research article a line-based adaptive-weight...

Research ArticleA Line-Based Adaptive-Weight Matching Algorithm UsingLoopy Belief Propagation

Hui Li1 Xiao-Guang Zhang1 and Zheng Sun2

1School of Mechanical and Electrical Engineering China University of Mining amp Technology Xuzhou 221116 China2College of Mechanical and Electrical Engineering Zao Zhuang University Zaozhuang 277160 China

Correspondence should be addressed to Xiao-Guang Zhang doctorzxghotmailcom

Received 22 July 2014 Accepted 22 March 2015

Academic Editor Hakim Naceur

Copyright copy 2015 Hui Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In traditional adaptive-weight stereo matching the rectangular shaped support region requires excess memory consumption andtime We propose a novel line-based stereo matching algorithm for obtaining a more accurate disparity map with low computationcomplexity This algorithm can be divided into two steps disparity map initialization and disparity map refinement In theinitialization step a new adaptive-weight model based on the linear support region is put forward for cost aggregation In thismodel the neural network is used to evaluate the spatial proximity and the mean-shift segmentation method is used to improvethe accuracy of color similarity the Birchfield pixel dissimilarity function and the census transform are adopted to establish thedissimilarity measurement function Then the initial disparity map is obtained by loopy belief propagation In the refinementstep the disparity map is optimized by iterative left-right consistency checking method and segmentation voting method Theparameter values involved in this algorithm are determined with many simulation experiments to further improve the matchingeffect Simulation results indicate that this new matching method performs well on standard stereo benchmarks and running timeof our algorithm is remarkably lower than that of algorithm with rectangle-shaped support region

1 Introduction

Stereo vision is a fundamental technique for extracting 3Dinformation of a scene from two or more 2D images Itis widely applied in robot navigation remote sensing andindustrial automation One of the key technologies of stereovision is stereo matching which produces a disparity mapThe stereo matching algorithm can be classified into twobroad categories global-based and local-based algorithms

Global-based matching algorithms follow the energyminimization principle First an energy function is establi-shed consisting of a data term and a smoothness term Nextthis function is minimized with a global optimizationmethodDynamic programming [1] loopy belief propagation(LBP) [2 3] and graph cut [4 5] are usually employed toidentify the minimum energy required for a global-basedalgorithm Comprehensive global constraint information canproduce a more accurate disparity map in a global-basedalgorithm

A local-based matching algorithm is a simple and effec-tive method for stereo matching that is commonly usedAn important underlying principle of local-based matchingis that pixels in a support region have an approximatelyequal disparity To satisfy this principle it is very importantto determine the support region size The support regionmust contain enough pixels for intensity variation and thesupport region must include only those pixels with thesame disparity Thus the traditional local-based matchingmethod is prone to false matching for pixels from the depthdiscontinuities region since those pixels are from differentdepths To ensure that a local-based matching algorithm per-forms well in practical applications various approaches havebeen proposed For example adaptive windows have beenused to improve matching results These methods search anappropriate support region for each pixel greatly improve theperformance of matching results and outperform standardlocal-based methods [6ndash10] However it is difficult to searcha support regionwith an arbitrary shape and size andmost of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 297392 13 pageshttpdxdoiorg1011552015297392

2 Mathematical Problems in Engineering

Left image

Right image

Computingcost with

newadaptive-

weightmethod

Matching costof left image

Matching costof right image

Loop beliefpropagation

Initial disparitymap of left image

Initial disparitymap of right image

Iterative left-right

consistencychecking

Segmentationvoting

Finaldisparity

map

Disparity map initialization Disparity map refinement

Figure 1 Block diagram of our algorithm

these methods have a high computational complexity Otherresearchers assign different support-weights to the pixels in asupport region keeping the size and shape of a support regionconstant [11ndash13]

In recent years several methods for acquiring satisfactoryeffect of stereo matching have been adopted Yang et al[17] presented a stereo matching algorithm which inte-grates color-weight belief propagation left-right checkingcolor segmentation plane fitting and depth enhancementMei et al [18] integrated the AD-census cost measurementfunction the cost aggregation method based cross-basedregion the scanline optimize method the multistep refine-mentmethod and the accelerative algorithmbased onCUDAinto their algorithm

The algorithm presented in this paper is inspired fromadaptive-weight matching algorithm In this paper the aim isto propose a low computation complexity and high accuracystereo matching algorithm So the rectangle-shaped supportregion is substituted for the line-shaped support regionLacking of enough pixel information is a main weakness ofthe line-shaped support region which is easy to cause errormatching Adaptive-weight can make full of limited pixelinformation by analyzing the characteristic of the adaptive-weight model proposed in [13] on disparity accuracy weuse neural network (NN) to determine the spatial proximityand mean-shift based segmentation method to effectivelydescribe the color similarity

In addition several approaches are applied to completethe algorithmWe develop a new pixel dissimilarity measure-ment function which combines Birchfield pixel dissimilaritymeasurement function and census transform to computethe matching cost The loopy belief propagation methodproposed in [2] is employed to estimate the initial disparitymap which is optimized with min convolution and imagepyramid There are several measurable improvements for theinitial disparity map To further improve the accuracy of theinitial disparity map iterative left-right consistency (LRC)checking and segmentation voting are used to refine thedisparity map by analyzing the features of the initial disparitymap

