Image Based Information Processing

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Image Based Information Processing

• Lecturer Steve Maybank
• School of Computer Science and Information
  Systems
• sjmaybank_at_dcs.bbk.ac.uk
• http//www.dcs.bbk.ac.uk/sjmaybank
• Spring 2009
• Week 2a Detection of Corners, Edges and Lines

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Grey Level Discontinuities
Grey level discontinuities often have
useful physical interpretations, e.g. as
boundaries of objects
http//hlab.phys.rug.nl/imlib/c1001_1200/index.htm
l
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Corners
Highlight Small feature on a surface
Corner of an object
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Uniform Regions and Edges
Edges
Uniform grey levels
Object boundary Surface marking Convex fold,
Concave fold Shadow boundary
Part of a single object Part of a uniform
region, e.g. road, field, sky
Image interpretation is difficult because a
single grey level patch usually has many possible
interpretations.
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Edges in an Image of a Natural Scene
Original image at http//hlab.phys.rug.nl/imlib/c1
001_1200/index.html
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Problem

• When is an image patch a corner, an edge or a
  line?
• Pragmatic solution design a plausible algorithm
  for detecting corners. Define anything it detects
  to be a corner.
• Theoretical solution make a mathematical model
  of an ideal corner. Define any image patch which
  matches the model to be a corner.
• The two solutions lead to similar results.

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Algorithm for Structure Detection

• Apply to the image one or more linear filters
  which have large responses to the structure in
  question.
• Choose a threshold.
• Detect the structure in those parts of the image
  where a well chosen function of the filter values
  exceeds the threshold.

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Linear Filter for Recognising Corners
Response8
Filter mask (Gon.W.)
Response3
Response5
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Responses to the Corner Detector
Filter mask
Original image
Filtered imageabsolute values of the
filter responses
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Top 1000 Responses
Threshold234
Original image
Locations of top 1000 responses shown by
3x3 blocks of peak white
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Top 100 Responses
Threshold384
Original image
Locations of top 100 responses shown by
3x3 blocks of peak white
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Histogram of Absolute Values of Filter Responses
Threshold for top 1000 responses 234 Threshold
for top 100 responses 384
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Edges

• An edge is an image region in which the grey
  level gradients have a common direction and large
  magnitudes.

• But which image regions are edges?

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Grey Level Gradient ideal version

• The grey levels vary smoothly over the image.
• The grey level gradient at an image point p is a
  vector v.
• The direction of v is the direction at p in which
  the grey level increases most rapidly.
• The magnitude of v is the rate of increase of
  grey level in the direction of v.

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Analogy

• Think of the image as a contour map and think of
  the grey level as height.
• Let v be the gradient at a point p on the map.
• The direction of v is uphill.
• The magnitude of v is the slope of the hill in
  the direction of v.

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Grey Level Gradient expensive practical version
Form the sum
Find a,b,c for which V is a minimum. Gradient at
(2,2) (a,b).
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Grey Level Gradient cheap practical version
Filter the image with the Sobel masks
Let the two responses at a pixel p be g1(p),
g2(p). The gradient at p is (g1(p), g2(p)). The
magnitude of the gradient at p is
(g1(p)2g2(p)2)1/2.
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Top 1000 Edge Responses
Top 1000 edges. Threshold245.6
Original image
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Top 100 Edge Responses
Top 100 edges. Threshold422.1
Original image
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Histogram of Sobel Edge Magnitudes
Threshold for top 1000 responses 245.6 Threshold
for top 100 responses 422.1
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Specification of a Line
y
Line not containing the origin give polar
coordinates (r,?) of the point on the line
closest to the origin.
p

r
x
Line containing the origin r0, ?slope of
normal to the line.
?
Equation of line (x, y).(cos(?), sin(?))r.
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Parameter Space for Lines in a Disk

• Radius of diskR.
• Origin of coordinates at the centre of the disk.
• Range of values of r 0ltrltR.
• Range of values of ? 0lt?lt2p.
• Parameter space for lines0,R)x0, 2p).

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Sobel Edge Magnitudes
Original image
Sobel edge magnitudes
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Hough Transform Method for Line Detection

• Divide 0,R)x0,2p) into small rectangles (bins,
  accumulators).
• Compute the Sobel edge magnitudes of the image.
• Find the locations (x1,y1), (x2,y2), of the
  largest Sobel edge magnitudes.
• For each i, increment all the accumulators
  containing lines which pass through (xi,yi).
• Find the accumulators with the largest values.
  The centres of these accumulators are the
  detected lines.

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Lines Passing Through (x,y)
Line (r,?)
The points (r,?) in 0,R)x0,2p) corresponding to
lines through (x,y) satisfy x cos(?)y sin(?)r

(x,y)

Origin
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Accumulator Cells
?
Accumulator cells which contain lines through
(x,y).
r
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Results of Hough Transform
Lines corresponding To the 10 accumulators with
the largest values
Largest 1000 Sobel magnitudes
Original image
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Problems with the Hough Transform

• Thresholds needed for the edge magnitudes and the
  accumulator values.
• How should the parameter space 0,R)x0,2p) for
  lines be divided into accumulators?
• Chance alignments in the image can lead to
  spurious detections of lines.

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References

• See chapter 15 of Computer Visiona modern
  approach by D.A. Forsyth and J. Ponce for an
  excellent discussion of the Hough transform.
• Also see Section 10.2.2 of Digital Image
  Processing by R.C. Gonzalez and R.E. Woods.