ME 453/553 – Digital Image Distortion Correction
Brian K. Bay – OSU School of Mechanical, Industrial & Manufacturing Engineering
Approach: Map a location within a regular rectangular region to a distorted
rectangular region, then return a value associated with the mapped location.
map the location …
°
… return a value
Among other uses this is an approach for distortion correction of images. If you setup a digital camera for measurements within a plane, you need undistorted images
(left), but at some level of precision the original images are always irregular (right).
We need a convenient and flexible technique for calibrating the measurement plane.
One approach is to create a template for the undistorted image (a blank pixel array),
and then fill-in this template using information from the distorted image. For any
given location within the template (open circle), if you identify an appropriate
location within the distorted image (open square) you can get a pixel value from
there. So the basic process is: map the location, return a value.
The following illustrations will guide the equation development.
u0x, u0y
u1x, u1y
d0x, d0y
d1x, d1y
d2x, d2y
u2x, u2y
corrected image domain
u3x, u3y
calibration target domain
d3x, d3y
Calibration
The left illustration shows four corners (u0, u1, u2, u3) of an undistorted region within
a new blank image (dashed line.) This image will ultimately be filled through the
map … return process. The right illustration shows corresponding points (d0, d1, d2,
d3) of a distorted region within an image. Points in the distorted image are actual
markers placed on a calibration target in a controlled way, with location and spacing
known to a high degree of precision. Markers do not appear at the expected
locations in the image due to irregularities (camera alignment imprecision, lens
distortions, etc.) in the optical system.
The first step is to choose a basis for the mapping process. A bilinear polynomial is
used here, although there are many other options. The basis defines equations for
the x,y coordinates of an arbitrary location within the distorted image (dx, dy) that
corresponds to an x,y location within the undistorted image (ux, uy):
dx  c 0 ux  c1uy  c 2 ux uy  c 3
dy  c 4 ux  c 5 uy  c 6 ux uy  c 7

Eq. 1
If we know a location within the undeformed space (ux, uy), we can calculate a
corresponding location within the deformed space (dx, dy). But clearly we need the
set of eight mapping coefficients (c0, c1, c2, c3, c4, c5, c6, c7).
To find the eight mapping coefficients we need eight equations, and these equations
can be developed with knowledge of the vertices (“corners”) in the undeformed and
deformed spaces. This is the role of the calibration target. We know what the target
is supposed to look like in an image (where the u0…u3 points ought to be), and we
can measure where they actually are (image the calibration target, threshold the
markers, find their centers to produce d0…d3 locations.)
This set of equations expresses mapping of the four vertex locations:
d0x  u0x
  
d0y   0
d1x  u1x
  
d1y   0
 
d2x  u2x
d2y   0
  
d3x  u3x


d3y 
  0

u0y
u0x u0y
1
0
0
0
0
u1y
0
u1x u1y
0 u0x
1 0
u0y
0
u0x u0y
0
0
u2y
0
u2x u2y
0
1
u1x
0
u1y
0
u1x u1y
0
0
u3y
0
u3x u3y
0 u2x
1 0
u2y
0
u2x u2y
0
0
0
0 u3x
u3y
u3x u3y
0c 0 
 
1c1 
0c 2 
 
1
c 3 
 
0c 4 

1c 5 
 
0c 6 
1
c 7 


Eq. 2
or simply d  uc. We can invert this system to find the mapping coefficients:
c  u d. Now any location within the corrected image domain can be mapped
1

to the distorted image by using Equation 1.

Image Correction
Once the calibration process is complete (determination of mapping coefficients)
any image collected by the system can be corrected for distortion. A good initial
check is to correct the calibration target image and verify that it is in fact
undistorted. But the important point is that any image can be subjected to the
map…return process with mapping controlled by the coefficients determined during
calibration and return based on content of the original image. Software rasters
through pixels of the corrected image from top-left to bottom-right (1, 2, 3, … , N)
mapping and returning values from the original image. Since a mapped location is
generally not at the precise center of an actual pixel in the original image, some
means of deciding what value to return is required. A nearest pixel approach is
simple and fast, but interpolation between pixel values improves smoothness of the
corrected image.
1,2,3,…
1,2,3,…
distortion corrected image
N
original image
N
Perspective, Lens Aberration, and Combined Distortions
The most obvious problem encountered in using a digital camera as a measurement
tool is alignment of the imaging plane (the CCD surface inside the camera) with the
measurement plane. It is impossible to achieve perfect alignment with physical
positioning adjustments, although it is best if you get good physical alignment. To
correct for the perspective distortion that results, a single mapped region of the type
described above is suitable.
However, there are always other more complex distortions in an imaging system.
Optical lenses all have aberrations that distort the image in ways that a single
bilinear polynomial will not adequately correct. A variety of methods have been
developed to manage specific types of common lens aberrations, but it is difficult to
accurately characterize a lens, and lens distortion always occurs together with some
degree of perspective shift. Both can be effectively managed together by simply
creating many bilinear mapping regions instead of one. Each region has its own set
of calibration coefficients and is mapped independently. The software
implementation is a bit more involved, but very complex distortions can be dealt
with using nothing more than a target with a grid of accurately spaced dots.
Perspective Distortion
Single Mapped Region
Lens Aberration
Combined
Multiple Mapped Regions