© Bill Amend

1. Convolution: the controlled mixing of data to serve a purpose.
   1. An Excel example.
   2. Notions borrowed from analysis of frequency-based signals.
   3. Concept: JPEG encodes images using frequency-based encoding; it's a "lossy" encodoing.
   4. Filters: low-pass, high-pass, and band-filters.
   5. Bluring. Accentuating low-pass filter.
   6. Edge-detection. High-pass filter.
   7. Sharpening. Enhancing edges, attenuating low-frequency components of the image.
2. GRAMMATICAL SYSTEMS: We recall phrase grammar systems.
   1. Traditional "phrase" grammars (Chomsky).
      1. Terminology: terminal and non-terminal symbols, start symbol, production, derivation, word, language, acceptance.
      2. Example: a simple English grammar
         1. Nonterminals: upper case letters
         2. Terminals: lower case words
         3. Start symbol: T.
         4. Grammar:
            1. T → S V O stop
            2. S → D N
            3. V → hugged | ate | chased | kissed | is
            4. O → D N
            5. O → D N C V O
            6. C → that | comma which | comma but | and
            7. D → a | the
            8. N → cat | dog | prince | frog | princess
         5. Some possible derivations:
            1. T ⇒ the dog chased the cat stop
            2. T ⇒ the cat hugged the dog stop
            3. T ⇒ the frog kissed the princess that is a cat stop
            4. T ⇒ a dog is a dog comma which is a prince that ate a cat stop
   2. Parallel string grammars are simply what happens when we rewrite everything in parallel. Recall our simple example when studying Turing machines:
      1. Nonterminals: upper case letters
      2. Start symbol: A
      3. Grammar:
         1. A → B
         2. B → BA
      4. First few words:

         Generation	Word	#A	#B
         0	A	1	0
         1	B	0	1
         2	BA	1	1
         3	BAB	1	2
         4	BABBA	2	3
         5	BABBABAB	3	5
         6	BABBABABBABBA	5	8
         7	BABBABABBABBABABBABAB	8	13
         8	BABBABABBABBABABBABABBABBABABBABBA	13	21

      5. Note the patterns.
   3. Notion of a "turtle-based" drawing system. Logo? An experimental applet
      1. "F" means "go forward".
      2. "-" means "turn left" by a specified amount.
      3. "+" means "turn right" by a specified amount.
      4. "[" means "remember position and orientation".
      5. "]" means "restore position and orientation".
   4. Use L-system to generate turtle commands. Basis for simple, stylized plants.
   5. Some exercises to think about (some in lab):
      1. Draw an oriented line.
      2. Draw a triangle.
      3. Draw a square.
      4. Draw like Spirograph! How?
      5. Draw a simple snowflake.
      6. Can you draw a binary tree without intersecting branches?
      7. Draw a simple tree.
      8. Draw a more complex tree.
      9. Draw an aperiodic comb.
      10. Draw a fractal snowflake.
      11. What do we get with the following grammar:
          1. Start: FX
          2. All turns 90 degrees.
          3. X → X+YF+
          4. Y → -FX-Y
   6. Fractal dimension: log(self-similar pieces)/log(magnification).
   Resources needed for this class:

   • The L-system applet.
   • Selections from the Algorithmic Beauty of Plants, by Przemyslaw Prusinkiewicz and Aristid Lindenmayer. We will be using this material in lab.
   • Turing's Chemical Basis for Morphogensis.

   For next lecture:

   • Lindenmayer's two papers on Mathematical Models for Cellular Interatictions in Development.
   • Donald Knuth's paper on song complexity.
   • A wonderful kolam website.

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