INTEGRATION OF INTENSITY EDGE INFORMATION INTO

THE REACTION-DIFFUSION STEREO ALGORITHM

Atsushi Nomura

Faculty of Education, Yamaguchi University, Yoshida 1677-1, Yamaguchi, Japan

Makoto Ichikawa

Faculty of Letters, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, Japan

Koichi Okada

Education Promoting Center, Yamaguchi University, Yoshida 1677-1, Yamaguchi, Japan

Hidetoshi Miike

Graduate School of Science and Engineering, Yamaguchi University, Tokiwadai 2-16-1, Ube, Japan

Keywords:

Stereo algorithm, Reaction-diffusion equation, Anisotropic diffusion, PDE approach, Visual integration.

Abstract:

The present paper proposes a visual integration algorithm that integrates intensity edge information into a

stereo algorithm. The stereo algorithm assumes two constraints of continuity and uniqueness on disparity

distribution. Since depth discontinuity around object boundaries does not satisfy the continuity constraint, it

causes numerous errors in stereo disparity detection. In order to reduce the errors due to the depth disconti-

nuity, we propose a new algorithm that integrates intensity edge information into the stereo algorithm. The

stereo algorithm utilizes reaction-diffusion equations, in which diffusion coefﬁcients control the continuity

constraint. Thus, we introduce anisotropic diffusion ﬁelds into the reaction-diffusion equations; that is, we

modulate the diffusion coefﬁcients according to results of edge detection applied to image intensity distribu-

tion. We demonstrate how the proposed algorithm works around areas having depth discontinuity and conﬁrm

quantitative performance of the algorithm in comparison to other stereo algorithms.

1 INTRODUCTION

The cooperative stereo algorithm proposed by Marr

and Poggio (Marr and Poggio, 1979) assumes two

constraints: uniqueness and continuity on stereo dis-

parity distribution. The uniqueness constraint states

that a particular point in a stereo disparity distribution

has only one stereo disparity level except for transpar-

ent object; the continuity constraint states that stereo

disparity varies continuously overobject surfaces. Al-

though most areas in stereo disparity distribution sat-

isfy the continuity constraint, areas having depth dis-

continuity such as object boundariesdo not satisfy the

continuity constraint. Thus, stereo algorithms assum-

ing the continuity constraint tend to provide numerous

errors in areas having the depth discontinuity.

The problem due to discontinuity is one of the cen-

tral issues in image processing and computer vision

research. We usually apply the Gaussian ﬁlter to dis-

tribution of visual properties such as intensity, dispar-

ity or motion around spatial neighboring points, in or-

der to smooth the distribution. We can utilize a dif-

fusion equation instead of the Gaussian ﬁlter. How-

ever, since a smoothing process or a diffusion process

across the discontinuity causes unexpected smoothing

or diffusion on the visual properties, there are also nu-

merous errors around areas having the discontinuity.

In order to solve the problem due to the discontinu-

ity, several algorithms introduced anisotropy into the

Gaussian ﬁlter or the diffusion process. An edge de-

tection algorithm proposed by Perona and Malik is

580

Nomura A., Ichikawa M., Okada K. and Miike H. (2009).

INTEGRATION OF INTENSITY EDGE INFORMATION INTO THE REACTION-DIFFUSION STEREO ALGORITHM.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 580-586

DOI: 10.5220/0001821005800586

 SciTePress

one of the most famous algorithms having anisotropic

diffusion (Perona and Malik, 1990).

When focusing on stereo disparity detection,

we can ﬁnd many state-of-the-art stereo algo-

rithms (Scharstein and Szeliski, 2002), some of which

handle the problem due to depth discontinuity or its

related occluding boundary problem. The cooperative

algorithm (Marr and Poggio, 1979) mentioned above

is one of the most widely known and primitive algo-

rithms. Zitnick and Kanade proposed a modern coop-

erative algorithm (Zitnick and Kanade, 2000), which

assumes the two constraints and also simultaneously

provides a solution to solve the occluding boundary

problem. The belief-propagation algorithm (Klaus

et al., 2006) and the graph-cuts algorithm (Deng et al.,

2007) are also attracting much attention from com-

puter vision researchers, since they achieved good

performance on stereo disparity detection. Some of

the algorithms can also detect occlusion areas and

thus can partially avoid the problem due to the depth

discontinuity.

