1Ellen L. Walker Edges Humans easily understand “line drawings” as pictures.

1 Ellen L. Walker

Edges

Humans easily understand “line drawings” as pictures.

2 Ellen L. Walker

What is an edge?

Edge = where intensity (or color or texture) changes quickly (large gradient)

Step edge

Roof edge

Noisy edge

In 2D, direction of max. gradient is perpendicular to the edge contour

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Edge Properties

Magnitude

Edge strength (how sharp is the change?)

Direction

Orientation along edge

Perpendicular to gradient (greatest change in brightness) direction

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Measuring Change

Difference (first derivative - S')

Look for large differences between nearby pixels

1D mask: -1 0 1 (vary # of 0’s)

Thick edges

Maximum absolute value at “center” of edge

Difference of differences (second derivative - S'')

Look for change from growing difference to shrinking difference

1D mask: 1 -2 1

Value crosses zero at “center” of edge

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Edge Detection (first derivative)

Roberts (“home” is upper-left, usually)

1 0 0 1 add the results of both 2x2

0 -1 -1 0 convolutions

Sobel (compute “vertical” and “horizontal” separately, take ratio v/h as estimate of tangent of angle of edge, v*v + h*h as estimate of magnitude of edge.

1 2 1 -1 0 1

0 0 0 -2 0 2

-1 -2 -1 -1 0 1

h v

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Edge Detection (second derivative)

Laplacian – symmetric differences of differences

0 1 0 1 -4 1 4-connected 3x3 0 1 0

1 1 1 1 -8 1 8-connected 3x3 1 1 1

Laplacian of Gaussian (LoG) (eq. 4.23)

Smooth first, then take second derivative

By properties of convolution, mix into one mask

Also called "Mexican Hat filter”

Difference of Gaussian (DoG) is often a good substitute (as in SIFT paper)

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Finding the zero-crossing

Look directly for 0’s -- not a good idea (why?)

Look for values near zero -- still not too great?

Look for small regions that contain both positive and negative values - assign one value in the region as zero

Zero-crossing images look like “plate of spaghetti”

http://library.wolfram.com/examples/edgedetection/Images/index_gr_87.gif

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Properties of Derivative Masks

Sum of coordinates of the mask is 0

Response to uniform region is 0

Different from smoothing masks, with output = input for uniform region

Response of first-derivative mask is large absolute value at points of high contrast

Positive or negative depending on which side of the edge is dark

Response of second-derivative mask is zero-crossing at points of high contrast

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Canny Edge Detector

Derived from “first principles”

Detect all edges and only edges (detection)

Put each edge in its proper place (localization)

One detection per edge (one response)

Mathematical optimization for first two, numerical optimization for all three

Assume edges are step edges corrupted by white noise

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Canny Edge Detector (Algorithm)

Convolve image with Gaussian of scale

Estimate local edge normal directions for each pixel

Find location of edges using one-dimensional Laplacian

(a form of non-maximal suppression)

Compute edge magnitudes

Threshold edges with hysteresis (as contours are followed)

(two thresholds - general and “border”)

Repeat for multiple scales & find persistent edges

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Scale

Features exist at different scales

Gaussian smoothing parameter chooses a scale

Edges can be associated with scale(s) where they appear

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Choosing Edge Pixels

First (higher threshold)

Find strong edge pixels

Next (lower threshold)

Find (weaker, but not too weak) neighboring edge pixels to follow the contours

This ensures connected edges

Non-maximal suppression

Given directed edges, remove all but the maximum responding pixel in the perpendicular direction to the edge

This effectively thins the edges

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Color Edges

Detect edges separately in each band (RGB, etc) – combine results for each point

Any edge detector can be used

Depending on combination method, issues can arise Adding – what if edges cancel? Or’ing – likelihood of thick edges.

Work in a higher dimension, e.g. 2x2x3

Estimate local color statistics in each band and make decisions

Estimate magnitudes and orientations in each band, then compute weighted average orientation

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Contour Following (General)

Assume a list of existing segments, a current segment, and a pixel

Examine every pixel

If "start", create a new segment with this pixel

If "interior", add it to the neighboring segment

If "end", add it to the neighboring segment and end that segment

If "corner", add to IN segment & end that segment, add pixel to OUT segment (or start one, if needed)

If "junction", add to IN segment and end that segment, also add to each OUT segment (creating new ones as necessary)

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Contour Tracking Issues

Starting a new segment

Adding a pixel to a segment

Need current segment #

Ending a segment

Need current segment #

Finding a junction

Need "in" segment # and "out" segment #s

Finding a corner

Need "in" segment # and "out" segment #

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Contour Tracking + Thresholding with Hysteresis

Original threshold (high) for selecting initial edge points

Second threshold (lower) for extending contours

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Encoding Contours

Chain code

Each segment is coded by the direction to its successor (N, NE, E, SE, S, SW, W, NW)

Not amenable to further processing

Arc length parameterization (fig. 4.38)

Curve is “unwrapped”; x and y plotted separately

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Matching Curves

First normalize:

Scale coordinates so total arc length = 1

Translate coordinates so centroid is (0,0)

Then match:

Similar curves will have similar coordinates, except Phase shift due to rotation Shift in starting point

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Fitting Lines to Contour

Given a contour (sequence of edgels)

Find a poly-line approximation (sequence of connected line segments)

For which all points lie sufficiently close to the contour.

Example:

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Curve Approximation Algorithm

Begin with a single segment (the endpoints of the curve)

Repeatedly subdivide the worst-fitting segment until all segments fit sufficiently well

Find the point on the segment furthest from the contour

Split the segment, by creating a new point on the contour.

Figure 4.43