Automata for bilevel image compression K. Culik II, V. Valenta: Finite Automata Based Compression of Bilevel images K. Culik II, V. Valenta: Finite Automata Based Compression of Bilevel and Simple Color Images M. Mindek: Finite State Automata and Image

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Automata for bilevel image compressionK. Culik
II, V. Valenta Finite Automata Based Compression
of Bilevel images K. Culik II, V. Valenta
Finite Automata Based Compression of Bilevel and
Simple Color ImagesM. Mindek Finite State
Automata and Image Recognition

• Alessandro Giusti
• April, 28 2006

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Quadtrees for multiresolution bw images

• Quadtree-based addressing scheme for image areas

3203

• Language specifies addresses of black pixels
• Multiresolution addresses can have arbitrary
  length (possibly infinite)

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Quadtrees for multiresolution bw images (2)

• A state represents an image.
• The image is defined quadrant by quadrant by
  outgoing transitions to other states (4
  sub-images)
• The color of an (infinite resolution) pixel is
  found by feededing the automaton with the
  (potentially infinite) pixel address
• If the pixel is white, the string will not be
  recognized
• The evaluation can be stopped early for a
  lower-resolution image (rough approximation)
• Zooming is straightforward
• just replace the initial state with the state
  obtained by feeding the automaton with the area
  address
• Note how language concatenation works
• Note how a black square is defined (final state)

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Quadtrees for multiresolution bw images (3)

• Another example

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Last example

• Re-use the triangle T and make a fractal out of
  it.

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Multiresolution black and white images

• Automata easily define
• images which can be defined vectorially, but not
  rasterized
• recursively-defined images (fractal-like),
  self-similar at various scales, with infinite
  resolution.

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Automaton construction procedure

• Goal given multiresolution image I as input,
  build the minimal automaton perfectly defining
  it.
• Approach recursively create new states for each
  subimage, unless that subimage is already
  represented in an existing state.
• Will not terminate when no automaton perfectly
  defines the input (e.g. a triangle with an
  irrational number as slope).

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Automaton construction procedure (example)
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Generalized finite automata

• The procedure becomes more powerful if we allow
  image transformations in transitions
• Rotation by 0, 90, 180, 270
• Optional mirroring
• Optional color negation
• Add one of the 4x2x216 possible transformations
  as an additional transition label

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Generalized finite automata (2)

• Meaning
• Quadrant 0
• Quadrant 1
• Quadrant 2
• Quadrant 3

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Full example
0
1,2,3,4
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Another example

• Divide et impera applied to images
• Note how the stripes are recursively defined, and
  how transformations partecipate in the definition
  (result are infinite resolution, perfect lines).
• Lines are not explicitly defined, and emerge from
  cross-resolution constraints ? fractal
  definition of lines.
• Find n small errors in the figure (n2)

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Generalized automaton construction procedure

• Goal given multiresolution image I as input,
  build an automaton approximating it.
• News recursively create new states for each
  subimage, unless the subimage is already
  approximated within a given error, by any
  transformation of an existing state.
• Will always terminate
• Interesting implementation-related observations

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About color images

• Consider only graphic images, not photographs ?
  no gradients.
• Quantize to colors (n3) ? n bilevel
  bitplanes
• Approach apply algorithm separately to each
  bitplane.
• News?
• Share the states between different bitplanes
  exploit cross-color self-similarity.

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Results showcase
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Results

• Remarkable (while not state of the art)
  compression ability, but
• Read the fine print! 8x8 subimages are vector
  quantized (traditional compression mechanism).
• Nothing more precise (e.g. number of quantized
  vectors) is stated about this last step this
  could even account for most of the compression
  power!
• Compression algorithm exploits
• Quadtree decomposition (Color-homogeneous areas)
• (Cross-resolution) self-similarity

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Critique

• Imprecise bilevel images are
• easily
• ? Lossy compression of bilevel images has very
  limited applications w.r.t. more general
  compression techniques
• The proposed method is an interesting application
  of automata, but
• very limited flexibility and generality
• Much better approaches exist in order to compress
  bilevel, non-fractal images (vectorial graphics,
  tracing...)

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Generality

• Self-similarity at different scales not frequent
  in everyday graphical images
• Quadrant-based notation is arbitrary and too
  rigid to be taken advantage of in practice
• Needs perfect alignment of replicas
• ? Even if self-similarity was present, most
  probably it would not be possible to exploit it
• Example images are artificial and ad-hoc. Almost
  everything else is not representable with the
  same elegance
• Why only 16 trasformations?
• Will the approach scale well if we improve it?

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Art (?)

• Also context-free grammars can generate beautiful
  pictures
• ? www.contextfreeart.org