Image Processing Reference
In-Depth Information
Fig. 9.8. The up-sampling of an image with the factor 2. Dots and crosses show where the
interpolation functions are placed. The dots represent zero values, whereas crosses represent
the data of the original grid, which has fewer samples than the new grid consisting of crosses
and dots. Two isocurves of the interpolation function are shown by two concentric circles
(dotted and solid)
define the resizing operator that effectuates the down-sampling as R , which is a fil-
tering followed by dropping appropriate number of pixels in each of the dimensions,
e.g., both rowwise and columnwise in 2D. For a Gaussian kernel g and an image f ,
R is defined as:
( R f )( r i )=
m
g ( r m ) f ( r i + r m ) ,
r i ∈N R ⊂N
with
.
(9.28)
Here
is the regular discrete grid on which the original image f is sampled and
from which r m is drawn. The points defining
N
obtained by
retaining every κ row and column of it. Consequently, r m tessellates a finer grid
compared to r i . In Fig. 9.7, which shows the case with the down-sampling rate of
κ =2, and in 2D,
N R
are a subset of
N
N R
is illustrated by the encircled grid points, whereas the points
of
N
are marked with crosses. Applying the operator R to fk times:
R k
f = R ···
R R f
(9.29)
yields the k th level in the pyramid, with k =0defining the original image f .
The Laplacian pyramid of an image is obtained from the Gaussian pyramid. It
consists of a hierarchy of images such that successive levels of the Laplacian pyra-
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