ee 4780 edge detection. bahadir k. gunturk2 detection of discontinuities matched filter example...

EE 4780

Edge Detection

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Detection of Discontinuities

Matched Filter Example>> a=[0 0 0 0 1 2 3 0 0 0 0 2 2 2 0 0 0 0 1 2 -2 -1 0 0 0 0];

>> figure; plot(a);

>> h1 = [-1 -2 2 1]/10;

>> b1 = conv(a,h1); figure; plot(b1);

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Detection of Discontinuities

Point Detection Example: Apply a high-pass filter. A point is detected if the response is larger than a positive

threshold.

The idea is that the gray level of an isolated point will be quite different from the gray level of its neighbors.

| |R T

Threshold

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Detection of Discontinuities

Point Detection

Detected point

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Detection of Discontinuities

Line Detection Example:

1R 2R 3R 4R

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Detection of Discontinuities

Line Detection Example:

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Detection of Discontinuities

Edge Detection: An edge is the boundary between two regions with relatively

distinct gray levels. Edge detection is by far the most common approach for

detecting meaningful discontinuities in gray level. The reason is that isolated points and thin lines are not frequent occurrences in most practical applications.

The idea underlying most edge detection techniques is the computation of a local derivative operator.

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Origin of Edges

Edges are caused by a variety of factors

depth discontinuity

surface color discontinuity

illumination discontinuity

surface normal discontinuity

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Profiles of image intensity edges

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Image gradient The gradient of an image:

The gradient points in the direction of most rapid change in intensity

The gradient direction is given by:

The edge strength is given by the gradient magnitude

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The discrete gradient How can we differentiate a digital image f[x,y]?

Option 1: reconstruct a continuous image, then take gradient Option 2: take discrete derivative (finite difference)

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Effects of noise

Consider a single row or column of the image Plotting intensity as a function of position gives a signal

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Solution: smooth first

Look for peaks in

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Derivative theorem of convolution

This saves us one operation:

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Laplacian of Gaussian Consider

Laplacian of Gaussianoperator

Zero-crossings of bottom graph

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2D edge detection filters

is the Laplacian operator:

Laplacian of Gaussian

Gaussian derivative of Gaussian

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Edge DetectionPossible filters to find gradients along vertical and horizontal directions:

This gives more importance to the center point.

Averaging provides noise suppression

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Edge Detection

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Edge Detection

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Edge Detection

The Laplacian of an image f(x,y) is a second-order derivative defined as

2 22

2 2

f ff

x y

Digital approximations:

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Edge Detection

One simple method to find zero-crossings is black/white thresholding:1. Set all positive values to white2. Set all negative values to black3. Determine the black/white transitions.

Compare (b) and (g):•Edges in the zero-crossings image is thinner than the gradient edges.•Edges determined by zero-crossings have formed many closed loops.

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Edge Detection

The Laplacian of a Gaussian filter

A digital approximation:

0 0 1 0 0

0 1 2 1 0

1 2 -16 2 1

0 1 2 1 0

0 0 1 0 0

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The Canny edge detector

original image (Lena)

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The Canny edge detector

norm of the gradient

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The Canny edge detector

thresholding

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The Canny edge detector

thinning (non-maximum suppression)

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Edge detection by subtraction

original

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Edge detection by subtraction

smoothed (5x5 Gaussian)

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Edge detection by subtraction

smoothed – original

Why doesthis work?

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Gaussian - image filter

Laplacian of Gaussian

Gaussian delta function