Optimized Regularized Edge Detectors

It is easy to derive an optimized regularized derivative operator with a signifi cantly lower error in the estimate of edges. A comparison of Eqs. (12.27) and (12.57) shows that the two fi lters have angle errors in op- posite directions. Thus it appears that the Sobel operator performs too

334                                                                                                                    12 Edges

 

a                                                                    b

0.2

0.1

0

-0.1

-0.2

 

0.2

 

 

0.4

 

0.6

 

 

0.8 ~

k 1 0

 

 

0.5

 

 

1.5

q

1

1°

0.5°

0

-0.5°

-1°

 

0.2

 

 

0.4

 

0.6

 

 

0.8

 

k

~ 1 0

 

 

0.5

 

 

1.5

q

1

Figure 12.12: a Anisotropy of the magnitude and b error in the direction of the gradient based on the optimized Sobel edge detector Eq. (12.62). Parameters are the magnitude of the wave number (0 to 1) and the angle to the x axis (0 to π /2).

 

many cross-smoothings, while the symmetric diff erence operator per- forms too few. Consequently, we may suspect that a combination of both operators may result in a much lower error. Indeed, it is easy to reduce the cross-smoothing by increasing the central coeffi cient. Jä hne et al. [85] show using a nonlinear optimization techniqe that the opera- tors

2                      1                    

3  0   − 3

                    

1/4 D _2x(3 B _y + I ) = 32          10  0  − 10    ,

3  0    − 3

(12.62)

2                      1                       

3   10   3

 

1/4 D _2y(3 B _x + I ) = 32  

0      0   0

− 3 − 10  − 3

have a minimum angle error (Fig. 12.12). Similar optimizations are pos- sible for larger-sized regularized derivative fi lters.

 

LoG and DoG Filter

Laplace fi lters tend to enhance the noise level in images considerably, be- cause the transfer function is proportional to the wave number squared. Thus, a better edge detector may be found by fi rst smoothing the image and then applying the Laplacian fi lter. This leads to a kind of regularized edge detection and to a class of fi lters called Laplace of Gaussian fi lters (LoG for short) or Marr-Hildreth operator [121].

In the discrete case, a LoG fi lter is approximated by fi rst smoothing the image with a binomial mask and then applying the discrete Laplace fi lter. Thus we have the operator LBp with the transfer function

Σ                                       Σ

Lˆ Bˆ p( k ˜ ) = − 4  sin²(π k˜ x/2) + sin²(π k˜ y /2)  cos^p(π k˜ x/2) cos^p(π k˜ y /2).

(12.63)

12.6 Edges in Multichannel Images‡                                                                335

For small wave numbers, this transfer function can be approximated

by

16

Lˆ Bˆ p(k˜, φ ) ≈ − (π k˜ )2 +    1

1      1

8

48

+ p +

cos(4φ )  (π k˜ )4.           (12.64)

 

B − I

In Section 12.4.2 we saw that a Laplace fi lter can be even better ap- proximated by operators of the type p. If additional smoothing is applied, this approximation for the Laplacian fi lter leads to the diff erence of Gaussian type of Laplace fi lter, or DoG fi lters:

4(Bq − I)Bp = 4(Bp+q − Bp).                                     (12.65)

The DoG fi lter 4(Bp+2 − Bp) has the transfer function 4(Bˆ p+2 − Bˆ p)( k ) = 4 cosp+2(π k˜ x/2) cosp+2(π k˜ y /2)

−  4 cosp(π k˜ x

/2) cosp(π k˜ y

(12.66)

/2),

which can be approximated for small wave numbers by

4(Bˆ p+2 − Bˆ p)(k˜, φ ) ≈ − (π k˜ )2 +    3  + 1 p −  1

cos(4φ )  (π k˜ )4.

