C05? ??????????????????????????????

1
C05???????????????????????????????

• ??????
• ?? ??,?? ??,????,(Arijit Bishnu)
• (?????????????)
• ??? ??(??????????)
• ?? ??(????????)

2
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• ?????????????????
• 1.??????????????????.
• 2.???????????????????????????.
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• ?????????????????????????????????????????????
• ??
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  ??.

3
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• ????????2??????????
• ??????????,2??????????????????????????
• ?????????????????---??????????????,????????????
  --gt ????????
• ???????????2???????---??????????????????.?????????
  ?????????????????.
• ???????.??????????????.?????????????????

4
Simple Thresholding
Binarize each pixel by the threshold 0.5
for i1 to N for j1 to N if a(i,j) lt
0.5 then b(i,j) 0
else b(i,j) 1
input image
output by simple threshold
Hard to deal with intermediate intensity levels
5
Ordered Dither
Using different thresholds for different places,
instead of using the same threshold over an image.
1 9 3 11 13 5 15 7 4 12 2 10 16 8 14 6
M(i,j) Dither matrix
Tile the entire image plane by the matrix M
pixel (i, j) ?M(i mod 4, j mod 4) if a(i,j) lt
M(i mod 4, j mod 4)/16 then b(i,j) 0
else
b(i,j) 1
6
Ordered Dither an output
7
Dither matrix
0
1
2
3
4
5
6
7
0 32 8 40 2 34 10 42
0

48 16 56 24 50 18 58 26
1
12 44 4 36 14 46 6 38
2
60 28 52 20 62 30 54 22
3
3 35 11 43 1 33 9 41
4
51 19 59 27 49 17 57 25
5
15 47 7 39 13 45 5 37
6
63 31 55 23 61 29 53 21
7
8x8 dither matrix
8
Dither matrix
0 32 8 40 2 34 10 42
48 16 56 24 50 18 58 26
12 44 4 36 14 46 6 38
60 28 52 20 62 30 54 22
3 35 11 43 1 33 9 41
51 19 59 27 49 17 57 25
15 47 7 39 13 45 5 37
63 31 55 23 61 29 53 21
Bayers dither matrix
Corresponding points should be uniformly
distributed. discrepancy measure
9
Evaluation of Bayers Dither Matrix
Recursive definition of the Bayers dither
matrix D0 0 Dk
111.1 111.1 . 111.1
4Dk-1 4Dk-1 2Uk-1 4Dk-1 3Uk-1
4Dk-1 Uk-1
Uk
Observation For any integer k, the matrix Dk
contains each integer from 0 through n2-1 exactly
once, where n 2k.
Observation The discrepancy of a Bayers dither
matrix of size nxn is O(n2).
10
Generalized Semi-magic square
0 63 4 59 8 55 12 51 31 32 27 36 23 40 19
44 1 62 5 58 9 54 13 50 30 33 26 37 22 41
18 45 2 61 6 57 10 53 14 49 29 34 25 38 21
42 17 46 3 60 7 57 11 52 15 48 28 35 24 39
20 43 16 47
0 62 2 60 4 58 6 56 55 9 53 11 51 13
49 15 16 46 18 44 20 42 22 40 39 25 37 27 35 29
33 31 32 30 34 28 36 26 38 24 23 41 21 43 19 45
17 47 48 14 50 12 52 10 54 8 7 57 5 59 3
61 1 63
Two zero-discrepancy matrices of order (2, 8)
11
Generalized Semi-magic square
7 9 23 61 63 77 34 36 50
29 40 51 2 13 24 56 67 78
57 71 73 30 44 46 3 17 19
25 0 14 79 54 68 52 27 41
47 31 42 20 4 15 74 58 69
75 62 64 48 35 37 21 8 10
16 18 5 70 72 59 43 45 32
38 49 33 11 22 6 65 76 60
66 80 55 39 53 28 12 26 1
A zero-discrepancy matrix of order (3, 9)
12
Results Obtained

• Let N(k, n) be the set of all zero discrepancy
  matrices of order (k, n)
• N(k, n) is not empty if k and n are both even.
• N(k, n) is empty if k and n are relatively prime.
• N(k, km) is not empty for any k, m gt 1.
• Characterization of zero-discrepancy matrices

13
Two kinds of halftoning

• Cluster-dot
• Dot sizes are modulated
• Called AM halftoning
• Offset print,
• Laser printer(Xerography)

• Disperse-dot
• Density of dots is modulated
• Called FM halftoning or stochastic halftoning
• Ink-jet printer

14
Conventional Cluster-dot
Cluster Region
Pixel
Halftone Image
12 15 4 5
9 8 5 4
6 8 4 2
2 4 2 1
















16 6 10 13
12 1 4 7
8 3 2 11
14 9 5 15
Input Image Data (Multi Levels)
Masking Tables

