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same address location, thus increasing the
total amount of memory by a factor of
four. This methodology can be mitigated
based on pixel bit width – that is, 8-bitwide
pixels can use this approach to fit into
a 32-bit-wide memory footprint.
Design Implementation with System Generator
Our example will implement image rotation
with the following assumptions:
• Input image size and pixel widths are
known at the time of implementation
• Use the Hotelling transform of the
form:
Sx = Dx * cos(t) + Dy * sin(t)
Sy = Dy * cos(t) – Dx * sin(t)
• Bi-linear interpolation
• Increased memory bandwidth is
accounted for by reading the data out
of the frame buffer four times faster
than the incoming pixel rate
Frame Buffer Read Address Generation
Because the data will be output in raster
scan format, you can create the destination
address with two counters representing the
two dimensions. These values must be center-adjusted
around zero before being
transformed. At this point, any adjustment
changes the center of rotation for the output
image.
The following two examples demonstrate
manipulating the output image by
using minimal extra logic:
• Two adders are all that are required to
add pan control for the output image.
• Using two multipliers, you can scale
the destination addresses to zoom in or
out from the center point.
As interpolation occurs at a later stage,
zooming and panning at this point will
result in an output image with each pixel
uniquely interpolated, versus a simple copy
or additional interpolation procedure later.
Figure 2 demonstrates a 45-degree counterclockwise
rotation with 8x zoom.
Once the destination address is correct
for center, pan, and zoom, the transform is
applied. The resulting addresses are reverse-
corrected for the centering function. The
numbers are of the form integer with a decimal
portion. The integer portions are the
required source address locations to read
from the frame buffer. The decimal por-
Figure 2 – Zoom and pan incorporated
into the algorithm
Figure 3 – Original picture
for subjective comparison
Figure 4 – Image rotated counter-clockwise 7.5
degrees, followed by rotation clockwise 7.5 degrees.
DIGITAL SIGNAL PROCESSING
tions are used to create weighting factors
used for the bi-linear interpolation.
Weighting the Data
Four pixels are fetched from the frame
buffer to create a 2 x 2 matrix containing
the source address pixel and its original
neighbors to the right, below, and belowright,
which we will identify as XY, XpY,
XYp, and XpYp, respectively. The source
address transform uses the decimal portions
of {Sx,Sy} represented as {Rx,Ry} to create
the following weighting equations:
• Weighting(XY) = Wxy = (1 – Rx) (1– Ry)
• Weighting(XpY) = Wxpy = (Rx) (1 – Ry)
• Weighting(XYp) = Wxyp = (1 – Rx) (Ry)
• Weighting(XpYp) = Wxpyp = (Rx) (Ry)
These equations can be shown to be
equal to the following equations to reduce
the number of multipliers necessary from
four to one:
• Wxy = (1 – Rx) – Wxyp
• Wxpy = (Rx) – Wxpyp
• Wxyp = (Ry) – Wxpyp
• Wxpyp = (Rx) (Ry)
The interpolated pixel value is the summation
of the 2 x 2 matrix pixels multiplied
by their respective weighting functions.
Compare the original image in Figure 3
with an image that was rotated through 7.5
degrees, followed by a -7.5 degree rotation
in Figure 4.
Conclusion
In this article, I’ve shown how to use System
Generator to explore and implement image
processing algorithms.
The use of a mathematical simulation
tool with image handling capabilities allows
you to easily investigate various options in
an intuitive capacity. Although this example
shows an image rotation implementation,
these same methodologies can help you
develop any image processing algorithm.
If you have any questions or suggestions,
call me, Daniel Michek, at (858)
431-5901, or send me an e-mail at
daniel.michek@xilinx.com.
Winter 2004 Xcell Journal 65