Computationally efficient nonlinear edge preserving smoothing of n-D medical images via scale-space fingerprint analysis Page: 2 of 5

                                
2000 IEEE Nuclear Science Symposium and Medical Imaging Conference Record (in press - preprint)

smoothed versions:

4

a
2
1

-20

8

f~x xi a )
a)  {[1- C(x)] f(x, as_1)
+ Cj (x)f(x,aj)

1

(1)

The blending functions {Cj (x); j  2, ... , J} play a role
similar to that of the spatially varying diffusion coefficients
used in typical implementations of edge preserving smoothing
via inhomogeneous diffusion (e.g., [2]). When C3 (xo) = 0,
smoothing stops in the neighborhood of xo and f(xo, aj)
remains unchanged from the value f(xo, aj_1) obtained using
nonlinear smoothing at the previous, finer scale. Conversely,
when Cj (xo)_ 1, smoothing is unabated and f(xo, aj) is set
to the value f(xo, aj) obtained using linear smoothing at the
current, coarser scale.
B. Defining the Blending Functions
The blending functions {C3 (x); j  2, ... , J} are defined
via the following analysis of the augmented scale-space
fingerprint for f(x). The augmented scale-space fingerprint
(Figure 1) is a graphical depiction of the locations of the
zero-crossings of the first two derivatives of the linearly
smoothed signal as a function of scale [8]. At a particular
scale aj, each zero-crossing location of f(2)(x, a) is labeled
as either a local maximum (edge) or local minimum (ledge)
in gradient magnitude, depending on its proximity to nearby
zero-crossing locations of f(l)(x, aj) (i.e., ridges and troughs).
For each of the resulting edge locations {xjk; k  1, . .. , Kj },
the distance Axjk separating the ridge, trough, or ledge
on either side of the edge is calculated.  The blending
function C3 (x) is then assigned values ranging between
zero and one at the edge locations, based on the separation
distances {Axjk; k  1, . . . , Kj}.
The value assigned to Cj(x) at the edge location xjk
is denoted by ryjk and is selected using a monotonically
decreasing function that maps larger separation distances
to smaller values. This heuristic mapping is based on the
observation that the separation distance Ax tends to be larger
for an isolated true edge, than it is for a random second
derivative zero-crossing associated with noise (Figure 1). For
simplicity, a piecewise linear mapping is used:

0
X

20

Figure 1: Augmented scale-space fingerprint for an isolated edge
of width four and a CNR of 2.5. Solid fingerprint lines depict the
zero-crossing locations of f(2) (x, a) (i.e., edge and ledge locations)
over a continuum of scales. Dashed lines depict the zero-crossing
locations of f(i) (x, a) (i.e., ridge and trough locations). The noiseless
signal is shown with the noisy signal below the fingerprint.
inversely with the scale aj, while the number of isolated true
edges will remain roughly constant.
Given the values {yik; k= 1, .. . , Kj} at the edge
locations, the blending function C3 (x) can be defined for
all x as follows. The blending function Cj(x) must be
continuous through at least its second derivative, in order for
the nonlinearly smoothed signal f(x, aj) to have continuous
first and second derivatives.  Rearranging the factors in
equation (1) and denoting the first and second derivatives
of Cj (x) by C(' (x) and C 2)(x), respectively, one obtains the
following expressions for the first and second derivatives of the
nonlinearly smoothed signal f(x, aj):
f () (x, aj)

J   1

f (x, a i)
f (x, aj-1 )
+ C,(x) If(l) (x, a)
+ C1 (x) [f (x, aj)

- f  (x, a 1)
f(x, aj_1)]

1
ryjk     1
0

Axjk < a
1j i  Axjk < )j
)i < AXjk,

(2)

where aj and Oi are selected as follows. The separation
distances {Axjk; k  1, ... , Kj } are first sorted in ascending
order, and then aj and Oi are set to values corresponding to a
lower and an upper percentile of the sorted values, respectively.
The lower and upper percentiles can be selected based on the
expected numbers of true and random edges at the jat scale. In
practice, the expected number of random edges due to noise
in the linearly smoothed signal f(x, a) will vary roughly

j= 2,..., J
(3)

f(2) (x, aj)
'(2) (x, a,)
f(2)(x, as _1)
-  +C () [(     (x, a) -f(2 (x, a  1)1
+ 2C') (x) !f(1) (x, aj) - (1) (x, aj1)1
+C 2 (x) [f(x,aj) -f(x,ap-)

j   1

j42,...,J.
(4)

To achieve the desired continuity in a relatively straightforward
fashion, the blending function C3(x) is defined to be the

I  I
III
 1        B       i

LBNL-46941

2,...,J.