Cost Adaptive Window for Local Stereo Matching

J. Navarro and A. Buades

Dpt. Matem

atiques Inform

atica, Universitat Illes Balears, Ctra Valldemossa km 7.5, Palma, Spain

Keywords:

Stereovision, Depth Estimation, Block-matching, Adaptive Windows.

Abstract:

We present a novel stereo block-matching algorithm which uses adaptive windows. The shape of the window

is selected to minimize the matching cost. Such a window might be the less distorted by the disparity function

and thus the optimal one for matching. Moreover, we introduce a coarse-to-ﬁne strategy to limit the number

of ambiguous matches and reduce the computational cost. The proposed approach performs as state of the art

local matching methods.

1 INTRODUCTION

The goal of stereovision is to estimate the depth of

the scene from at least two images taken from dif-

ferent viewpoints. The depth estimation is equivalent

to computing the apparent motion of corresponding

points in the two images. For an epipolar rectiﬁed im-

age pair, all pixels have horizontal motion (which is

called disparity) and the problem is then reduced into

a 1D correspondence problem.

Over last years, several approaches have been pro-

posed to solve the stereo matching problem. The

strategies can be divided into local and global meth-

ods (Scharstein and Szeliski, 2002). Local methods

compute the disparity d of a point (x,y) by means of

a block-matching (also called area-based) approach in

which a small window or patch around (x,y) in the left

image is compared with windows in the same epipolar

line in the right image. The comparison is done by as-

signing a matching cost c to each candidate window

in the second image. The global methods overcome

the main limitation of block based methods, that is,

the non presence of enough distinctive information in

the block. For global methods, the disparity estima-

tion is formulated by means of an optimization prob-

lem where the solution is constrained to satisfy some

smoothness assumption. These methods, being very

similar to optical ﬂow methods, mainly differ in the

energy minimization method being used, belief prop-

agation (Sun et al., 2003), graph cuts (Kolmogorov

and Zabih, 2001), etc. Global methods depend on ad-

ditional parameters difﬁcult to ﬁx in general, its value

being different for each stereo pair.

In order to reliably match a block, the depth

should vary as less as possible inside the block. Oth-

erwise, the block might be distorted by the effect of

the disparity function and its retrieval in the second

image not be an easy task. Block matching methods

tend to identify depth discontinuities with image color

ones, and adapt the shape of the window to the color

of the image (Patricio et al., 2004; Yoon and Kweon,

2006; Wang et al., 2006; Rhemann et al., 2011). This

solution might be effective for scenes with uniform

color objects being at different depth planes, however

this is not the case for a general scenario with textured

objects and slanted surfaces.

We propose to adapt the shape of the window to

the unknown disparity function instead of the image

color. Windows sharing the same depth for all pixels

will be less distorted by the application of the dis-

parity and therefore will be matched with minimal

cost. Similarly, the authors in (Buades and Facciolo,

2015; Hirschm

uller et al., 2002) proposed a similar

approach but limiting the choice of the matching win-

dow to belong to a small pre existing set, containing

mainly directional and corner windows. As we will

see, the choice of a correct window is not straighfor-

ward and some regularity on the shape of the window

might be demanded.

The proposed approach is able to deal with depth

discontinuities and slanted surfaces. Compared to

state-of-the-art, the proposed approach is able to pre-

cisely identify depth discontinuities avoiding the well

known fattening problem (Blanchet et al., 2011). The

algorithm is embedded into a coarse-to-ﬁne strategy

in which the disparity computed at a coarser scale

is used to restrict the search disparity range at ﬁner

scales reducing both the match ambiguity and com-

Navarro J. and Buades A.

Cost Adaptive Window for Local Stereo Matching.

DOI: 10.5220/0006100503690376

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 369-376

ISBN: 978-989-758-227-1

369

putational time.

This paper is organized as follows. In Section 2

we present the state-of-the-art of local methods for

disparity estimation. In Section 3 it is explained

the new block-matching algorithm with adaptive win-

dows. Finally, Section 4 shows the performance of

the proposed method by means of a comparison with

state-of-the-art approaches.

