Image and Object Representations

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• Image and Object Representations

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Representing Objects in Space

• Cells primitive elements dividing space
• E.g. pixels and voxels
• Object-based decomposition
• Represent the object in space
• Image(or cell)-based decomposition
• Represent the space and describe objects therein

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Queries on Objects

• Feature query
• Given an object, what are its constituent cells?
• Location query
• Given a cell, determine the object (or objects)
  to which it belongs
• Synergy between representation and efficiency
  queries

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Explicit Representation

• B(5,1),(6,0),(6,1),(7,0),(7,1)
• Object-based
• How to
• locate unoccupied cells
• location query

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Array Access Structures

• Implicit representation (cell-based)
• How to
• Feature query

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Tiling

• Requirements
• Infinitely repetitive
• Infinitely decomposable (resolution changes)
• Tessellation polygonal tiling of the plane
• Polygonal for computational simplicity linear
  boundaries, indexing and reverse lookup
• Nomeclature
• (a) 44 (b) 63 (c) 4.82 (d) 4.6.12 (e) 36

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Properties

• Regular (of regular polygons)
• Similar (same shape at all levels)
• Isohedral (all tiles have equivalent symmetry)
• Unlimited (corollary to similar)
• 44 and 63 are the only to meet all definitions

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Properties

• Uniform orientation
• All tiles similar without rotation or reflection
• Uniform adjacency
• All neighbors same distance
• Neither (but 44 has 2 distance and 63 has 3
  distances)

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Tree-Access Structures

• Array access can be wasteful of space
• Two approaches
• Use a multidimensional-tree (point or MX
  quadtree, kd-tree)
• Use a space-filling curve and map to a B-tree

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Morton-Order Tree
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Aggregation

• Describe objects based on collections
• Using decomposition rules
• Arbitrary decomposition
• Simplest
• Into axis-orthogonal rectangles

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Medial-Axis Transform (MAT)

• Representation of arbitrary decomposition
• Early technique
• Compact for simple object shapes
• Corner MAT (CMAT) Fig b.

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MAT and Distance Metrics

• Skeleton locations that completely describe an
  object
• Given a Minkowski distance Lp
  ((x1-x2)p(y1-y2)p)1/p
• For Manhattan L1 (c,e) and Chessboard Linf (d,f)

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Irregular Grid

• Organize the space not the objects
• Still arbitrary
• Array representation

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Region Quadtree (resp. Octree)

• Restricted decomposition rule
• Recursive into 4 (resp. 8)
• Regular decomposition
• Both implicit and explicit representations
  possible
• Location codes
• Describe placement and size
• Special case of MAT

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Region Quadtree

• Implicit representation

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Region Quad in a B-Tree
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K-d Region Tree