1. School of Computing Faculty of Engineering Meshing with Grids: Toward Functional Abstractions for Grid-based Visualization Rita Borgo & David Duke Visualization & Virtual Reality Group School of Computing University of Leeds, UK Colin Runciman & Malcolm Wallace Department of Computer Science University of York, UK

2. Overview Why functional programming (still) matters Project: a lazy polytypic grid Marching cubes Streaming Making it generic Performance Looking back, looking forwards

3. Why Functional Programming Still Matters Academic arguments J. Hughes, Why Functional Programming Matters Problem decomposition program composition Absence of side-effects Higher-order functions Laziness Practical arguments: Natural progression: OO service-orientation Tower of Babel Novel solutions come from working against the OO grain!

4. Introduction to FP and Haskell Functional building blocks square :: Int -> Int square x = x*x map :: (a -> b) -> [a] -> [b] map _ [] = [] map f (a:as) = (f a):(map f as) (.) :: (b -> c) -> (a -> b) -> (a -> c) (f. g) x = f(g(x)) sqList ls = map square ls sqsqList ls = map sqList (map sqList ls) = (map sqList. map sqList) ls = map (sqList. sqList) ls fibs = [0,1] ++ [ a+b | (a,b) <- zip fibs (tail fibs) ] Further information: see www.haskell.org Note: loop fusion law encoded as a rule in GHC compiler

5. Why FP matters (to grid-enabled vis) pipeline architecture widespread in visualization supports distribution and streaming However Streaming is ad-hoc and coarse grained Algorithms depend on mesh type Data traversed multiple times readerozone levelsisosurfacenormals isosurfacereadertemperature displaygeo-reference ? Note: analogy of pipeline composition and function composition: f. g

6. A Lazy Polytypic Grid Grid enabling: distribution of the run-time system and on-demand streaming of arbitrary data. Through fusion laws, multiple traversals on a single resource are folded into one pass. 2 readerozone levelsisosurfacenormals isosurfac e reade r temperatu re geo-referencedisplay Algorithms: written once, based on generic pattern of data types, then instantiated for any type. 1 3 Specialization: adapt programs to utilize resources available – data or computational.

7. Isosurfaces Widely-used technique for both 2D and 3D scalar data Two general approaches: Contour tracking: follow a feature through the dataset Marching: traverse dataset, processing each cell as encountered in-core versus out-of-core variations 2D examples: skull cross-section; isoline for t=5

8. Marching Squares Input: a dataset, and a threshold value to be contoured Output: line segments representing contour's path within dataset Algorithm: For each cell, compare field value at point with threshold Sixteen possible cases: index into case-table to find edges Interpolate along edges to find intersection points Note ambiguity in cases 5 and 10!

9. Marching Cubes... and beyond 3D surface generalizes 2D case: Isolines become surfaces composed of triangles 16-case lookup table becomes 256-case table (15 cases if we use symmetry) Tetrahedral cells also common Other cell types possible common pattern of processing need appropriate case-table

10. Implementation 1: Functional Arrays Basic types type XYZ = (Int,Int,Int) type Num a => Dataset a = Array XYZ a type Cell a = (a,a,a,a,a,a,a,a) Top-level traversal isoA :: (Ord a, Intgeral a) => a -> Dataset a -> [Triangle] isoA th sampleArr = concat. zipWith1 (mcubeA th lookup) addrs where addrs = [ (i,j,k) | k <- [1..ksz-1], j <- [1..jsz-1], i <- [1..isz-1]] lookup arr (x,y,z) = (arr!(x,y,z), arr!(x+1,y,z),.., arr!(x+1,y+1,z+1)) Worker function mcubeA :: (Ord a, Intgeral a) => a -> (XYZ -> Cell a) -> XYZ -> [Triangle] mcubeA th lookup xyz = group3. map (interp th cell xyz). mctable!. toByte. map8 (>th) $ cell where cell = lookup xyz

11. Problems Entire dataset must be resident in memory Vertex shared by n cells threshold comparison repeated n times > 1 triangle in a cell => edge interpolation repeated within cell Edge shared by m cells interpolation repeated m times

12. Thinking differently - streaming mkStream :: XYZ -> [a] -> [Cell a] mkStream (isz,jsz,ksz) origin = zip8 origin (drop 1 origin) (drop (line+1) origin) (drop line origin) (drop plane origin) (drop (plane+1) origin) (drop (planeline+1) origin) (drop planeline origin) where line = isz plane = isz * jsz planeline = plane + line line plane 8-tuple...