2 STATE OF THE ART

Local methods mainly differ in the choice of the

matching cost and the size and shape of the match-

ing window. The most common costs are the sum

of squared differences, the normalized cross cor-

relation (Hannah, 1974), the summed normalized

cross correlation (Einecke and Eggert, 2010), the mu-

tual information (Viola and Wells III, 1997) and the

census transform (Zabih and Woodﬁll, 1994), see

(Hirschm

uller and Scharstein, 2009) for a review.

While the choice of a different cost might be impor-

tant in order to be robust to local differences in color,

noise, shadows or transparencies, the shape and size

of the window turn to be the most important in order

to overcome the effect of disparity in the image.

Block-matching algorithms typically assume that

the depth is the same for all pixels inside the matching

window. This assumption may not hold for slanted

surfaces or near depth discontinuities unless the shape

of the window is locally adapted. Kanade et al.

(Kanade and Okutomi, 1994) were the ﬁrst to address

this problem. The authors proposed the use of rectan-

gular adaptive windows whose shape and size is se-

lected locally at each pixel by looking the differences

between gray level values. Approaches with ﬁxed

size and squared windows were presented in (Fusiello

et al., 1997; Kang et al., 2001) where it is performed

the correlation with different windows containing the

reference pixel and then it is taken the disparity giv-

ing the smallest cost, arguing that a window yielding

the smallest error is more likely to correspond to a

constant depth region. There are also methods that

base the shape and size of the correlation window on

an image segmentation (Gerrits and Bekaert, 2006;

Wang and Zheng, 2008). Other approaches, instead of

adapting the window make use of varying weights for

the pixels inside the ﬁxed window (Yoon and Kweon,

2006; Wang et al., 2006; Rhemann et al., 2011). See

(Hosni et al., 2013) for an extensive review. All these

methods identify differences in pixels’ gray level with

differences in pixels’ depth, which is not always the

case.

All the methods mentioned so far are expected to

cope with depth discontinuities but not with slanted

surfaces. However, proposals have arisen with the

aim of handling the presence of slanted surfaces in

the scene. There are approaches trying to estimate

depth by means of planes at each region (Lu et al.,

2013; Bleyer et al., 2011). Other approaches cope

with slanted surfaces by selecting the most appropri-

ate window at each pixel depending on cost, using a

pre existent set of windows. In (Hirschm

uller et al.,

2002) the shape of the window was adapted by divid-

ing the correlation window into sub-windows and se-

lecting the ones yielding the minimum matching cost.

The recent method presented in (Buades and Facci-

olo, 2015) consists of a parameter-less approach in

which multiple elongated windows with different ori-

entations are tested in the matching process. At each

point, the one providing minimum matching cost is

selected. The authors also introduce a set of validation

criteria in order to provide a mask of invalid or am-

biguous matches. Furthermore, they use a coarse-to-

ﬁne strategy in which disparity computed at a coarser

scale is used to restrict the search disparity range at

the current scale.

One of the main problems of local methods is the

errors that can be produced in non-textured regions.

The lack of information in this patches lead to ambi-

guities in the matching process. The authors in (Man-

duchi and Tomasi, 1999) target this problem by com-

puting the distinctiveness of a pixel as the dissimi-

larity in color between the pixel and the most similar

other point in the search window. The recent approach

proposed in (Sabater et al., 2012) measures reliabil-

ity from the number of false alarms to each match

and matches whose patches are slightly different are

discarded. Also, they use the distinctiveness measure

presented in (Manduchi and Tomasi, 1999).

Finally, the sampling of the disparity space is also

an important question since it can produce incorrect

matches. In (Birchﬁeld and Tomasi, 1998) the authors

proposed a matching cost based on the SSD that is in-

sensitive to the sampling of the correlation window.

However, in (Buades and Facciolo, 2015) the authors

show that similar results can be obtained with a sub-

pixel SSD cost.

