Disc Covering Problem with Application to Digital Halftoning

Disc Covering Problem with Application to Digital Halftoning

Tetsuo AsanoSchool of Information Science,

JAIST Japan Advanced Institute of Science and Technology

Joint Work withShinji Sasahara, Fuji Xerox Co.Ltd.

　　　　　　　 Peter Brass, CUNY, U.S.A.

Tetsuo AsanoSchool of Information Science,

JAIST Japan Advanced Institute of Science and Technology

Joint Work withShinji Sasahara, Fuji Xerox Co.Ltd.

　　　　　　　 Peter Brass, CUNY, U.S.A.

Dagstuhl 　 Workshop, 2004

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

How hard is it? NP-hard?

Approximation algorithm with guaranteed ratio

One-dimensional problem still hard? or an efficient algorithm?

Motivation application to digital halftoning: conversion from continuous-tone images to binary-tone images

Algorithmic questions and Motivation:

NP-hard or polynomial-time algorithm? openOne-dimensional problem still hard? or an efficient algorithm? polynomial-time algorithms 1. graph-based approach 2. plane-sweep and dynamic programming

How hard is it? NP-hard?

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.

input set of discs accepted discs by Algorithm 1

Lemma 5: Algorithm 1 finds a 9-approximate solution.

Proof:

S: a set of all input discsS’: a solution (set of discs) obtained by Algorithm 1.all the discs in S’ are disjoint.

(1) r(Cv) r(C≧ u) : larger discs are examined firstIf we enlarge Cv by a factor 3, the all the discs rejected by Cv are completely contained in the enlarged disc.

Cu: a disc which has not been accepted by Algorithm 1there must be some disc Cv such that (1) Cv has been examined before Cu, and (2) Cv intersects Cu.

Cu

Cv

Approximation algorithm with better performance

some terminologies and notations: Cu: a disc with center at u r(Cu): radius of the disc Cu

R(Cu): a disc contracted by , 0<<1, which is called the core of the disc Cu.

In this caseCv violates Cu

but Cu does notviolates Cv

Cu

R(Cu)

R(Cv)

Cv

Cv violates Cu

if Cv intersects the core R(Cu) of Cu.

Otherwise, two discs safely intersect.

Algorithm 2:・ Sort all the discs in the decreasing order of their radii.・ for each disc Cu in the order do・ if Cu is not violated by any previously accepted disc (i.e., Cu does not intersect the core of any previously accepted disc)

・ then accept it else reject it・ Output all the accepted discs.

Cu

R(Cu)

R(Cv)

Cvin this caseCu can be accepted

input set of discs accepted discs

Lemma 6: Algorithm 2 finds a 5.83-approximate solution.

Proof:

Cu: a disc which has not been accepted by Algorithm 2there must be some disc Cv such that (1) Cv has been examined before Cu, and (2) Cv violates Cu i.e., Cv intersects the core of Cu.

Cu

R(Cu)

R(Cv)

Cv

Enlarge each accepted disc by a factor 2 +

Cu

Enlarge each accepted disc by a factor 2 +

Cv

R(Cv)

R(Cu)

)()2(

)(2)(

)(2))((

v

vv

vv

Cr

CrCr

CrCRrr

Every disc rejected by Cv is completely contained inthe enlarged disc.

How much area is covered exactly once by accepted discs?

How much area is covered exactly once by accepted discs?

a set of accepted discs

additively weighted Voronoi diagram of the discs

truncate each Voronoi cell of an accepted disc by its blown-up copy

evaluate how much area in each cell is covered by the disc ratio > 1 / 5.83.

accepted discs additively weightedVoronoi diagram of the discs

blow up a disc by 2+ truncate each cell by the copy

partition into angular sectors

Voronoi edge outside disc

Voronoi edge inside disc

In this angular sector,at least 1/(2+)2 is occupiedby the disc.

2+

1

The core is not overlappedby any other disc. So, the ratiois at least 2.

Voronoi edge outside disc

Voronoi edge inside disc

In this angular sector,at least 1/(2+)2 is occupiedby the disc.

The core is not overlappedby any other disc. So, the ratiois at least 2.

We chooseso that 2 ＝ 1 / (2+)2.

Then, the ratio of singly-covered area is at least 1/(2+)2.approximation ratio is (2+)2=5.83.

NP-hardness proof (uncompleted)

reduction from planar 3SAT

variable part

we have to chooseeither all of red circlesor all of blue circles.

NP-hardness proof(continued)

path part

NP-hardness proof(continued)

clause part

Open Problems

• Improve the approximation ratio

approximation ratio 2 or (1+)? O(n log n) time and O(n) space for practic

al applications

• Complete NP-hardness proof

Concluding Remarks

considered a geometric optimization problem to choose discs among n given discs to maximize the singly-covered area.Application: adaptive digital halftoning. Proposed some heuristic algorithms and verified their effectiveness by experiments.

Heuristic Algorithm 1

(1) Fix a contraction factor appropriately. (2) D = a set of all discs in the raster order(3) for each disc C in D (in raster order) accept C if core(C) does not intersect the core of any previously accepted disc and reject it otherwise.

Cu

R(Cu)

R(Cv)

Cv

In this algorithm, the first disc in the raster order is alwaysaccepted, which may be a bad choice for future selection.

Heuristic Algorithm 2

(1) Fix a contraction factor appropriately. (2) D = a set of all discs in the decreasing order of their radii(3) for each disc C in D (large discs first) accept C if core(C) does not intersect the core of any previously accepted disc and reject it otherwise.

Cu

R(Cu)

R(Cv)

Cv

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(contracted discs)

Output of Heuristic Algorithm 1(discs of original sizes)

Voronoi diagram for the set of circle centers

Output of Heuristic Algorithm 2(contracted discs)

Output of Heuristic Algorithm 2(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

enlarged picture of a part