pyramid levels. From a conceptual point of view, our work
and Fattal’s are complementary. Whereas he designed pyramids in which edges do not generate correlated coefficients,
we seek to better understand this correlation to preserve it
during filtering.
Li et al. 21 demonstrate a tone-mapping operator based
on a generic set of spatially invariant wavelets, countering
the popular belief that such wavelets are not appropriate for
edge-aware processing. Their method relies on a corrective
scheme to preserve the spatial and intrascale correlation
between coefficients, and they also advocate computing
each level of the pyramid at full resolution to prevent aliasing. However, when applied to Laplacian pyramids, strong
corrections are required to avoid halos, which prevents a
large increase of the local contrast. In comparison, in this
work, we show that Laplacian pyramids can produce a
wide range of edge-aware effects, including extreme detail
amplification, without introducing halos.
Gaussian pyramids are closely related to the concept of
Gaussian scale-space defined by filtering an image with a
series of Gaussian kernels of increasing size. While these
approaches are also concerned with the correlation between
scales created by edges, they are used mostly for purposes of
analysis (e.g., Witkin37 and Witkin et al. 38).
Background on Gaussian and Laplacian pyramids. Our
approach is based on standard image pyramids, whose
construction we summarize briefly (for more detail, see
Burt and Adelson6). Given an image I, its Gaussian pyramid is a set of images {Gl} called levels, representing
progressively lower resolution versions of the image, in
which high-frequency details progressively disappear. In
the Gaussian pyramid, the bottom-most level is the original image, G0 = I, and Gl+ 1 = downsample(Gl) is a low-pass
version of Gl with half the width and height. The filtering
and decimation process is iterated n times, typically until
the level Gn has only a few pixels. The Laplacian pyramid
is a closely related construct, whose levels {Ll} represent details at different spatial scales, decomposing the
image into roughly separate frequency bands. Levels of
the Laplacian pyramid are defined by the details that distinguish successive levels of the Gaussian pyramid, Ll = Gl
− upsample(Gl + 1), where upsample(×) is an operator that
doubles the image size in each dimension using a smooth
kernel. The top-most level of the Laplacian pyramid, also
called the residual, is defined as Ln = Gn and corresponds
to a tiny version of the image. A Laplacian pyramid can be
collapsed to reconstruct the original image by recursively
applying Gl = Ll + upsample(Gl+ 1) until G0 = I is recovered.
3. DEALING WITH EDGES IN LAPLACIAN PYRAMIDS
The goal of edge-aware processing is to modify an input
signal I to create an output I¢, such that the large discon-
tinuities of I, that is, its edges, remain in place, and such
that their profiles retain the same overall shape. For exam-
ple, the amplitude of significant edges may be increased
or reduced, but the edge transitions should not become
smoother or sharper. The ability to process images in this
edge-aware fashion is particularly important for techniques
that manipulate the image in a spatially varying way, such
as image enhancement or tone mapping. Failure to account
for edges in these applications leads to distracting visual
artifacts such as halos, shifted edges, or reversals of gradi-
ents. In the following discussion, for the sake of illustration,
we focus on the case where we seek to reduce the edge
amplitude—the argument when increasing the edge ampli-
tude is symmetric.
In this work, we characterize edges by the magnitude
of the corresponding discontinuity in a color space that
depends on the application; we assume that variations due
to edges are larger than those produced by texture. This
model is similar to many existing edge-aware filtering tech-
niques (e.g., Aubert and Kornprobst1 and Paris et al. 27); we
will discuss later the influence that this assumption has
on our results. Because of this difference in magnitude,
Laplacian coefficients representing an edge also tend to
be larger than those due to texture. A naive approach to
decrease the edge amplitude while preserving the texture
is to truncate these large coefficients. While this creates an
edge of smaller amplitude, it ignores the actual “shape” of
these large coefficients and assigns the same lower value to
all of them. This produces an overly smooth edge, as shown
in Figure 2.
Intuitively, a better solution is to scale down the coefficients that correspond to edges, to preserve their profile,
and to keep the other coefficients unchanged, so that
only the edges are altered. However, it is unclear how to
separate these two kinds of coefficients since edges with
different profiles generate different coefficients across
scales. On the other hand, according to our model, edges
(a) Step edge (b) First pyramid level
(c) Second pyramid level
input
(sharp edge)
our approach
(sharp edge)
clipped
Laplacian coeffs
(rounded edge)
ground truth
compressed edge
(sharp edge)
Figure 2. Range compression applied to a step edge with fine
details (a). The different versions of the edge are offset vertically
so that their profiles are clearly visible. Truncating the Laplacian
coefficients smooths the edge (red), an issue which Li et al. 21 have
identified as a source of artifacts in tone mapping. In comparison,
our approach (blue) preserves the edge sharpness and very
closely reproduces the desired result (black). Observing the shape
of the first two levels (b, c) shows that clipping the coefficients
significantly alters the shape of the signal (red vs. orange). The
truncated coefficients form wider lobes whereas our approach
produces profiles nearly identical to the input (blue vs. orange).