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3.5 Pyramids and wavelets 135

space: − =

frequency: − =

low-pass lower-pass

Figure 3.35 The difference of two low-pass filters results in a band-pass filter. The dashed blue lines show the
close fit to a half-octave Laplacian of Gaussian.

image, which can be stored away for further processing. The resulting pyramid has perfect
reconstruction, i.e., the Laplacian images plus the base-level Gaussian (L 2 in Figure 3.34b)
are sufficient to exactly reconstruct the original image. Figure 3.33 shows the same com-
putation in one dimension as a signal processing diagram, which completely captures the
computations being performed during the analysis and re-synthesis stages.
Burt and Adelson also describe a variant on the Laplacian pyramid, where the low-pass
image is taken from the original blurred image rather than the reconstructed pyramid (piping
the output of the L box directly to the subtraction in Figure 3.34b). This variant has less
aliasing, since it avoids one downsampling and upsampling round-trip, but it is not self-
inverting, since the Laplacian images are no longer adequate to reproduce the original image.
As with the Gaussian pyramid, the term Laplacian is a bit of a misnomer, since their
band-pass images are really differences of (approximate) Gaussians, or DoGs,

) ∗ I. (3.84)
DoG{I; σ 1 ,σ 2 } = G σ 1 ∗ I − G σ 2 ∗ I =(G σ 1 − G σ 2
A Laplacian of Gaussian (which we saw in (3.26)) is actually its second derivative,
2
2
LoG{I; σ} = ∇ (G σ ∗ I)=(∇ G σ ) ∗ I, (3.85)
where
∂ 2 ∂ 2
2
∇ = + (3.86)
∂x 2 ∂y 2
is the Laplacian (operator) of a function. Figure 3.35 shows how the Differences of Gaussian
and Laplacians of Gaussian look in both space and frequency.
Laplacians of Gaussian have elegant mathematical properties, which have been widely
studied in the scale-space community (Witkin 1983; Witkin, Terzopoulos, and Kass 1986;
Lindeberg 1990; Nielsen, Florack, and Deriche 1997) and can be used for a variety of appli-
cations including edge detection (Marr and Hildreth 1980; Perona and Malik 1990b), stereo
matching (Witkin, Terzopoulos, and Kass 1987), and image enhancement (Nielsen, Florack,
and Deriche 1997).
A less widely used variant is half-octave pyramids, shown in Figure 3.36a. These were
first introduced to the vision community by Crowley and Stern (1984), who call them Dif-
ference of Low-Pass (DOLP) transforms. Because of the small scale change between adja-