Digital Signal Processing Reference
In-Depth Information
Stability of the two-dimensional recursive filters is also an important issue and is much more
complicated than for 1D filters. For 2D filters, in general, it is quite difficult to take stability
constraints into account during approximation stage (O'Connor & Huang, 1978). Therefore,
various techniques were developed to separate stability from approximation. If the designed
filter becomes unstable, some stabilization procedures are needed (Jury et al., 1977). Various
stability conditions for 2D filters have been found (Mastorakis, 2000).
The medical image processing field has known a rapid development due to imaging value in
assisting and assessing clinical diagnosis (Semmlow, 2004; Berry, 2007; Dougherty, 2011). In
particular, the currently used vascular imaging technique is x-ray angiography, mainly in
diagnosing cardio-vascular pathologies. A frequent application of cardiac imaging is the
localization of narrowed or blocked coronary arteries. Fluorescein angiography is the best
technique to view the retinal circulation and is useful for diagnosing retinal or optic nerve
condition and assessing disorders like diabetic retinopathy, macular degeneration, retinal vein
occlusions etc. There are many papers approaching various methods and techniques aiming
at improving angiogram images. In papers like (Frangi et al., 1998) the multiscale analysis is
used, with the purpose of vessel enhancement and detection. Usual approaches include
Hessian-based filtering, based on the multiscale local structure of an image and directional
features of vessels (Truc et al., 2007). In cardio-vascular imaging, an essential pre-processing
task is the enhancement of coronary arterial tree, commonly using gradient or other local
operators. In (Khan et al., 2004) a decimation-free directional filter bank is used. An adaptive
vessel detection scheme is proposed in (Wu et al., 2006) based on Gabor filter response.
Filtering is an elementary operation in low level computer vision and a pre-processing stage
in many biomedical image processing applications. Some edge-preserving filtering techniques
for biomedical image smoothing have been proposed (Rydell et al., 2008; Wong et al., 2004).
At the end of this chapter some simulation results are given for biomedical image filtering
using some of the proposed 2D filters, namely the directional narrow fan-filter with specified
orientation and the zero-phase circular filter.
2. Analog and digital 1D prototype filters used in 2D filter design
This section presents the types of analog and digital 1D recursive prototype filters which will
be further used to derive the desired 2D filter characteristics. An analog IIR prototype filter of
order
N
has a transfer function in variable
s
of the general form:
M
N
P s
( )
å
å
i
j
H s
( )
=
=
p s
×
q s
×
(1)
P
i
j
Q s
( )
i
=
0
j
=
0
This general transfer function can be factorized into simpler rational functions of first and
second order. Such a second-order rational function (biquad) can be written:
