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

functions

ψ
1
(x,y) =
∂θ(x,y)

∂x

and ψ
2
(x,y) =
∂θ(x,y)

∂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

1

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

1

s
2

1

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

and

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

Number of edge pixels

This probe function returns the total number of pixels in a subimage belonging to an edge

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