2 Algorithm Description

The algorithm presented in this paper can be divided intotwo steps a disparity map initialization step and a refine-ment step The framework of the algorithm is shown in

p

Figure 2 The assumption of segmentation-based stereo matching

x

wx

120573

120573

minus15

minus1

1

1

1

f

fIT

OMn

fc(Δgpq)

IMn

Figure 3 Spatial proximity described by neural network

Figure 1 A detailed description of this algorithm is givenin the following sections

21 Adaptive-Weight BasedCost AggregationMethod Assum-ing the two pixels119901 and 119902 the disparity of center pixel119901wantsto be computed 119878(119901) is the support region of pixel 119901 while 119902is a neighboring pixel of119901 in the support regionThe support-weight of 119902 is assigned by the following according to [12]

119908 (119901 119902) = 119891119888(Δ119888119901119902) 119891119888(Δ119892119901119902) (1)

where 119891119888(Δ119892119901119902) represents the spatial proximity 119891

119888(Δ119888119901119902)

represents the color similarity and 119908(119901 119902) is the support-weight Our algorithm is designed on the basis of thisframework The list of variables used in this paper is shownin the end of the paper

211 The Model of Line-Based Adaptive-Weight Wang et al[13] noted that when the support region is large enough colorsimilarity plays a major role in computing the center pixeldisparity within a certain range As shown in Figure 2 redrepresents the support region 119878(119901) We used the pixels in 119878(119901)to compute the disparity of 119901 The effects of spatial proximitycan be neglected in the pale blue region according to [13] Wecall this region the transition area represented by 119879(119901) Tosatisfy this principle the neural network can be applied in thedesign of this spatial proximity model

Figure 3 shows the spatial proximitymodel established byneural networkThe position of a pixel is the input the spatial

Mathematical Problems in Engineering 3

minus15 minus12 minus9 minus6 minus3 0 3 6 9 12 15035

04

045

05

055

06

x

fc(Δgpq)

Figure 4 Spatial proximity proposed in this paper when 120572 = 1 120573 = 8 and 119871119904= 15

proximity is the output and the connect weights are shown asin the figure In fact the distance 119909minus119909

119901is the input of neural

network To simplify the notations suppose that the centerpoint 119901 is at 119909

119901= 0 the distance 119909minus119909

119901can be simplified into

119909 which represents the position of a pixel The concrete formof the spatial proximity is expressed by

1198681198721= 119909 + 120573

1198681198722= minus119909 + 120573

119874119872119894= 119891 (119868

119872119894)

119868119879=

2

sum

119894=1

119874119872119894minus 15

119891119888(Δ119892119901119902) = 119891 (119868

119879)

(2)

where 119891(119909) = 1(1 + 119890minus120572119909) is the sigmoid function Figure 4demonstrates the varied trend of the spatial proximityaccording to the position of a pixel

In Figure 4 the space between the two blue lines is thetransition area and the support region is represented by thewhole 119909-axis It can be seen from Figure 4 that (1) the spatialproximity of pixel in the transition area is significantly greaterthan that of pixel in other area which accords with the spatialproximity model of traditional adaptive-weight theory (2)there is not much difference between these pixels in thetransition area for the spatial proximity whichmeans that theinfluence of spatial proximity can be neglected

According to the segmentation-based stereo matchingprinciple [14 19] a new model of color similarity is estab-lished by the following in [20]

119891119888(Δ119888119901119902) =

1 if seg (119902) = seg (119901)

exp(minusΔ2

119888119901119902

120574119888

) otherwise(3)

Equation (3) shows that color similarity contributesenormously tomeasure the dissimilarity between center pixel

119901 and its neighbor pixel 119902 when 119901 and 119902 belong to the samesegmentation

That color similarity model based on image segmentationcan achieve good performance as introduced in [13] Mean-shift is a nonparametric estimation iterative technique andits application domains include computer vision clusteringand image processing [21] In this work we use mean-shift asthe segmentation method

212 Cost Aggregation The matching cost of pixel 119901 withdisparity 119889 is represented by

119864 (119901 119889) =

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889) 119890 (119902 119902

119889)

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889)

(4)

where 119901119889and 119902

119889are the corresponding pixels of 119901 and 119902

respectively when the disparity of the center pixel is 119889The pixel dissimilarity measurement function 119890(119902 119902

119889)

in (4) is very important for cost aggregation The absolutedifference and Birchfield function [22] are widely used incost aggregation To improve the matching accuracy of thetextureless and repetitive regions the pixel dissimilaritymeasurement function is described by combing Birchfieldfunction and census transform

119890 (119902 119902119889) = min (119862Birch (119902 119902119889 119868119877 119868119871) 119879Birch)

+min (119862Census (119902 119902119889) 119879Census) (5)

To validate the effect of this cost aggregation methodsimulation results on Teddy and Cones with Birchfieldmethod and our method are shown in Figure 5

22 Initial Disparity Determination The winner-take-all(WTA) searching strategy is a commonmethod for determin-ing disparity which can be expressed by

119863(119901) = argmin119889

119864 (119901 119889) (6)


