BCTCS 2006

1. BCTCS 2006 The Theory of Spatial Data Types and Constructive Volume Geometry

2. Constructive Volume Geometry Is an algebraic framework for the high-level • Specification, • Representation, • Manipulation, and • Rendering of spatial data types. We use CVG to build complex scenes using algebraic operations on fully 3D spatial objects. CT planet, Min Chen, 1999 M. Chen and J. V. Tucker, Constructive volume geometry, Computer Graphics Forum, Vol.19, No.4, 281-293, 2000.

3. CVG Spatial Objects A spatial object is a tuple o = (O, A1, A2, …, Ak) of scalar fields defined in R3, including an opacity field O: R3 [0,1] and possibly other attribute fields: A1, A2, …, Ak: R3 R, k ≥ 0. O: sphereR: noiseG: constantB: constant Visibility of every point in R3 specified by opacity attribute where 0 is transparent and 1 is opaque.

4. Constant • Volume Dataset • Colour-separated image • Built-in mathematical scalar field • Procedural scalar field • plus various mappings CVG Scene with Several Objects scene object object ...... object field O R G B Ge Ka Kd Ks N Dn Rfl Rfr Txt ...

5. o1=(O1, R1, G1, B1) o2=(O2, R2, G2, B2) CVG Operations: 4 Colour Channel Model (I) (o1, o2)

6. CVG Operations: 4 Colour Channel Model (II) (o1, o2) (o1, o2) (o1, o2) (o2, o1)

7. { s1 s1 s2s2 s1 < s2 max(s1, s2) = { t1 s1 s2t2 s1 < s2 select(s1, t1, s2, t2) = CVG Operations: 4 Colour Channel Model (III) operations on spatial objects (o1, o2) = ( Max(O1, O2),Select(O1, R1, O2, R2),Select(O1, G1, O2, G2),Select(O1, B1, O2, B2) ) operations on scalar fields Max(O1,O2)(x) = max(o1(x),o2(x))Select(S1,T1,S2,T2)(x) = select(s1(x),t1(x),s2(x),t2(x)) operations on scalars

8. Algebraic Specification and CVG • Equational axioms on scalars extend to axioms on spatial objects. • Example: Associativity law for the CVG operation: (o1, (o2,o3)) = ( (o1,o2),o3) defined using Max, Select Associativity of Max, Select on scalar fields pointwise-lifting of operations Associativity of max, select on scalars

9. Hierarchical Data Representation composite volume object (o1, o2) o1 o2 convex volume object convex volume object 1

10. CVG Tree (I)     – – – –  –

11. CVG Syntax and Semantics • Define algebraic terms and CVG term evaluation (= structural induction) in the usual way. CVG Term:

12. Expressiveness of CVG CVG is an algebraic framework for volume graphics. Question: Do the CVG terms define all the spatial objects one wants? We can show the answer is: Yes • Any continuous spatial object can be approximatedby a CVG term to arbitrary accuracy. • Given any computable spatial object o and an ε > 0 onecan compute a CVG term toε which approximates o towithin the error margin ε. In fact we prove this by a general theory of spatial objects…

13. Validity Theorem:For any Σ equation t = t’ the following are equivalent: t = t’ is valid in A, t = t’ is valid in F(X,A), Corollary:If A satisfies a set E of Σ equations then F(X,A) satisfies E. Examples: Axioms for commutative rings, Axioms for CVG. General Theory of Spatial Data Types: IAlgebra The Σ-operations on A are pointwise lifted to F(X,A): FF(X,A)(φ1,…, φn)(x) = fA(φ1(x),…, φn(x)) which preserves the equivalence of term evaluation: [t]F = [t]A

14. General Theory of Spatial Data Types: IITopology • We use topology to define the notion of approximation. • C(X,A) = the set of all continuous total spatial data types from X to A • X is “any” topological space and A is a topological Σ-algebra. • C(X,A) has the compact-open topology: Basic open sets are W(J,U) = {f | f[J] U}, J compact subset of X and U open in A. • For X compact and A = R, the compact-open topology on C(X,R) is equivalent to the topology defined by the sup norm: || f – g || = sup{z in X}|f(z) – g(z)| with open sets: B(f, ε) = {g in C(X,R) : || f – g || < ε } U

15. R We want a “nice” function g such that ║ φ – g ║ < ε φ + ε φ g φ - ε K compact

16. Dense Subsets of C(X,R) • We say that a set D is dense in C(X,R) if for any W(J,U) we have D W(J,U) ≠ 0. • For K compact, there is a spatial object g in D arbitrarily close to any given φ in C(K,R): || φ – g || < ε. • If D is dense on C(X,R) then spatial objects in D can approximate all continuous spatial objects. • We use the Stone-Weierstrass Theorem when we deal with approximation on a compact K, but what about non-compact X? U

17. Is it possible to find a collection of spatial operators on C(X,R) Σ = {F1,…,Fk} and a subset B of C(X,R) consisting of continuous basic spatial objects b1,… bn such that every continuous spatial object in C(X,R) can be approximated by an object t(b1,… bn) from the subalgebra <B>Σ generated by the operations Σ on B? Completeness of Spatial Data Types

18. Basic Spatial Objects and Primitive Operations Basic spatial objects in B should have the properties: • separate the points of space X • contain a non-zero spatial object Operations we need in Σsw • Addition • Multiplication • Scalar Multiplication

19. Approximation Theorem for Spatial Objects Theorem: Let C(X,R) be the set of all continuous total spatial objects, and B is a subset with Properties 1 and 2. Using only the operations in Σsw the subalgebra <B>Σsw generated by the repeated application of operations in Σsw on B can approximate all spatial objects in C(X,R). • Proof Idea: Localize the problem to a compact space K, find an approximate, and then show it works on C(X,R) • Pick any W(K,U) in C(X,R). • We want a term t in <B>Σsw such that [t]C(X,R)(b1,…bn) is in W(K,U).

20. Inverse Limit Construction • Sequences of restrictions directed under set inclusion by the family Z of all compact sets of X. • Shows that the compact-open topology “behaves the same way”on both sets. • Each coordinate Φ(f)(K) is the restriction fK in C(K,R). Theorem: LimI in ZC(I,R) ≈ C(X,R) Homeomorphism: Φ(f) = (…,fK,…,fJ,…fI,…)

21. C(X,R) LimI in ZC(I,R) πK ΦK C(K,R) Proof Sketch: Restriction Homomorphism Φ The mapping ΦK = πK ◦ Φ is a homomorphism and restricts C(X,R) to C(K,R) The set ΦK(B) of basic spatial objects restricted to a compact K satisfies Properties 1 and 2 on the coordinate space C(K,R).

22. Proof Sketch: Approximation on C(K,R) • We can approximate on a compact K, so by Stone-Weierstrass <ΦK(B)>Σsw is dense in C(K,R). • Meaning that there is a term t in <ΦK(B)>Σsw such that [t]C(K,R)(ΦK(b1),…, ΦK(bn)) is in W(K,U). • Since ΦK is a homomorphism: [t]C(K,R)(ΦK(b1),…, ΦK(bn)) = ΦK([t]C(X,R)(b1,…,bn)) i.e. the term t in <B>ΣSW evaluates to the same points in C(X,R) and so [t]C(X,R) is in W(K,U).

23. Goals • Extensions of the Stone-Weierstrass and spatial object approximating theorems • Computability and effectivity issues; ‘While’ programs over C(X,A) • Graphics applications: • Discretisation D : C(R3,A) → C(Z3,A) • Rendering R: C(R3,A) → C(Z2,A)