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Quadtrees

Quadtrees. Raster and vector. Quadtrees. Finkel and Bentley, 1974 Raster structure: divides space, not objects Form of block coding: compact storage of a large 2-dimensional array Vector versions exist too. Quadtrees, the idea. NW. NE. SW. SE. NW. NE. SW. SE.

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Quadtrees

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  1. Quadtrees Raster and vector

  2. Quadtrees • Finkel and Bentley, 1974 • Raster structure: divides space, not objects • Form of block coding: compact storage of a large 2-dimensional array • Vector versions exist too

  3. Quadtrees, the idea NW NE SW SE NW NE SW SE 1, 4, 16, 64, 256nodes

  4. Quadtrees, the idea NW NE Choropleth raster map SW SE NW NE SW SE

  5. Quadtrees • Grid with 2k times 2k pixels • Depth is k +1 • Internal nodes always have 4 children • Internal nodes represent a non-homogeneous region • Leaves represent a homogeneous region and store the common value (or name)

  6. Quadtree complexity theorem • A subdivision with boundary length r pixels in a grid of 2k times 2k gives a quadtree with O(kr) nodes. • Idea: two adjacent, different pixels “cost” at most 2 paths in the quadtree.

  7. Overlay with quadtrees Acid rain with PH below 4.5 Water

  8. Overlay with quadtrees

  9. Result of overlay

  10. Overlay algorithm • If color Q1 is specified, and Q2 is a leaf,thenreturn a leaf for Q3 with as color the combination of those of Q1 and Q2 • If color Q2 is specified, and Q1 is a leaf,thenreturn a leaf for Q3 with as color the combination of those of Q1 and Q2 • IfQ1 and Q2 are both a leaf,thenreturn a leaf for Q3 with as color the combination of those of Q1 and Q2

  11. Overlay algorithm • If color Q1 is specified or Q1 is a leaf, and Q2 is an internal nodethen recurse in the four subtrees of Q2(with the color of Q1 as specified) • If color Q2 is specified or Q2 is a leaf, and Q1 is an internal nodethen recurse in the four subtrees of Q1(with the color of Q2 as specified) • IfQ1 and Q2 are both internal nodesthen recurse in the four corresponding subtrees of Q1 and Q2

  12. Overlay, efficiency • Result is quadtree Q3 that represents overlay • Assume n nodes in Q1 and m nodes in Q2: O(n+m) time • If Q3 need only contain the overlay where Q1 has a particular theme, then it can be done more efficiently by pruning

  13. Quadtree construction from raster, I • Construct complete quadtree on all pixels • Merge bottom-up groups of four with same attribute • Disadvantage: Temporarily a lot of storage needed (quadtree on all pixels), and not efficient in time

  14. Quadtree construction from raster, II • Construct quadtree top-down and recursively • Merge four subtrees immediately (when returning from recursion) into a leaf if they are four leaves with the same attribute

  15. Various queries • Point location: trivial • Windowing: descend into subtree(s) that intersect query window • Traversal boundary polygon: up and down in the quadtree

  16. Vector quadtree • PM quadtree in naming of Samet • Divide each square if there are:- 2 or more vertices inside- a vertex and an edge inside that are not incident- 2 (or more) edges inside, unless these edges are all incident to the same vertex and this vertex is also in the square

  17. Leaf in vector quadtree • Empty (inside a region; pointer to cell) • One edge (pointer to edge in leaf) • One vertex with incident edges (pointer to vertex in leaf)

  18. Example vector quadtree

  19. Point location in vector quadtree • Go down quadtree to leaf that contains the query point • At the leaf, resolve the query e f v O(1) time O(degree(v)) time O(1) time

  20. Complexity vector quadtree • Cannot be expressed in the number of stored objects only • Depends on inter-distance of vertices compared to region size • Depends on “accidental” distance of a vertex to a side of a square

  21. Alleviating the problem • Allow a small number (e.g. 3 or 4) of points or independent edges in a square • Do not split squares beyond a certain size

  22. Summary • Raster quadtree: simple structure, simple map overlay, simple queries • Vector quadtree: possibly useful for efficient access, no performance guarantees

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