Image Processing Reference
In-Depth Information
The edge based probe functions integrated in the NEAR system incorporate an implemen-
tation of Mallat's Multiscale edge detection method based on Wavelet theory (Mallat and
Zhong, 1992). The idea is that edges in an image occur at points of sharp variation in pixel
intensity. Mallat's method calculates the gradient of a smoothed image using Wavelets, and
defines edge pixels as those that have locally maximal gradient magnitudes in the direction
of the gradient.
Formally, define a 2-D smoothing function θ(x,y) such that its integral over x and y is
equal to 1, and converges to 0 at infinity. Using the smoothing function, one can define the
ψ 1 (x,y) = ∂θ(x,y)
and ψ 2 (x,y) = ∂θ(x,y)
which are, in fact, wavelets given the properties of θ(x,y) mentioned above.
Next, the
dilation of a function by a scaling factor s is defined as
s 2
ξ( x
s , y
ξ s (x,y) =
s ).
Thus, the dilation by s of ψ 1 , and ψ 2 is given by
s 2
s 2
ψ 1 s (x,y) =
ψ 1 (x,y)
and ψ 2 s (x,y) =
ψ 2 (x,y).
Using these definitions, the wavelet transform of f(x,y)∈L 2 (
R 2 ) at the scale s is given by
W 1 s f(x,y) = f∗ψ 1 s (x,y)
and W 2 s f(x,y) = f∗ψ 2 s (x,y),
which can also be written as
W 1 s f(x,y)
W 2 s f(x,y)
= s ∂x (f∗θ s )(x,y)
= s ~ ∇(f∗θ s )(x,y).
∂y (f∗θ s )(x,y)
Finally, edges can be detected by calculating the modulus and angle of the gradient vector
defined respectively as
M s f(x,y) = p |W 1 s f(x,y)| 2 +|W 2 s f(x,y)| 2
A s f(x,y) = argument(W 1 s f(x,y) + iW 2 s f(x,y)),
and then finding the modulus maximum defined as pixels with modulus greater than the
two neighbours in the direction indicated by A s f(x,y) (see (Mallat and Zhong, 1992) for
specific implementation details). Examples of Mallatt's edge detection method obtained
using the NEAR system are given in Fig. 7.9.
Edge present
This prob function simply returns true if there is an edge pixel contained in the subimage
(see, e.g., Fig. 7.10).
Number of edge pixels
This probe function returns the total number of pixels in a subimage belonging to an edge
(see, e.g., Fig. 7.11).
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