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the intersection of two planes corresponding to the locations along the x
and y axes.)
8.9 The k -bit Gray code G k for k
1 is a sequence of 2 k binary codewords
2 k
{
G k (
1
)
,
...
, G k (
) }
constructed recursively as follows. First, we initialize G 0
as the empty set,
{∅}
. Then G k is built from G k 1 as:
2 k 1
2 k 1
G k ={
0 G k 1 (
1
)
,
...
,0 G k 1 (
)
,1 G k 1 (
)
,
...
,1 G k 1 (
1
)) }
(8.25)
That is, the first 2 k 1 codewords are the same as G k 1 with a 0 prefix, and
the second 2 k 1 codewords are the codewords of G k 1 in reverse order with
a 1 prefix.
a) Construct G 4 .
b) Prove that each of the 2 k possible binary codewords appears exactly
once in G k .
c) Prove that each pair of adjacent entries of G k (i.e., G k (
i
)
and G k (
i
+
1
)
)
2 k
).
8.10 How many patterns would be needed to resolve 600 unique vertical stripe
indices using patterns of red, green, blue, and white stripes?
8.11 In Figure 8.19 we assumed the on/off decision was based on the grayscale
intensity of the two images at a given pixel. Generalize the decision criterion
in the case where a color and its RGB complement are projected onto a
colorful scene surface.
8.12 Construct a stripe boundary code similar to Figure 8.20 of two patterns
containing thirteen stripes each, such that (a) each of the twelve pairs of
on-off transitions occurs exactly once, (b) no stripe is continuously on or
off for more than two time units, and (c) at least one stripe changes at every
time step.
8.13 Determine a de Bruijn sequence of order 3 over an alphabet of three
symbols.
8.14 Verify the three-image phase recovery Equation ( 8.8 ).
8.15 This problem is meant to suggest some of the issues that can occur when a
scene is changing as it is being scanned. Consider the scenario illustrated in
Figure 8.41 . A fixed-stripe laser scanner is mounted on a vehicle, pointed at
a right angle to the direction of motion. Suppose the scanner vehicle moves
forward at a constant rate of tenmeters per second, and the laser acquires a
vertical stripe of range data every 0.25 seconds. We assume that the scanner
knows where to correctly place the range samples it acquires in 3D space
(e.g., using an inertial measurement unit).
a) The scanner vehicle passes a 3m long pickup truck with the profile
sketched in Figure 8.41 traveling at nine meters per second. Howmany
stripes from the scanner will hit the truck, and where will the resulting
range samples be in 3D? What will be the apparent length and direction
of the truck?
b) What if the scanner vehicle is passed by a truck traveling at twelve
meters per second?
differs in exactly one bit (including the cyclic pair G k (
)
and G k (
1
)
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