(a) (b) (c)

Figure 5 (a) Part of the origin image (b) Disparity results which are computed with our cost aggregationmethod (c) Disparity results whichare computed with Birchfield pixel dissimilarity measurement function

(1) Computing matching cost 119864 by (4)Initializing pyramid level 119897 and down sampling factor 120578 = 05

(2) Pyramid initializationPyramid(1)cost = 119864

(3) Establishing pyramid structure with 119897 and 120578(4) For each level 119894(5) For each iteration 119895(6) Updating left right up and down direction message for each pixel(7) End(8) End(9) Determining disparity for each pixel by max-product principle

Algorithm 1 The procedure of initial determination procedure

WTA tends to produce a low accuracy disparity mapTherefore we adopt an efficient LBP algorithm proposed in[2] in this paper In this LBP algorithm FFT convolution andimage pyramid are integrated into LBP which can effectivelydecrease the complexity of LBP and increase the matchingeffect The flowchart of the initial determination procedureis shown in Algorithm 1

23 Disparity Refinement It is inevitable that initial disparitymaps will contain many error-matched pixels To refinethe disparity map a two-step postprocessing method is putforward in this section

231 Left-Right Consistency Check The disparity map119863119871for

the left image is computed by previous steps The disparitymap119863

119877for the right image is computed in a similar manner

In [15 17] pixels are classified into several types according to119863119871 119863119877 and 119864

119877to remove outliers Then different strategies

are designed for different type of pixels to determine itsdisparity

In this work we analyzed the property of initial disparitymap Figure 6 shows the distribution of bad pixels afterexecuting the disparity initialization step of our algorithmfor Tsukuba It can be seen from this figure that most of thebad pixels concentrate in the occluded region According tothis result most of the pixels match correctly and the initial


(1) Establishing the table of pixel type 119878119905119894according to (7)-(8)

(2) 119878119905larr 119878119905119894

(3) For each iteration 119896(4) For each undependable pixel 119901

119894which is determined by 119878

119905

(5) Searching the nearest dependable pixel 119901119871119894and 119901119877

119894

(6) 119889(119901119894) larr min(119889

119900(119901119871

119894) 119889119900(119901119877

119894)) where 119889

119900(sdot) represents initial disparity of pixel

(7) If 119889(119901119894) = 119889119900(119901119894)

(8) Set 119901119894as dependable pixel in 119878

119905

(9) End(10) End(11) End(12) End

Algorithm 2 The disparity map after execution of left-right consistency checking

Table 1 The error percentages in different region for disparityrefinement step

Nonocc All DiscInitial disparity map 121 320 642Left-right consistency 117 240 630Segmentation voting 095 162 507

Figure 6 The distribution of bad pixels for initial disparity map ofTsukuba Bad pixels are marked with red points

disparity in the nonocc region should be trustedTherefore aniterative left-right consistency check is proposed for handlingthis

Pixels can be divided into two types undependable pixelsand dependable pixels Pixel 119901 is classified as dependablewhen it meets the following condition

119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (7)

Pixel 119901 is considered to be undependable if it fits thefollowing condition

119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (8)

Thenewdisparity of the undependable pixel can be computedas in Algorithm 2

Figure 7 shows the result after the execution of iterativeleft-right consistency checking Table 1 shows the detaileddata for the role of iterative left-right consistency checking

Figure 7The disparity map after execution of left-right consistencychecking

p

Figure 8 Pixelwise region for segmentation voting

232 Segmentation Voting The pixelwise region shown inFigure 8 is established according to color consistency Thisregion can be represented by 119880(119901) Pixels in 119880(119901) satisfy thefollowing condition

10038161003816100381610038161003816119868119902minus 119868119901

10038161003816100381610038161003816lt 120591 119902 isin 119880 (119901)

10038161003816100381610038161003816119909119902minus 119909119901

10038161003816100381610038161003816lt 119871 sv

10038161003816100381610038161003816119910119902minus 119910119901

10038161003816100381610038161003816lt 119871 sv

(9)

Let119867(119889) represent the frequency distribution of disparity119889 in 119880(119901) The new disparity is updated by Algorithm 3


Figure 9 The disparity map of Tsukuba obtained by segmentation voting

0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)

Figure 10 (a) Performance evaluation of the proposed method when varying 119871119904(Tsukuba and Venus) (b) Performance evaluation of the

proposed method when varying 119871119904(Teddy and Cones)

(1)119867(119889) larr 0

(2) For each 119902 isin 119880(119901)(3) For each 119889 isin [119889min 119889max]

(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)

Algorithm 3 Disparity determination of segmentation voting

The 3 times 3 median filter is applied to the left disparitymap Figure 9 shows the effect of segmentation voting on theaccuracy of the disparity map

3 Results and Discussion

31 Parameters Determining The parameters involved in ouralgorithm greatly affect the performance of the algorithm Inthis section we present the parameter settings

We considered eight main parameters 119871119904 120572 120573 120574

119888 ℎ119878 ℎ119868

119871 sv and 120591 which are kept constant for all benchmarksFigure 10 shows the influence of 119871

