Extracting Math from PostScript Documents

1
Extracting Math from PostScript Documents

• Michael Yang
• Univ. Calif., Irvine
• Richard Fateman
• Univ. Calif, Berkeley

2
Why Extract Math from Documents?

• The current and recent past publications of
  scholarly journals in mathematics are not
  adequately indexed.
• Imagine a query Find papers that involve this
  differential equation
• x2 yxy(x2-m2)y0
• Or Is there a common name for this equation?
  Ans yes, Bessels

3
Why Extract Math from Documents?

• Find papers that may be relevant to a formula or
  a proof of a related theorem.
• Find out if a discovery is actually novel or a
  rediscovery of a previous result.
• Even Is this formula true?

4
How can we search, anyway?

• Search in integral tables using hashing, flexible
  pattern matching.
• Example TILU (Fateman, Einwohner)
• The general problem looks like a huge challenge
  of unification with simplifications of analytic
  functions. Is af(b) the same as f-1(a)b ?

5
These are obviously hard questions

• But we are much better off if we can start with
  a few decades of the most recent math papers and
  their formulas to search.
• Prerequisite encoding of formulas with semantic
  markup, the point of this paper.

6
Why start with PostScript or PDF?

• We have many papers, including math journals,
  online, some of them free, with essentially all
  markup removed, stored for printing as PS or PDF.
• Automation of inserting the markup, even if only
  partly successful, can help enable further work
  to make it possible to index and search for math.

7
Is this easier or harder than OCR?

• It should be easier, because all the characters
  are known as error-free glyphs.
• OCR tends to make erroneous symbol
  identifications if there is inadequate word-based
  context.
• For example o0Oº, 1lI!i , Illinois (!),
  -_
• Well-known sources of PS provide stereotypes for
  the font/glyph/location mapping.
• But it could be harder if the PostScript is truly
  obscure (PS is Turing equivalent, after all)

8
An Example
From a paper by Cyril Banderier et al, Random
Maps, Coalescing Saddles, Singularity Analysis,
and Airy Phenomena,'' Random Structures and
Algorithms, 19 3-4, 194--246 (2001) only
slightly edited by inserting newlines. explain
origin ....0.002 0.0025 200 400 600 800 1000 k
Figure 3. Left The standard Airy distribution.
Right Observed frequencies of core sizes k 2
20 1000 in 50,000 random maps of size 2,000,
showing the bimodal character of the
distribution. variety of integral or power
series representations including (see 1, 45)
1) Ai(z) 1 2 Z 1 1 e i(zt t 3 3) dt 1 3 23 1
X n0 3 13 z n ( n 1) 3) n sin 2(n 1) 3
Equipped with this de nition, we present the
main character of the paper, a probability
distribution closely related to the Airy
function. De nition 1. The standard ....
9
What is this really?
In this particular case, extraction of the
document image shows two formulas in the middle
of the citation
10
How could we encode this image?
Recognize the characters on the page as
equivalent to a expression, for
example \mbox Ai(z) 1\over2 \pi\int
_-\infty\infty ei(ztt3/3)dt
1 \over \pi 32/3\sum_n0\infty
(31/3z)n \Gamma((n1)/3) \over n! \sin
2(n1)\pi\over 3. or some alternative in
MathML or OpenMath. What are the barriers to
getting to this point?
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Detecting Math in the first place

• Look for changes in font, italics, font size
  changes, altered baselines.
• Consider the density of text (formulas are low
  density).
• Notice the presence of special characters unusual
  in text is common in math, but not in text
  (Also , -, parens).

12
Implementation

• Run PostScript through a modified Ghostscript
  (PS interpreter) to output text file information
  suitable for geometric/math processing.
• Run this file through previously developed
  OCR-based technology (in Lisp) for using
  bounding-boxes, contents, positions, to create a
  geometric 2-D relative position tree. Process
  further to identify semantic relationships if
  possible and output a hierarchical
  tree-representation of math formulas.
• Convert this to TeX (could be MathML equally
  well).

13
Possible Future Work

• Better font tools
• Look at more producers of PS (not just TeX and
  dvips), e.g. Acrobat Distiller.
• Run some tests (NEC) to see if we can extract
  sufficient formulas to add to the indexing
  information.
• Examine the issue of formula similarity e.g.
  parameter substitution, simplification,
  rearrangement. (relatively easy in the context
  of integration because there is a designated
  variable of integration.)

14
Conclusions

• Its possible to automatically revisit previously
  typeset documents and invent plausible versions
  of TeX source-code for some, perhaps much, of
  published TeX.
• This provides an additional link to a chain which
  may eventually lead to more widespread semantic
  encoding of math for index and retrieval.
• Given the difficulties, a better route for the
  future is to have authors or editors use semantic
  mark-up for digital mathematical documents for
  born digital documents. Publishers should
  encourage this kind of work, although standards
  are currently disappointing.

15
Another paper, not included

• Submitted to ISSAC-2004
• Author R. Fateman

16
Rational Function Computing with Poles and
Residues

• Heres the idea consider 2 forms for the same
  rational expression.

17
Which form is better?

• Generality of representation
• Complexity (Cost) of operations
• Arithmetic (, , /)
• Integration, derivatives, limits, series,
• Numerical evaluation
• Display for human viewing

18
Keep constant numerators over (powers of) linear
denominators ( polynomial)

• Works for encoding arbitrary rational functions
  (over complex numbers) in one variable.
• Plausibly requires high-precision floats if you
  start with ratio of polynomials where the roots
  of the denominator cannot be expressed as exact
  rational numbers.

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PRO Once you have this representation

• Addition of rational functions is essentially
  free, compared to standard representation since
  no polynomial GCD is required.
• a/b c/d is already simplified except for
  sorting and the possibility that bd
• Multiplication of rational functions is
  inexpensive also, again no GCD needed.

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CON Do you want to use this representation?

• Division is not fast, so it is more appropriate
  if division is infrequent.
• If the input is not already in residue/pole form,
  or if you have to do division, finding zeros
  introduces approximations maybe for the first
  time in a problem.
• Output forms may look longer.

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Examples

• Ordinary addition orders of magnitude faster.
  E.g 45,000 times faster.
• Ordinary multiplication maybe 2X faster
• What about mixtures of and together? What
  important algorithms are there?
• Sparse determinant calculation.

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A determinant benchmark

• Consider matrices with entries of this form
• Determinant of 8X8 matrix in Macsyma 2.4, on a
  2.6GHz Pentium 4 computer.
• Using Gaussian Elimination 112 sec
• Using Minor Expansion 109 sec
• Using Residues/Poles (75 in bignum
  arithmetic) 41 sec
• Using Residues/Poles and double-floats
  1.6sec

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Conclusions

• No surprise that avoiding GCDs is a winner.
• Using approximate calculations can provide huge
  speedups. Do we really need exact computation
  everywhere we provide it?
• We have a potential application for
  high-precision zero-finding, as well as
  non-overflowing software floats (GMP, ARPREC)