Quad Trees Region data vs. point data. Roads and rivers in a country/state.  Which rivers flow through Florida?  Which roads cross a river? Network firewalls.  (source prefix, destination prefix, action)  (01*, 110*, drop packet) source dest

Quad Trees Binary images.  Image region is divided into cells called pixels.  Each pixel is either black (0, background) or white (1).

Example Binary Image Remaining entries are 0.

Image Operations Rotation. Rotate 90 degrees clockwise.

Scaling Expansion.  Each pixel is replaced by a 2 k x 2 k window of pixels of the same intensity. Shrinking.  Each 2 k x 2 k window is replaced by a pixel whose intensity is some function of that of the window pixel intensities.

Shrinking Example k = 1. >= 2 white pixels in 2 1 x 2 1 window => white pixel.

Union & Intersection Image1 = roads. Image2 = rivers. Union(Image1, Image2) = roads and rivers. Intersection(Image1,Image2) = places where a road crosses a river => bridge or tunnel.

Image Representation n x n matrix.   (n 2 ) space.   (n 2 ) time for rotation, scaling, union, intersection, and so on. Quad tree.  O(n 2 ) space.  O(n 2 ) time for rotation, scaling, union, intersection, and so on.  For some images, the space and time could be as little as O(1) (e.g., all white or all black images).

Quad Tree Degree 4 tree (each node has up to 4 children). Each node represents a portion of the image. Root node represents entire 2 k x 2 k image. The children of a node that represents a 2 q x 2 q region represent its 4 2 q-1 x 2 q-1 subregions.

Quad Tree Each node has one of the colors white, black, gray.  White (black) => all pixels in node’s region are white (black).  Gray => at least one black and at least one white. White and black nodes have no children. Gray nodes have 4 children each.

Quad Tree Example NWSW NW SWSE NE SE

Quad Tree Example NW SWSE NE

Quad Tree Example NW SWSE NE

Quad Tree Example NW SWSE NE

Oct Tree Extension of quad tree to the representation of 3-d images. Degree 8 tree (each node has up to 8 children). The children of a node that represents a 2 q x 2 q x 2 q region represent its 8 2 q-1 x 2 q-1 x 2 q-1 subregions.

From Matrix To Quad Tree 2 k x 2 k binary matrix. If k = 0, return a single-node quad tree.  Root is white if pixel is 1.  Root is black if pixel is 0.

From Matrix To Quad Tree If k > 0, recursively construct quad trees for the four quadrants. NW SWSE NE

From Matrix To Quad Tree If the 4 quad tree roots are all black (white), return a single node quad tree with a black (white) root. Otherwise, return a quad tree with a gray root whose subtrees are the quadrant quad trees. NW SWSE NE

Complexity Let t(k) be the time to construct the quad tree of a 2 k x 2 k binary image (matrix). t(0) = c, where c is some constant. t(k) = 4t(k–1) + d, where d is some constant and k > 1. t(k) =  (4 k ) =  (#pixels in matrix).

From Quad Tree To Matrix Run divide-and-conquer algorithm for matrix to quad tree transformation backwards. t(k) =  (4 k ) =  (#pixels in matrix).

Clockwise Rotation By 90 Degrees NENW SW

Clockwise Rotation By 90 Degrees NENW SW

Recursive Algorithm If root is a leaf, return. Rotate the children of the root.  (NW, NE, SE, SW)  (SW, NW, NE, SE) Recursively perform the rotation in the subtrees of the root. Complexity is  (size of quad tree).  Quad tree size = # nodes. Quad tree size is usually substantially less than matrix size.

Other Rotations Clockwise rotations by 180 and 270 degrees are similar. Counterclockwise rotations by 90, 180, and 270 degrees are similar. Rotation by 360 degrees is null.

Shrinking k = 1. >= 2 white pixels in 2 1 x 2 1 window => white pixel.

Shrinking k = 1. >= 2 white pixels in 2 1 x 2 1 window => white pixel.

Shrinking k = 1. >= 2 white pixels in 2 1 x 2 1 window => white pixel.

Shrinking k = 1. >= 2 white pixels in 2 1 x 2 1 window => white pixel. NW SWSE NE

Algorithm To Shrink (k = 1) Root represents a 1 x 1 region => return. Root represents a 2 x 2 region.  Root is a leaf => return.  Color the root white if it has >= 2 white children.  Otherwise color the root black.  Make root a leaf and return. Root represents a 2 k x 2 k, k > 1, region.  Shrink subtrees (if any) of root.  Recolor root if all subtrees now have same color. Complexity is  (size of quad tree).

Union Rules.  Do a pairwise union of corresponding pixels in the two images.  Union is white iff at least one pixel of the pair is white.

Recursive Union(A,B) root(A) (root(B)) is white => return A (B). root(A) (root(B)) is black => return B (A). Both have a gray root.  Pairwise union the subtrees of A and B.  If all four of the returned subtrees have a white root, return a single node tree with a white root.  Otherwise return a tree with a gray root and the four returned subtrees as its subtrees. Complexity is O(min{|A|, |B|}). AB

Intersection Rules.  Do a pairwise intersection of corresponding pixels in the two images.  Intersection is white iff both pixels of the pair are white.

Recursive Intersection(A,B) root(A) (root(B)) is black => return A (B). root(A) (root(B)) is white => return B (A). Both have a gray root.  Pairwise intersect the subtrees of A and B.  If all four of the returned subtrees have a black root, return a single node tree with a black root.  Otherwise return a tree with a gray root and the four returned subtrees as its subtrees. Complexity is O(min{|A|, |B|}). AB