32  8     96

(12.67)

B −   I− L B

+

The transfer function of the LoG and DoG fi lters are compared in Fig. 12.13. It is obvious that the DoG fi lter is signifi cantly more isotropic. A fi lter with even less deviation in the isotropy can be obtained by com- paring Eqs. (12.64) and (12.67). The anisotropic cos 4φ terms have dif- ferent signs. Thus they can easily be compensated by a mix of LoG and DoG operators of the form 2/3DoG 1/3LoG, which corresponds to the operator (8/3 2 8/3 1/3 ) p.

DoG and LoG fi lter operators have some importance for the human visual system [120].

 

12.6 Edges in Multichannel Images‡

In multichannel images, it is signifi cantly more diffi cult to analyze edges than to perform averaging, which was discussed in Section 11.8. The main diffi culty is that the diff erent channels may contain confl icting information about edges. In channel A, the gradient can point to a diff erent direction than in channel B. The simple addition of the gradients in all channels

.

P

∇ g_p( x )                                                 (12.68)

p=1

is of no use here. It may happen that the gradients in two channels point in opposite directions and, thus, cancel each other. Then, the sum of the gradi- ent over all channels would be zero, although the individual channels would

336                                                                                                                    12 Edges

 

A                                                                    b

0

-0.5

-1

-1.5

-2

-1

 

 

-0.5

0

 

0.5

 

~

k x 1 -1

 

0

 

-0.5

 

1

 

0.5

~

k y

 

0

-0.5

-1

-1.5

-2

-1

 

 

-0.5

0

 

 

0.5  ~

k x

 

1 -1

 

0

 

-0.5

 

1

 

0.5

~

k y

Figure 12.13: Pseudo 3-D plot of the transfer function of a the LoG fi lter LB2

and b the DoG fi lter 4(B4 − B2).

 

have non-zero gradients and we would be unable to distinguish this case from constant areas in both channels.

Thus, a more suitable measure of the total edge strength is the sum of the squared magnitudes of gradients in all channels:

P                              P   W                     2

. | ∇ g_p|2 =.

.. ∂ g p Σ

.                           (12.69)

p=1

p=1 w=1

∂ x_w

×

While this expression gives a useful estimate of the overall edge strength, it still does not handle the problem of confl icting edge directions. An analysis of how edges are distributed in a W -dimensional multichannel image with P channels is possible with the following symmetric W W matrix S (where W is the dimension of the image):

S = J T  J,                                                   (12.70)

where J is known as the Jacobian matrix. This P × W matrix is defi ned as







J = 

 

∂ g₁

∂ x₁

∂ g₂

∂ x₁

.

 

∂ g₁

∂ x₂

∂ g₂

∂ x₂

 

∂ g₁

··· ∂ xW

∂ g₂

··· ∂ xW

.





 .                        (12.71)



.                . ..    . 

 

 

∂ g_P

∂ x₁

∂ g_P

∂ x2 ···

∂ g_P

∂ x_W

Thus the elements of the matrix S are

.  k     l

P

 

∂ g_p ∂ g_p

S_kl =      ∂ x  ∂ x .                                         (12.72)

p=1

12.7 Further Readings‡                                                                                        337

As S is a symmetric matrix, we can diagonalize it by a suitable coordinate trans- form. Then, we can write

2

 .. ∂ g p Σ

0         ···      0       

p  ∂ x1'                                 

                                ∂ g

2  ..                             

S ' = 

0        ..

 

p

∂ x₂'

 

Σ

 .          (12.73)



0                ...         ...           0

2



0                ···    ···





                                                          .. ∂ g p Σ  

 

In the case of an ideal edge, only one of the diagonal terms of the matrix will be non-zero. This is the direction perpendicular to the discontinuity. In all other directions it will be zero. Thus, S is a matrix of rank one in this case.

In contrast, if the edges in the diff erent channels point randomly in all direc- tions, all diagonal terms will be non-zero and equal. In this way, it is possible in principle to distinguish random changes by noise from coherent edges. The trace of the matrix S

W                      W  P                      2

trace( S ) =. S_ww =.

