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

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 ThresholdingSimple Thresholding

Binarize each pixel by the threshold 0.5for 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 DitherOrdered 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 1016 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) < M(i mod 4, j mod 4)/16 then b(i,j) = 0 else b(i,j) = 1

Ordered Dither: an outputOrdered Dither: an output

Dither matrixDither matrix

63 31 55 23 61 29 53 21

15 47 7 39 13 45 5 37

51 19 59 27 49 17 57 25

3 35 11 43 1 33 9 41

60 28 52 20 62 30 54 22

12 44 4 36 14 46 6 38

48 16 56 24 50 18 58 26

0 32 8 40 2 34 10 420

1

2

3

4

5

7

6

0 1 2 3 4 5 6 7

8x8 dither matrix

Dither matrix

63 31 55 23 61 29 53 21

15 47 7 39 13 45 5 37

51 19 59 27 49 17 57 25

3 35 11 43 1 33 9 41

60 28 52 20 62 30 54 22

12 44 4 36 14 46 6 38

48 16 56 24 50 18 58 26

0 32 8 40 2 34 10 42

threshold=16

threshold=24

Corresponding points should be uniformly distributed. discrepancy measure

Bayer’s dither matrix

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

Uk=

111….1111….1……….111….1

Observation: For any integer k, the matrix Dk contains each integerfrom 0 through n2-1 exactly once, where n = 2k.

Observation: The discrepancy of a Bayer’s dither matrix of sizenxn is O(n2).

Generalized Semi-magic square

0 63 4 59 8 55 12 5131 32 27 36 23 40 19 44 1 62 5 58 9 54 13 5030 33 26 37 22 41 18 45 2 61 6 57 10 53 14 4929 34 25 38 21 42 17 46 3 60 7 57 11 52 15 4828 35 24 39 20 43 16 47

0 62 2 60 4 58 6 56 55 9 53 11 51 13 49 1516 46 18 44 20 42 22 4039 25 37 27 35 29 33 3132 30 34 28 36 26 38 2423 41 21 43 19 45 17 4748 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

A zero-discrepancy matrix of order (3, 9)

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

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-dotDot sizes are modulatedCalled AM halftoningOffset print, Laser printer(Xerography)

Disperse-dotDensity of dots is modulatedCalled FM halftoning or

stochastic halftoningInk-jet printer

16 6 10 13

12 1 4 7

8 3 2 11

14 9 5 15

Conventional Cluster-dot12 15 4 5

9 8 5 4

6 8 4 2

2 4 2 1

Input Image Data(Multi Levels)

Masking Tables

Halftone Image

Drawbacks of the conventional cluster-dotCluster-dots are arranged periodically

Artifacts due to Moire patternEach cluster region has a fixed size

hard to achieve good balance between spatial resolution and tone scale resolution

Pixel

Cluster Region

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

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

Key Idea of Adaptive Cluster-dotKey Idea of Adaptive Cluster-dot

Want to achieve good balance between tone scale resolution and spatial resolution

Effect of cluster size

Low-tone scale resolution ->16 steps

High-spatial resolution-> 300lpi

Cluster- regionCluster- region

Low-tone scale resolution ->64 steps

High-spatial resolution-> 150lpi

ClusterCluster

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.

Problem Specification

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

Example:

a set of input discsgiven by a matrix

a set of discs that maximizes the totalsingly-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 cellaccording to the grey levelat the center point of the cellby 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.

• 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).

Trademark, seal, personal identification.

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

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

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

FCV 2002NIST 14

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.