Image Processing Reference
In-Depth Information
schemes which remain relevant today. Given the requirement for convolution of large
templates, attention quickly focused on frequency domain implementation (Huertas, 1986),
and speed improvement was later considered in some detail (Forshaw, 1988). Later, schemes
were developed to refine the edges produced via the LoG approach (Ulupinar, 1990).
Though speed and accuracy are major concerns with the Marr-Hildreth approach, it is also
possible for zero-crossing detectors to mark as edge points ones which have no significant
contrast, motivating study of their authentication (Clark, 1989). More recently, Gunn studied
the relationship between mask size of the LoG operator and its error rate (Gunn, 1999).
Essentially, an acceptable error rate defines a truncation error which in turn gives an
appropriate mask size. Gunn also observes the paucity of studies on zero-crossing detection
and offers a detector slightly more sophisticated than the one here (as it includes the case
where a zero-crossing occurs at a boundary whereas the one here assumes that the zero-
crossing can only occur at the centre). The similarity is not coincidental: Mark developed
the one here after conversations with Steve Gunn, with whom he works!
4.4
Other edge detection operators
There have been many approaches to edge detection. This is not surprising since it is often
the first stage in a vision process. The most popular operators are the Sobel, Canny and
Marr-Hildreth operators. Clearly, in any implementation there is a
compromise
between
(computational) cost and efficiency. In some cases, it is difficult to justify the extra complexity
associated with the Canny and the Marr-Hildreth operators. This is in part due to the
images: few images contain the adverse noisy situations that complex edge operators are
designed to handle. Also, when finding shapes, it is often prudent to extract more than
enough low-level information, and to let the more sophisticated shape detection process
use, or discard, the information as appropriate. For these reasons we will study only two
more edge detection approaches, and only briefly. These operators are the
Spacek
and the
Petrou
operators: both are designed to be optimal and both have different properties and a
different basis (the
smoothing
functional in particular) to the Canny and Marr-Hildreth
approaches. The Spacek and Petrou operators will be reviewed briefly, by virtue of their
optimality. Of the other approaches, Korn developed a unifying operator for symbolic
representation of grey level change (Korn, 1988).
4.4.1
Spacek operator
Canny derived an operator to satisfy performance measures describing maximum signal to
noise ratio and with good localisation and chose a filter functional which maximised a
composite measure of these parameters, whilst maintaining the suppression of false maxima.
Spacek used a performance measure that included all three factors (Spacek, 1986). Essentially,
whilst Canny maximised the ratio of the signal to noise ratio with the localisation, Spacek
maximised the ratio of the product of the signal to noise ratio and the peak separation with
the localisation. In Spacek's work, since the edge was again modelled as a step function,
the ideal filter appeared to be of the same form as Canny's. After simplification, this
resulted in a one-dimensional optimal noise smoothing filter given by:
f
(
r
) = (
C
1
sin(
r
) +
C
2
cos(
r
))
e
r
+ (
C
3
sin(
r
) +
C
4
cos(
r
))
e
-
r
+ 1 (4.26)
By numerical solution, Spacek determined optimal values for the constants as
C
1
= 13.3816,
