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5.2. EDGE DETECTION

-1 0 1

1 1 1

-1 -1 -1

0 1 0

-1 0 1

0 0 0

-1 8 -1

-1 0 1

-1 -1 -1

0 -1 0

Figure 5.2: Masks Used by Faler for Edge Detection

transition point in the edge. Convolving [-1 0 1] with a line performs a type

of dierentiation or edge detection.

The number of masks used for edge detection is almost limitless. Re-

searchers have used dierent techniques to derive masks and then experi-

mented with them to discover more masks. Figure 5.3 shows four masks used

in the source code and examples in this chapter. The rst three masks are

the Kirsch, Prewitt, and Sobel masks as given in Levine’s text [5.2] (there are

dierent masks bearing the same name in the literature) [5.3]. The fourth

mask, the \quick" mask, is one I \created" while working on this process

(there is no doubt that someone else created this mask before me).

The Kirsch, Prewitt, and Sobel masks are \compass gradient" or direc-

tional edge detectors. This means that each of the eight masks detects an

edge in one direction. Given a pixel, there are eight directions to travel to

aneighboring pixel (above, below, left, right, upper left, upper right, lower

left, and lower right). Therefore, there are eight possible directions for an

edge. The directional edge detectors can detect an edge in only one of the

eight directions. To detect only left to right edges, use only one of the eight

masks. To detect all of the edges, perform convolution over an image eight

times using each of the eight masks. The \quick mask" is so named because

it can detect edges in all eight directions in one convolution. This has obvious

speed advantages. There are, however, occasions when detecting one type of

edge is desired, so use a directional mask for those occasions.

There are two basic principles for each edge detector mask. The rst

is that the numbers in the mask sum to zero. If a 3 x 3 area of an image

contains a constant value (such as all ones), then there are no edges in that

area. The result of convolving that area with a mask should be zero. If the

numbers in the mask sum to zero, convolving the mask with a constant area

will result in the correct answer of zero. The second basic principle is that

the mask should approximate dierentiation or amplify the slope of the edge.

The simple example [-1 0 1] given earlier showed how to amplify the slope of

the edge. The rst Kirsch, Prewitt, and Sobel masks use this idea to amplify

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CHAPTER 5. BASIC EDGE DETECTION

Kirsch

Prewitt

Sobel

Quick mask

5 5 5

1 1 1

1 2 1

-1 0 -1

-3 0 -3

1 -2 1

0 0 0

0 4 0

-3 -3 -3

-1 -1 -1

-1 -2 -1

-1 0 -1

-3 5 5

1 1 1

2 1 0

-3 0 5

1 -2 -1

1 0 -1

-3 -3 -3

1 -1 -1

0 -2 -2

-3 -3 5

1 1 -1

1 0 -1

-3 0 5

1 -2 -1

2 0 -2

-3 -3 5

1 1 -1

1 0 -1

-3 -3 -3

1 -1 -1

0 -1 -2

-3 0 5

1 -2 -1

1 0 -1

-3 5 5

1 1 1

2 1 0

-3 -3 -3

-1 -1 -1

-1 -2 -1

-3 0 -3

1 -2 1

0 0 0

5 5 5

1 1 1

1 2 1

-3 -3 -3

-1 -1 1

-2 -1 0

5 0 -3

-1 -2 1

-1 0 1

5 5 -3

1 1 1

0 1 2

5 -3 -3

-1 1 1

-1 0 1

5 0 -3

-1 -2 1

-2 0 2

5 -3 -3

-1 1 1

-1 0 1

5 5 -3

1 1 1

0 1 2

5 0 -3

-1 -2 1

-1 0 1

-3 -3 -3

-1 -1 1

-2 -1 0

Figure 5.3: Masks for Edge Detection

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5.3. IMPLEMENTING EDGE DETECTORS

an edge ramping up from the bottom of an image area to the top.

5.3 Implementing Edge Detectors

Listing 5.1 shows source code that will implement the four edge detectors

shown in Figure 5.3. The rst section of code declares the masks shown in

Figure 5.3. The functions detect

edges and perform

convolution implement

the Kirsch, Prewitt, and Sobel edge detectors. The detect

edges function

calls perform

convolution to detect the edges. Next, it \xes" the edges of

the output image (more on this later) and writes it to the output image le.

The function perform

convolution does the convolution operation eight

times (oncefor each direction)to detect all the edges. First, it calls setup

masks

to copy the correct masks. The parameter detect

type determines which

masks to use. The convention is type 1=Prewitt, 2=Kirsch, and 3=Sobel.

The function perform

convolution clears the output image, sets several max-

imum values, and does the convolution eight times over the entire image

array. At each point, the code checks to see if the result of convolution is

greater than the maximum allowable value or less than zero, and corrects for

these cases.

After convolution, there is the option of thresholding the output of edge

detection. Edge detectors produce results that vary from zero to the maxi-

mum gray level value. This variation shows the strength of an edge. An edge

that changes from 10 to 200 will be stronger than one that changes from 10

to 50. The output of convolution will indicate this strength. It is often desir-

able to threshold the output so strong edges will appear relatively bright (or

dark) and weak edges will not appear at all. This lowers the amount of noise

in the edge detector output and yields a better picture of the edges. The

detect

edges and perform

convolution functions pass a threshold parameter.

If threshold == 1, perform

convolution goes through the output image and

sets any pixel above thehigh parameter to the maximum and any pixel below

the high parameter to zero.

