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 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

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.

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