119904on the accuracy of

disparity map obtained by our algorithm When 119871119904varies

from 35 to 65 our algorithm is insensitive to 119871119904

Figure 11 demonstrates that the performance of our algo-rithm varies with 120572 and 120573 From these figures it can benoted that (1) generally speaking when 120573 is larger than 15the influence of 120572 over algorithm performance is small forTsukuba Venus and Teddy (2) Our algorithm shows goodperformance for Tsukuba and Teddy when 120573 isin [0 45] and120572 isin [03 1] (3)The error percentage tends to decrease with adecreasing120572when120573 is smaller than 15 (4) In regard toCones


0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)

Figure 11 (a) Performance evaluation of the proposedmethod according to 120572 and 120573 for Tsukuba (b) Performance evaluation of the proposedmethod according to 120572 and 120573 for Venus (c) Performance evaluation of the proposedmethod according to 120572 and 120573 for Teddy (d) Performanceevaluation of the proposed method according to 120572 and 120573 for Cones

0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)

Figure 13 (a) Performance evaluation of the proposed method according to ℎ119878and ℎ

119868for Tsukuba (b) Performance evaluation of the

proposed method according to ℎ119878and ℎ

119868for Venus (c) Performance evaluation of the proposed method according to ℎ

119878and ℎ

119868for Teddy

(d) Performance evaluation of the proposed method according to ℎ119878and ℎ

119868for Cones

the performances of this algorithm improve as 120572 increasesand when 120573 lies between 15 and 30

Figure 12 shows the influence of 120574119888on performance The

trend of the error percentage is U-shaped for Tsukuba when120574119888is smaller than 20The error percentage bottomoutwhen 120574

119888

is between 7 and 13 The error percentage for Venus follows adownward trend For Teddy and Cones the error percentageis inversely proportional to 120574

119888 When 120574

119888is larger than 12 the

error percentage is insensitive to 120574119888

Figure 13 shows the performance of our algorithmaccording to ℎ

119878and ℎ119868 From these figures our algorithm has

a good ability of robustness with different values of ℎ119878 When

ℎ119868isin [3 8] the error percentage of the disparitymap obtained

by our algorithm is still fairly lowFor the disparity map refinement step two main parame-

tersmust be set119871 sv and 120591 are previously introduced Figure 14shows the influence of 119871 sv and 120591 on performanceWhen 119871 sv isin[5 20] and 120591 isin [7 10] all data sets show good performance

Table 2 Parameter setting for Middlebury benchmark

119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12

32 Experimental Results We evaluate our algorithm onMiddlebury benchmarks [23] with error threshold 1 The testplatform hardware consists of T9600CPU and 5GBmemorySoftware consists of MATLAB 2014a and VS2012 Parametersare shown in Table 2

Simulation results on Middlebury data sets are presentedin Figure 15 The quantitative performance of our algorithmis shown in Tables 3 and 4 Our algorithm ranks 5th in theMiddlebury data set (July 1 2014)

These results demonstrate our algorithm has a good per-formance However it is difficult to decrease the error per-centages in the three regions (nonocc all and disc) at thesame timeThis is because many pixels with correct disparity


05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)

Figure 14 (a) Performance evaluation of the proposed method according to 119871 sv and 120591 for Tsukuba (b) Performance evaluation of theproposed method according to 119871 sv and 120591 for Venus (c) Performance evaluation of the proposed method according to 119871 sv and 120591 for Teddy(d) Performance evaluation of the proposed method according to 119871 sv and 120591 for Cones

Table 3 Quantitative evaluation results for Middlebury benchmark

Algorithm Tsukuba Venus Teddy ConesNonocc All Disc Nonocc All Disc Nonocc All Disc Nonocc All Disc

AdaptingBP 11122 1379 57924 0104 0217 1448 42217 70615 11818 24822 79228 73227CoopRegion [14] 0875 1162 4614 0115 0215 15414 51630 83120 13025 27939 71814 80147Our algorithm 0898 15923 4789 0115 03624 14911 43318 99332 11217 28141 84645 78540RDP [15] 09711 13911 50011 02139 03829 18924 48422 99433 12622 25325 76920 73828MultiRBF [16] 13347 15620 60232 01311 0172 18421 50928 6368 13429 29048 6768 71024

Table 4 The results of whole performance evaluation for Middlebury benchmark

Algorithm Average rank Average percent bad pixelsAdaptingBP 165 423CoopRegion 122 441Our algorithm 228 448RDP 229 457MultiRBF 232 439


(a) (b) (c)

Figure 15 (a) The final disparity map obtained by our algorithm (b) Error maps (c) Ground truth


Table 5 Comparison between standard LRC checking and iterative LRC checking presented in Section 231

Algorithm Nonocc All DiscInitial disparity 121 320 642Standard LRC checking 143 210 754Iterative LRC checking 117 240 630

Table 6 The average running time of each part of our algorithm for Tsukuba

Mean-shift Cost aggregation LBP Iterative LRC Segmentation voting737 1923 078 024 030

Table 7 The variation of average running time of cost aggregation of our rectangle-based algorithm under different size of support region

The size of support region 11 times 11 15 times 15 19 times 19 23 times 23 27 times 27