Several researchers have proposed to utilize

reaction-diffusion systems or equations in image pro-

cessing and computer vision research. Kuhnert et

al. found that a reaction-diffusion system described

with reaction-diffusion equations works as an opti-

cal memory device and visualizes edges and segments

of patterns from image intensity distribution (Kuh-

nert, 1986; Kuhnert et al., 1989). Adamatzky et

al. proposed novel computer architecture that per-

forms image processing with a reaction-diffusion

system (Adamatzky et al., 2005); they also pro-

posed computer algorithms utilizing the reaction-

diffusion equations and named a class of the algo-

rithms ’reaction-diffusion algorithm’. Suzuki et al.

realized a reaction-diffusion system with large-scale

integrated circuits for an application to ﬁnger-print

identiﬁcation (Suzuki et al., 2005). Ueyama et al.

proposed a model described with a reaction-diffusion

equation for explaining ﬁgure-ground separation ob-

served in the human motion perception (Ueyama

et al., 1998). The authors applied the reaction-

diffusion equations to edge detection and segmenta-

tion in image processing (Nomura et al., 2003).

The previous stereo algorithm proposed by the

authors also utilizes the reaction-diffusion equa-

tions (Nomura et al., 2009). The algorithm con-

sists of multi-sets of the reaction-diffusion equations;

each set governs areas of its corresponding dispar-

ity level, in accordance with the cooperative algo-

rithm. Diffusion processes in the reaction-diffusion

equations realize the continuity constraint; a mutual

inhibition mechanism built in the multi-sets realizes

the uniqueness constraint. However,the algorithm did

not achieve satisfactory performance on disparity de-

tection, in particular, in areas having depth disconti-

nuity.

Reaction-diffusion equations were originally pro-

posed as mathematical models for explaining pattern

formation or signal propagation observed in natural

systems such as chemical and biological systems. The

equations couple diffusion equations with reaction

terms describing chemical reaction or biological phe-

nomena; they are composed of time-evolving partial-

differential equations. For example, the FitzHugh-

Nagumo type reaction-diffusion equations are a sim-

pliﬁed model of equations describing signal propaga-

tion along a nerve axon (FitzHugh, 1961; Nagumo

et al., 1962). By modeling visual functions and realiz-

ing their computational algorithms with the reaction-

diffusion equations, we are trying to support the al-

gorithms in their biological background, even if we

do not have direct evidence that connects each of the

algorithms with the human visual perception. The au-

thors believe that such the biologically motivated al-

gorithms are interesting from both scientiﬁc and en-

gineering points of view.

In this position paper, we focus on the problem

due to depth discontinuity and propose an idea of a

visual integration algorithm that integrates intensity

edge information into the reaction-diffusion stereo al-

gorithm. Previous psycho-physical studies provided

several evidences showing that the human vision sys-

tem reconstructs depth distribution from combination

of several kinds of visual information such as binoc-

ular stereopsis, motion, texture and shading (Landy

et al., 1995). We believethat integration of edge infor-

mation into the stereo algorithm brings better perfor-

mance, also in the case of stereo disparity detection.

Since the reaction-diffusion stereo algorithm realizes

the continuity constraint with diffusion processes,

weak diffusion prevents the stereo disparity informa-

tion from diffusing. Thus, in order to realize the al-

gorithm of the integration, we introduce anisotropic

diffusion ﬁelds (Perona and Malik, 1990; Black et al.,

1998) into the reaction-diffusion equations; we can

expect that anisotropic diffusion ﬁelds modulated by

depth edge information prevent disparity information

from diffusing across depth edges. However, it is dif-

ﬁcult to detect areas having depth discontinuity prior

to stereo disparity detection. Thus, we utilize areas

having intensity edges instead of areas having the

depth discontinuity; another reaction-diffusion algo-

rithm designed for edge detection provides the in-

tensity edge information (Nomura et al., 2008). We

realize the full reaction-diffusion system that detects

stereo disparity and intensity edges; we provide re-

sults of the intensity edge detection to anisotropic dif-

INTEGRATION OF INTENSITY EDGE INFORMATION INTO THE REACTION-DIFFUSION STEREO

ALGORITHM

581

fusion ﬁelds in the proposed stereo algorithm. We

conﬁrm how the proposed algorithm works around

the areas having depth discontinuity for test stereo

image pairs provided on the Middlebury website

(http://vision.middlebury.edu/stereo/) (Scharstein and

Szeliski, 2002). In addition, we evaluate the quan-

titative performance of the stereo algorithm, in com-

parison to the previous reaction-diffusion stereo algo-

rithm and a state-of-the-art stereo algorithm.