3 BLOCK-MATCHING WITH

ADAPTIVE WINDOWS

Block-matching methods assume that disparity does

not change drastically inside the window. A proper

choice of the window shape is important in order to

satisfy such a condition. This optimum shape has to

be adapted locally for each pixel.

VISAPP 2017 - International Conference on Computer Vision Theory and Applications

370

The optimal window cannot be known a pri-

ori, however the analytical study of correlation in

(Blanchet et al., 2011) permits to characterize the

pixel minimizing such a cost for a ﬁxed window. Fol-

lowing (Blanchet et al., 2011), the optimum match

writes as a weighted average of disparities of pixels in

the window, being this average weighted by the gra-

dient of the image. Such an average would coincide

with the true disparity when the disparity is constant

and the cost be minimal.

We propose to select at each pixel a matching win-

dow sharing the same depth, indirectly by choosing

the one matching with minimal cost. This window

has to be adapted independently for each pixel and

for each possible disparity, then the disparity and win-

dow with minimal cost have to be chosen. In order

to balance the shape of the window around the ref-

erence pixel we penalize the distance of the barycen-

ter of the window to the reference pixel. We do not

penalize the distance of the chosen pixels to the ref-

erence pixel since we allow for elongated windows.

Although marginally, in some cases it may be neces-

sary to take into account the intensity or color of the

pixels belonging to the selected window. Thereby, we

select only pixels having a similar color to the refer-

ence pixel.

The selection of the adapted window W

for a cer-

tain pixel p and disparity d writes as a minimization

C (p,d) = min

∑

∈W

c(p

,d) + βkp − b

+λ

∑

∈W

(p) − I

))

, (1)

being c(p

,d) the cost of assigning disparity d to the

pixel p

, b

the barycenter of the chosen window

and λ,β > 0.

Then, for each pixel p we compute the disparity

D(p) as

D(p) = argmin

C (p,d). (2)

3.1 Adaptive Window Selection

Since not all possible window conﬁgurations can be

tested, we perform a greedy algorithm making use

of a cost volume structure, similarly to (Rhemann

et al., 2011). First, in order to be robust we use

3 × 3 squared windows for building the cost volume

c(p,d) as the three dimensional array which stores

the zero-mean sum of squared differences (ZSSD)

cost (Hirschm

uller and Scharstein, 2009) for choos-

ing disparity d at pixel p = (x,y). The ZSSD cost re-

moves the average intensity of the window rendering

the comparison independent of the mean intensity:

ZSSD(p,q) :=

∑

t∈B



(p + t)− I

(q + t)− I

p+B

+ I

q+B



(3)

where q = p +(d,0)

, B

denotes the matching win-

dow and I

p+B

is the mean of pixel intensities in im-

age I

inside the window with reference pixel p.

Then, we select at each pixel p the window W

that

minimizes (1). This window is composed by neigh-

boring 3 × 3 patches of p. For a given n, we select

the n − 1 neighboring patches that minimize such an

energy. These patches are selected with an iterative

process, selecting at each step the patch yielding min-

inum cost and being connected to the already selected

patches. That is, by construction the window is con-

nected. Moreover, in order to control the elongation

of the optimal window, we introduce a parameter M

to limit the distance between the reference pixel p and

the center pixel of the joined patch.

Let b

1,...,k

be the barycenter of points p,p

,..., p

At iteration k we add the neighboring patch centered

at p

such that

= arg min

∈N

1,...,k−1

kp−p

k≤M

c(p

,d) + βkp − b

1,...,k

+λ



p+B

− I



(4)

where b

1,...,k

is the updated barycenter taking into ac-

count p

and N

1,...,k−1

the set of connected neighbor-

ing patches to the already selected ones. We repeat

this process until we add n − 1 patches centered at

,..., p

n−1

The difference between the average color of the

3 × 3 candidate patches and the reference one is used

to penalize windows across image edges. The use of

the average value is more robust than comparing only

the value of pixel p.

