Nonselective Derivation and Isotropy

Intuitively, we expect that any derivative operator amplifi es smaller scales more strongly than coarser scales, because the transfer function of an ideal derivative operator goes with kw (Eq. (12.1)). Consequently, we could argue that the transfer function of a good discrete derivative op- erator should approximate the ideal transfer functions in Eq. (12.1) as close as possible.

However, this condition is a too strong restriction. The reason is the following. Imagine that we fi rst apply a smoothing operator to an image before we apply a derivative operator. We would still recognize the joint operation as a derivation. The mean gray value is suppressed and the operator is still only sensitive to spatial gray value changes.

Therefore, the ideal transfer function in Eq. (12.1) could be restricted to small wave numbers:

ˆ

2 ˆ

.k  =0

ˆ   .          ∂ h( k ).               p ∂ h( k ).

 

h( k ) w = 0,

∂ k_w

.k_w =0

= (ik_w),

∂ k2

.k_w =0

= 0.    (12.11)

For good edge detection, it is important that the response of the op- erator does not depend on the direction of the edge. If this is the case, we speak of an isotropic edge detector. The isotropy of an edge detector can best be analyzed by its transfer function. The most general form for an isotropic derivative operator of order p is given by

hˆ ( k ) = (ik_w)pbˆ (| k ˜ |)  with  bˆ (0) = 1  and   ∇ _kbˆ ( k ) = 0.                            (12.12)

 

The constraints for derivative operators are summarized in Appen- dix A (± R24 and ± R25).

 

12.3 Gradient-Based Edge Detection†

Principle

In terms of fi rst-order changes, an edge is defi ned as an extreme (Fig. 12.1). Thus edge detection with fi rst-order derivative operators means to search for the steepest changes, i. e., maxima of the magnitude of the gradient vector (Eq. (12.2)). Therefore, fi rst-order partial derivatives in all direc- tions must be computed. In the operator notation, the gradient can be

320                                                                                                                    12 Edges

 

written as a vector operator. In 2-Dand 3-Dspace this is

Σ Dx Σ

 Dx 

D= Dy

or  D= D_y

 

D_z

  .                   (12.13)

 

Because the gradient is a vector, its magnitude (Eq. (12.3)) is invariant upon rotation of the coordinate system. This is a necessary condition for isotropic edge detection. The computation of the magnitude of the gradient can be expressed in the 2-Dspace by the operator equation

1/2

|D| = Σ D_x · D_x + D_y · D_yΣ          .                      (12.14)

D     D

|D|

The symbol · denotes a pointwise multiplication of the images that re- sult from the fi ltering with the operators _x and _y, respectively (Sec- tion 4.1.4). Likewise, the square root is performed pointwise in the space domain. According to Eq. (12.14), the application of the operator to the image G means the following chain of operations:

1. fi lter the image G independently with D_x and D_y,

2. square the gray values of the two resulting images,

3. add the resulting images, and

4. compute the square root of the sum.

At fi rst glance it appears that the computation of the magnitude of the gradient is computationally expensive. Therefore it is often approx- imated by

. .

|D|≈ |D_x|+  D_y  .                                          (12.15)

However, this approximation is anisotro√ pic even for small wave num-

bers. It detects edges along the diagonals 2 times more sensitively than

along the principal axes. The computation of the magnitude of the gra- dient can, however, be performed as a dyadic point operator effi ciently by a look-up table (Section 10.4.2).

 

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