Edge-Preserving Solution - Hyperspectral Image Fusion - page 43

differentials, and the space-frequency analysis has been proposed in [25]. Chen
combined the detection of contextual and local discontinuities for preservation of fea-
tures during adaptive smoothing of image [34]. These quantities have been detected
via the inhomogeneity and the local spatial gradient, respectively.
Bilateral filter is another edge-preserving non-linear filter. The kernel of bilat-
eral filter is a product of the spatial domain kernel, and the range (or intensity)
domain kernel [175]. The filtering operation, thus, weighs the closeness in intensity
along with conventional spatial closeness. Therefore, the filtered images preserve
edges which are characterized by dissimilar intensity contents. The SUSAN filter
developed by Smith and Brady [169] is similar to bilateral filter which operates in
the brightness and spatial domains for the purpose of edge detection, and thus, the
structure-preserving noise reduction.
It may be noted that the aforementioned three approaches toward the edge pre-
serving smoothing described in [130, 156, 175] are related to each other [11]. The
relation between anisotropic diffusion and adaptive smoothing has been discussed
[11]. It has also been shown that the extension of adaptive smoothing results in a
bilateral filter. The next section describes the bilateral filter in detail.

3.3 Basics of Bilateral Filter

Bilateral filter was introduced by Tomasi and Manduchi [175]. This filter combines
the pixel intensities for fusion based on their geometric as well as photometric close-
ness. The closeness of two pixels can be related to (i) nearby spatial location, and (ii)
similar intensity/color values which are possibly useful in human perception. The
basic idea behind bilateral filter is to employ an additional filtering kernel in range
domain similar to a traditional filtering kernel in the spatial domain. The spatial
domain filtering used in bilateral filter is essentially similar to the traditional filtering
where the weights decay as one moves away from the pixel under consideration. In
the range domain filtering, the weights decay with the dissimilarity in the intensities.
That is, the weights are somewhat inversely proportional to the difference in the
intensities. If the intensities of two neighboring pixels are very different, these pixels
are likely to be parts of two different objects in the scene. These pixels, although may
lie spatially close to each other, should not have much influence during the smoothing
operation. This makes the filtering operation edge preserving, and eliminates only
the finer textures while smoothing.
For an image I to be processed using a bilateral filter, let G σ S be theGaussian

spatial kernel similar to the traditional Gaussian filter. The spatial extent of the kernel
is decided by the value of

( x , y )

σ S , the filter operates over
a larger neighborhood of the pixel to be filtered, and more is the smoothing. Let G σ R
be the Gaussian range kernel where

σ S . Higher the spread parameter

σ R decides how the difference in the intensities of
the pixels gets converted into the fusion weight. If two pixels are parts of dissimilar
regions, then these regions are probably separated by an edge. The range kernel
parameter

σ R helps quantifying the strength of the edge in order to compute the