666 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 5, MAY 2008
compare it with that of the OUM and the bilateral filter. Finally,
we provide conclusions in Section VI.
II. B
ILATERAL
FILTER AND
ITS
PROPERTIES
The bilateral filter proposed by Tomasi and Manduchi in 1998
is a nonlinear filter that smoothes the noise while preserving
edge structures [23]. The shift-variant filtering operation of the
bilateral filter is given by
(1)
where
is the restored image, is the
response at
to an impulse at , and
is the degraded image. For the bilateral filter, see (2),
shown at the bottom of the page, where
is
the center pixel of the window,
, and
are the standard deviations of the domain and range Gaussian
filters, respectively, and
(3)
is a normalization factor that assures that the filter preserves
average gray value in constant areas of the image.
The edge-preserving de-noising bilateral filter adopts a low-
pass Gaussian filter for both the domain filter and the range filter.
The domain low-pass Gaussian filter gives higher weight to
pixels that are spatially close to the center pixel. The range low-
pass Gaussian filter gives higher weight to pixels that are similar
to the center pixel in gray value. Combining the range filter and
the domain filter, a bilateral filter at an edge pixel becomes an
elongated Gaussian filter that is oriented along the edge. This
ensures that averaging is done mostly along the edge and is
greatly reduced in the gradient direction. This is the reason why
the bilateral filter can smooth the noise while preserving edge
structures.
Fig. 1(b) shows that a bilateral filter with
and
removes much of the noise that appears in the degraded image
shown in Fig. 1(a) and preserves the edge structures. In Fig. 1(c),
where the spatial domain Gaussian with
is applied alone,
the edges are significantly blurred. The range filter with
at the edge pixel A is shown in Fig. 1(e). Combining the
spatial domain Gaussian filter [Fig. 1(d)] and the range Gaussian
filter [Fig. 1(e)] results in the bilateral filter at pixel A shown
in Fig. 1(f). The transfer function of the bilateral filter shown
in Fig. 1(g) demonstrates that the bilateral filter at pixel A is
low pass in one direction, and almost all-pass in the orthogonal
direction. This explains, from a frequency domain perspective,
why this filter is able to preserve edges while removing noise.
On the other hand, the bilateral filter is essentially a smoothing
filter. It does not sharpen edges. As shown in Fig. 1(h) and (i),
the edge rendered by the bilateral filter has the same level of
blurriness as in the original degraded image, although the noise
is greatly reduced.
The results of the bilateral filtering are a significant improve-
ment over a conventional linear low-pass filter. However, in
order to enhance the sharpness of an image, we need to make
some modifications to this filter.
III. A
DAPTIVE BILATERAL
FILTER
(ABF) FOR
IMAGE
SHARPENING AND
DE-NOISING
In this section, we present a new sharpening and smoothing
algorithm: the adaptive bilateral filter (ABF). The response at
of the proposed shift-variant ABF to an impulse at
is given by (4), shown at the bottom of the page, where
and are defined as before, and the normaliza-
tion factor is given by
(5)
The ABF retains the general form of a bilateral filter, but
contains two important modifications. First, an offset
is in-
troduced to the range filter in the ABF. Second, both
and the
width of the range filter
in the ABF are locally adaptive. If
and is fixed, the ABF will degenerate into a con-
ventional bilateral filter. For the domain filter, a fixed low-pass
Gaussian filter with
is adopted in the ABF. The com-
bination of a locally adaptive
and transforms the bilateral
filter into a much more powerful filter that is capable of both
smoothing and sharpening. Moreover, it sharpens an image by
increasing the slope of the edges. To understand how the ABF
works, we need to understand the role of
and in the ABF.
else
(2)
else
(4)