Visualization of Multidimensional Functions

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Visualization of Multidimensional Functions

• Team Members
• Mrinmayee Kulkarni
• Reenal Mahajan
• Priya Shastri

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Motivation

• Difficulty in visualizing functions of higher
  dimensions.
• Inspiration from Sanjinis work.

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Our Approach
Range of x,y and z 40 (-20 to 20)
(n-1) dimensional hypercube
Y
X
Z
-Y
Y
Focal point
X
-X
-Z
rays
Z
Radius 400
Terms used Rays, Focal Point, Range
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Approach in detail.

• Function to screen mapping
• Steps involved
• Step 1
• Calculate the number of slices using the formula
• No. of Slices(n)2Dimensions
• Step 2
• Angle of each slice(theta) 360/No.of slices
• Step 3
• No. of rays(r) 4002Pi/2n

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• Step 4
• Each (n-1) dimensional face of the hypercube is
  divided into smaller cubes using the formula
• 40(n-1) r x(n-1) where x is the dimension
  of each hypercube.
• Step 5
• Order the hypercubes row-wise and later
• layer-wise.
• Step 6
• Divide the line joining the center of the
  hypercube and the focal point into 400 points and
  compute the function values for each.
• Step 7
• Plot the function points on the screen with
  intensity computed using the formula
• intensity function value 255/maxvalue of
  function

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Stage 1
All 400 points.
First line drawn
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Stage 2
X Surface
Angle theta
Co-ordinates 400,400
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Stage 3
X Surface
White portion which was due to out of range
color values
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Stage 4
Background Changed To Black for Better effect.
Out of range color values removed
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Current stage
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Some interesting Functions
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Screen to function mapping

• Step 1
• Store the intensity values calculated for each
  point (a,b) plotted in 2 arrays green800800
  and red800800
• Step 2
• Get screen co-ordinates (a,b) on mouse click
• Step 3
• Search in green,red arrays and find intensity
  at (a,b)
• Step 4
• Shift coordinate axis to focal point Here
  (400,400)
• Step 4
• Shift coordinate axis to focal point Here
  (400,400)

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• Step 5
• Find angle and quadrant of shifted
  coordinateshence find surface which they map to
  in hypercube
• Step 6
• Substitute known values in the intensity
  function and get the value of function
• Step 7
• Sign of function is obtained from plotted
  color
• (red- negative green positive)

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Function gt 0
Value 16.941
Green576264intensity value
Red5762640
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Pros and Cons

• Pros
• Visualization of functions of higher dimensions.
• Ability to scale up to 20 dimensions.
• Ordering of cubes can be altered to accommodate
  more rays.
• Easier to classify the type of function
  (symmetric, asymmetric).
• Cons
• More processing time.
• Aggregation of function points.
• Limited interaction.

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Future Work

• Addition of mouse interactions.
• Enable the user to change focal point.
• Display function values at different points.
• Change the range of display of function values.
• Use of a different method to order the rays on
  the face of cube.
• Make a generalized module to extend to n-
  dimensions.( part of the work going on).

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Acknowledgements

• Thanks to Dr. North Sanjini for their help

Questions???