Semantics For Pictures COS 441 Princeton University Fall 2004

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

Concepts Point – A location on R 2 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

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

Picture Syntax

Points (1,1) (0,0)

Shapes: Everything

Shapes: Nothing

Shapes: Ellipse (0,0) 2 1 ellipse((0,0),1,2)

Shapes: Half Planes (1,1) (0,0) halfplane((0,0),(1,1))

Shapes: Half Plane (1,1) (0,0) halfplane((1,1),(0,0))

Composite Shapes (0,0) 1 1 (0,1) intersect(ellipse((0,0),1,1),halfplane((0,0),(0,1))

Polygon as Shapes intersect(halfplane((0,0),(0,1), intersect(halfplane((0,1),(1,1), halfplane((1,1),(0,0)))

Red Texture

Layer (0,0) 1 1 layer(ellipse((0,0),1,1),red)

Picture 1 layer(ellipse((0,0),1,1),red) B layer(ellipse((1,0),1,1),green)

Miscellaneous Operators Set operators on shapes –Intersection, Union, Difference Scaling and Translation on shape coordiates

Formal Semantics

Formal Semantics (cont.)

A Simple Theorem

More Theorems

Scaling and Translation Thms.

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