• Drawbacks of the conventional cluster-dot
• Cluster-dots are arranged periodically
• Artifacts due to Moire pattern
• Each cluster region has a fixed size
• hard to achieve good balance between spatial
  resolution and tone scale resolution

15
Key Idea of Adaptive Cluster-dot

• The Portion of rapid tone change
• Spatial resolution is important
• ? Small Cluster-Regions

• The Portion of smooth tone change
• Tone scale resolution is important
• ? Large Cluster-Regions

Want to achieve good balance between tone scale
resolution and spatial resolution
16
Effect of cluster size
Cluster
Cluster- region
































Low-tone scale resolution -gt16 steps
High-spatial resolution -gt 300lpi
Low-tone scale resolution -gt64 steps
High-spatial resolution -gt 150lpi
17
Problem Specification
Problem R r11, r12, ... , rnn a matrix of
n2 positive real numbers. Each rij is a radius
of a disc at (i,j). Choose disks so as to
maximize the total singly-covered area.
2.4 3.3 3.6 4.1 2.5
2.5 3.5 3.8 3.3 1.9
2.6 3.7 3.2 2.5 1.5
2.2 3.3 2.2 1.2 1.0
1.9 1.7 3.5 3.6 4.2
R
circle of radius 1.2
singly-covered area
18
Example
a set of input discs given by a matrix
a set of discs that maximizes the
total singly-covered area
19
a set of discs
singly-covered area
20
Approximation algorithm with guaranteed
performance
Cu a disc with center at u r(Cu) radius
of the disc Cu
Cu
u
Algorithm 1 Sort all the discs in the
decreasing order of their radii. for each disc
Cu in the order do if Cu does not intersect
any previously accepted disc then accept
it else reject it Output all the accepted discs.
21
Experimental Results
Input images small 106 x 85, and large 256 x
320 ? enlarge them into 424 x 340 and 1024 x
1280
?
22
Running time Heuristic 1 0.06 sec. for the
small image 0.718 sec. for
the large image on PC
DELL Precision 350 with Pentium 4. Heuristic
2 0.109 sec. for the small image
1.031 sec. for the large image
23
Output of Heuristic Algorithm 1(discs of original
sizes)
24
Voronoi diagram for the set of circle centers
25




















Fill out each Voronoi cell according to the grey
level at the center point of the cell by cubic
interpolation
26
Output halftoned image
27
1S. Sasahara, T. Asano, " A new halftone
technique to eliminate ambiguous pixels for
stable printing", Proc Electronic Imaging
Science and Technology, Color Imaging IX
Processing, Hardcopy, and Applications,
pp.490-497. January 2004. 2 T. Asano, P. Brass,
S. Sasahara "Disc Covering Problem with
Application to Digital Halftoning", Proc. of the
Workshop on Computational Geometry and
Applications (CGA 04) LNCS 3045, part III, pp.
11-21, 3 T. Asano, Naoki Katoh, Hisao Tamaki,
and Takeshi Tokuyama "On Geometric Structure of
Global Roundings for Graphs and Range Spaces ",
Proc. of the Scandinavian Workshop on Algorithm
Theory (SWAT 04) , Denmark, 2004. 4 B.
Aronov, T. Asano, Y. Kikuchi, S. C. Nandy, S.
Sasahara, and T. Uno "A Generalization of Magic
Squares with Applications to Digital Halftoning,"
Proc. ISAAC 2004, Hong Kong, 2004. 5 T. Asano
"Computational Geometric and Combinatorial
Approaches to Digital Halftoning," Prenary Talk
at International Conference on Computational
Science and Its Applications, Singapore, May,
2005. 6 T.Asano, S. Choe, S. Hashima, Y.
Kikuchi, and S.-C. Sung "Distributing Distinct
Integers Uniformly over a Square Matrix with
Application to Digital Halftoning," Invited Talk
at 7th Hellenic European Conference on Computer
Mathematics and its Applications, 2005, Athens,
Greece. 7 T.Asano "Computational Geometric and
Combinatorial Approaches to Digital Halftoning,"
Computing The Australasian Theory Symposium, 2006
28
????????????

• ?????2????????.???????????????,2??????????????????
  ??
• ?????????,??????????????????(????????????)
• ????????????????????????(??????????????????)
• ??????????????????.??????????????????.????????????
  ?????(?????)
• ????????????????????

29
Fingerprint

• Fingerprint is a strong biologic information for
  recognizing a people.
• Fingerprint applications.
• Fingerprint has been studied for a long time.

Trademark, seal, personal identification.

1. First scientific paper (Nehemiah Grew, 1684)
2. Accepted as evidence by law enforcement
   departments (the Home Ministry, UK, 1893)
3. Starting investigation of Automatic Fingerprint
   Identification System (FBI, Home Office in UK,
   Paris Police Department, from 1960s).