Now, the cost of assigning disparity d to the pixel

p with the selected window is given by

A({p,p

,..., p

n−1

},d) =

n−1

∑

i=0

c(p

,d)

+ βkp −b

1,...,n−1

+ λ

n−1

∑

i=1



p+B

− I



(5)

being b

1,...,n−1

the barycenter of the chosen window.

For notation purposes, notice that we have denoted p

as p

to introduce it into the sum. The shape resulting

Cost Adaptive Window for Local Stereo Matching

371

Aloe Wood Fountain

Book Village1 Village2

Figure 1: Images used in the experiments section to evaluate the performance of our method. We show the left image of the

stereo pair and the ground truth disparity with the true occlusion mask overimposed.

from accumulating the smallest matching costs coin-

cides with the one having the disparity function most

uniform.

The volume of costs A({p,p

,..., p

n−1

},d) turns

to be an approximation of the cost C (p, d). Then, for

each pixel p we ﬁnally compute the disparity D(p) as

D(p) = argmin

A({p,p

,..., p

n−1

},d). (6)

3.2 Multi-scale Approach

As in (Buades and Facciolo, 2015), our approach is

embedded into a coarse-to-ﬁne strategy. This permits

to reduce computational cost and match ambiguity.

A large search range may lead to mismatches as in-

creases the possibility of matching with a repetitive

pattern. For this reason, at each scale we adapt the

search range locally at each pixel by looking at the

minimum and maximum disparity values of a neigh-

borhood obtained at the previous coarser scale. With

this step we compute the images Dmin and Dmax

which determine respectively the minimum and maxi-

mum values of the disparity range at each pixel. Each

level of the pyramid is obtained by a convolution with

a Gaussian kernel with standard deviation σ = 1.2 and

subsampling by a factor of two from the initial stereo

pair. The disparity computed at coarser scales is up-

sampled by bicubic spline interpolation.

3.3 Detection of Invalid Matches

In order to reject possible incorrect matches we use

the common left-right consistency check. Let D

and

be the left-based and right-based disparities. This

is,

(p) = I

(p+D

(p)) and I

(p+D

(p)) = I

(p).

(7)

Then, a match is rejected if D

does not coincide with

the inverse mapping of D

, i. e. when

(p + D

(p)) + D

(p)| > ε (8)

with ε > 0. In practice it is set to ε = 1.

4 EXPERIMENTS

For evaluating the proposed multi-scale adaptive win-

dow (MSAW) approach we use two databases: the

set of images presented in Figure 1 and the new Mid-

dlebury stereo benchmark version 3 (Scharstein and

Hirschm

uller, 2014).

In this section we compare the proposed algorithm

with several state-of-the-art methods. The Adaptive

Support Weights (ASW) approach (Yoon and Kweon,

2006) uses bilateral weights with the aim of assigning

large weights to close pixels having a similar color to

the reference one, expecting to use pixels only from

the same physical object. A window size of 35 × 35

is used. This method provides only pixel precision.

The Cost-Volume Filtering (CVF) method (Rhe-

mann et al., 2011) builds a volume c

(p,d) measuring

the cost for selecting the disparity d at pixel p. Then,

the guided ﬁlter (He et al., 2010) is used to ﬁlter the

volume using the left image as a guidance. The patch

size used in this approach is 19 × 19 and the method

provides pixel precision.

Finally, the Multi-Scale and Multi-Window

(MSMW) method (Buades and Facciolo, 2015) con-

sists of a multi-scale approach which uses multiple

elongated windows with different orientations in or-

der to cope with slanted surfaces. We run the algo-

rithm with three scales, a window size of 5 × 5 and

1/4 of disparity precision. For this algorithm we

only apply the matching strategy and not the poste-

rior validation criteria deﬁned in the method. We do

VISAPP 2017 - International Conference on Computer Vision Theory and Applications

372

Simple ASW CVF MSMW MSAW

Figure 2: Disparity estimation and error, both with ground truth mask and left-right mask overimposed, for Aloe, Wood,

