 5508464.4323527571 3340425.4078937685
iR = -0.96635634448923746 -0.16918164442572045 0.1937343197650008 -0.23427205529446918 0.26804084264169648 -0.93448954557235941 0.10616976770014927 -0.94843643849513648 -0.29865750042675621
speed = 7700
mean_earth_radius = 6371000
mean_surface_elevation = 4000
motion_compensation_factor = 1.0
scan_dir = left

Here, the image size and center are given in pixels, with the width followed by the height. The pixel pitch and focal length f are in meters. The scan time is seconds, the forward tilt is in radians, the speed is in meters per second, and the Earth radius and mean surface elevation are in meters. The initial camera center iC is in meters, and the rotation matrix iR stores the absolute pose. scan_dir must be set to left or right. The values scan_dir and use_motion_compensation control how the sensor model accounts accounts for the motion of the satellite during the image scan. Without the benefit of detailed historical documents it may require experimentation to find the good initial values for these cameras. When using bundle_adjust, the intrinsic parameters that are solved for are speed, motion_compensation_factor, and scan_time.

20.3. CSM frame camera

ASP supports the CSM (Section 8.12) frame camera model. This behaves about the same as the Pinhole model (Section 20.1), but it has different lens distortion models. Those include the OpenCV radial-tangential distortion model (it has 5 parameters, in the order k1, k2, p1, p2, k3), transverse distortion, which is a pair of full polynomials of degree 3 in both x and y (20 coefficients), and various other specialized models.

ASP’s cam_gen program (Section 16.8.1.4) can find the best-fit CSM frame camera model with the OpenCV radial-tangential distortion and with the transverse (3rd degree polynomial) distortion model. Then bundle adjustment can be used to refine the intrinsic and extrinsic camera parameters (Section 12.2.3).

The sat_sim program (Section 16.60) can create CSM frame cameras simulating a satellite in orbit.