30
Fingerprint representation

• Fingerprint patterns (global level)
• Fingerprint minutiae (local level)
• Ridge ending (termination)
• Ridge bifurcation

31
Binarization of fingerprint image

• OBSERVATION The fingerprint images have almost
  equal width ridges and valleys.
• A combinatorial algorithm for binarization of
  fingerprint images where optimal threshold is
  based on equal widths of ridges and valleys.
• PROBLEM Measuring the width of arbitrary shapes
  is a non-trivial task.
• Euclidean distance
  transform (EDT)

32
Euclidean distance transform
Euclidean distance transform of a binary image is
an assignment to each non-zero pixel the
Euclidean distance between it and the closest
zero pixel. (It is same to compute the zero
pixels distance value.)
0 0 0 0 1 1.414 2.236
0 0 1 1 1.414 2.236 1.414
0 0 1 2 2.236 1.414 1
0 1 1.414 2.236 1.414 1 0
1 1.414 2.236 1.414 1 0 0
2 2.236 1.414 1 0 0 0
2.236 2 1 0 0 0 0
0 0 0 0 1 1 1
0 0 1 1 1 1 1
0 0 1 1 1 1 1
0 1 1 1 1 1 0
1 1 1 1 1 0 0
1 1 1 1 0 0 0
1 1 1 0 0 0 0
A part of binary matrix

Hirata,T.,and Katoh, T., An Algorithm for
Euclidean distance transformation, SIGAL
Technical Report of IPS of Japan, 94-AL-41-4,
pp.25-31, 1994
33
Binarization results
Ratha et als
Coetzee and Botha
Moayer andFus
34
Denoising of fingerprint image

• Impulsive noise.
• (salt and pepper noise)
• Useless components.
• (Useless component is an object disjoint from
  other objects and whose largest width is less
  than the mean width of fingerprint ridges.)
• Mathematical Morphology

35
Minutiae detection results
36
Distortion correction

• Conventional methods
• Using bigger tolerance box
• Using mass experimental parameters.
• Our method
• Providing a higher accuracy
• Using much fewer parameters.

37
Distortion correction
38
A combined RBF model

• Rigid region This is the closest contact region,
  in which skin slippage normally does not exist.
  In our method, the radii of region I is 1/3 of
  radius of whole fingerprint region.
• Non-rigid region The main elastic distortion is
  located in this region.

39
Distortion correction results
40
1 Xuefeng Liang and T. Asano "A Fast Denoising
Method for Binary Fingerprint Image ", Proc.
IASTED Conference on Visualization, Imaging, and
Image Processing, Paper No. 452-168, pp.
309-313, Marbella, Spain, September, 2004. 2
X. Liang, A. Bishnu, and T. Asano "A Near-Linear
Time Algorithm for Binarization of Fingerprint
Images Using Distance Transform," Proc. 10th
International Workshop, IWCIA 2004, pp.197-208,
2004. 3 X. Liang, K. Kotani and T. Asano
"Automatically Choosing Appropriately-Sized
Structuring Elements to Eliminate Useless
Components in Fingerprint Image" Proc. Visual
Communications and Image Processing 2005, pp.
284-293, 2005. 4 X. Liang, A. Bishnu, and T.
Asano, Distorted Fingerprint Indexing Using
Minutia Detail and Delaunay Triangle, Proc.
International Symposium on Voronoi Diagram in
Science and Engineering, July, Banff, Canada,
pp.8-17, 2006. 5 X. Liang, T. Asano, and H.
Zhang, A Combined Radial Basis Function Model
for Fingerprint Distortion, Proc. ICIAR
Intel. Cnf. on Image Analysis and Recognition,
Portugal, September, pp.286-296, 2006. 6 X.
Liang and, A. Bishnu, and T. Asano, Fingerprint
Matching Using Minutia Polygons, Proc. ICPR
18th Intl. Conf. on Pattern Recognition, Hong
Kong, pp.1046-1049, 2006
41
????????????? 1 S. Teramoto, T. Asano, B.
Doerr, and N. Katoh "Inserting Points Uniformly
at Every Instance," Proc. 2005 Korea Japan Joint
Workshop on Algorithms and Computation, pp.3-9,
2005, Seoul, Korea. ????????? 2 T. Asano, J.
Matousek, and T. Tokuyama Zone Diagram
Existence, Uniqueness and Algorithmic
Challenge, SIAM-ACM Symposium on Discrete
Algorithms, 2007. 3 T. Asano, J. Matousek, and
T. Tokuyama The Distance Trisector Curve, ACM
Symposium on Computing Theory, Seatle, USA,
pp.336-343, May, 2006.