3-D Computer Vision CSc 83020

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3-D Computer VisionCSc 83020

• Feature Detection and Grouping

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Finding Corners
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What Is a Corner?
Large gradients in more than one direction.
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A Simple Corner Detector
Find eigenvalues of C
If the smaller eigenvalue is above a threshold
then we have a corner.
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A Simple Corner Detector
Find eigenvalues of C
If the smaller eigenvalue is above a threshold
then we have a corner.
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A Simple Corner Detector
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Sobel Operator
-1 -2 -1 0 0 0 1 2 1
-1 0 1 -2 0 2 -1 0 1
S1
S2
Edge Magnitude
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S1 S2
Edge Direction
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Sobel in Matlab
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Canny Edge Detector
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Comparison
Canny
Sobel
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Canny Edge Detection

• Steps
• Apply derivative of Gaussian
• Non-maximum suppression
• Thin multi-pixel wide ridges down to single
  pixel width
• Linking and thresholding
• Low, high edge-strength thresholds
• Accept all edges over low threshold that are
  connected to edge over high threshold

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Non-Maximum Supression
Non-maximum suppression Select the single
maximum point across the width of an edge.
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Linking to the Next Edge Point
Assume the marked point q is an edge point.
Take the normal to the gradient at that point
and use this to predict continuation points
(either r or p).
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Edge Hysteresis

• Hysteresis A lag or momentum factor
• Idea Maintain two thresholds khigh and klow
• Use khigh to find strong edges to start edge
  chain
• Use klow to find weak edges which continue edge
  chain
• Typical ratio of thresholds is roughly
• khigh / klow 2

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Canny Edge Detection (Example)
Strong connected weak edges
Original image
Strong edges only
Weak edges
courtesy of G. Loy
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Canny Edge Detection (Example)
Using Matlab with default thresholds
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Bridge Example Again
edge(im,canny)
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Summary Canny Edge Detection

• Most commonly used method
• Traces edges, accommodates variations in contrast
• Not a linear filter!
• Problems with corners

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Towards Global Features
Local versus global
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From Edges to Lines
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Finding Lines

• Assume edge detection
• Each pixel is either edge or not
• How do we find the line?

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Finding Lines

• Assume edge detection
• Each pixel is either edge or not
• How do we find the line?

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Least Squares Fitting of Lines
Minimize E
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Least Squares Fitting of Lines
Minimize E
Problem E must be formulated carefully!
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Least Squares Fitting of Lines
Minimize E
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Hough Transform
y
x
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Hough Transform Quantization
y
x
m
Detecting Lines by finding maxima / clustering in
parameter space
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Hough Transform Algorithm

• For each image point, determine
• most likely line parameters b,m (direction of
  gradient)
• strength (magnitude of gradient)
• Increment parameter counter by strength value
• Cluster in parameter space, pick local maxima

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Hough Transform Results
Hough Transform
Image
Edge detection
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Finding Lines Using the Hough Transform
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Algorithm

• Discretize the parameter spaces ? and ?.
• Create Accumulator array A(1..R,1..T).
• Set A(k,h)0 for all k and h.
• For each image edge E(i,j)1
• For h1T
• ? i cos?d(h)j sin?d (h)
• Find index k ?d is closest to ?
• Increment A(h,k) by one.
• Find all local maxima (kp, hp) such that A (kp,
  hp)gtt

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Finding Lines Using the Hough Transform
Strong local peaks correspond to lines.
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Finding Lines Using the Hough Transform
Resolution Issues

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From Forsyth and Ponce
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Summary Hough Transform

• Smart counting
• Local evidence for global features
• Organized in a table
• Careful with parameterization!
• Problem Curse of dimensionality
• Works great for simple features with 3 unknowns
• Will fail for complex objects
• Problem Not a local algorithm

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Finding Circles by Hough Transform
y
r
b0
(xi,yi)
x
a0
Equation of Circle
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Finding Circles by Hough Transform
y
r
b0
(xi,yi)
x
a0
Equation of Circle
Circles!
If radius r is known
b
y
(xi,yi)
x
a
Accumulator array A(a,b)
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Finding Circles by Hough Transform
y
r
b0
(xi,yi)
x
a0
If r is not known Use accumulator array
A(a,b,r) For each (xi,yi) increment A(a,b,r)
such that
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Using Gradient Information
Can save lot of computations!
Given location (xi,yi) Edge direction fi
y
x
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Using Gradient Information
Can save lot of computations!
Given location (xi,yi) Edge direction fi
y
x
Assume r is known
ax-rcosf by-rsinf Need to increment only one
point in Accumulator Array.
(xi,yi)
y
fi
(a,b)
x
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Hough Transform for Curves

• Curve yf(x,a)
• aa1, , ap the parameters of the curve.
• Limitation size of parameter space wrt of
  parameters.
• Solution variable-resolution parameter space.

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Hough Transform

• Pattern Matching.
• More efficient than template matching.
• Handles occlusion.
• Finds all instances of pattern.
• Handling inaccurate edge locations?
• Drawbacks?