Detecting Edges in Images

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Detecting Edges in Images

• by
• Dr. Niels Lobo
• UCF EXCEL Applications of Calculus

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Overview of talk

• We will first do three Examples of problems
• from the book, and then address the
• application to Computer Vision.

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Calculus Problem A from the Book

• Example A Textbooks Section 3.1 Example 6
• Let D (t) be the U.S. National
• debt at time t. The table
• gives approximate values
• of this function by providing
• end of year estimates, in
• billions of dollars, from
• 1980 to 2000. Interpret and
• estimate the value of

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Calculus Problem A from the Book

• The derivative means the
  rate of change of D with respect to t when t
  1990,
• i.e, the rate of increase of national debt in
  1990.
• To compute this, refer to Eqn. 3 in section 3.1,
• So, according to this eqn., we have
• 
  

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Calculus Problem A from the Book

• Eqn. 3 in section 3.1,
• According to this eqn., we have
• 
  
• So we compute and tabulate values of the
  difference quotient as follows (the difference
  quotient records the average rate of change).

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Calculus Problem A from the Book

• So we compute and
• tabulate values of the
• difference quotient as
• follows (the difference
• quotient records the
• average rate of change).

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Calculus Problem A from the Book

• Let us see how the table
• gets its values.

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Calculus Problem A from the Book

• Remember that the For t 1980,
  compute
• original data was
• 
  which is
• 
  3233.3 930.2
• 
  divided by 10 (years)
• 
  
• 
  which is 230.31.

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Calculus Problem A from the Book

• So, that is the first entry.

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Calculus Problem A from the Book

• Similarly, get the rest.

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Calculus Problem A from the Book

• From the table we see that
  lies
• somewhere between 257.48 and 348.14
• billion dollars per year. (Here we are
• making the reasonable assumption that the
• debt didnt fluctuate wildly between 1980
• and 2000.)

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Calculus Problem A from the Book

• We estimate that the rate of increase of the
• national debt of the United States in 1990
• was the average of these two numbers, namely
• 303 billion
  dollars per year.
• (because 303 is the average of 257.48 and 348.14)

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Calculus Problem B from the Book

• Example B Textbooks Section 3.1 Problem 31
• Let T be the temperature (in ) in
  Dallas t hours after midnight on June 2, 2001.
  The table shows values of this function recorded
  every two hours. What is the meaning of
  ? Estimate its value.

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Calculus Problem B from the Book

• Solution
• means the rate of change of
  temperature
• in Dallas at 10 am.
• To estimate its value, first we calculate the
• difference quotient values .
• Need table of

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Calculus Problem B from the Book

• Note the use of the word DIFFERENCE.
• Due to the fact that we do not have continuous
• data, (meaning we do not have the function
• values at all possible times), the best we can
• do is approximate the derivative by a
• DIFFERENCE. We would like to have been able to
  set the gap to being infinitesimally small.
• But we cannot. So, we settle for a larger gap.

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Calculus Problem B from the Book

• Input
• gives a Diff Quotient table

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Calculus Problem B from the Book

• To estimate , given by
  ,
• we take the average of the difference
• quotient values at t8, and t12, to get
• the result (4.5 3.5)/2, i.e.,
  4.

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Calculus Problem C from the Book

• Example C Textbooks Section 3.1 Problem 32
• Life expectancy improved dramatically in
  the
• 20th century. The table gives values of E(t),
  the life
• expectancy at birth (in years) of a male born in
  the
• year t in the United States. Interpret and
  estimate
• the values of and
  .

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Calculus Problem C from the Book

• .

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Calculus Problem C from the Book

• Solution
• and
  refer to the rate of change (in this
  case, increase) of life expectancy at birth for
  males born in 1910 and 1950 respectively.
• To estimate these quantities, we build the
  difference quotient table for each of the
  required years.

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Calculus Problem C from the Book

• .

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Calculus Problem C from the Book

• .

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Calculus Problem C from the Book

• Original table is So, first
  entry is
• 
  (48.3 51.1)/ (-10)
• 
  

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Calculus Problem C from the Book

• .
• Do we need whole table?

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Calculus Problem C from the Book

• .

