Edge Detection

Phil Mlsna, Ph.D.

Dept. of Electrical EngineeringNorthern Arizona University

Some Important Topics in Image Processing

• Contrast enhancement• Filtering (both spatial and frequency domains) • Restoration• Segmentation• Image Compression

etc.EE 460/560 course, Fall 2003

(formerly CSE 432/532)

Edge Detection uses spatial filtering to extract important information from a scene.

Types of Edges

• Physical Edges– Different objects in physical contact– Spatial change in material properties– Abrupt change in surface orientation

• Image Edges– In general: Boundary between contrasting regions

in image– Specifically: Abrupt local change in brightness

Image edges are important clues for identifying and interpreting physical edges in the scene.

Goal: Produce an Edge Map

Original Image Edge Map

Edge Detection Concepts in 1-D

)(xf

)(xf

)(xf

Edges can be characterized as either:• local extrema of • zero-crossings of

)(xf )(xf

Continuous Gradient

jy

yxfix

yxfyxf ˆ),(ˆ),(),(

But is a vector.

We really need a scalar that gives a measure of edge “strength.”

22 ),(),(),(

y

yxfx

yxfyxf

f

This is the gradient magnitude. It’s isotropic.

Classification of PointsLet points that satisfy be edge points.Tyxf ),(

PROBLEM:

Tf

Non-zero edge width

Stronger gradient magnitudes produce thicker edges.

To precisely locate the edge, we need to thin. Ideally, edges should be only one point thick.

Practical Gradient Algorithm

1. Compute for all points.2. Threshold to produce candidate edge points.3. Thin by testing whether each candidate edge point

is a local maximum of along the direction of . Local maxima are classified as edge points.

ff

ff

Cameramanimage

ThresholdedGradient

Thresholded andThinned

Directional Edge Detection

Tx

yxf

),(

Ty

yxf

),(

Horizontal operator(finds vertical edges)

Vertical operator(finds horizontal edges)

Ty

yxfx

yxf

sin),(cos),(

finds edges perpendicular to the direction

Horizontal DifferenceOperator

Vertical DifferenceOperator

Directional Examples

Discrete Gradient Operators

Pixels are samples on a discrete grid.Must estimate the gradient solely from these samples.

STRATEGY: Build gradient estimation filter kernels and convolve them with the image.

Two basic filter conceptsFirst difference:

Central difference:

11

101

Simple Filtering Example in 1-D

11

[ 5 5 5 8 20 25 25 22 12 4 3 3 ]

Convolving with

11

[ 0]

 

Simple Filtering Example in 1-D

11

[ 5 5 5 8 20 25 25 22 12 4 3 3 ]

Convolving with

11

[ 0 0]

 

Simple Filtering Example in 1-D

11

[ 5 5 5 8 20 25 25 22 12 4 3 3 ]

Convolving with

11

[ 0 0 3]

 

Simple Filtering Example in 1-D

11

[ 5 5 5 8 20 25 25 22 12 4 3 3 ]

Convolving with

[ 0 0 3 12 5 0 -3 -13 -8 -1 0 ]produces:

11 

Gradient Estimation

),(),(),(ˆ21

2221

2121 nnfnnfnnf

1. Create orthogonal pair of filters,

),( 211 nnh ),( 212 nnh

2. Convolve image with each filter:

3. Estimate the gradient magnitude:

),(),(),( 21121211 nnhnnfnnf

),(),(),( 21221212 nnhnnfnnf

Roberts Operator

• Small kernel, relatively little computation• First difference (diagonally)• Very sensitive to noise• Origin not at kernel center• Somewhat anisotropic

10

01),( 212 nnh

0110

),( 211 nnh

Noise

• Noise is always a factor in images.• Derivative operators are high-pass filters.• High-pass filters boost noise!

• Effects of noise on edge detection:– False edges– Errors in edge position

Key concept: Build filters to respond to edges and suppress noise.

Prewitt Operator

• Larger kernel, somewhat more computation• Central difference, origin at center• Smooths (averages) along edge, less sensitive to

noise• Somewhat anisotropic

101101101

111000111

• 3 x 3 kernel, same computation as Prewitt• Central difference, origin at center• Better smoothing along edge, even less sensitive to

noise• Still somewhat anisotropic

Sobel Operator

101202101

121000121

Discrete Operators Compared

Original Roberts

Roberts Prewitt

Prewitt Sobel

T = 5 T = 10

T = 20 T = 40T = 40

Roberts

Continuous Laplacian

2

2

2

22 ),(),(),(),(

yyxf

xyxfyxfyxf

This is a scalar. It’s also isotropic.

Edge detection: Find all points for which

0),(2 yxfNo thinning is necessary.Tends to produce closed edge contours.

Discrete Laplacian Operators

010141010

111181111

121242121

• Origin at center• Only one convolution needed, not two• Can build larger kernels by sampling Laplacian of

Gaussian

),(),(),(ˆ212121

2 nnhnnfnnf

Laplacian of Gaussian(Marr-Hildreth Operator)

2

22

4

2222

2exp2),(),(

yxyxyxgyxh

2

22

2exp),(

yxyxgGaussian:

),(),(),( 22 yxfyxgyxf

Let:

Then:

),(),( yxfyxh

LoG Filter Impulse Response

),(2 yxg

LoG Filter Frequency Response

)},({ 2 yxg

Laplacian of Gaussian Examples

= 1.0 = 2.0

= 1.5

LoG Properties• One filter, one convolution needed• Zero-crossings are very sensitive to noise (2nd deriv.)• Bandpass filtering reduces noise effects• Edge map can be produced for a given scale• Scale-space or pyramid decomposition possible• Found in biological vision!!

Practical LoG Filters:• Kernel at least 3 times width of main lobe, truncate• Larger kernel more computation

Summary

• Edges can be detected from the derivative:– Extrema of gradient magnitude– Zero-crossings of Laplacian

• Practical filter kernels; convolve with image• Noise effects

– False edges– Imprecise edge locations– Correct filtering attempts to control noise

• Edge map is the goal

Questions?