3.2.6 Refinement of Photo Co-ordinates: - Lens distortion deteriorates the positional accuracy of image points located on the Image plane. Therefore in order to determine the exact Image/Photo Co-ordinates, it is necessary to apply necessary lens distortion correction to photo co-ordinates so that accurate ray that existed at the instant of exposure be re-created.

After applying the lens distortion correction; the photo co-ordinates are being refined. The refined photo co-ordinates then clubbed with a third dimension (-f) from Exposure station to Image Plane (Available from CCC) to make it a triplet.

Figure illustrates the difference between radial and tangential lens distortion

There are two types of lens distortion error Radial Lens distortion Error (r) Tangential Lens distortion Error (t)   t being much smaller r in magnitude is treated as negligible and correction not applied in photogrammetric processing.

The lens distortion pattern is given in CCC in the form a table of ‘Radial distance’ and ‘Distortion’ or in equation form.

Radial lens distortion causes imaged points to be distorted along radial lines from the principal point o. The effect of radial lens distortion is represented as Δr. Radial lens distortion is also commonly referred to as symmetric lens distortion. Tangential lens distortion occurs at right angles to the radial lines from the principal point. The effect of tangential lens distortion is represented as Δt. Because tangential lens distortion is much smaller in magnitude than radial lens distortion, it is considered negligible. The effects of lens distortion are commonly determined in a laboratory during the camera calibration procedure.

Radial distortion can be determined by: Calibration procedure As a function of the angle from the optical axis, since the angle α between the optical axis and the ray is a function of the focal length f and the distance d of the point from the optical axis:

Lens Distortion

Lens Distortion Radial distortion causes inward (barrel or negative) or outward (positive or pincushion) displacement of a given image point from its ideal location negative radial displacement >> barrel distortion positive radial displacement >> pincushion distortion

For the point ‘a’ let the refined photo co-ordinate is (xa, ya) adding third dimension (-f); it becomes a triplet (xa ya –f) which is nothing but the desired co-ordinate in Image Space system to formulate the vector Oa to represent the ray for image point ‘a’ that existed at the instant of exposure as per Co-linearity Condition.

3.2.3 Lens Distortion  Lens distortion deteriorates the positional accuracy of image points located on the image plane. Two types of radial lens distortion exist: radial and tangential lens distortion. Lens distortion occurs when light rays passing through the lens are bent, thereby changing directions and intersecting the image plane at positions deviant from the norm.

The effects of radial lens distortion throughout an image can be approximated using a polynomial. The following polynomial is used to determine coefficients associated with radial lens distortion:

So the distortion can be evaluated from the calculated distances. Lens Distortion So from the previous equation, the calibrated based on minimizing the distortions values by least squares. A method to express the distortion as a polynomial function of odd powers of radial distance: k1, k2, k3 = polynomial coefficients for radial lens distortions. So the distortion can be evaluated from the calculated distances.

Lens Distortion example We are given a photograph with lens whose radial coefficients are

Understanding Least Squares Adjustment Residual Difference between observation and MPV e.g. i = MPV – obsi Principle of LS ERE 371

Measure distance five times Every observation has associated residual 100.04, 100.15, 99.97, 99.94, 100.06 Every observation has associated residual M = 100.04 + 1  1 = M – 100.04 M = 100.15 + 2  2 = M – 100.15 M = 99.97 + 3  3 = M – 99.97 M = 99.94 + 4  4 = M – 99.94 M = 100.06 + 5  5 = M – 100.06 LS selects M to minimize sum of squared residuals

Example cont. Want to minimize sum of residuals Recall from calculus Function minimum occurs when derivative is zero Take derivate with respect to M and equate to zero

Example for a table with distortion values: 0.0 0.000 each line: radius [mm], distortion [mm] 1.0 0.007 2.0 0.013 3.0 0.020 4.0 0.026 (etc.) A click onto the OK button store the selected parameters. If no distortion cor- rection should be used, click onto the Reset button.