Notes for Computer Graphics Class

Peering into the depths of color

Color Depth

• The amount of data used to represent a single pixel

size	color options
1 bit	1 color, on | off
2 bit	1 color with intensity
3 bit	Red, Green, Blue, on | off
4 bit	RGB with intensity
6 bit	RGB, each color has its own intensity
3 byte	RGB, each with 256 possible intensities

Other Color Spaces

• RGBA
  • Red, Green, Blue + Alpha (transparency)
• HSB
  • Hue, Saturation, Brightness
  • 

Image File Formats

• Raster vs. Vector
  • Vector formats represent images as a series of drawing instructions
    • Infinitely scalable
    • SVG (Scalable Vector Graphics)
  • Raster formats represent images as a grid of pixels
• Uncompressed vs. Compressed (Raster)
  • Uncompressed formats contain data for each pixel
    • BMP, TIFF, RAW <!------- Thursday, January 30, 2020 ------->
  • Compressed formats use a compression algorithm to minimize file size
  • Lossless vs. Lossy
    • Lossless compression algorithms contain enough information to exactly recreate the image
      • Run Length Encoding
        • RRRRRGGBBBB --> R5G2BB4
      • PNG (Portable Network Graphics), GIF (Graphics Interchange Format)
    • Lossy compression algortihms do not retain all the details of the original image
      • JPEG (Joint Photographic Experts Group)

PPM (Portable PixMap)

• Uncompressed raster format
• Part of NETPBM Graphics family
• Pixel data is represented by RGB triplets in either ASCII or binary
• All whitespace is equivalent
• Ex ASCII ppm file (pic.ppm)

P3
5 5
255
255 255 0 255 255 0 255 255 0 255 255 0 255 255 0
255 255 0 255 255 0 255 255 0 255 255 0 255 255 0
255 255 0 255 255 0 255 255 0 255 255 0 255 255 0
255 255 0 255 255 0 255 255 0 255 255 0 255 255 0
255 255 0 255 255 0 255 255 0 255 255 0 255 255 0 

• Making a PPM file (line-by-line, but all whitespace is equivalent anyways):
  • Must start with P3 if in ASCII, P6 if in binary
  • Specify width then height
  • Specify max value (will map 0->255 to 0->100 if you use 100)
  • Then your pixel triplets
  • Note-to-self: leave the trailing space after last pixel; if not there, will give error about end-of-file when converting

Line Algorithm

Input: 2 endpoints

Ideas?:

• test potential pixels
• iterate through potential pixels
• one pixel per x-value*

Octant I: 0 < m < 1

Steps of Algorithm:

1. Loop through x_0 --> x_1
2. Test(x+1, y)
3. Test(x+1, y+1)
4. Pick best point

Pseudocode for Picking best point/pixel (OCTANT I) v0:

f(x, y) = Ax + By + C

x = x_0
y = y_0

while x <= x_1
    plot(x, y)
    d0 = f(x+1, y+1)
    d1 = f(x+1, y)
    
    if (abs(d0) < abs(d1))
        y = y+1
    x = x+1

Pseudocode for Picking best point/pixel (OCTANT I) v1:

f(x, y) = Ax + By + C

x = x_0
y = y_0

while x <= x_1
    plot(x, y)
    d = f(x+1, y+(1/2))
    
    if d > 0
        y = y+1
    x = x+1

Pseudocode for Picking best point/pixel (OCTANT I) v2:

f(x, y) = Ax + By + C

x = x_0
y = y_0
d = A + (1/2)B
while x <= x_1
    plot(x, y)
    
    if d > 0
        y = y+1
        d = d+B
    x = x+1
    d = d+A

Pseudocode for Picking best point/pixel (OCTANT I) v3:

f(x, y) = Ax + By + C

x = x_0
y = y_0
d = A + (1/2)B
while x <= x_1
    plot(x, y)
    
    if d > 0
        y = y+1
        d = d+B
    x = x+1
    d = d+A

Pseudocode for Picking best point/pixel (OCTANT I) v4:

f(x, y) = Ax + By + C

x = x_0
y = y_0
d = 2A + B
while x <= x_1
    plot(x, y)
    
    if d > 0
        y = y+1
        d = d+2B
    x = x+1
    d = d+2A

Pseudocode for Picking best point/pixel (OCTANT II):

x = x_0
y = y_0
d = A + 2B
while y <= y_1
    plot(x, y)
    
    if d <= 0
        x = x+1
        d = d+2A
    y = y+1
    d = d+2B

Matrices in Graphics

• we will store all the significant endpoints of our images in a single list/matrix
• we will maintain an edge matrix, where each pair of entries defines an edge
• points are (x,y) or (x,y,z) coordinates
• Matrix Multiplication (Review)

Transformations

• scale, translate, rotate

• applied as matrix multiplication

• Scale

  • (x,y,z) --- S_{a,b,c} ---> (ax,by,cz)

      [ a 0 0 0 ][ x ]   [ ax ]
      | 0 b 0 0 || y |   | by |
      | 0 0 c 0 || z | = | cz |
      [ 0 0 0 1 ][ 1 ]   [ 1  ]
    

• Translate

  • (x,y,z) --- T_{a,b,c} ---> (x+a,y+b,z+c)

      [ 1 0 0 a ][ x ]   [ x+a ]
      | 0 1 0 b || y |   | y+b |
      | 0 0 1 c || z | = | z+c |
      [ 0 0 0 1 ][ 1 ]   [ 1   ]
    

• Rotate

  • About the Z-Axis

    • (x,y,z) --- R_{theta,z-axis} ---> (xcosθ-ysinθ, xsinθ+ycosθ, z)

        [ cos(theta) -sin(theta) 0 0 ][ x ]   [ xcosθ-ysinθ ]
        | sin(theta) cos(theta)  0 0 || y |   | xsinθ+ycosθ |
        | 0          0           1 0 || z | = | z           |
        [ 0          0           0 1 ][ 1 ]   [ 1           ]
      

  • About the X-Axis

    • (x,y,z) --- R_{theta,x-axis} ---> (x,ycosθ-zsinθ,ysinθ+zcosθ)

        [ 1 0          0           0 ][ x ]   [ x           ]
        [ 0 cos(theta) -sin(theta) 0 ]| y |   | ycosθ-zsinθ |
        | 0 sin(theta) cos(theta)  0 || z | = | ysinθ+zcosθ |
        [ 0 0          0           1 ][ 1 ]   [ 1           ]
      

  • About the Y-Axis

    • (x,y,z) --- R_{theta,z-axis} ---> (xcosθ+zsinθ,y,zcosθ-xsinθ)

        [ cos(theta)  0 sin(theta) 0 ][ x ]   [ xcosθ+zsinθ ]
        [ 0           1 0          0 ]| y |   | y           |
        | -sin(theta) 0 cos(theta) 0 || z | = | zcosθ-xsinθ |
        [ 0           0 0          1 ][ 1 ]   [ 1           ]
      

• Combining Transformations

  • E0: Edge Matrix, T: Translate Matrix, R: Rotate Matrix, S: Scale Matrix
  • T * E0 = E1 (Translated)
  • R * E1 = E2 (Translated, Rotated)
  • S * E2 = E3 (Translated, Rotated, Scaled)
  • E3 = S * R * T * E0