Running time 3702 6981 10653 15441 21394

Figure 16 Erroneous disparity distribution determined by (7)-(8)for the Tsukuba image

are classified as undependable according to (8) Tsukuba canbe used as an example Figure 16 represents the bad pixelsdetected by (7)-(8)

Figure 6 shows the bad pixels obtained by comparing theinitial disparity map and ground truth Figure 6 indicatesthat true bad pixels are distributed in the occlusion regionHowever it can be inferred from Figure 16 that many pixelsin nonocc and disc regions are mistakenly classified asundependable When standard LRC checking is used mis-classified pixels in the disc region may be assigned a wrongdisparity as shown in Figure 17 Table 5 shows quantitativecomparison results for standard LRC checking and iterativeLRC checking

Thus it can be seen that the error percentage in the discregion greatly increases after applying standard LRC check-ing Because the disc region is part of the nonocc region theerror percentage in the nonocc region also increases Ouriterative LRC checking method can effectively improve theperformance of our stereo matching algorithm

33 Running Time of Our Algorithm In this section weinvestigate the computation running time of our algorithmThe running time directly reflects computational complexityWithout a loss of generality our algorithm runs 50 times and

Figure 17 Erroneous disparity distribution after executing standardLRC checking for Tsukuba

average running time was calculated Results are shown inTable 6

The new rectangle-based adaptive-weight method canbe obtained by imposing the 119910-direction constraint on thenew weight model Table 7 displays the running time ofcost aggregation of our algorithm with the rectangle-shapedsupport region under different sizes of support region Therectangle-based algorithm also runs 50 times under each sizeof support windows

Our algorithm with the rectangle-shaped support regionperformed best when the size of the support region is 27 times27 the error percentages of the initial disparity map innonocc all and disc regions are 145 342 and 708respectively Tables 5ndash7 show that our algorithm producednotable results in decreasing computation complexity andimproving performance

4 Conclusions

In this work we proposed a new line-based adaptive-weightstereo matching algorithm that integrates several methodsThe main conclusions that can be drawn from our results areas follows


(1) Cost aggregation is the most time-consuming partof stereo matching algorithm Using a line-shapedsupport region can dramatically reduce the elapsedtime of cost aggregation

(2) The adaptive-weight model proposed in this papercan produce a rather satisfactory initial disparity mapin the absence of enough pixel information

(3) Experimental results show that the algorithm pro-posed in this paper can attain a better matching effectwith less running time

Although our algorithm has a good performance onMid-dlebury data sets there is still much room for improvement

(1) There are too many parameters in our algorithmto accommodate different image pairs In furtherresearch we will analyze the intrinsic relationshipamong the parameters and reduce the number ofparameters

(2) Figure 13 indicates that image characteristics have asignificant impact on the optimum values of 120572 and 120573In future studies we will explore how the optimumvalues of 120572 and 120573 vary according to the texture ofimage

List of Variables

119871119904 The half-length of support region

120572 Shape controlling parameter120573 The half-length of transition regionℎ119878 Spatial radius of mean-shift

ℎ119868 Color radius of mean-shift

119871 sv The half-length of segment forsegmentation voting

120591 Confidence level of color similarity forsegmentation voting

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors express their appreciation for the financialsupport of the Shandong Natural Science Foundation Grantno ZR2013FL033 They also extend their sincere gratitude toreviewers for their constructive suggestions

References

[1] J C Kim K M Lee B T Choi and S U Lee ldquoA dense stereomatching using two-pass dynamic programming with gen-eralized ground control pointsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 1075ndash1082 San Diego Calif USAJune 2005

[2] P F Felzenszwalb and D P Huttenlocher ldquoEfficient beliefpropagation for early visionrdquo International Journal of ComputerVision vol 70 no 2 pp 41ndash54 2006

[3] J Sun Y Li S B Kang and H-Y Shum ldquoSymmetric stereomatching for occlusion handlingrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 399ndash406 San Diego Calif USAJune 2005

[4] V Kolmogorov and R Zabih ldquoWhat energy functions canbe minimized via graph cutsrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 26 no 2 pp 147ndash1592004

[5] H Li and G Chen ldquoSegment-based stereo matching usinggraph cutsrdquo in Proceedings of the IEEE Computer Society Conf-erence on Computer Vision and Pattern Recognition (CVPR rsquo04)vol 1 pp 74ndash81 Washington DC USA June 2004

[6] T Kanade andMAOkutomi ldquoStereomatching algorithmwithan adaptive window theory and experimentrdquo IEEE Transac-tions on Pattern Analysis and Machine Intelligence vol 16 no9 pp 920ndash932 1994

[7] X-Z Zhou G-J Wen and R-S Wang ldquoFast stereo matchingusing adaptive windowrdquo Chinese Journal of Computers vol 29no 3 pp 473ndash479 2006

[8] Y Boykov O Veksler and R I Zabi ldquoA variable window app-roach to early visionrdquo IEEETransactions on PatternAnalysis andMachine Intelligence vol 20 no 12 pp 1283ndash1294 1998

[9] O Veksler ldquoFast variable window for stereo correspondenceusing integral imagesrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo03) vol 1 pp 556ndash561 Madison Wis USA June 2003

