SCALED Pattern Matching

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SCALEDPattern Matching

• Amihood Amir Ayelet Butman
• Bar-Ilan University Moshe
  Lewenstein
• and
• Johns Hopkins University
  Bar-Ilan University

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Motivation

• Searching for Templates in
• Aerial Photographs
• Input Aerial photo
• Template
• Task Search for all locations where the template
  appears in the image.

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(No Transcript)
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Model

• Low level (pixel level)
• avoid costly processing
• Asymptotically efficient solutions.
• Serial, exact algorithms.

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Types of Approximations

• Local errors Level of detail
• Occlusion
• Noise
• results O(n² log m) mismatches
• O(n²k²( edit distance, k
  errors,
• 
  rectangular patterns.
• O(n²kv(m log m) v(k log k)
• edit distance, k
  errors,
• 
  half rectangular patterns

AL-88
AF-95
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Types of Approximation

• Orientation.
• results O(n²m ) FU-98
• O(n²m³) ACL-98
• Scaling Natural scales
• results O(n) 1-d EV-88
• O(n² log S) 2-d ALV-92
• O(n²) dictionary
  AC-96
• Real scales
• this result O(n) 1-d,
  truncation

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It seems daunting, but
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CPM 2003 Morelia, Mexico
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Problem inherently inexact

• What if occurrence is 1½ times bigger?
• What is the meaning of ½ a pixel?
• Solutions until now Natural Scales -
• Consider only discrete scales
• 1, 2, 3, 4, 5, . . .

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Definition

• Text
  Pattern
• 
  
• 
  
• Find all occurrences of the pattern in the text
  in all discrete sizes.

n
m
m
n
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Discrete exact Scaled Matching

• T
  P
• A A A A A A A A A A A A A
  A A A
• A A A A A A A A A A C C A
  A C A
• A A A C C A A A A A C C A
  A A A
• A A A C C A A A A A A A A
• A A A A A A A A A A A A A
• A A A A A A A A A C C A A
• A A A A A A A A A C C A A
• A A A C C C A A A A A A A
• A A A C C C A A A A A A A
• A A A C C C A A A A C A A
• A A A A A A A A A A A A A
• A A A A A A A A A C C A C
• A A A A A A A A A A A A A

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Discrete exact Scaled Matching
P³
Z Z Z U U U Y Y Y Z Z
Z U U U Y Y Y Z Z Z U U U Y Y Y K
K K V V V S S S K K K V V V S S S
K K K V V V S S S X X X E E E T T T
X X X E E E T T T X X X E E E T T T
P Z U Y K V S
X E T
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Idea Fix a scale s
s

• Constant amount of work for each square (s-block)

s
n/s
n
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Algorithm time

• Time for scale s
• Total time
• 
  converges to a constant
• Making the total time O(n²)

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Problem Real scales

• Was open even for strings
• How do we define?
• aabcccbb
• Scaled to 2 aaaabbccccccbbbb
• Scaled to 1½ aaab cccc bbb
• truncate
  truncate
• ½b
  ½c

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Formally
r times
r
Denote a aaa . . .
a Problem Definition 1 Input Pattern
Text Output All text locations where
appears for some
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Remark

• a 1 means we only scale up
• Reasons Avoid conceptual problem of loss of
  resolution.
• From far enough away everything looks the same.
• By our definition, for klt1/m there is a match at
  every text location.

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Simplify definition
Definition 2 Look for in
the text. Example Paabcccbbbb Match
by definition 2 daaabccccbbbbbbe Match by
definition 1 but not by def 2
daaaabccccbbbbbbbe
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Why are definitions equivalent?

• Split text and pattern to
• symbol part Ts , Ps and
• length part TL , PL.
• Example P aabcccbbbb
• Psabcb
• PL2134
• Tdaaabccccbbbbbbe
• Tsdabcbe
• TL131461

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Time

• Time for split O(nm)
• Finding Ps in Ts O(nm) (e.g. KMP)
• HARD PART Finding PL in TL.

