Transcript C05班 幾つかの画像関連問題の計算複雑度の解析と効率的な解決法の

Ｃ０５班
幾つかの画像関連問題の計算複雑度
の解析と効率的な解決法の提案
構成メンバー
浅野 哲夫，上原 隆平，元木光雄，(Arijit Bishnu)
(北陸先端科学技術大学院大学)
小保方 幸次（一関工業高等専門学校）
河村 泰之（愛媛大学教育学部）
研究目的・目標
研究対象
画像に関連する幾つかの問題について
１．本質的な計算複雑度を明らかにすること．
２．近似解法を含めて効率の良い現実的な解決法を提案す
ること．
具体的な問題
①画像の領域分割：画像に含まれる対象物を最適化の基準
に基づいて背景から切り出す問題
②一様な点配置生成：合理的な尺度の下に指定された長方
形領域内に指定個数の点を一様に配置する問題
③適応型メッシュ生成：変形量に応じたサイズのメッシュを生
成する問題
④高信頼度指紋同定：入力画像に一致する指紋をデータ
ベースから高い信頼性で効率的に探索する問題
目標
計算量の理論に基づいて計算困難度を数学的に解析し、そ
れに基づいて効率の良いアルゴリズムを開発すること．
ディジタル・ハーフトーニングに関する研究
• 入力の多値画像を２値画像に変換する方式
• 人間の錯覚を利用して，２値でも入力の多値画像と同じよう
に見えるように工夫する
• 最も単純な方式はマスクを用いるもの---どのようにマスクを
設計すれば，多様な画像に対応できるか --> 組合せ最適
化問題
• 入力画像に応じて最適な２値化を行う方式---基準を決めると
やはり組合せ最適化問題．解法としてはネットワークフロー
に基づく方法などがある．
• 近似解法の利用．平面の円による被覆問題と関連．ボロノイ
図と組み合わせた方式の提案
Simple Thresholding
Binarize each pixel by the threshold 0.5
for i=1 to N
for j=1 to N
if a(i,j) < 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
Ordered Dither
1 9 3 11
13 5 15 7
4 12 2 10
16 8 14 6
M(i,j)
Dither matrix
Using different thresholds for different places,
instead of using the same threshold over an image.
Tile the entire image plane by the matrix M
pixel (i, j) ⇔M(i mod 4, j mod 4)
if a(i,j) < M(i mod 4, j mod 4)/16 then b(i,j) = 0
else b(i,j) = 1
Ordered Dither: an output
Dither matrix
0
0 0
1 48
1
32
16
2
8
56
3
40
24
4
2
50
5
6
7
34 10 42
18 58 26
2 12
3 60
4 3
44
28
35
4
52
11
36
20
43
14
62
1
46
30
33
6
54
9
38
22
41
5 51
6 15
7 63
19
47
31
59
7
55
27
39
23
49
13
61
17
45
29
57
5
53
25
37
21
8x8 dither matrix
Dither matrix
0
48
32
16
8
56
40
24
2 34 10 42
50 18 58 26
12
60
3
44
28
35
4
52
11
36
20
43
14 46
6 38
62 30 54 22
1 33
9 41
51
15
63
19 59 27 49 17 57
47
7 39 13 45
5
31 55 23 61 29 53
Bayer’s dither matrix
threshold=16
25
37
21
threshold=24
Corresponding points should be uniformly distributed.
discrepancy measure
Evaluation of Bayer’s Dither Matrix
Recursive definition of the Bayer’s dither matrix
D0 = [0]
Dk=
4Dk-1
4Dk-1 + 2Uk-1
4Dk-1 + 3Uk-1 4Dk-1 + Uk-1
111….1
111….1
Uk=
……….
111….1
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 Bayer’s dither matrix of size
nxn is O(n2).
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)
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)
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 > 1.
• Characterization of zero-discrepancy matrices
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
Conventional Cluster-dot
Cluster Region
Pixel
Halftone Image
12 15
4
5
9
8
5
4
6
8
4
2
16
6
10
13
2
4
2
1
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
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
Effect of cluster size
Cluster- region
Low-tone scale resolution
->16 steps
High-spatial resolution
-> 300lpi
Cluster
Low-tone scale resolution
->64 steps
High-spatial resolution
-> 150lpi
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.
R=
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
circle of
radius 1.2
singly-covered area
Example:
a set of input discs
given by a matrix
a set of discs that
maximizes the total
singly-covered area
a set of discs
singly-covered area
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.
Experimental Results
Input images
small: 106 x 85, and large: 256 x 320
 enlarge them into 424 x 340 and 1024 x 1280

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
Output of Heuristic Algorithm 1(discs of original sizes)
Voronoi diagram for the set of circle centers
Fill out each Voronoi cell
according to the grey level
at the center point of the cell
by cubic interpolation
Output halftoned image
[1]S. 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
指紋認識・同定に関する研究
• 入力画像の２値化が最初の問題．グローバルな閾値は存在
しないが，２分探索により局所的に最適な閾値が決定可能
• 距離変換を用いると，指紋の幅をうまく推定することができる
（線形時間の距離変換を利用）
• 構造を維持しつつ雑音を除去する方法についても検討（幾何
形状を利用した効率よい算法の開発）
• 最も困難な問題は指圧による変形の扱い．柔らかい物体を
扱う計算幾何学の必要性．指圧に対して頑健な特徴抽出法
の開発（指圧の推定）
• 指圧を考慮したパターンマッチング法の開発
Fingerprint
• Fingerprint is a strong biologic information for
recognizing a people.
• Fingerprint applications.
Trademark, seal, personal identification.
• Fingerprint has been studied for a long time.
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).
Fingerprint representation
• Fingerprint patterns (global level)
• Fingerprint minutiae (local level)
– Ridge ending (termination)
– Ridge bifurcation
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)
Euclidean distance transform
00
0
0
0
1
1.414
1
2.236
1
00
0
1
1
1.414
1
2.236
1
1.414
1
00
0
1
2
1
2.236
1
1.414
1
1
00
1
1.414
1
2.236
1
1.414
1
1
0
11
1.414
1
2.236
1
1.414
1
1
0
0
12
2.236
1
1.414
1
1
0
0
0
2.236
1
12
1
0
0
0
0
A part of binary matrix
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 pixel’s distance
value.)
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
Binarization results
Moayer andFu’s
Ratha et al’s
Coetzee and Botha’
Denoising of fingerprint image
1. Impulsive noise.
(salt and pepper noise)
2. 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
Minutiae detection results
NIST
14
FCV
2002
Distortion correction
• Conventional methods
– Using bigger tolerance box;
– Using mass experimental parameters.
• Our method
– Providing a higher accuracy;
– Using much fewer parameters.
Distortion correction
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.
Distortion correction results
[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
点配置問題・ディスクレパンシ
[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.
トライセクター曲線
[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.