Robust estimation

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Robust estimation

• Problem we want to determine the displacement
  (u,v) between pairs of images.
• We are given 100 points with a correlation score
  computed for each one.
• Using the Kalman Filter
• For each matrix we fit a Gaussian
• We now know how to combine the information from
  all of the 100 points.
• Problem Even if we have a single outlier, we
  will find a wrong solution !

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First try

• For each point, choose the displacement with the
  biggest score (Like in optical flow)
• Compute the median of the horizontal and vertical
  displacements.
• Problem We will get a correct result only if
  more than 50 of the points are inliers.

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What will be the median in this case?
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Second try

• For each point, choose the displacement with the
  biggest score
• Perform a vote take the winner.
• Advantages
• we can receive a good solution even with less
  then 50 good points.
• The weight of each point is bounded by 1.
• Problems
• We need to determine a grid.
• We give the same weight to points in smooth
  regions or in 1D regions.

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A possible solution

• For each point, calculate a matrix of
  probabilities for each displacement.
• Perform a vote sum all the weights of each
  displacement and take the best one.
• Points located in more informative regions will
  have more effect on the resulting displacement.
• Yet the weights of each point are bounded by 1
  (so outliers have limited effect).

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RANSAC- randomized algorithm

• Lotter K points.
• Solve the problem using the K points.
• Check the solution on the rest of the points.

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Finding shapes using the Hough Transform(proposed
by Hough in 1959)

• Straight Lines
• Rectangles
• Circles
• Others

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• Applications
• Find signs in the images (rectangles, circles).
• 3D reconstruction of buildings (lines).
• Find the eye in a face image.
• Tracking
• More

• The challenge
• The image contains a lot of irrelevant
  information.
• There are partial occlusions.
• The image might be noisy

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Use a voting scheme

• Find lines of the form yaxb (slope-intercept
  representation)
• Apply an edge detector on the image (gradient
  threshold, or using Cannys method)
• Prepare a table h(a,b) for a,b in a certain
  range.
• Loop over the image- for each pixel (x,y) on an
  edge, vote for the cells (a,b) which satisfy the
  equation yaxb. (increase the cell accumulator-
  h(a,b))
• Choose the cell with the maximal value (or the
  cells which pass a given threshold)

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Some properties of this representation

• A point corresponds to the set of lines passing
  through it.
• Lines are represented by a set of points in the
  transformed representation.
• All these points lie on a line (in the
  transformed representation) b -axy
• So with have Duality A point is mapped to a
  line, and a line is mapped to a point.

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Example Slope-Intercept Representation
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Problems

• Vertical lines are not covered by the
  representation yaxb.
• Selecting the range of values for the table (a
  and b are not bounded!)
• Quantization choosing a grid.
• Thresholding Similar lines will get similar
  scores.

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Solutions Representation

• Use a polar representation instead

- The line normal
d - The distance from (0,0)
In this representation, each point (x,y) will be
mapped to a sinusoidal curve in the (d,?) space.
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Solutions Range of Values and Quantization

• In the polar representation it is easier to
  determine the range
• The values of d and of ? are now bounded !
• The actual range of image can be application
  dependent. We have natural units (pixels for d,
  and degrees for ?)

• Quantization
• Option1 Vote for the nearest cell (f.g for
  each ? take the nearest d).
• Option2 Perform a weighted vote.

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Solutions Thresholding

• How can we find the two most dominant lines in
  the image ?
• The problem if (r,?) is the cell with the
  maximal value, then (r?, ? ?) will get a high
  score too.
• Solution 1 After finding the first line, omit
  its neighborhood from the search.
• Solution 2 Search only for local maxima.
• Solution 3 Back Mapping- Perform a second
  vote, where each point votes only for the best
  line intersecting it.

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Detecting Complex Shapes - Circle

• Naïve solution
• Construct a table h(a,b,r).
• For each pixel (x,y) which is edge in the image,
  vote for all the cells satisfying
• The problem
• The size of the table O(N3).
• The voting for each pixels takes O(N2).
• Solution 1 Use the gradient information -
  Solve for (a,b) and then for r.
• Solution 2 Randomized Hough Transform.