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Calculus Problem C from the Book

• and we proceed to use the approach
• that is the average
  of
• 0.28 and 0.41, for an answer of 0.345.
• Similarly, for , the
  difference
• quotient table is (on the next slide)

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Calculus Problem C from the Book

• .
• 
  
• 
  
• 
  

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Calculus Problem C from the Book

• and is taken to be the
  average of
• 0.31 and 0.1, which is 0.205, indicating that
• the rate of life expectancy gain was
• slower in 1950 than in 1910.

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Images and their Edges

• Computer Vision
• Good for substituting machine in place of eye
• Can assist with recognition
• Can assist with navigation
• Can assist with manipulation

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Computer Vision Helpful Mirror

• .

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Computer Vision Driverless Cars
Automated Driver Console
Driverless Taxis
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Surveillance for Safety
Camera Network
Airport Security
Title
Your Text here
Speaker
Crowd
Border Control
Monitor U.S. assets abroad
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Computer Vision

• Crime Watch

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Computer Vision

• Airport Security

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Computer Vision

• Monitor U.S. Assets Abroad

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Computer Vision

• Border Security

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Medical Imaging
3D Models
Image Guided Surgery
Revolutionizing Medical Science
Automated Cancer Scans
Computerized Fracture Estimation
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Computer Vision

• A Basic Task Detect Edges of Regions

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Detecting Edges in an image

• This is an example of a picture you might see on
  a computer screen.
• 20 X 20 pixel image of black box
• on square white background.
  
• 
  Pixel Values for image

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Black Rectangle on White Background

• A

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Computer Vision

• To

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Plot values from a row

• To

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Find jumps in the plot

• Denote the plot by , then we Compute
• We can think of two values, A and B, moving
• along the row.
• So we get the calculation being merely (B-A)/1

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Again, the plot of a row
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Plot of difference of pairs B-A
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Absolute Value of B-A
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How to find strong edges

• To find an edge from this derivative plot,
• use a threshold.

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Effect of Thresholding
Threshold Bar
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Back to the other example
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Back to the other example
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This one has a drop and then a rise

• .

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Difference of pairs, B-A

• .

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Absolute Value of (B-A)

• .

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Thresholding

• .

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QUESTION 6

• Clicker question 6
• To find an edge in an image
• A. Compute absolute difference, then Thresh
• B. Threshold, then take difference
• C. Differences divide us, so compute similarity
• D. First non-absolute difference, then Thresh

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Back to Complete Image

• A Basic Task Detect Edges of Regions

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Consider a typical image

• .

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For this typical image

• We know we can
• find the edges
• as we proceed along the
• horizontal direction
• i.e., along a row
• i.e., the x-direction

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For this typical image

• What about the vertical direction???
• i.e., along a column? i.e., along the
  y-direction?

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The y-direction

• .

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Compute the Vertical Edges

• So, just as for the x-direction, we can compute
• the difference quotient for the y-direction
• which means we are to compute the difference
• between two neighboring points that are
• vertical.

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Computer Vision

• So, at all points on the image, we have 2 answers
• (one from x-direction and one from y-direction.)
  How to give a unified answer at
• all the points on the image?

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Computer Vision

• So, denote image
• by
• Then, the two
• Difference quotients are
• and

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What to call these two?

• So, get the notion of the Gradient.
• The two quantities combine to give the Gradient
  Vector. See Page 1095 of text.

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The Gradient

• .

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Question 7
Clicker Questions

• Clicker Question 7
• In two dimensions, the derivative is the
• Image
• Gradient
• Only one Partial derivative
• 15 partial derivatives

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Back to the complete image

• The Gradient Vector is a physical descriptor of
  two dimensional functions
• The symbol for the gradient vector is
  ,
• and to repeat, it has two parts, the partial
  derivatives
• and

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Back to the complete image

• The magnitude of the gradient vector can be
  obtained by squaring the individual components,
  adding them, and taking the square root, to get
  one scalar number. This concept is introduced in
  your Calculus textbook on page 1095, Chapter 17.

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Back to the complete image

• The magnitude of the gradient vector can be
  obtained by squaring the individual components,
  adding them, and taking the square root,
• s magn

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Computer Vision

• Apply the gradient
• magnitude
• computation to this
• image

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Gradient Magnitude

• .

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Use a Threshold

• .

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Use a LOWER Threshold

• 
  
• 
  Edges too Thick !!

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Thickness of edges

• Next week, will see how to get thin edges.
• In the meantime, look at some cool stuff.

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Computer Vision

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Computer Vision

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