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SCALED Pattern Matching

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Title: SCALED Pattern Matching

1
SCALEDPattern Matching

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

2
Motivation

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

3
(No Transcript)
4
Model

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

5
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
6
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

5
7
It seems daunting, but
8
CPM 2003 Morelia, Mexico
9
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, . . .

10
Definition

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

n
m
m
n
11
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

12
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
13
Idea Fix a scale s
s

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

s
n/s
n
14
Algorithm time

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

15
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

16
Formally
r times
r
Denote a aaa . . .
a Problem Definition 1 Input Pattern
Text Output All text locations where
appears for some
17
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.

18
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
19
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

20
Time

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

21
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)).
22
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
23
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

24
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.

25
Improvement Parameterized Matching
Introduced Baker 1994. Motivation
copying code.
26
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
27
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).

28
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
29
Possibility 1
w.l.o.g. c a1
TL
a
c?a
PL
b
b
For c a1 (smallest possible)
30
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.
31
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

32
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
33
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).

34
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)

35
Proof of Lemma