The quick

edge function performs edge detection using the single 3 x 3

quick

mask. Itperforms convolution overtheimage array usingthequick

mask.

It thresholds the output image if requested, and xes the edges of the out-

put image. All these operations are the same as in the detect

edges and

perform

convolution functions.

Several short utility functions make up the remainder of Listing 5.1. The

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CHAPTER 5. BASIC EDGE DETECTION

setup

masks function copies the desired type of mask (Kirsch, Prewitt, or So-

bel) into the mask arrays for the perform

convolution function. The x

edges

function corrects the output image after convolution (x

edges is shown in

this listing, but resides in source code le utility.c). Convolving a 3 x 3 mask

over an image array does not process the pixels along on the outer edge of

the image. The result is a blank line around the image array. The x

edges

function goes around the edge of the image array and copies valid values out

to the edge. This removes the distracting lines.

Theseedgedetectors arecalled by themain routine ofthe medge program.

The medge program ties these and the edge detectors described in the next

chapter into one convenient program. That program is presented in the next

chapter.

5.4 Results

Let’s close with examples of the edge detectors in action. Figure 5.4 shows

a house image. Figure 5.5 shows the result of applying the Kirsch edge

detector masks. Figure 5.6 shows the result of the Prewitt masks and Figure

5.7 shows the result of the Sobel masks. Figures 5.5, 5.6, and 5.7 are outputs

that were thresholded. Edge values above a threshold of 33 were set to 255

and all others were set to zero. This gives a clear picture of edges and non-

edges. Figure 5.8 shows the result of applying the Sobel masks and not

thresholding the result. If you look closely, you can see some variations in

gray level indicating some edges are stronger than others. Figure 5.9 shows

the result of applying the quick mask. The results of the quick mask are as

good as the other masks, and it operates in one-eighth the time.

5.5 Conclusion

This chapter discussed basic edge detection. The next chapter continues the

discussion of edge detection. There are many creative methods of detect-

ing edges in images. The next chapter discusses the homogeneity operator,

the dierence operator, contrast-based edge detection, and edge ltering by

varying the size of the convolution mask.

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5.5. CONCLUSION

Figure 5.4: The House Image

Figure 5.5: The Result of the Kirsch Masks

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CHAPTER 5. BASIC EDGE DETECTION

Figure 5.6: The Result of the Prewitt Masks

Figure 5.7: The Result of the Sobel Masks

5.5. CONCLUSION

Figure 5.8: The Result of the Sobel Masks Without Thresholding

Figure 5.9: The Result of the Quick Mask

CHAPTER 5. BASIC EDGE DETECTION

5.6 References

5.1 \Image Manipulation By Convolution," Wesley Faler, The C Users Jour-

nal, Vol. 8, No. 8, August 1990, pp. 95-99.

5.2 \Vision in Man and Machine," Martin D. Levine, McGraw-Hill, 1985.

5.3 \Digital Image Processing," Kenneth R. Castleman, Prentice-Hall, 1979.

Chapter 6

Advanced Edge Detection

6.1 Introduction

There are many dierent methods of edge detection. Chapter 5 discussed

some basic techniques. This chapter discusses some unusual and advanced

ideas and presents four edge detectors. The rst two do not use the convo-

lution operation | they use only subtraction. The third edge detector can

vary the level of detail of the edges it will detect. The fourth edge detector

will detect edges in unevenly lit images. Finally, an edge detector is used to

enhance the appearance of an original image. Figure 6.1 shows the original

image used by all the operators.

Figure 6.1: Original House Image

CHAPTER 6. ADVANCED EDGE DETECTION

6.2 Homogeneity Operator

The rst edge detector is the homogeneity operator [6.1] which uses subtrac-

tion to nd an edge. Figure 6.2 shows an example of this operator. The

operator subtracts each of the pixels next to the center of a 3x3 area from

the center pixel. The result is the maximum of the absolute value of these

subtractions. Subtraction in a homogeneous region (one that is a solid gray

level) produces zero and indicates an absence of edges. A region containing

sharp edges, such as area 2 of Figure 6.2, has a large maximum.

The rst section of Listing 6.1 shows the homogeneity function. This

function is similar in form to the edge detectors discussed in Chapter 5. In

the loop over rows and cols, the code performs the subtraction and nds

the maximum absolute value of the subtractions. The homogeneity operator

requires thresholding (which you can specify). A perfectly homogeneous 3x3

area is rare in an image. If you do not threshold, the result looks like a faded

copy of the original. Thresholding at 30 to 50 for a 256 gray level image gives

good results.

Figure 6.3 shows the result of the homogeneity operator. This operator

gives a good rendition of the edges in the original house image. This is a

quick operator that performs only subtraction | eight operations per pixel

|and no multiplication.

6.3 Dierence Operator

The next edge detector is the dierence operator, another simple operator

that uses subtraction. Recall that edge detection is often called image dier-

entiation (detecting the slope of the gray levels in the image). The dierence

operator performs dierentiation by calculating the dierences between the

pixels that surround the center of a 3x3 area.

Figure 6.4 shows an example of the dierence operator. The dierence

operator nds the absolute value of the dierences between opposite pixels,

the upper left minus lower right, upper right minus lower left, left minus

right, and top minus bottom. The result is the maximum absolute value.

The results shown in Figure 6.4 are similar but not exactly equal to those

from the homogeneity operator in Figure 6.2.

The second part of Listing 6.1 shows the dierence

edge function, which

is similar to the homogeneity function. The dierence

edge function loops

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