GE 178 Lecture 6:

Distortion and Displacement

Relief Displacement

Tilt Displacement

DISTORTION VS. DISPLACEMENT

Distortion shift in the location of an object, which

changes the perspective characteristics of

the photo

Types of Distortion

1. Film and Print Shrinkage – negligible effect*

2. Atmospheric Refraction of Light Rays – negligible effect*

3. Image Motion

4. Lens Distortion

*Except for precise mapping projects

Lens Distortion

small effects due to the flaws in the optical components (lens) of camera systems leading to distortions

typically more serious at the edges of photo

radial from the principal point

makes objects appear either closer to, or farther from the principal point than they actually are

may be corrected using calibration curves

examples: car windows/windshields, carnival mirrors

Lens Distortion

Lens Distortion

Displacement

shift in the location of an object in a photo,

which does not change the perspective

characteristics of the photo

fiducial distance between an object's image and

it's true plan position, caused by change in

elevation

Types of Displacement

1. Curvature of the Earth – negligible effect*

2. Relief Displacement – radial from the nadir

3. Tilt Displacement – radial from the isocenter

*Except for precise mapping projects

Major Causes of Non-uniformity in

Scale within a Single Photograph 1. Relief Displacement

2. Tilt Displacement

Relief

Displacement

Relief Displacement

Error in the position of the point in a photograph because of relief

The position of a point in the photograph (which has a central projection) is different from its corresponding position on the map (which has an orthogonal projection) due to relief

Radial from the nadir (assuming a vertical photograph, therefore, nadir = center of photo)

Relief Displacement

Relief Displacement

The farther a point is from the nadir, the greater the

displacement

Relief Displacement

Relief Displacement

Relief Displacement

f

Hmge (flying height)

datum plane

∆h

r’

∆r

CASE 1:

Point is above the datum plane

datum plane

Relief Displacement

f

Hmge (flying height)

∆h

r’

∆r

CASE 2:

Point is below the datum plane

Relief Displacement

r'

a‘ a

A’

A

Class

Exercise: Derive the

equation for relief

displacement Dr

Dr

Formula for Relief Displacement

Where:

r’ = erroneous radial distance from the center of photo

h = height/elevation of the point above/below the datum

plane

H = flying height above the datum plane

H

hrr

'D

General Conclusion: Elevation and Relief Displacement

The higher the point is above the datum plane

(or the lower it is below the datum plane), the

greater the relief displacement

The higher the flying height, the lesser the

relief displacement

H

hrr

'D

Corrected Radial Distance

If the point on the ground is ABOVE the datum,

the corrected position will be towards the center

Otherwise, if the point is BELOW the datum, the

corrected position will be away from the center

'r r r D

'r r r D

Occlusion

Occlusion

How can we minimize ∆r?

Use only the central part of the photograph

(discard the edges)

Fly higher but this would yield a smaller

photoscale

Fly higher, and use a camera with a larger focal

length (for example, use a normal angle camera

instead of a wide-angle camera)

H

hrr

'D

Example

A 1:15000 aerial photograph was taken using a

wide-angle camera. A point on the photograph

was identified and its measured distance from the

center is 5.4 centimeters. If the corresponding

point on the ground is elevated from the datum by

60 meters, determine the displacement due to

relief and the correct radial distance of the point

from the center of the photo.

Solution

6 inches

16*2.54*1 100

15000

2286 meters

' (0.054)(60)

2286

0.001417322 meters

0.1417322 cms.

f

H

H

r hr

H

r

r

DD

D

D

'

5.4 0.1417322

5.2582678 cms.

r r r

r

r

D

Quiz 1 (1/4 Sheet of paper)

The top and bottom of a utility pole in an

image are 129.8 mm and 125.2 mm,

respectively, from the principal point of a

vertical photograph. What is the height of the

pole if the flying height above the base of the

pole is 875m?

Tilt

Displacement

b’’

n

p

i

t

a’

a’’

b’

Tilt Displacement

An error in the position of a point on the

photograph due to indeliberate tilting of the aircraft

Due to instability of aircraft

May be due to tilting of the aircraft along the flight

line and/or perpendicular to the flight line

Increases radially from the isocenter

∆ta

b’’

Tilt Displacement

n

p

i

t

a’

a’’

ya

yb

b’

∆tb

Principal Line Line of maximum tilt

Line connecting the principal point, isocenter and nadir

All lines perpendicular to this line are lines of zero inclination

or zero phototilt

this means that all points along a perpendicular line

have uniform scale

∆ta

b’’

Tilt

Displacement

n

p

i

t

a’

a’’

ya

yb

b’

∆tb

Phototilt (t)

Amount of tilt of the aircraft

(and thus the camera lens)

with respect to the vertical

axis

Angle of tilt between the line

perpendicular to the horizontal

datum and the line

perpendicular to the lens

Formula for Phototilt

Where:

t = phototilt

Sa = scale of first point, projected to the principal line

Sb = scale of second point, projected to the principal line

y = distance between a and b along the principal line

Hmge = flying height with respect to the mean ground

sin b amge mge

S SdSt H H

y y

Locating the Nadir and Isocenter

Nadir – radial center of relief displacement

Isocenter – radial center of tilt displacement

distance between p and n (pn) tan

distance between p and i (pi) tan2

f t

tf

Formula for Tilt Displacement

Formula for Tilt Displacement

Where:

i = isocenter

y = projection of erroneous radial distance from the isocenter (i) to the point along the principal line

f = focal length

t = phototilt

2 sin

sin

y tt

f y tD

Corrected Radial Distance

if the point on the ground is above the horizontal 'r r t D

'r r t D if the point on the ground

is below the horizontal

Auxiliary Tilted Photo Coordinate System

'sincos

fy t

tSH h

Scale of a Tilted Photograph

Tilt Displacement Practical Solution PROBLEM:

may cause large errors in determining scale and

distances

SOLUTION:

use 2 known or measurable ground distances that are:

About the same elevation

Equal distances from the photo center

Diametrically opposite from the center

END OF LECTURE