1. A Spreadsheet-based Manipulative
Support for Teaching Convolution
STEM Teaching and Learning Conference
March 7, 2014
Georgia Gwinnett College
School of Science and Technology
Dr. Jim Rowan

2. Sections of today’s talk
-A moment about GGC and Digital Media
-About blurring images before digital images
Camera
Artistically
-About blurring digital images
-The powerpoint manipulative

3. GGC and Digital Media
GGC has been built from the ground up since it’s founding in 2005
I arrived in 2007 before any classes were being offered to Freshmen or Sophomores
This was an extraordinary opportunity
Digital Media, a sophomore level class, was my choice

4. GGC and Digital Media Two parts: Projects and Theory Projects:
Using Open Source software…
Audacity for audio
Inkscape for 2D vector graphics
GIMP for bitmapped graphics
Blender for 3D animation
Theory
look under the covers
know a little about what is going on
strip away the “magic”
banish the wizards
empowering students

5. GGC and Digital Media In this class we deal with bitmapped images
how they are stored on a computer
how to manipulate them how to do editing
Part of the editing involves the use of filters
One of the filters we use on a project is blur
So… The question is
How does a digital image get blurred?

6. Image Blur… How does this happen?
Before the advent of digital cameras there were a variety of ways to blur an image
Using a film camera
an image could become blurred if the lens wasn’t focused on the subject of interest
You could intentionally blur background while keeping the subject in focus using the F stop to affect the depth of field
Aging actresses might insist the “soft” focus caused by applying vaseline to the lens

7. Pre-Digital Image Blur: The entire image
Photographically you can de-focus the lens
If you are an aging diva you can insist that vaseline be applied to the lens to get this “soft” focus

8. Pre-Digital Image Blur: Just the background
Photographically you can adjust the F stop
imageSource

9. Pre-Digital Image Blur: Artistically
You can use your fingers to smooth and spread out the charcoal
imageSource

10. Image Blur: Digital?
You can’t reach into the computer to smear the charcoal… What do you do?
The simple answer is use some image software…
But how does
that work?
Before the advent of digital cameras there were a variety of ways to blur an image
Using a film camera
an image could become blurred if the lens wasn’t focused on the subject of interest
You could intentionally blur background while keeping the subject in focus using the F stop to affect the depth of field
Aging actresses might insist the “soft” focus caused by applying vaseline to the lens

11. Digital Blur
Blur (as well as many other digital image filters) use a process known as convolution
Convolution is simple…
it’s just multiplication and addition
Convolution is hard…
there’s a lot to keep up with

12. Underlying any digital image…
Is a huge collection of numbers
In the case of 24 bit color depth there are 3 bytes
-one for red
-one for green
-one for blue
for each pixel!
This image measures
1983 X 2252 pixels
Doing the math you get
4,465,716 pixels and
a file size of 4,465,716 X 3 or
13,397,148 bytes

13. Homing in on the car’s front grill
If you drill down you can finally see the pixels and the numbers that define the colors

14. Digital Blur
Blur is accomplished by applying a “convolution matrix” to every pixel of the image to create a new blurred image
This convolution matrix is a square matrix
that contains numbers that frequently add
up to 1
can be negative
can be as small as 3 X 3
can be much larger
The choice of convolution matrix determines
the affect that it has on the image

15. GIMP Convolution Matrix
GIMP allows up to a 5 X 5 convolution matrix as well as showing a preview.
Design and test your own. BUT how does it work?

16. Showing how it works
To show the underlying workings of convolution simplifications must be made
-Use a 3 X 3 convolution matrix
-Operate on black/white/grey scale images
-each pixel then has only one number
associated with it
-Select a very small but representative
sample of the original image
-Calculate only a few but representative
pixels in the new blurred image
-Extrapolate the result to the rest of the image

17. A simple example: A two pixel wide horizontal line The image
255
The image
The underlying numbers

18. The convolution matrix
1/3
To calculate a single pixel (X) in the new blurred image you need center the matrix over the pixel of interest and then multiply each cell of the matrix by the associated color value of it’s underlying pixel and then add all 9 of these up
255 X

19. The calculations are blinding!
1/3
255 X

28. That’s a lot of work to make my point
That’s a lot of work to make my point! Students get lost in the trees and miss the forest!

29. Instead use a powerpoint slide to do the calculation
1/3
Instead use a powerpoint slide to do the calculation
It is much easier to keep track of where you are…
short enough to make the point
255 X

30. Made easier with powerpoint!
1/3
Select the convolution matrix
255 X

31. Made easier with powerpoint!
Drag and center the matrix over the pixel of interest in the original image…
You can “see” the multiplications and the additions
255 X
1/3

32. Made easier with powerpoint!
You can “see” the multiplications!
0X /3X X255 +
0X0 + 1/3X0 + 0X0 = = 170
255
170
1/3

33. This method made a HUGE difference in student success

34. One Complete Example!

35. 255`
255
0/9
3/9 X

36. 255`
255
0/9
3/9 X

37. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

38. 255
0/9
3/9
255 X

39. 255
0/9
3/9
255
170
0/9 x /9 x /9 x 255 +
0/9 x 0+ 3/9 x 0 + 0/9 x 0=
= 170

40. 255
0/9
3/9
255
170 X

41. 255
0/9
3/9
255
170
0/9 x /9 x /9 x 255 +
0/9 x /9 x 0+ 0/9 x 0+
0/9 x /9 x /9 x 255 =
= 170

42. 255
0/9
3/9
255
170 X

43. 255
0/9
3/9
255
170
0/9 x 0 + 3/9 x 0 + 0/9 x 0 +
0/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 170

44. 255
0/9
3/9
255
170 X

45. 255
0/9
3/9
255
170
0/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

46. This matrix blurs horizontal lines (but not vertical ones!)
255
0/9
3/9
255
170
This matrix blurs horizontal lines
(but not vertical ones!)

47. Another complete example… Sometimes a mask has no effect

48. 255
0/9
3/9

49. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
3/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

50. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
3/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

51. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
3/9 x 0+ 3/9 x 0+ 3/9 x 0+
0/9 x /9 x /9 x 255 =
= 0

52. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
3/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

53. 255
0/9
3/9
255
0/9 x /9 x /9 x 255 +
3/9 x /9 x /9 x 255 +
0/9 x /9 x /9 x 255 =
= 255

54. This matrix does nothing to a horizontal line…
255
0/9
3/9
255
This matrix does nothing to a horizontal line…
What about a vertical one?

55. Other example setups Try your hand at it!

56. What values would be in the turquoise cells?
255
1/3
Diagonal Image,
Diagonal Mask:
What values would be in the
turquoise cells?

57. A different diagonal Mask: What values would be in the
255
1/3
Diagonal Image,
A different diagonal Mask:
What values would be in the
turquoise cells?

58. A wider horizontal line, The mask can be signed numbers:
255 -1 9
A wider horizontal line,
The mask can be signed
numbers:
What values would be in the
turquoise cells?

59. Further information
A collection of related images and convolution matrices

60. Further information
My GGC wiki site

61. Further information
My eBook text used to support the Digital Media course

62. Contact