Graphics Reference
In-Depth Information
3.1.2
Multiresolution Blending with a Laplacian Pyramid
Burt and Adelson [ 78 ] made a key observation about blending across a seam, rooted
in a frequency-domain interpretation of the images to be combined. The idea is sim-
ple: low-frequency components (i.e., smooth, gradual intensity variations) should be
blended acrosswide transition regions, while high-frequency components (i.e., edges
and regions with fine detail) should be blended across narrow transition regions.
These goals can be easily, simultaneously accomplished using a Laplacian pyramid ,
a common multiresolution representation for images. In this section, we assume
that the images are grayscale, and that for color images each channel is processed
independently and recombined.
For a given image I , the first step is to blur it at different scales, by successively
filtering it with a Gaussian (or more generally, low-pass) kernel. At each step, the res-
olution of the image is halved in both dimensions, so that successive images appear
as smaller, blurrier versions of the original. The top row of Figure 3.4 illustrates sev-
eral steps of this process, which is called a Gaussian pyramid . That is, we create a
hierarchy of images given by
G i
= (
K
G i 1
) 2 ,
i
=
1,
...
, N
(3.2)
where
indicates two-dimensional convolution,
2 indicates downsampling by 2
in both dimensions, and G 0
I , the original image. K is an approximate Gaussian
kernel, or a low-pass filter whose elements sum to 1, such as
=
] [−
K
=[−
0.05, 0.25, 0.6, 0.25,
0.05
0.05, 0.25, 0.6, 0.25,
0.05
]
(3.3)
For compositing, we're interested in the edges that are significant at every scale,
which can be obtained by taking the difference of Gaussians at each scale:
L i =
G i (
K
G i )
,
i
=
0,
...
, N
1
(3.4)
The images L i form what is called a Laplacian pyramid , since the shape of the two-
dimensional Laplacian operator (also known as the “Mexican hat” function) is similar
to a difference of Gaussians at different scales (we'll discuss this property more in
Section 4.1.4 ). As illustrated in the bottom row of Figure 3.4 , each image in the Lapla-
cian pyramid can be viewed as a bandpass image at a different scale. The smallest
image L N in the pyramid is defined to be a small, highly blurred version of the original
image, given by G N , while the other images contain edges prevalent at different image
scales (for example, L 0 contains the finest-detail edges). Therefore, we can write the
original image as the sum of the images of the pyramid:
N
I
=
L i
(3.5)
=
i
0
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