[10] K Zhang J Lu andG Lafruit ldquoCross-based local stereomatch-ing using orthogonal integral imagesrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 7 pp 1073ndash1079 2009

[11] Y Xu D Wang T Feng and H-Y Shum ldquoStereo computationusing radial adaptive windowsrdquo in Proceedings of 16th Interna-tional Conference on Pattern Recognition (ICPR rsquo02) vol 3 pp595ndash598 Quebec Canada August 2002

[12] K-J Yoon and I S Kweon ldquoAdaptive support-weight approachfor correspondence searchrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 28 no 4 pp 650ndash656 2006

[13] F Wang and D Huang ldquoImproved Yoon stereo matchingalgorithm based on adaptive weightrdquo Journal Electronic Mea-surement and Instrument vol 24 no 7 pp 632ndash637 2010

[14] A Klaus M Sormann and K Karner ldquoSegment-based stereomatching using belief propagation and a self-adapting dis-similarity measurerdquo in Proceedings of the 18th InternationalConference on Pattern Recognition (ICPR rsquo06) pp 15ndash18 HongKong China August 2006

[15] X Sun X Mei S-H Jiao M-C Zhou and H Wang ldquoStereomatching with reliable disparity propagationrdquo in Proceedings ofthe International Conference on 3D Imaging Modeling Process-ing Visualization and Transmission (3DIMPVT rsquo11) pp 132ndash139Hangzhou China May 2011

[16] X-Z Zhou and P Boulanger ldquoNew eye contact correction usingradial basis function for wide baseline videoconference systemrdquoin Proceedings of 13th Pacific-Rim Conference on Multimedia(PCM 12) pp 68ndash79 Singapore December 2012

[17] Q X Yang L Wang R Yang H Stewenius and D NisterldquoStereo matching with color-weighted correlation hierarchicalbelief propagation and occlusion handlingrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 3 pp492ndash504 2009

[18] X Mei X Sun M-C Zhou S-H Jiao H Wang and XZhang ldquoOn building an accurate stereo matching system on


graphics hardwarerdquo in Proceedings of the IEEE InternationalConference on Computer Vision Workshops (ICCV rsquo11) pp 467ndash474 Barcelona Spain November 2011

[19] Z-F Wang and Z-G Zheng ldquoA region based stereo matchingalgorithm using cooperative optimizationrdquo in Proceedings ofthe 26th IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo08) pp 1ndash8 Anchorag Alaska USA June2008

[20] F Tombari S Mattoccia and L D Stefano ldquoSegmentation-based adaptive support for accurate stereo correspondencerdquoin Proceedings of 2nd Pacific-Rim Symposium on Image andVideo Technology (PSIVT rsquo07) pp 427ndash438 Santiago ChileDecember 2007

[21] D Comaniciu and P Meer ldquoMean shift a robust approachtoward feature space analysisrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 24 no 5 pp 603ndash6192002

[22] S Birchfield and C Tomasi ldquoDepth discontinuities by pixel-to-pixel stereordquo International Journal of Computer Vision vol 35no 3 pp 269ndash293 1999

[23] ldquoMiddlebury StereoVisionPagerdquo httpvisionmiddleburyedustereo

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Left image

Right image

Computingcost with

newadaptive-

weightmethod

Matching costof left image

Matching costof right image

Loop beliefpropagation

Initial disparitymap of left image

Initial disparitymap of right image

Iterative left-right

consistencychecking

Segmentationvoting

Finaldisparity

map

Disparity map initialization Disparity map refinement

Figure 1 Block diagram of our algorithm

these methods have a high computational complexity Otherresearchers assign different support-weights to the pixels in asupport region keeping the size and shape of a support regionconstant [11ndash13]

In recent years several methods for acquiring satisfactoryeffect of stereo matching have been adopted Yang et al[17] presented a stereo matching algorithm which inte-grates color-weight belief propagation left-right checkingcolor segmentation plane fitting and depth enhancementMei et al [18] integrated the AD-census cost measurementfunction the cost aggregation method based cross-basedregion the scanline optimize method the multistep refine-mentmethod and the accelerative algorithmbased onCUDAinto their algorithm

The algorithm presented in this paper is inspired fromadaptive-weight matching algorithm In this paper the aim isto propose a low computation complexity and high accuracystereo matching algorithm So the rectangle-shaped supportregion is substituted for the line-shaped support regionLacking of enough pixel information is a main weakness ofthe line-shaped support region which is easy to cause errormatching Adaptive-weight can make full of limited pixelinformation by analyzing the characteristic of the adaptive-weight model proposed in [13] on disparity accuracy weuse neural network (NN) to determine the spatial proximityand mean-shift based segmentation method to effectivelydescribe the color similarity

In addition several approaches are applied to completethe algorithmWe develop a new pixel dissimilarity measure-ment function which combines Birchfield pixel dissimilaritymeasurement function and census transform to computethe matching cost The loopy belief propagation methodproposed in [2] is employed to estimate the initial disparitymap which is optimized with min convolution and imagepyramid There are several measurable improvements for theinitial disparity map To further improve the accuracy of theinitial disparity map iterative left-right consistency (LRC)checking and segmentation voting are used to refine thedisparity map by analyzing the features of the initial disparitymap

