Lecture 4Edge Detection

Slides by:David A. Forsyth

Clark F. OlsonSteven M. Seitz

Linda G. Shapiro

2

Image edges

Points of sharp change in an image are interesting:• changes in reflectance• changes in object• changes in illumination• noise

These are sometimes called edge points or edge pixels.We want to find the edges generated by scene elements and not by noise.

Convert a 2D image into a set of curves:– Extracts salient features of the scene– More compact than pixels

3

Edge detection

Edges are caused by a variety of factors.

depth discontinuity

surface color discontinuity

illumination discontinuity

surface normal discontinuity

4

Origins of edges

How can you tell whether a pixel is on an edge?

5

Edge detection

6

Edge detection

Basic idea: look for a neighborhood with lots of change.

81 82 26 2482 33 25 2581 82 26 24

Questions:

• What is the best neighborhood size?

• How should change be detected?

7

Finding edges

General strategy:− Determine image gradients after smoothing

(Gradients are directional derivatives computed using finite differences.)

− Mark points where the gradient magnitude is large with respect to neighboring points

− Ideally this yields curves of edge points.

8

Image gradients

We use the image gradient to determine whether a pixel is an edge.

• Two components: [gx, gy]• Both components use finite differencing to approximate derivatives• Gradients have magnitude and orientation• Vertical edges respond strongly to the x component• Horizontal edges respond strongly to the y component• Diagonal edges will respond less strongly, but to both components• Overall magnitude should be the same (on edge of same contrast)

9

Sobel operator

The Sobel operator is a simple example that is common.

-1 0 1 1 2 1Sx = -2 0 2 Sy = 0 0 0 -1 0 1 -1 -2 -1

On a pixel of the image I:• let gx be the response to Sx• let gy be the response to Sy

g = (gx + gy ) is the gradient magnitude. = atan2(gy, gx) is the gradient direction.

2 2 1/2

Then the gradient is I = [gx gy]T

10

Smoothing and differentiation

Issue: noise• Need to smooth image before determining image gradients• Should we perform two convolutions (smooth, then differentiate)?• Not necessarily: we can use a derivative of Gaussian filter

− Differentiation is convolution and convolution is associative− D * (G * I) = (D * G) * I What are D, G, and I?

Gaussian Gaussian derivative in x Plot of Gaussian derivative

11

Smoothing and differentiation

Shape of Gaussian derivative:• Light on one side (positive values)• Dark on other side (negative values)• Values fall off from horizontal center line• After initial peaks, values fall off from vertical center line

Gaussian derivative in x Plot of Gaussian derivative

12

Smoothing and differentiationImportant implementation trick – we don’t need to convolve by a 2D kernel!• A 2D Gaussian function is “separable.”

− Gσ(x, y) = Gσ(x) * Gσ(y) • This means we can convolve the image with two 1D

functions (rather than one 2D function).• This results in considerable savings for an n x n image

and k x k kernel:− 1 2D kernel: approximately n2k2 multiplications and additions− 2 1D kernels: approximately 2n2k

• The gradient operator is convolved with the appropriate 1D kernel or applied in succession.

13

As the scale (sigma) increases, finer features are lost, but diffuse edges are gained.

Note that the gradient magnitude encompasses horizontal, vertical, and diagonal edges.

Original Sigma = 1 Sigma = 5

Gradient magnitudes after smoothing

14

There are three major issues: 1) The gradient magnitude at different scales is different; which should we choose? 2) The gradient magnitude is large along thick trail; how do we identify the

significant points? 3) How do we link the points up into curves?

Original Sigma = 1 Sigma = 5

Gradient magnitudes after smoothing

15

We wish to mark points along the curve where the gradient magnitude is largest.We can do this by looking for a maximum along a slice along the gradient direction. These points should form a curve. There are two algorithmic issues: at which point is the maximum, and where is the next one along the curve?

Non-maxima suppression

16

At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.

Non-maxima suppression

17

Non-maxima suppression

• At q, the gradient Gq is a vector perpendicular to the edge direction.

• The locations p and r are one pixel in the direction of the gradient and the opposite direction.

• One pixel in the gradient direction is: g = [Gx/Gmag, Gy/Gmag].

• Recall that Gmag is the length of the gradient vector [Gx, Gy].r = q + g and p = q - g

18

Non-maxima suppression

At p and r, the gradient magnitude should be interpolated from the surrounding four pixels.

If the gradient magnitude at q is larger than the interpolated value at p and r, then q is marked as an edge.

19

Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).

Only necessary if following edges.

Predicting the next edge pixel

20

Remaining issues

Must check that the gradient magnitude is sufficiently large.

A common problem is that at some points along the curve the gradient magnitude will drop below the threshold, but not at others.

• Use hysteresis: a high threshold to start edge curves and a lower threshold to continue them.

Accuracy at corners is poor.

21

Canny edge detector

The Canny edge detector (1986) is still used most often in practice. It is essentially what we have discussed:

• Smooth and differentiate the image using derivative of Gaussian filters in x and y

• Detect initial candidates by thresholding the gradient magnitude• Apply non-maxima suppression at the candidates• Aggregate edge pixels into contours by following edges

perpendicular to the gradient • When aggregating, allow contour gradient magnitude to fall

below initial threshold, but must remain above lower threshold

Note that this detector (and others) is sensitive to the parameters used (sigma, thresholds)

22

Zero-crossing detectors

Edge detection using the zero-crossing of the 2nd derivative is historically important.

Performance at corners is poor, but zero-crossings always form closed contours.

step edge

smoothed

1st derivative

2nd derivativezero crossing

23

Originalimage

Example

24

Fine scale(sigma=1),Medium threshold,No hysteresis

Much detail (and noise) that disappears at coarser scales.

Example

25

ExampleCoarse scale(sigma=4),High threshold,No hysteresis

Curves are often broken, not closed contours.

26

Coarse scale(sigma=4),Low threshold,No hysteresis

Additional edges found are questionable.

Example