Semantics For Pictures

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Semantics For Pictures

• COS 441
• Princeton University
• Fall 2004

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A Simple Picture Language

• Provide rigorous denotational semantics
• Resolution and device independent specification
• Not about how to draw a picture but about what a
  picture is
• Start with informal concepts then dive into
  semantics

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Concepts

• Point A location on R2
• Shape A set of points
• Color red, blue, green, etc
• Texture An assignment of color to every point
  on the plane
• Layer A partial assignment of colors to points
  on the plane

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Concepts (cont.)

• Picture An order collection of layers
• Image A total assignment of colors to every
  point on the plane. e.g. A picture with a default
  background

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Picture Syntax
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Points
(1,1)
(0,0)
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Shapes Everything
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Shapes Nothing
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Shapes Ellipse
1
2
(0,0)
ellipse((0,0),1,2)
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Shapes Half Planes
(1,1)
(0,0)
halfplane((0,0),(1,1))
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Shapes Half Plane
(1,1)
(0,0)
halfplane((1,1),(0,0))
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Composite Shapes
1
1
(0,0)
(0,1)
intersect(ellipse((0,0),1,1),halfplane((0,0),(0,1)
)
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Polygon as Shapes
intersect(halfplane((0,0),(0,1),
intersect(halfplane((0,1),(1,1),
halfplane((1,1),(0,0)))
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Red Texture
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Layer
1
1
(0,0)
layer(ellipse((0,0),1,1),red)
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Picture
1
layer(ellipse((0,0),1,1),red) B
layer(ellipse((1,0),1,1),green)
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Miscellaneous Operators

• Set operators on shapes
• Intersection, Union, Difference
• Scaling and Translation on shape coordiates

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Formal Semantics
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Formal Semantics (cont.)
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A Simple Theorem
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More Theorems
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(No Transcript)
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Scaling and Translation Thms.
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Scaling and Translation Thms.
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Optimizations

• We can systematically remove every scaling and
  translation operation to obtain an equivalent
  picture
• Removing scaling and translation can speed up
  rasterizing of image by removing unneeded
  computations