In the future direction of the present research

work, we are planning to dynamically integrate edge

information into the reaction-diffusion stereo algo-

rithm. That is, we ﬁrst detect edge information

from both distribution maps on image intensity and

initially-detected disparity, which is obtained from an

output of a matching cost function. Next, we com-

pute tentative disparity distribution by integrating the

ﬁrstly detected edge information into the reaction-

diffusion stereo algorithm having anisotropic diffu-

sion. By repeating the two steps of edge detection and

stereo disparity detection alternately, we update stereo

disparity distribution so as to achieve better perfor-

mance of the reaction-diffusion stereo algorithm. The

present position paper is the ﬁrst step in this future

direction.

2 REACTION-DIFFUSION

The FitzHugh-Nagumo type reaction-diffusion equa-

tions consist of two time-evolving partial differen-

tial equations with two variables: activator and in-

hibitor. The equations were proposed as a model

of signal propagation along a nerve axon (FitzHugh,

1961; Nagumo et al., 1962). Let us consider two-

dimensional space (x,y) and time t; let u(x,y,t) be the

activator variable and v(x,y,t) be the inhibitor vari-

able. Then, the reaction-diffusion equations are de-

scribed as follows:

∂

u = D

∇

u+ [u(u− a)(1− u) − v]/ε, (1)

∂

v = D

∇

v+ (u− bv), (2)

where ∂

denotes ∂/∂t and ∇ denotes the two-

dimensional spatial gradient operator; D

is a diffu-

sion coefﬁcient on the activator u(x,y,t), D

is a dif-

fusion coefﬁcient on the inhibitor v(x,y,t), and ε is a

positive small constant (0 < ε ≪ 1).

The diffusion-less system derived from Eqs. (1)

and (2) behaves as two different types of system: bi-

stable system and mono-stable system, depending on

the parameter values of a and b. The bi-stable system

has two stable steady states and the mono-stable sys-

tem has the one stable steady state; the stable steady

states satisfy u(u− a)(1− u) − v = 0 and u− bv = 0.

u(x,t=0)

u(x,t=1.0)

v(x,t=1.0)

u,v

Propagation

(a)

u(x,t=0)

u(x,t=1.0)

v(x,t=1.0)

0 2010 30 40

0.0

0.5

1.0

1.5

-0.5

0 2010 30 40

0.0

0.5

1.0

1.5

-0.5

u,v

Propagation

(b)

(c)

(d)

u(x,t=0) u(x,t=0)

u(x,t=10.0)

v(x,t=10.0)

u,v u,v

0 2010 30 40

0.0

0.5

1.0

1.5

-0.5

0 2010 30 40

0.0

0.5

1.0

1.5

-0.5

Figure 1: Numerical results on the reaction-diffusion

Eqs. (1) and (2) in one-dimensional space x. The param-

eter values utilized here are as follows. (a) D

= 0.0, ε =

1.0× 10

−2

, b = 10; (b) D

= 0.0, ε = 1.0× 10

−2

, b = 1.0;

= 10.0, ε = 1.0 × 10

−3

, b = 10; (d) D

= 10.0,

ε = 1.0× 10

−3

, b = 1.0. We ﬁxed the other parameter val-

ues of D

and a at D

= 1.0 and a = 0.05 and discretized

space x with the ﬁnite difference δx = 1/5 and time t with

the ﬁnite difference δt = 1/1000 for the numerical compu-

tation. The system of Eqs. (1) and (2) is bi-stable in (a)

and (c), and mono-stable in (b) and (d). A solid black line

indicates u(x,t = 1.0) or u(x,t = 10.0), and a broken line in-

dicates v(x,t = 1.0) or v(x,t = 10.0) in each ﬁgure. A solid

gray line indicates an initial condition for u(x,t = 0) in each

ﬁgure; an initial condition for v(x,t = 0) was ﬁxed at v = 0

over the entire space x. A wave front or a wave pulse prop-

agates towards a right arrow in each of (a) and (b); a wave

front or a wave pulse does not propagate but remains at the

center edge position in the one dimensional space x in (c)

and (d).