2 Algorithm Description

The algorithm presented in this paper can be divided intotwo steps a disparity map initialization step and a refine-ment step The framework of the algorithm is shown in

p

Figure 2 The assumption of segmentation-based stereo matching

x

wx

120573

120573

minus15

minus1

1

1

1

f

fIT

OMn

fc(Δgpq)

IMn

Figure 3 Spatial proximity described by neural network

Figure 1 A detailed description of this algorithm is givenin the following sections

21 Adaptive-Weight BasedCost AggregationMethod Assum-ing the two pixels119901 and 119902 the disparity of center pixel119901wantsto be computed 119878(119901) is the support region of pixel 119901 while 119902is a neighboring pixel of119901 in the support regionThe support-weight of 119902 is assigned by the following according to [12]

119908 (119901 119902) = 119891119888(Δ119888119901119902) 119891119888(Δ119892119901119902) (1)

where 119891119888(Δ119892119901119902) represents the spatial proximity 119891

119888(Δ119888119901119902)

represents the color similarity and 119908(119901 119902) is the support-weight Our algorithm is designed on the basis of thisframework The list of variables used in this paper is shownin the end of the paper

211 The Model of Line-Based Adaptive-Weight Wang et al[13] noted that when the support region is large enough colorsimilarity plays a major role in computing the center pixeldisparity within a certain range As shown in Figure 2 redrepresents the support region 119878(119901) We used the pixels in 119878(119901)to compute the disparity of 119901 The effects of spatial proximitycan be neglected in the pale blue region according to [13] Wecall this region the transition area represented by 119879(119901) Tosatisfy this principle the neural network can be applied in thedesign of this spatial proximity model

Figure 3 shows the spatial proximitymodel established byneural networkThe position of a pixel is the input the spatial



04

045

05

055

06

x

fc(Δgpq)








1198681198721= 119909 + 120573

1198681198722= minus119909 + 120573

119874119872119894= 119891 (119868

119872119894)

119868119879=

2

sum

119894=1

119874119872119894minus 15

119891119888(Δ119892119901119902) = 119891 (119868

119879)

(2)




119891119888(Δ119888119901119902) =

1 if seg (119902) = seg (119901)

exp(minusΔ2

119888119901119902

120574119888

) otherwise(3)





119864 (119901 119889) =

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889) 119890 (119902 119902

119889)

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889)

(4)

where 119901119889and 119902



119889)


119890 (119902 119902119889) = min (119862Birch (119902 119902119889 119868119877 119868119871) 119879Birch)




119863(119901) = argmin119889

119864 (119901 119889) (6)


(a) (b) (c)

















(2) 119878119905larr 119878119905119894



119905


119894

(6) 119889(119901119894) larr min(119889

119900(119901119871

119894) 119889119900(119901119877

119894)) where 119889


(7) If 119889(119901119894) = 119889119900(119901119894)


119905

(9) End(10) End(11) End(12) End







119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (7)


119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (8)




p



10038161003816100381610038161003816119868119902minus 119868119901

10038161003816100381610038161003816lt 120591 119902 isin 119880 (119901)

10038161003816100381610038161003816119909119902minus 119909119901

10038161003816100381610038161003816lt 119871 sv

10038161003816100381610038161003816119910119902minus 119910119901

10038161003816100381610038161003816lt 119871 sv

(9)




0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)



(1)119867(119889) larr 0


(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)






119888 ℎ119878 ℎ119868







0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











04

045

05

055

06

x

fc(Δgpq)








1198681198721= 119909 + 120573

1198681198722= minus119909 + 120573

119874119872119894= 119891 (119868

119872119894)

119868119879=

2

sum

119894=1

119874119872119894minus 15

119891119888(Δ119892119901119902) = 119891 (119868

119879)

(2)




119891119888(Δ119888119901119902) =

1 if seg (119902) = seg (119901)

exp(minusΔ2

119888119901119902

120574119888

) otherwise(3)





119864 (119901 119889) =

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889) 119890 (119902 119902

119889)

sum119902isin119871119901119902119889

isin119871119901119889

119908 (119901 119902)119908 (119901119889 119902119889)

(4)

where 119901119889and 119902



119889)


119890 (119902 119902119889) = min (119862Birch (119902 119902119889 119868119877 119868119871) 119879Birch)




119863(119901) = argmin119889

119864 (119901 119889) (6)


(a) (b) (c)

















(2) 119878119905larr 119878119905119894



119905


119894

(6) 119889(119901119894) larr min(119889

119900(119901119871

119894) 119889119900(119901119877

119894)) where 119889


(7) If 119889(119901119894) = 119889119900(119901119894)


119905

(9) End(10) End(11) End(12) End







119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (7)


119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (8)




p



10038161003816100381610038161003816119868119902minus 119868119901

10038161003816100381610038161003816lt 120591 119902 isin 119880 (119901)

10038161003816100381610038161003816119909119902minus 119909119901

10038161003816100381610038161003816lt 119871 sv

10038161003816100381610038161003816119910119902minus 119910119901

10038161003816100381610038161003816lt 119871 sv

(9)




0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)



(1)119867(119889) larr 0


(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)






