Edge Detection
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Transcript of Edge Detection
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
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[ 5 5 5 8 20 25 25 22 12 4 3 3 ]
Convolving with
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[ 0]
Simple Filtering Example in 1-D
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[ 5 5 5 8 20 25 25 22 12 4 3 3 ]
Convolving with
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[ 0 0]
Simple Filtering Example in 1-D
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[ 5 5 5 8 20 25 25 22 12 4 3 3 ]
Convolving with
11
[ 0 0 3]
Simple Filtering Example in 1-D
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[ 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:
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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?