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).