119888 ℎ119878 ℎ119868







0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

















(2) 119878119905larr 119878119905119894



119905


119894

(6) 119889(119901119894) larr min(119889

119900(119901119871

119894) 119889119900(119901119877

119894)) where 119889


(7) If 119889(119901119894) = 119889119900(119901119894)


119905

(9) End(10) End(11) End(12) End







119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (7)


119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (8)




p



10038161003816100381610038161003816119868119902minus 119868119901

10038161003816100381610038161003816lt 120591 119902 isin 119880 (119901)

10038161003816100381610038161003816119909119902minus 119909119901

10038161003816100381610038161003816lt 119871 sv

10038161003816100381610038161003816119910119902minus 119910119901

10038161003816100381610038161003816lt 119871 sv

(9)




0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)



(1)119867(119889) larr 0


(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)






119888 ℎ119878 ℎ119868







0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(2) 119878119905larr 119878119905119894



119905


119894

(6) 119889(119901119894) larr min(119889

119900(119901119871

119894) 119889119900(119901119877

119894)) where 119889


(7) If 119889(119901119894) = 119889119900(119901119894)


119905

(9) End(10) End(11) End(12) End







119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (7)


119863119871(119901) = 119863

119877(119901 minus 119863

119871(119901)) (8)




p



10038161003816100381610038161003816119868119902minus 119868119901

10038161003816100381610038161003816lt 120591 119902 isin 119880 (119901)

10038161003816100381610038161003816119909119902minus 119909119901

10038161003816100381610038161003816lt 119871 sv

10038161003816100381610038161003816119910119902minus 119910119901

10038161003816100381610038161003816lt 119871 sv

(9)




0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)



(1)119867(119889) larr 0


(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)






119888 ℎ119878 ℎ119868







0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 10 20 30 40 50 60 703

42

54

Erro

r per

cent

age i

n al

l reg

ion

()

13

17

21

Ls

TsukubaVenus

(a)

0 10 20 30 40 50 60 70

Erro

r per

cent

age i

n al

l reg

ion

()

105

11

115

12

125

13

135

14

145

TeddyCones

Ls

105

115

125

135

145

155

165

175

185

(b)



(1)119867(119889) larr 0


(4) If 119889 == 119863119871(119902)

(5) 119867(119863119871(119902)) larr 119867(119863

119871(119902)) + 1

(6) End(7) End(8) End(9)119863

119871(119901) = argmax

119889

119867(119889)






119888 ℎ119878 ℎ119868







0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

005

115

23

32

34

36

38

Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

37

36

35

34

33

32

(a)

132

134

136

138

14

142

144

13

135

14

145

15

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(b)

12

125

13

135

14

126

128

13

132

134

136

138

0 10 20 30 40 50

005

115

2Erro

r per

cent

age i

n al

l reg

ion

()

120572

120573

(c)

010

2030

4050

0

1

105

11

115

12

109

1095

11

1105

111

1115

112

1125

113

1135

114

05

15

Erro

r per

cent

age i

n al

l reg

ion

()

2

120572

120573

(d)


0 2 4 6 8 10 12 14 16 18 2032

34

36

38

4

Erro

r per

cent

age i

n al

l reg

ion

()

120574c

TsukubaVenus

16

15

14

13

12

(a)

0 2 4 6 8 10 12 14 16 18 2011

12

13

14

15

16

17

18

Erro

r per

cent

age i

n al

l reg

ion

()

10

105

11

115

12

125

13

135

TeddyCones

120574c

(b)




05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05

1015

20

05

1015

203

4

5

6

7

8

35

4

45

5

55

6

65

7

75Er

ror p

erce

ntag

e in

all r

egio

n (

)

hS

h I

(a)

0 5 10 1520

5

10

15

201

15

2

25

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

21

2

19

18

17

16

15

14

13

12

(b)

2 4 6 8 10 12 14 165

10

15

2012

125

13

135

14

145

122

124

126

128

13

132

134

136

138

14

Erro

r per

cent

age i

n al

l reg

ion

()

hS

hI

(c)

05

1015

20

5

10

1520

105

11

115

12

125

11

112

114

116

118

12

Erro

r per

cent

age i

n al

l reg

ion

()

hS

h I

(d)





119878and ℎ

119868for Teddy


119868for Cones




119888


119888 When 120574










119871119904

120572 120573 120574119888

ℎ119878

ℎ119868

119871 sv 120591

40 1 15 8 15 5 7 12





05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05

1015

20

05

1015

2025

15

2

25

3

18

2

22

24

26

28

Erro

r per

cent

age i

n al

l reg

ion

()

120591SV

L SV

(a)

05

1015

20

05

1015

2025

02

04

06

08

1

03

035

04

045

05

055

06

065

07

075

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(b)

05

1015

20

010

2030

95

10

105

11

115

98

10

102

104

106

108

11

Erro

r per

cent

age i

n al

l reg

ion

()

120591SVL SV

(c)

05

1015

20

010

20

308

85

9

95

10

105

11

85

9

95

10

105

Erro

r per

cent

age i

n al

l reg

ion

()

120591L SV

(d)








(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)









Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Running time 3702 6981 10653 15441 21394










4 Conclusions









List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















List of Variables








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a line-based adaptive-weight...

Documents