Edge detection Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Feb 15, 2016
Edge detection
Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
2Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Image edges
• Points of sharp change in an image are interesting:– changes in reflectance– changes in object– changes in illumination– noise
• Sometimes called edge points or edge pixels• We want to find the edges generated by scene elements
and not by noise
Edge detection
• Convert a 2D image into a set of curves– Extracts salient features of the scene– More compact than pixels
Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
3
Origin of edges
• Edges are caused by a variety of factors
depth discontinuity
surface color discontinuity
illumination discontinuity
surface normal discontinuity
4Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Edge detection
How can you tell whether a pixel is on an edge?
5Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
6Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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?
7Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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.
8Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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
9Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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
10Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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
11Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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
12Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Smoothing and differentiation• Important 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
13Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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.
Gradient magnitudes after smoothing
Original Sigma = 1 Sigma = 5
14Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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?
Gradient magnitudes after smoothing
Original Sigma = 1 Sigma = 5
15Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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?
16Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Non-maximumsuppression
At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.
17Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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, p = q - g
18Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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
19Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
Predictingthe nextedge point
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.
20Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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.• Performance at corners is poor.
21Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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)
22Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
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 edgesmoothed
1st derivative
2nd derivativezero crossing
23Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
originalimage
24Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
fine scale(sigma=1),medium threshold,no hysteresis
Much detail (and noise) that disappears at coarser scales
25Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
coarse scale(sigma=4),high threshold,no hysteresis
Curves are often broken, not closed contours
26Computer Vision Set: Edge detection
Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro
coarse scale(sigma=4),low threshold,no hysteresis
Additional edges found are questionable.