General Properties of Edge Filters

Averaging fi lters suppress structures with high wave numbers. Edge de- tection requires a fi lter operation that emphasizes the changes in gray values and suppresses areas with constant gray values. Figure 12.1 illus- trates that derivative operators are suitable for such an operation. The fi rst derivative shows an extreme at the edge, while the second deriva- tive crosses zero where the edge has its steepest ascent or descent. Both criteria can be used to detect edges.

A pth-order partial derivative operator corresponds to multiplication by (ik)p in the wave number space (Section 2.3, ± R4):

  ∂    

∂ 2                                  2 2

 

w

∂ xw  ◦    • 2π ik_w,

∂ x2

◦ • − 4π

k_w.             (12.1)

The fi rst-order partial derivatives into all directions of a W -dimensional signal g( x ) form the W -dimensional gradient vector:

∂ x1

∂ x2

∂ xW

T

∇ g( x ) =   ∂ g ,  ∂ g ,..., ∂ g   .                                 (12.2)

The magnitude of the gradient vector,

 

 

1/2

Σ

1/2

..             . T

 W. ∂ g Σ 2

. ∇ g. = ∇ g 2 =

∇ g ∇ g

=  w.=1

∂ x_w 

,         (12.3)

12.2 General Properties of Edge Filters                                         317

is invariant on rotation of the coordinate system.

×

All possible combinations of second-order partial derivates of a W - dimensional signal form a symmetric W W matrix, known as the Hesse matrix:





H = 



 

∂ g2

 

1

∂ x2

 ∂ g2

∂ x₁x₂

..

∂ g2

∂ x₁x_W

∂ g2

 

∂ x₁x₂

∂ g2

2

∂ x2

..

∂ g2

∂ x₂x_W

...  ∂ g2

∂ x₁x_W

...  ∂ g2

∂ x₂x_W

...      ..

...  ∂ g2

W

∂ x2





 .              (12.4)



 

The trace of this matrix, i. e., the sum of the diagonal is called the Lapla- cian operator and is denoted by ∆:

w

.◦ •  −        k

W  ∂ 2                               W

∆ = trace H =

w.=1

∂ x2

2

w

w=1

= − k2.          (12.5)

As the magnitude of the gradient, the Laplace operator is invariant upon rotation of the coordinate system.

In Sections 12.2.1–12.2.4, we discuss the general properties of fi lters that form the basis of edge detection. This discussion is similar to that on the general properties of averaging fi lters in Sections 11.2.1–11.2.4.

 

Zero Shift

With respect to object detection, the most important feature of a deriv- ative convolution operator is that it must not shift the object position. For a smoothing fi lter, this constraint required a real transfer function and a symmetric convolution mask (Section 11.2.1). For a fi rst-order derivative fi lter, a real transfer function makes no sense, as extreme val- ues should be mapped onto zero crossings and the steepest slopes to extreme values. This mapping implies a 90° phase shift. Therefore, the transfer function of a fi rst-order derivative fi lter must be imaginary. An imaginary transfer function implies an antisymmetric fi lter mask. An antisymmetric convolution mask is defi ned as

h_{− n} = − h_n.                                                   (12.6)

For a convolution mask with an odd number of coeffi cients, this implies that the central coeffi cient is zero.

A second-order derivative fi lter detects curvature. Extremes in func- tion values should coincide with extremes in curvature. Consequently, a second-order derivative fi lter should be symmetric, like a smoothing

318                                                                                                                    12 Edges

 

fi lter. All the symmetric fi lter properties discussed for smoothing fi lters also apply to these fi lters (Section 11.2.1).

 

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