Fountain and Village2 images of the proposed method (MSAW) compared with a classic block matching with 5 × 5 squared

windows (Simple), Adaptive Support Weights (ASW) (Yoon and Kweon, 2006), Cost-Volume Filtering (CVF) (Rhemann

et al., 2011) and Multi-Scale and Multi-Window algorithm (MSMW) (Buades and Facciolo, 2015). Image error between

computed disparities and ground truth is displayed in range [0, 2.5].

this in order to fairly compare all algorithms, since

all block-matching algorithms could beneﬁt of such

criteria. For completeness in the comparison, we in-

clude a matching algorithm with a ﬁxed squared 5×5

window with 1/4 of precision. We will refer to this

algorithm as Simple.

Regarding our method, we use a ﬁxed set of pa-

rameters in all the experiments. We take n = 25 as

the number of patches to be joined. The maximum

allowed distance when joining a new pixel is M = 4,

the penalization parameters are λ = 0.05 and β = 0.5,

we use 1/4 of precision in the disparity map and

we perform a total of three scales in the pyramidal

scheme. For all methods we use only the left-right

consistency check as validation. Figure 1 displays the

image pairs used for testing in this section. For each

pair we dispose of a different initial disparity range:

[−150.50,−21.50] for the Aloe pair, [−108,−28] for

Wood, [−84, 200] for Fountain, [−60,−21] for Book,

[−22,23] for Village1 and the range [−9,10] for the

Village2 stereo pair.

Figure 2 visually compares the described methods

Cost Adaptive Window for Local Stereo Matching

373

ASW CVF MSAW

Figure 3: Comparison of the proposed method (MSAW) with Adaptive Support Weights (ASW) (Yoon and Kweon, 2006)

and Cost-Volume Filtering (CVF) (Rhemann et al., 2011). It is shown the error with the ground truth mask and left-right mask

overimposed for Fountain and Village2 pairs, displayed in range [0, 2.5].

Table 1: Comparison of our method (MSAW) with a classic block matching with 5 × 5 squared windows (Simple), Adaptive

Support Weights (ASW) (Yoon and Kweon, 2006), Cost-Volume Filtering (CVF) (Rhemann et al., 2011) and the Multi-Scale

and Multi-Window algorithm (MSMW) (Buades and Facciolo, 2015). We show the density (D), the percentage of pixels with

an error above one pixel (E1) and above three pixels (E3), being all values computed for all the pixels.

Simple ASW CVF MSMW

image D E1 E3 D E1 E3 D E1 E3 D E1 E3

Aloe 82.16 4.08 2.26 86.33 3.59 1.85 84.34 3.08 1.61 83.33 2.98 1.70

Wood 79.41 4.34 1.98 84.74 3.63 1.10 82.91 2.71 0.87 83.15 2.63 0.90

Fountain 81.99 12.26 4.66 87.14 13.03 4.9 85.19 10.49 3.21 84.98 10.26 3.24

Book 87.03 6.02 1.79 89.38 3.59 0.15 87.57 3.41 0.14 87.60 5.13 1.25

Village1 92.71 2.62 1.46 94.36 2.92 0.72 93.81 2.22 0.56 93.01 2.07 1.10

Village2 97.16 0.61 0.42 97.57 0.65 0.61 97.48 0.61 0.61 97.31 0.47 0.35

average 86.74 4.99 2.10 89.92 4.57 1.56 88.55 3.75 1.17 88.23 3.92 1.42

MSAW

image D E1 E3

Aloe 84.49 2.58 1.29

Wood 84.29 2.17 0.44

Fountain 85.25 9.73 2.52

Book 84.79 4.85 0.88

Village1 93.65 1.24 0.53

Village2 97.82 0.26 0.24

average 88.38 3.47 0.98

for images Aloe, Wood, Fountain and Village2. In

the ﬁgure we show the estimated disparity map and

its error with the ground truth displayed in the range

[0,2.5]. We over-impose the true occlusion mask

and the one resulting from the left-right consistency

check. Near the contour of the leaves of the Aloe and

in the contours of the buildings in Village2 we can

appreciate how our method is the one that yields low-

VISAPP 2017 - International Conference on Computer Vision Theory and Applications

