Transcript On Mathematical Expression Analysis in Arabic Handwriting

On Mathematical
Expression Analysis
in Arabic
Handwriting
Elena Smirnova and Stephen Watt
ORCCA, UWO,
Feb 2007
Categories of Math Notations
• Writing direction
– Math flows against text
– Math is written in the same direction as text (right to left)
• Use of alphanumerics and math symbols
– Variables
• Use of Latin and Greek alphabet
• Use of Arabic alphabet
– Numerals
• Use of Western Arabic notation for numbers
• Use of Arabic - Indic or Eastern Arabic-Indic numbers
– Math operators and function names
• Western notation
• Mirrored glyphs
• Special Arabic glyphs
Directions in Arabic Math
• Dual direction (Persian and Moroccan Styles)
<text 2> math <text 1>
• Single direction (Maghreb and Machrek Styles)
<text 2> math <text 1>
٠<
٢
۱+ (‫ب‬
– ‫ ب )ا‬، ‫ ا‬
Numerals in Arabic Notations
• Western Arabic (Europe)
0 1 2 3 4 5 6 7 8 9
• Arabic – Indic (Most of Arabic counties)
٠۱٢٣٤٥٦٧٨٩
• Eastern Arabic-Indic (Iran, Urdu)
٠۱٢٣۴۵۶٧٨٩
Math Variables
• Latin and Greek alphabets
• Arabic alphabet
Math Operators and Functions
• European Notation – Persian Style
• Mirrored glyphs – Maghreb “Western” style
• Arabic glyphs – Machrek “Eastern” style
Typeset Arabic Math
• Related projects in rendering typeset math:
– DADTeX, a TeX environment supporting Arabic
– Dadzilla, a MathML browser supporting Arabic
– Arabic Unicode, with respect to directionality
New Challenges in HWR
• Stroke segmentation in text fragments
• If Moroccan or Persian notations are used,
structure recognizers has to handle
bidirectional input.
• In Maghreb notation recognizer has to
handle mirrored glyphs.
• In Machrek notation a special recognition
technique is needed for handling ligatures.
Influence on Expression Analysis
• Arabic notation affects methods not only
for analyzing the structure, but also
for interpreting the results of recognition
• Special attention to be paid to
– Implicit directionality
– Mirrored expressions
– Special container glyphs
– Stretched ligatures
Implicit Directionality
Statement “A2>0 if A>0” written in Farsi
• Recognizer determined the glyphs as
{ A,>,٠, ‫اگر‬, A, ٢, >,٠ }.
• Persian notation  mathematical content
flows from left to right
• Naïve structure analyzer may translate this to
A > 0 if A2 > 0 WRONG!!!
(A >٠ ‫ اگر‬A٢ > ٠)
Careful Mirroring
• Every asymmetric operator is assigned its
mirrored glyph: “(“  “)”, “>”  “<“, etc.
٢
٠< ۱+ (‫ ب )ا – ب‬، ‫ ا‬
 a, b (a –
2
b)
+ 1> 0
• Some mirrored glyphs have not only opposite,
but very different mathematical meaning
– For ex. pair “\”, “/” is direction sensitive:
• “A / B” means division in Left to Right notation
• “B / A” means set subtraction in Right to Left notation
New Container Glyphs
notation for “5!”
• The notation for factorial introduces one
more case of a container symbol, in
addition to the symbols for radical and long
division.
• New set of rules to the structural analyzer
must be added, i.e. the layout of the
expression “n!” will be detected as nested
rather than linear.
Advantages
• Stretched large operators allow to avoid
ambiguities in structure recognition
Examples
Maghreb
N-ary
Summation
vs.
Machrek
N-ary
Product
vs.
Farsi
Limit
vs.
Context Assistance
• Extra challenge:
characters
lots
of
ambiguous
math
– ‫"( ا‬ALEF") and ١ ("1");
– ٠("0") and a dot;
– ٥("5") and ‫"(ه‬HEH") or the symbol for degree "".
• Ex:
• Suggested strategy for character disambiguation:
use of Math Context Database for Arabic notations
Conclusions
• Recognition of Arabic handwritten math introduces
new classes of problems, mainly dealing with
– stroke segmentation
– structure analysis in bidirectional notations.
• However, many methods developed for European
style of math handwriting analysis are applicable
to Arabic notations.
• Moreover, certain things that are easier with
Arabic notations:
– clearer structure organization in case of large delimiters
– more explicit distinction between mathematical and text
fragments (in bidirectional notations).
Future Work
• Identifying suitable source of Arabic training
material for building Context DB and for training
the structure analyzer
• Merging our Mathink framework with existing
recognizers for Arabic script (for text fragments)
• Enhancing char recognizers to handle very
stretched glyphs
• Adding direction awareness to the structure
analyzer
• Developing tools for automated notational profile
detection
References
[1] Azzeddine Lazrek, Mustapha Eddahibi, Khalid
Sami, Cadi Ayyad, Bruce R. Miller. Arabic
mathematical notation. W3C Interest Group Note,
January 2006. http://www.w3.org/TR/arabic-math/
[2] T. Sari and M. Sellami,Cursive Arabic Script
Segmentation and Recognition System. International
Journal of Computers and Applications, Vol. 27,
2005.
[3] Al-Emami, S. and Usher, M., On-Line Recognition
of Handwritten Arabic Characters. Pattern Analysis
and Machine Intelligence, IEEE Transactions (PAMI)
Vol. 12, No. 7, 1990, pp. 704-710.