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Definitions are Equivalent
Claim Solving def 2 in time O(f(n))
Solving def 1 in time O(f(n)). Why? - Find
in time O(f(n)) - For each
match verify 1st and last symbol in
constant time in Ts and TL. Total time
O(f(n)n)O(f(n)).
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Naïve algorithm for matching PL in TL
For each text location, position pattern starting
at that location and calculate interval t/p,
(t1)/p) for each resulting lttext, patterngt
pair. This is the interval of possible scales
since t/p?p t for every a lt t/p, ap lt
t (t1)/p ?p t1 for every a t/p,
ap gt t
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Check intersection

• If intersection of all intervals is not empty
  then there is a match.
• Time O(nm)
• Example
• PL 2 1 2 3 2
• TL 2 4 2 4 7 4 5 3
• 1,3/2) 4,5)
• The intersection is empty thus no scaled match in
  location 1. But

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Check intersection

• If intersection of all intervals is not empty
  then there is a match.
• Time O(nm)
• Example
• PL 2 1 2 3 2
• TL 2 4 2 4 7 4 5 3
• 2,5/2) 2,3) 2,5/2)7/3,8/3)2,5
  /2)
• The intersection is 7/3,5/2) thus there is a
  scaled match in location 2.

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Improvement Parameterized Matching
Introduced Baker 1994. Motivation
copying code.
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Parameterized Matching

• Input two strings s and t st, over
  alphabets ?s and ?t.
• s parameterize matches t if bijection
  ?s ?t , such that (s) t.

Example
a
a
b
b
b
(a)x
x
x
y
y
y
(b)y
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Parameterized Matching

• Claim (AFM-94)
• For S that can be sorted in linear time (e.g.
  S1, . . . , n)
• Parameterized matching can be done in time O(n).

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The reduction
Lemma for which PL
matches TL at location i scaled to a only if PL
p-matches TL at i. Proof Assume PL does not
p-match TL at location i. The possible
situations are
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Possibility 1
w.l.o.g. c a1
TL
a
c?a
PL
b
b
For c a1 (smallest possible)
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Possibility 2
TL
a
a
w.l.o.g. c b1
PL
b
c?b
Intersection not empty only if
(a1)/(b1) gt a/b i.e.
abb gt aba
bgta But this can never happen if a
1.
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Algorithm for Real Scaled String Matching

• Let Pi1, Pi2, . . ., Pij be the different
  numbers in PL.
• P-match PL in TL.
• For each match, chack intersection of intervals
  between Pi1, . . . , Pij and corresponding
  symbols in TL.
• End Algorithm

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Example
PL 2 3 2 3 2
Pi12 Pi23 p-matches TL 5 6 5 6 5 6
10 6 10 6 10 7
scaled match
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Important Fact

• So there are at most O(vm) different Piks.
• Time O(n) for parameterized matching
• 
  (S1,2,,n).
• O(vm) verification for each
• location.
• Total O(nvm).

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Tighter analysis

• Upper bound number of possible p-matches.
• Lemma Let Pm, Tn, Pi1, Pi2, . . ., Pij
  be the different numbers in PL.
• Then there are at most n/2j p-matches of PL in
  TL.
• Meaning Since verification time is O(j) per
  p-match, the lemma implies that total
  verification time is
• O((n/2j) j) O(n)

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Proof of Lemma

• 1st appearance of Pi1, . . . , Pij
• PL Pi1
  Pi2 Pij
• TL a1 a2
  aj
• m-match

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Lemmas proof (cont.)

• Let x be the total number of p-matches in the
  text.
• The sum of all text elements that match 1st
  occurrences of Piks in the pattern
• (xj²)/2
• But There are overlaps!
• How many?

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Lemmas proof (cont.)

• For each text location, at most j matches will
  count it. Therefore
• Total count without overlaps
• Clearly xj/2 n thus
• x (2n)/j

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Open Problem

• Give 1-d algorithm linear in run-length
  compressed text and pattern.