As time proceeds, any solution converges to either of

the two stable steady states in the bi-stable system,

and to the stable steady state in the mono-stable sys-

tem.

The set of the reaction-diffusion Eqs. (1) and (2)

generates a traveling wave under the condition D

. Figure 1(a) shows a propagating wave front gen-

erated by the bi-stable system; Fig. 1(b) shows a

propagating pulse wave generated by the mono-stable

system. In the both cases, a wave front or a pulse

wave propagates at almost constant velocity in one-

dimensional space. The propagation velocity depends

on the parameter values of D

, a and ε.

When the set of Eqs. (1) and (2) is in the condition

≪ D

, it generates a static pattern (Turing, 1952).

Under the condition D

≪ D

, the inhibitor v diffuses

more rapidly than the activator u. Since the increasing

inhibitor variable v prevents the activator variable u

from increasing, it preventsthe wave front or the pulse

wave from propagating. That is, we obtain a static

pattern of the wave front or the pulse wave.

The authors found that the set of Eqs. (1) and

(2) has two functions applicable to image process-

ing (Nomura et al., 2003). They found that the bi-

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

582

stable system can detect segments and the mono-

stable system can detect edges from image intensity

distribution given to the initial condition of u(x,y,t =

0). The parameter a in Eq. (1) works as a thresh-

old level for segmentation and edge detection. Fig-

ures 1(c) and 1(d) show one-dimensional examples of

segmentation and edge detection on a step function

given to the initial condition of u(x,t = 0).

In addition to segmentation and edge detection,

the authors proposed a stereo algorithm that utilizes

multi-sets of the FitzHugh-Nagumo type reaction-

diffusion equations (Nomura et al., 2009). They con-

nected the bi-stable systems of the reaction-diffusion

equations through the parameter a as follows:

∂

= D

∇

+ µC

+[u

(1− u

)(u

− a

max

)) − v

]/ε,(3)

∂

= D

∇

+ (u

− bv

), (4)

where u

(x,y,t) and v

(x,y,t) are activator and in-

hibitor variables in each set of the reaction-diffusion

equations, and µ is a constant that controls an external

stimulus of a matching cost function C

(x,y) with a

disparity level d. Thus, each of the sets governs areas

associated with the disparity level d. A wave front

associated with the disparity level d propagates into

undeﬁned disparity areas, according to the diffusion

of the activator variable u; thus, the wave front prop-

agation realizes the continuity constraint imposed on

the stereo algorithm. A mutual inhibition mechanism

built into the multi-sets of the equations through the

next function a

max

) realizes the uniqueness con-

straint as follows:

max

) = a

max

[1+ tanh(d

− a

)],

max

= max

′

∈Θ

′



d − argmax

′

∈Θ

′



(5)

where a

and a

are constants and Θ denotes the inhi-

bition area for the uniqueness constraint (Zitnick and

Kanade, 2000).

The ﬁnite difference method with the spatial ﬁnite

differences of δx and δy and the temporal ﬁnite dif-

ference of δt discretizes partial differential operators

∇ and ∂/∂t. For example, the Gauss-Seidel method

provides a solution to a set of linear equations derived

from Eqs. (3) and (4). After enough duration of time

t, the next equation provides a disparity map M(x,y,t)

as follows:

M(x,y,t) = argmax

d∈∆

(x,y,t), (6)

where ∆ denotes a set of possible disparity levels.

3 PROPOSED INTEGRATION

ALGORITHM

Depth discontinuity causes numerous errors in the

previous stereo algorithm utilizing multi-sets of the

reaction-diffusion equations. As mentioned in the

previous section, a propagating wave front ﬁlls in un-

deﬁned disparity areas; this ﬁlling-in process realizes

the continuity constraint. However, the ﬁlling-in pro-

cess causes error around areas having the depth dis-

continuity. In order to reduce errors around the areas,

we need to prevent the ﬁlling-in process, that is, to

prevent the wave front from propagating across depth

edges.

Areas having depth discontinuity are generally un-

known prior to stereo disparity detection. Poggio et

al. proposed a visual integration algorithm that de-

tects depth discontinuity, by integrating intensity edge

information and stereo disparity distribution (Pog-

gio et al., 1988). Thus, instead of taking account

of the areas having the depth discontinuity, we uti-

lize edges detected for image intensity distribution in

the stereo algorithm. The authors also proposed an-

other reaction-diffusion algorithm designed for edge

detection (Nomura et al., 2008). Thus, the two vari-

ables u(x,y,T) and v(x,y,T) obtained by the reaction-

diffusion algorithm after ﬁnite duration of time T

are useful as intensity edge information. Let u

(x,y)

be u(x,y,T) and v

(x,y) be v(x,y,T); high values in

(x,y) and v

(x,y) denote the existence of edges, as

shown in Fig. 1(d).