13. Discontinuities Two solutions: Rewrite mkStream, considering dataset boundaries; or Strip phantom cells from output of mkStream disContinuities :: XYZ -> [b] -> [b] disContinuities (isz,jsz,ksz) = step (0,0,0) where step (i,j,k) (x:xs) | i==(isz-1) = step (0,j+1,k) xs | j==(jsz-1) = step (0,0,k+1) (drop (isz-1) xs) | k==(ksz-1) = [] | otherwise = x : step (i+1,j,k) xs cellStream = disContinuities size. stream

14. Implementations 2 & 3: Streams Version 2: replace array lookup with stream access isoS th samples = concat. zipWith2 (mcubeS th) addrs cells where cells = stream size samples mcubeS :: a -> XYZ -> Cell a -> [Triangle] mcubeS th xyz cell = group3. map (interp th cell xyz). mctable!. toByte. map8 (>th) $ cell Version 3: share vertex comparison by creating a stream of case-indices isoT th samples = concat. zipWith3 (mcubeS th) addrs cells indices where indices = map toByte. stream. map (>th) mcubeT :: a -> XYZ -> Cell a -> Byte -> [Triangle] mcubeT th xyz cell index = group3. map (interp th cell xyz). mctable! $ index Further improvements explored in IEEE Visualization paper

15. From generic cells... Functions already polymorphic.. generic over one or more type variables constraints may limit instantiation isoA :: (Ord a, Intgeral a, Fractional b) => a -> Dataset a -> [Triangle b]... and abstracting from Cell type is (nearly) straightforward mcubeRec :: (Num a, Floating b) => a -> XYZ -> CellR a -> [Triangle b] mcubeRec th xyz cell = group3. map (interp th cell xyz ). mcTable!. toByte8. map8 (>th) $ cell mcubeTet :: (Num a, Floating b) => a -> CellT a -> CellT b -> [Triangle b] mcubeTet th g cell verts = group3. map (interp th cell verts). mtTable!. toByte4. map4 (>th) $ cell Can capture general pattern within a type-class class Cell T where patch :: (Num a, Floating b) => a -> T a -> T b -> [Triangle b]

16. ... to generic meshes Dealing with different mesh-type organizations is harder... Regular meshes: implicit geometry and topology Irregular meshes: implicit geometry Unstructured meshes: geometry and topology explicit Polytypic functions are independent of data organization Haskell data constructions isomorphic to sum-of-products type Foundation on categorical model of data type structure Examples data List a = Nil | Cons a (List a) --> List = 1 + (a x List) data Tree a = Leaf a | Node (Tree a) a (Tree a) --> Tree = a + (Tree x a x Tree)

17. Polytypism in practice From Generic Haskell: Practice & Theory, Hinze & Jeuring, 2001 define generic function by induction over type structure generic version can then be instantiated for any SoP type mapG {|t::kind|} :: Map {|kind|} t t mapG {|Char|} c = c mapG {|Int|} i = i mapG {|Unit|} Unit = Unit mapG {|:+:|} mapa mapb (InL a) = InL (mapa a) mapG {|:+:|} mapa mapb (InR b) = InR (mapb b) mapG {|:*:|} mapa mapb a :*: b = mapa a :*: mapb b data Tree a = Leaf a | Node (Tree a) a (Tree a) mapList = mapG {| List |} -- standard Haskell map mapTree = mapG {| Tree |} -- apply function to each node in the tree Research question: can we actually apply this idea to mesh traversal? surface t = concat. mapG {| Mesh |} (\c -> patch t c)

18. Sample Results

19. Performance Performance difference decreases with surface size D.J. Duke, M. Wallace, R. Borgo, and C. Runciman, Fine-grained visualization pipelines and lazy functional languages, to appear in Proc. IEEE Vis06

20. Conclusions and Future Work What we've achieved: re-constructed fundamental visualization algorithms Implemented fine-grained streaming demonstrated that FP can be (surprisingly) efficient What we're doing now: generalizing from specific types of mesh exploring capabilities of generic programming Where we are going next: grid-enabled Haskell pipelines some prior work: GRID-GUM, Michaelson, Trinder, Al Zain build into York Haskell Compiler (bytecode) RTS want simple, lightweight grid tools!

21. Finally... Thanks to EPSRC Fundamental Computing for e-Science Programme Further information: hackage.haskell.org/trac/PolyFunViz/ Any Questions?