Chapter 2: Image Analysis

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Chapter 2 Image Analysis

• Edge and Line Detection

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Edge/Line Detection

• There are various types of operators that can be
  used to perform edge/line detection.
• Many edge/line detection operators are
  implemented with convolution mask.
• The concept behind edge detection is to measure
  the rate of change in brightness.

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Edge/Line Detection

• A large change of brightness over a short spatial
  distance indicates the presence of an edge.
• Edge detection operators return information
  regarding the edge magnitude.
• A larger change in brightness would return a
  higher edge magnitude.
• Some edge detection operators even return
  information regarding the direction of the edge.

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Edge/Line Detection

• The output of an edge detection process can be
  visualized in two ways
• Using a binary image where the edges are
  highlighted in white (1) and everything else in
  black (0).
• Using a gray level image where the edges are
  highlighted using white color of varying
  brightness. Edges with higher magnitude will
  produce brighter white.

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Edge/Line Detection
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Edge/Line Detection

• Before edge detection is performed, some
  pre-processing may need to be done.
• For example, the existence of noise can create
  problems in performing edge detection.
• It is best to preprocess an image to
  eliminate/minimize noise effects using mean or
  median filters.

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Edge/Line Detection
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Edge/Line Detection
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Edge/Line Detection

• When dealing with noise, there is a tradeoff
  between the sensitivity and the accuracy of an
  edge detector.
• Sensitivity refers to how easy it is for the edge
  detector to detect edges.
• A sensitive edge detector can detect edges more
  easily.
• Making an edge detector too sensitive or less
  sensitive has its own pros and cons.
• If an edge detector is too sensitive, it may
  return many edge points which are noise.
• If it is less sensitive, it may miss valid edges.

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Edge/Line Detection

• The sensitivity of an edge detection operator
  depends on
• Size of the edge detection mask.
• The value of the gray-level threshold.
• Mask size
• A larger mask size makes the edge detector less
  sensitive.
• Value of gray-level threshold
• A higher threshold makes the edge detector less
  sensitive.

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Roberts Operator

• Mark edge points only does not return any
  information about edge orientation.
• The simplest operator and work best with binary
  images.
• There are two forms of Roberts operator.
• Square root of the sum of the difference of the
  diagonal neighbors squared.

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Robert Operators

• The sum of the magnitude of the differences of
  the diagonal neighbors.

• The second form is often used in practice due to
  its computational efficiency.
• It is typically faster for a computer to find an
  absolute value than to find square roots.

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Sobel Operator

• Look for edges in both the horizontal and
  vertical directions and then combine this
  information into a single metric.

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Sobel Operator

• Each of the masks are convolved with the image.
• At each pixel location, there will now be two
  numbers
• s1 corresponding to the result from row mask.
• s2 corresponding to the result from column mask.
• These two numbers are then used to compute the
  edge magnitude and direction.

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Sobel Operator

• The edge direction is defined as the degree from
  the vertical axis in clockwise direction.

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Prewitt Operator

• Similar to the Sobel operator, except that it has
  different mask coefficients.

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Prewitt Operator

• Each of the masks is convolved with the image.
• At each pixel location, there will now be two
  numbers
• p1 corresponding to the result from row mask.
• p2 corresponding to the result from column mask.
• These two numbers are then used to compute the
  edge magnitude and direction.

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Prewitt Operator

• The edge direction is defined as the degree from
  the vertical axis in clockwise direction.

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Kirsch Compass Masks

• The masks are defined by taking a single mask,
  and rotating it to eight major compass direction
• North, Northwest, West, Southwest, South,
  Southeast, East, and Northeast.
• The masks are defined as follows

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Kirsch Compass Masks
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Kirsch Compass Masks

• The edge magnitude is defined as the maximum
  value found by the convolution of each of the
  masks with the image.
• The edge direction is defined by the mask that
  produces the maximum magnitude. For example
• Mask K0 corresponds to a vertical edge.
• Mask K5 corresponds to a diagonal edge in the
  northwest/southeast direction.

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Robinson Compass Mask

• Used in a similar manner to the Kirsch masks, but
  are easier to implement.
• Only has coefficients of 0, 1 and 2.
• Symmetrical about their directional axis the
  axis with zeros.
• Only need to compute the result on four of the
  masks. The results of the other four can be
  obtained by negating the first four results.

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Robinson Compass Masks
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Robinson Compass Masks

• The edge magnitude and direction are defined in
  the same way as in Kirsch masks.
• Notice that r0 and r6 are the same as the Sobel
  masks.
• Actually, any of the edge detection masks can be
  extended by rotating them in a manner similar to
  these compass masks.
• Example One may rotate Prewitt operator.

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Laplacian Operators

• The laplacian masks are rotationally symmetric
  edges at all direction contribute to the result.
• They are applied by selecting one mask and
  convolving it with the image.
• The sign of the result (positive or negative)
  from two adjacent pixel locations tells which
  side of the edge is brighter.

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Laplacian Operators

• Notice that the laplacian masks here are
  different from the enhancement filters used in
  preprocessing.
• Center coefficient is decreased by one.
• Sum of the mask coefficients becomes zero.
• Does not retain original brightness, only return
  edge information.

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Frei-Chen Masks

• They form a complete set of basis vectors.
• We can represent any 3x3 subimage as a weighted
  sum of the nine Frei-Chen masks.
• These weights are found by projecting a 3x3
  subimage onto each of these masks.
• The projection process is similar to convolution
  process.
• Overlay the mask on the image, multiply
  coincident terms and sum the results.