.. ∂ g p Σ

(12.74)

w=1

w=1 p=1

∂ x_w

 

gives a measure of the edge strength which we have already defi ned in Eq. (12.69). It is independent of the orientation of the edge since the trace of a symmetric matrix is invariant to a rotation of the coordinate system.

 

12.7 Further Readings‡

 

A vast body of literature about edge detection is available. We will give here only a few selected references. The development of edge detection based on fi rst-order diff erence fi lters can nicely be followed by a few key papers. Canny

[18] formulated an optimal edge detector based on derivatives of the Gaussian, Deriche [30] introduced a fast recursive implementation of Canny’s edge de- tector, Lanser and Eckstein [104] improved the isotropy of Deriche’s recursive fi lter, and Jä hne et al. [85] provide a nonlinear optimization strategy for edge de- tectors with optimal isotropy. Edge detection based on second-order diff erence (zero crossings) was strongly infl uenced by biological vision. The pioneering work is described by Marr and Hildreth [121] and Marr [120]. More recent work towards unifi ed frameworks for neighborhood operators can be found in Koen- derink and van Doorn [101] and Danielsson et al. [24].

338                                                                                                                    12 Edges

 

 

Simple Neighborhoods

Introduction

In the last two chapters we became acquainted with neighborhood op- erations for performing averaging and detecting edges. In fact, we only studied the very simplest structures in a local neighborhood: constant areas and discontinuities. However, a local neighborhood could also contain patterns. In this chapter, we discuss the simplest class of such patterns, which we will call simple neighborhoods. As an introduction, we examine what types of simple patterns can be used to make an object distinguishable from a background for the human visual system.

Our visual system can easily recognize objects that do not diff er from a background by their mean gray value but only by the orientation or scale of a pattern, as demonstrated in Fig. 13.1. To perform this recog- nition task with a digital image processing system, we need operators that determine the orientation and the scale of the pattern. After such an operation, a gray scale image is converted into a feature image. In the feature image, we can distinguish patterns that diff er by orientation or scale in the same way we can distinguish gray values.

We denote local neighborhoods that can be described by an orienta- tion as simple neighborhoods. The development of suitable operators for orientation and scale is an important and necessary requirement for analysis of more complex structures. It is interesting to observe that the meaning of one and the same local structure may be quite diff erent, as illustrated in Fig. 13.2 for 2-Dimages:

•

In the simplest case, the observed scene consists of objects and a background with uniform radiance (Fig. 13.2a). Then, a gray value change in a local neighborhood indicates that an edge of an object is encountered and the analysis of orientation yields the orientation of the edge.

•

In Fig. 13.2b, the objects diff er from the background by the orienta- tion of the texture. Now, the local spatial structure does not indicate an edge but characterizes the texture of the objects. The analysis of texture will be discussed in Chapter 15.

•

In image sequences, the local structure in the space-time domain is determined by motion, as illustrated by Fig. 13.2c for a 2-D space-

 

339

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

340                                                                                  13 Simple Neighborhoods

 

A                                           b                                           c

Figure 13.1: An object can be distinguished from the background because it diff ers in a gray value, b the orientation of a pattern, or c the scale of a pattern.

 

A                               b                               c

y                                y                              t

 

 

x                                 x                                x

 

Figure 13.2: Three diff erent interpretations of local structures in 2-D images: a edge between uniform object and background; b orientation of pattern; c orien- tation in a 2-D space-time image indicating the velocity of 1-D objects.

 

time image. Motion is an important feature, just like any other, for identifying objects and will be treated in detail in Chapter 14.

Although the three examples refer to entirely diff erent image data, they have in common that the local structure is characterized by an ori- entation, i. e., the gray values change locally only in one direction. In this sense, the concept of orientation is an extension of the concept of edges.

 

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