374

Table 2: Evaluation on Middlebury online benchmark version 3. Comparison between our method and SGM (Hirschm

uller,

2008), SNCC (Einecke and Eggert, 2010), IDR (Kowalczuk et al., 2013), Cens5 (Hirschm

uller et al., 2002), TMAP (Psota

et al., 2015), CVF (Rhemann et al., 2011), ASW (Yoon and Kweon, 2006) and MSMW (Buades and Facciolo, 2015). The

numbers represent the proposed weighted average for the training dataset over non occluded pixels.

method resolution density (%) bad 0.5 (%) bad 1.0 (%) bad 2.0 (%)

SGM H 84.04 38.5 17.7 7.66

SNCC H 71.29 29.8 13.4 6.14

IDR H 76.10 31.9 12.8 4.67

Cens5 H 76.38 35.3 17.5 8.37

TMAP H 69.80 37.3 15.4 5.97

CVF

H 62.62 34.13 17.13 12.03

ASW H 66.05 42.19 26.78 21.73

MSMW H 66.90 28.37 14.52 9.08

MSAW Q 80.50 19.48 11.10 7.12

est errors near depth discontinuities. Larger errors of

AWS and CVF are due to the different sampling of

the disparity space. In general AWS and CVF behave

similar, in Figure 3 we can see the comparison of our

method and these two methods with cropped regions

of Fountain and Village2 images. The regions of large

error that we see in both images for AWS and CVF

disappear in our result. At the top-left part of the Vil-

lage2 image there is a slanted roof in which disparity

is well estimated by our method while the two others

fail.

In Table 1 we show a quantitative comparison with

the mentioned methods. In both tables we show the

density (D) of each method obtained from the left-

right consistency check, the percentage of pixels with

an error above one pixel (E1) and with an error above

three pixels (E3). We use all the pixels (occluded and

non occluded) in the evaluation. The proposed algo-

rithm is the one achieving in general lower errors, ob-

taining the lowest average in both cases, all and non

occluded pixels.

Finally, we apply our method MSAW to

the new Middlebudy benchmark version 3

(http://vision.middlebury.edu/stereo/eval3/). In

Table 2 we show the results provided by other pub-

lished approaches that are in the highest positions of

the ranking (September 2016): SGM (Hirschm

uller,

2008), SNCC (Einecke and Eggert, 2010), IDR

(Kowalczuk et al., 2013), Cens5 (Hirschm

uller et al.,

2002) and TMAP (Psota et al., 2015). Additionally,

we also include the results given by the methods used

in the previous comparison: CVF (Rhemann et al.,

2011), AWS (Yoon and Kweon, 2006) and MSMW

(Buades and Facciolo, 2015). Our method gives

a considerable number of estimated pixels while

yielding low errors.

5 CONCLUSIONS

We proposed a new local matching algorithm to com-

pute the disparity from a stereo rectiﬁed image pair.

Our method adapts for each pixel the shape of the

window in order to select the one with minimal

matching cost. This window being the less distorted

by the disparity coincides with the one for which the

depth varies the least and then the optimal one for lo-

cal matching.

The proposed algorithm makes use of a cost vol-

ume for selecting the minimal cost window. Addi-

tional criteria is used to balance the shape of the win-

dow and avoid windows containing more than one

physical object. Moreover, we use a pyramidal strat-

egy in which the search disparity range is restricted

at each scale reducing the match ambiguity and the

computational time.

The experiments show how the proposed method

outperforms state-of-the-art local matching methods

being able to deal with depth discontinuities and

slanted surfaces.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge support by Min-

isterio de Economia y Competitividad under grant

TIN2014-53772-R and CNES research and technol-

ogy project DAJ/AR/IB/16-10117037.

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