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Frei-Chen Masks
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Frei-Chen Masks
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Frei-Chen Masks
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Frei-Chen Masks

• One can reconstruct the image with the weight and
  also the masks.
• Take the weight, multiply with the masks and sum
  the corresponding values.

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Frei-Chen Masks

• To use Frei-Chen masks for edge detection
• Select a particular subspace of interest
  (horizontal edge, vertical edge, etc) and find
  the relative projection of the image onto the
  particular subspace.
• The advantage of this method is that we can
  select particular edge or line masks of interest
  and consider the projection of those masks only.

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Frei-Chen Masks

• The set e consist of the marks of interest.
• The (Is, fk) notation refers to the process of
  overlaying the mask on the subimage, multiplying
  the coincident terms and summing the results (a
  vector inner product).

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Frei-Chen Masks

• The output of Frei-Chen Masks operation is a
  theta (?) value.
• This value ranges from 0 to 90 degree.
• This value acts like an edge magnitude.
• Smaller theta (?) corresponds to higher magnitude.

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Edge Operator Performance

• To evaluate the performance of different edge
  detection operators, we need to use both
  objective and subjective evaluation.
• Objective evaluation measure using fixed
  analytical methods.
• Subjective evaluation let human view the image.

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Edge Operator Performance

• In order to develop a performance metric for edge
  detector, we need to define
• What constitutes success.
• What types of errors can occur.
• The types of errors are
• Missing valid edge points.
• Classifying noise pulses as valid edge points.
• Smeared edges.

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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance
One metric called Pratt Figure of Merit Rating
Factor is define as follow
IN the maximum of II and IF II the number of
ideal edge points in the image IF the number of
edge points found by the edge detector ? a
scaling constant that can be adjusted to adjust
the penalty for offset edges d the distance of
a found edge point to an ideal edge point
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Edge Operator Performance

• For this metric, R will be 1 for a perfect edge.
• In general, this metric assigns a better rating
  to smeared edges than to offset or missing edges.
• This is done because there exist techniques to
  thin smeared edges, but it is difficult to
  determine when an edge is missed.

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Edge Operator Performance

• The objective metrics are often of limited use in
  practical applications.
• The subjective evaluation by human visual system
  is still superior than any computer vision system.

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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance

• All operators return almost similar results
  except the laplacian operators.
• Laplacian returns positive and negative numbers
  that get linearly remapped to 0 to 255.
• The background value of 0 is mapped to some
  intermediate gray level.
• Only the magnitude is used for displaying the
  results.

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Edge Operator Performance

• If we add noise to the image, the results are not
  as good.
• There are a number of ways to solve this
• Preprocess the image with spatial filters to
  remove the noise.
• Expand the edge detection operators.

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Edge Operator Performance
Extended Prewitt Edge Detection Mask
Extended Sobel Edge Detection Mask
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Edge Operator Performance

• We can also define a truncated pyramid operator.
• This operator provides weights that decrease as
  we get away from the center pixel.

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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance
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Edge Operator Performance

• On images with noise, the extended operators
  exhibit better performance than the 3x3 masks.
• However, they have a number of setbacks
• Requires more computation.
• Tends to slightly blur the edges.

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Hough Transform

• Hough transform is used to find lines.
• A line is defined as a collection of edge points
  that are adjacent and have the same direction.
• Hough transform will take a collection of edge
  points found by an edge detector, and find all
  the lines on which these edge points lie.

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Hough Transform

• In order to understand Hough transform, consider
  the normal (perpendicular) representation of a
  line
• ? r cos ? c sin ?
• Each pair of ? and ? corresponds to a possible
  line equation.
• The range of ? is ? 90o
• The range of ? is from 0 to ?2N (N is image size)

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Hough Transform
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Hough Transform

• A line is a path where points that lie on it have
  the same direction (share the same line
  equation).
• The idea is to find the number of points that lie
  in each possible line within an image plane.
• The line that accommodates the most points shall
  be the candidate line.

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Hough Transform

• However, in practical, there tend to be many
  lines.
• Therefore, in practice, a line is selected if the
  number of points that lie on it is more than
  certain threshold (user-defined).
• We cannot have infinite precision for ? and ? or
  we will have infinite line equations.
• Therefore, we need to quantize the ?? parameter
  space.

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Hough Transform
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Hough Transform

• Each block in the quantized space corresponds to
  a line, or group of possible lines.
• The algorithm used for Hough Transform consists
  of three primary steps
• Define the desired increments on ? and ?, ?? and
  ??, and quantize the space accordingly.

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Hough Transform

• For every point of interest (edge points), plug
  the values for r and c into the line equation
• ? r cos ? c sin ?
• Then, for each value of ? in the quantized
  space, solve for ?.
• For each ?? pair from step 2, record the rc pair
  in the corresponding block in the quantized
  space. This constitutes a hit for that particular
  block.

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Hough Transform

• When this process is completed, the number of
  hits in each block corresponds to the number of
  pixels on the line as defined by the values ? and
  ? in that block.
• Next, select a threshold and select the
  quantization blocks that contain more points than
  the threshold.

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Hough Transform

• When this process is completed, the lines are
  marked in the output image.
• There is a tradeoff in choosing the size of the
  quantization blocks
• Large blocks will reduce search time.
• However, it may reduce the line resolution in the
  image space.

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Hough Transform
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Hough Transform
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Hough Transform
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Hough Transform