Now, we modify the reaction-diffusion Eqs. (3)

and (4) in order to integrate edge information into

the stereo algorithm. Let us recall that a single set

of the reaction-diffusion Eqs. (1) and (2) generates

a wave front propagating at a velocity. The veloc-

ity depends on the diffusion coefﬁcient D

and de-

creases as D

decreases, when other parameter values

are ﬁxed. Thus, we propose to introduce the inten-

sity edge information denoted by u

and v

into the

diffusion terms as follows:

∂

= D

∇· [(1 − u

)∇u

] + µC

+[u

(1− u

)(u

− a

max

)) − v

]/ε,(7)

∂

= D

∇· [(1 − v

)∇v

] + u

− bv

, (8)

where ∇·[(1− u

)∇u

] and ∇· [(1− v

)∇v

] describe

the anisotropic diffusion ﬁelds. The two terms (1 −

) and (1 − v

) weaken the diffusion of u

and v

around the edge areas.

INTEGRATION OF INTENSITY EDGE INFORMATION INTO THE REACTION-DIFFUSION STEREO

ALGORITHM

583

Table 1: Quantitative performance evaluations on the four

stereo image pairs: CONES, TEDDY, TSUKUBA and

VENUS. We evaluated stereo disparity maps with the

bad-match-percentage error measure having the thresh-

old level of 1.0 (pixel) in non-occlusion area (nonocc.),

all area (all) and area having depth-discontinuity (disc.).

Stereo algorithms evaluated here are as follows: the

reaction-diffusion algorithm (RD), the reaction-diffusion

algorithm with the integration of intensity edge informa-

tion (RD+IE), the state-of-the-art algorithm (AdaptingBP:

ABP) (Klaus et al., 2006). We ﬁxed the parameter

values of the reaction-diffusion stereo algorithms across

the four stereo image pairs. The Middlebury website

(http://vision.middlebury.edu/stereo/) provides the stereo

image pairs, the ground truth data and deﬁnitions of the ar-

eas as well as the scores on the state-of-the-art algorithm.

See also Figs. 2 and 3 on the results of the reaction-diffusion

algorithms.

RD RD+IE ABP

nonocc. 5.53 5.36 2.48

CONES all 12.57 12.98 7.92

disc. 15.18 14.45 7.32

nonocc. 14.85 14.68 4.22

TEDDY all 20.86 20.91 7.06

disc. 30.15 29.06 11.8

nonocc. 5.98 8.46 1.11

TSUKUBA all 7.86 10.19 1.37

disc. 19.70 20.13 5.79

nonocc. 2.75 2.93 0.10

VENUS all 3.92 4.09 0.21

disc. 21.23 20.18 1.44

4 EXPERIMENTAL RESULTS

This section presents experimental results on

the well-known four test stereo image pairs:

CONES, TEDDY, TSUKUBA and VENUS, all

of which are provided on the Middlebury website

(http://vision.middlebury.edu/stereo/) with their

ground truth data and deﬁnitions of areas having

depth discontinuity. We applied two reaction-

diffusion stereo algorithms to the four stereo image

pairs; one of the two stereo algorithms is the previous

algorithm proposed by the authors (Nomura et al.,

2009) and the other one is the proposed algorithm

integrating intensity edge information.

Figures 2 and 3 show results of stereo disparity

detection on CONES and TEDDY. Let us focus on

areas surrounded by red broken lines in Figs. 2 and

3. The proposed algorithm detected disparity values

correctly along the areas having the depth disconti-

nuity, in comparison to the results detected by the

previous reaction-diffusion algorithm. In these areas,

detected intensity edges coincide quite well with the

edges having the depth discontinuity.

