SCALED PATTERN MATCHING

1
SCALED PATTERN MATCHING

• A.Amir Bar-Ilan Univ. Georgia Tech
• A.Butman Holon College
• M.Lewenstein Bar-Ilan Univ.
• E.Porat Bar-Ilan Univ.

2
SEARCHING FOR TEMPLATES IN AERIAL PHOTOGRAPHS
INPUT
TASK Search for all locations where the
template appears in the image.
3
Theoretically, need to consider

• Noise
• Occlusion
• Scaling (size)
• Rotation (orientation)

We are interested in asymptotically efficient
algorithms in pixel space.
4
MODEL

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

5
TYPES OF APPROXIMATIONS
Local Errors Level of detail Occlusion Nois
e
6
TYPES OF APPROXIMATIONS
Orientation
7
EVEN WITHOUT ERRORS AND ROTATIONS

• DIGITIZING NEWSPAPER STORIES
• IDEA Keep dictionary of fonts
• Search for appearances in all size.

8
PROBLEM

• INHERENTLY INEXACT
• What if appearance is 1½ times bigger ?
• What is ½ a pixel ?

SOLUTIONS UNTIL NOW NATURAL SCALES Consider
only discrete scales
9

• How does one model for
• real scales?

10
Step 1 Define grid pixel centers.
Example Unit pixel array for a 7?7 array.
11
Step 2 Define scaling. Example 3?3 array.
Scaled To 1?
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
Scaled To 1?
17
Remark

• We only scale up
• Reasons
• Avoid conceptual problems of loss of resolution.
• From far enough away everything looks the same.

18
Let P be a m?m pattern and T an n?n text.
How many different scaled patterns of P are
there?
19
In fact, can there be two different scaled
patterns of P of size k?k?
20
Example





55
21
4 x 1.1 4.4
Scaled by 1.1 to 6x6
22
4 x 1.125 4.5
Scaled by 1.125 to 6x6
23
3 x 1.17 3.51
Scaled by 1.17 to 6x6
24
2 x 1.25 2.5
Scaled by 1.25 to 6x6
25
Let P be a m?m pattern and T an n?n text.
How many different scaled patterns of P are
there?
26
Claim

• There are nm different scaled patterns
  representing all the occurrences of P.

27
Proof
Each one has at most m possible matrices
representing it
m?m, (m1)?(m1), , n?n
n-m different possible sizes
28
Proof
Each one has at most m possible matrices
representing it
Why?
29
Distance 1
30
Pattern P scaled to size kk,
31
(No Transcript)
32
Therefore

• There are nm different scaled patterns
  representing all the occurrences of P.

33
Algorithm outline for 2-D scaled matching
34
Straightforward Idea

• Construct dictionary of O(nm) possible scaled
  occurrences of P.
• Use 2-dimensional dictionary matching algorithm
  to scan the text in linear time and find all
  occurrences.

35
Space and Time Analysis
Each one has at most m possible matrices
representing it
m?m, (m1)?(m1), , n?n
Dictionary size O(n3m)
36
Solution

• Our idea is to keep the dictionary in compressed
  form.
• The compression we use is run-length of the rows.

37
Run-length
aabcccbb
a2b1c3b2
38
The compressed dictionary
C C B B B A A
C C B B B A A
F F E E E D D
F F E E E D D
F F E E E D D
I I H H H G G
I I H H H G G
C B A
F E D
I H G
Scaled To 2?
39
C C B B B A A
C C B B B A A
F F E E E D D
F F E E E D D
F F E E E D D
I I H H H G G
I I H H H G G
2 C2 B3 A2
3 F2 E3 D2
2 I2 H3 G2
Compressed form
40
2 C2 B3 A2
3 F2 E3 D2
2 I2 H3 G2
Size of Array mxm
of diff. scaled patterns (n-m) x m
Dictionary size O(nm3)
41
The Idea behind the text searching

• For every text location i,j, we assume that
  there is a pattern scaled occurrence beginning at
  that location.
• Subsequently, we establish the number of times
  this row repeats in the text.
• This allows us to an appropriately scaled pattern
  row from the dictionary.

42
Example for text searching
43

c150
b
a10
1
aaaaaaaaaa
44
Look in the text location 1,10
45
(No Transcript)
46

c150
b
a10
1
47

c150
b
a10
1
aaaaaaaaaab
48

c150
b
a10
1
Look in the text location 1,10
aaaaaaaaaabccccccccccccccc
49
(No Transcript)
50
A scale range of 1,1¼)
51
Last symbol may repeat in the text more time
than the scaled pattern need.
52
What about the number of times the first subrow
repeats?
53
The range of 1,1¼) is valid since
?102?1¼?128lt132.
54
Look in the text location 1,9
55
(No Transcript)
56
A scale range of 1¼,1¾)
57
Too large, it requires the c to repeat 175 times.
1¾
58
(No Transcript)
59
(No Transcript)
60
The maximum scale valid for both horizontal and
vertical scales produces the pattern whose first
row is a2bc129 and which repeats 132 times.
61
(No Transcript)
62
OPEN PROBLEM

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

63
END