Geometric Transformations
Geometric Transformations
consist of two basic operations:
(1) a spatial transformation - defines the
“rearrangement” of pixels on the image plane
(2) a grey-level interpolation method determines the assignment of intensity values
to pixels in the spatially transformed image
Geometric Transformations
- scaling
- rotation
- translation
- perspective transformations
- image registration
- warping / morphing
Interpolation Methods:
- nearest neighbor
- bilinear interpolation
- cubic B-spline
Interpolation Methods:
NEAREST NEIGHBOR (zero-order interpolation) :
assigns pixel value to that of nearest neighbor
advantages:
- simple and efficient implementation
- enables image zooming
disadvantages:
- may produce distortion of edges
- undesirable sawtooth artifacts (jaggies)
Interpolation Methods:
BILINEAR INTERPOLATION (first-order interpolation)
assigns pixel value as a weighted interpolation
of the values of its four defined neighbors
advantages:
- relatively simple and efficient
- produces more desirable (smoother)
results than nearest neighbor
disadvantages:
- derivatives of piecewise bilinear surfaces
are discontinuous at neighborhood boundaries
Interpolation Methods:
CUBIC B-SPLINE (third-order interpolation) :
assigns pixel value as a weighted interpolation
of the values of a larger set of defined
neighbors
advantages:
- produces most desirable (smoothest)
results
disadvantages:
- computationally costly
Geometric Transformations: Example
SCALING
(1) a spatial transformation:
X’= S X,
X
Y ’= S Y
y
(2) a grey-level interpolation method nearest neighbor (pixel replication)
bilinear interpolation (average of 4 neighbors)
cubic b-spline
Geometric Transformations: SCALING
When Sx = Sy = 2k, all three methods of
grey level interpolation may be performed by
1) inserting zero values into image for every
other row and every other column
2) performing a convolution with the proper
convolution kernel
Nearest neighbor
*
2X
1 1
1 1
Nearest neighbor
4X
Bilinear Interpolation
1
*
2X
4
1 2 1
2 4 2
1 2 1
Bilinear Interpolation
4X
Cubic B-Spline
*
1
64
2X
1 4 6 4 1
4 16 24 16 4
6 24 36 24 6
4 16 24 16 4
1 4 6 4 1
Cubic B-Spline
4X
Nearest Neighbor
Bilinear Interpolation
Bilinear Interpolation
Bi-cubic Spline Interpolation
Geometric Transformations: SCALING
When Sx = Sy = 2k, the computation of pixel
values for the new image is more complex.
Two different schemes are possible for
mapping pixel locations from the source image
to pixel locations within the new resized image
1) Source-to-target mapping
2) Target-to-source mapping
Geometric Transformations: SCALING
When Sx = Sy = 2k, the computation of pixel values
for the new image is more complex.
1) Source-to-target mapping
X ’ = SX X,
Y ’= S y Y
Using this mapping certain pixels in the new image
will have nothing mapped to it.
Geometric Transformations: SCALING
X ’ = S X,
Y ’= S Y
X
y
With a scaling factor Sx=3, Sy=3
pixel(3,5) -> newpixel(9,15)
pixel(4,5) -> newpixel(12,15)
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Geometric Transformations: SCALING
X ’ = S X,
Y ’= S Y
X
y
With a source-to-target mapping how does
newpixel(10,15) get its value?
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Geometric Transformations: SCALING
When Sx = Sy = 2k, the computation of pixel
values for the new image is more complex.
1) Source-to-target mapping
X ’ = SX X,
Y ’= S y Y
2) Target-to-source mapping
X = X’/ SX ,
Y = Y’/ S y
Geometric Transformations: SCALING
X = X’/ S ,
Y = Y’/ S y
X
With a scaling factor Sx=3, Sy=3
newpixel(9,15) -> pixel(3,5)
newpixel(10,15) -> pixel(3.3,5)
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Geometric Transformations: SCALING
newpixel(10,15) -> pixel(3.3,5)
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1) nearest neighbor
round the x,y values and take
the new pixel value from that
location in the source image
2) bilinear interpolation
compute the new pixel value
as a weighted average of the
neighboring pixels in the
source image
Geometric Transformations
- scaling
- rotation
- translation
- perspective transformations
- image registration
- warping / morphing
perspective transformations
perspective transformations
Tie points are identified in each image to
compute a transformation matrix which
can be used to compute remapping
locations in the new image
perspective transformations
Calibration images can be used to automate tie
point identification