Finding Line Orientation Dr' Niels Lobo Computer Science

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Finding Line OrientationDr. Niels
LoboComputer Science
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Finding Line Orientation

• 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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Helpful Mirror

• Computers Highly Interactive
• Imaging Sensors Seamlessly Integrating into Our
  Environment
• Image Processing to Make Computers See
• Fusion is Key!

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Driverless Cars
Automated Driver Console
Driverless Taxis
DARPA Grand Challenge To Urban Challenge
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SafeCam Force
Camera Network
Airport Security
Title
Your Text here
Speaker
Crowd
Monitor U.S. assets abroad
Border Control
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Medical Imaging
Revolutionizing Medical Science
3D Models
Image Guided Surgery
Computerized Bone Age Estimation
Automated Cancer Scans
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Let us examine a specific subtask

• Monitor an elderly persons medication intake

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Monitoring Taking Medicines
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An Even Simpler Task

• Track babys hands to keep him safe

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Let us see how to do this

• First, break movie into single frames
• Next, work on each frame separately

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Frames in a video
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Suppose have detector for skin

• d

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Measure Orientation of Each Arm
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Orientation is best line thru region
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Best line is measured by distancesDistance
between point and line
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Total Squared Distances

• Now, we can add up the total of all distances
• 
  (1)

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Find line that minimizes total distances

• Our task then is to find the line that
• minimizes

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This is Optimization

• This brings us to the Calculus topic of
• Minimization, which is special case of
• Optimization.

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How your book does Optimization

• Example from your book
• Section 4.7, 3
• Find two positive numbers, whose product
• is 100, and whose sum is a minimum.

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Solution

• Have x y 100, giving y 100/x.
• Now, Sum x y
• or Sum x (100/x) i.e. (x² 100)/x
• d/dx Sum (2x² - x² - 100)/(x²) (x² - 100)/x²
• When we set to zero, we get x 10 or -10
• Problem asked for positive, so x 10, then y
  10.

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Back to our application

• Our task was to find the line that
• minimizes
• where
  (1)
  
• 
  

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Small Test

• Write expression for
• if the problem also specified that each point
• has weight wi attached to it.

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Proceed to Our Minimization
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Alternative Equation for Line

• Normalized Equation of Line
• Obtainable from A x B y C 0.
• It is
  (2)

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Alternate Equations Figure

• This equation can be used for ANY line

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Fearlessly, moving on

• Combining Equation 2 and Figure 1a, get

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Obtain an equation to minimize

• So, Equation 1 now becomes
• 
  (3)
• Need to miminize this new equation
• 
  

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Minimizing equations of 2 variables

• Our equation has two variables
  and
• We want specific choices of and
• that will give a minimum value for

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Minimizing Equations of 2 variables

• When have 2, need to minimize for first
• variable, obtain a new equation that does
• not contain the first, and then minimize
• for second variable.
• We will first minimize for , and then
  
• for

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Minimize for

• Take derivative of with respect to
• and then set the answer to zero, and solve
• for .

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Taking Derivative w.r.t.

• To repeat,
• So,
• Hence,
• Giving,

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Simplifying to get Intuition

• Dividing all by N
• we get,
• meaning this is the Centroid. It lies on the line.

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Yay!! Removed

• Substituting this new value of gives
• This completes the first stage of
  differentiation.
• Note that new equation has no

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Change origin to centroid

• Substituting and
• gives
  (6)
• This represents a change of coordinates.
• If the new coordinate origin is moved to the
  centroid,
• the only unknown left is the angle of rotation.

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Need to minimize for

• Equation 6 can be expanded as
• 
  (7)
  
  
  
  
  
  
• where

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Minimize for second variable

• We need to differentiate w.r.t. , to
  solve for
• However, the differentiation will be easier if
• equation 7 can be converted further, as
  follows
• Use
• and
• get

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To differentiate w.r.t.

• simplifies to
• When differentiated w.r.t. , and set to
  zero
• get
• Could solve for , but complications with
  ARCTAN need further steps.

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A way to not take ArcTan 2

• Using double angle formula for tan,
• get
• gives
• or

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Get expression for tan

• Use Quadratic Formula Solution,
• and the negative option (need minimum)

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Finally

• Finally get,
• Phew!! Now can get answer for
• Hence, have final equation for winning line

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Small Test 2

• Summarize the major ideas in minimizing
• a two-variable expression.

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In Practice, Just Need Final Steps

• 1. Compute Centroid Position (average x and y)
• 2. Compute new positions
  for all points
• 3. For new positions, compute a, b, c.
• 4. Compute value for
• 5. Obtain value for
• 6. Plot line for the specific two numbers.