We quantitatively evaluated the stereo disparity maps

with respect to the bad-match-percentage error mea-

sure. Table 1 shows the results of the evaluations. Al-

though the performance of the proposed algorithm is

rather better than the performance of the previous al-

gorithm for the areas having the depth discontinuity

(disc.) on the stereo image pairs: CONES, TEDDY

and VENUS, it becomes worse for the all areas (all)

and on the image pair TSUKUBA. Table 1 also shows

the results evaluated for the top-ranked state-of-the-

art algorithm (Klaus et al., 2006). We recognize that

the quantitative performance of the reaction-diffusion

algorithm is much worse than that of the state-of-the-

art algorithm.

5 CONCLUSIONS

In the present paper, we have proposed a visual inte-

gration algorithm that integrates intensity edge infor-

mation into a reaction-diffusionstereo algorithm (No-

mura et al., 2009) for improving its performance

around areas having depth discontinuity. We pro-

vide intensity edge information obtained by another

reaction-diffusion algorithm designed for edge detec-

tion to anisotropic diffusion ﬁelds of multi-sets of

the reaction-diffusion equations utilized in the stereo

algorithm. Stereo disparity maps obtained for test

stereo image pairs show that success of the proposed

algorithm was observed in part around the areas hav-

ing depth discontinuity. However, the quantitative

performance evaluated for the proposed algorithm

was not superior but almost similar to the previous

reaction-diffusion stereo algorithm. There are sev-

eral areas in which the proposed algorithm does not

work, even if detected intensity edges coincide with

depth edges. Thus, we need to observe the situation

causing errors and to consider how to suppress the er-

rors around areas having the depth discontinuity. This

is the next work required to establish the proposed

stereo algorithm.

ACKNOWLEDGEMENTS

The present study was supported in part by the Grant-

in-Aid for Scientiﬁc Research (C) (No. 20500206)

from the Japan Society for the Promotion of Science.

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

584

(a) (b)

(e) (f)

(g) (h)

0 31 63

Disparity

(pixel)

0.0 1.0 2.0

Absolute

error

(pixel)

Figure 2: Experimental results on the stereo image pair

CONES. (a) Left image of the pair and (b) ground truth dis-

parity map. Image size is 450× 375 (pixels

) and possible

disparity levels are ∆ = {0,1, ··· , 59} (pixels). (c) Stereo

disparity map M(x, y,t = 100) detected by the previous

reaction-diffusion stereo algorithm; (d) that detected by the

proposed algorithm. (e) Absolute error map on the de-

tected map (c); (f) that on (d). (g) Deﬁnition of area hav-

ing depth discontinuity (white area) and non-occlusion area

(gray area). (h) Edge detection result obtained for the left

image (a) by the reaction-diffusion algorithm designed for

edge detection (Nomura et al., 2008); black dots and lines

indicate detected intensity edges. Parameter values of the

two stereo algorithms were ﬁxed as δx = δy = 1/5, δt =

1/100, D

= 1.0, D

= 3.0, a

= 0.13, a

= 1.5, b = 10, ε =

1.0× 10

−2

and µ = 3.0. See the literature (Nomura et al.,

2008) for parameter values utilized in (h). Table 1 shows re-

sults of the quantitative performance evaluations. The Mid-

dlebury website (http://vision.middlebury.edu/stereo/) pro-

vides the stereo image pair, the ground truth data and the

deﬁnitions of the areas.

(a) (b)

(e) (f)

(g) (h)

0 31 63

Disparity

(pixel)

0.0 1.0 2.0

Absolute

error

(pixel)

Figure 3: Experimental results on the stereo image pair

TEDDY. (a) Left image of the pair and (b) ground truth dis-

parity map. Image size is 450× 375 (pixels

) and possible

disparity levels are ∆ = {0,1, ··· , 59} (pixels). (c) Stereo

disparity map M(x, y,t = 100) detected by the previous

reaction-diffusion stereo algorithm; (d) that detected by the

proposed algorithm. (e) Absolute error map on the de-

tected map (c); (f) that on (d). (g) Deﬁnition of area hav-

ing depth discontinuity (white area) and non-occlusion area

(gray area). (h) Edge detection result obtained for the left

image (a) by the reaction-diffusion algorithm designed for

edge detection (Nomura et al., 2008); black dots and lines

indicate detected intensity edges. Parameter values of the

two stereo algorithms were ﬁxed as δx = δy = 1/5, δt =

1/100, D

= 1.0, D

= 3.0, a

= 0.13, a

= 1.5, b = 10, ε =

1.0× 10

−2

and µ = 3.0. See the literature (Nomura et al.,

2008) for parameter values utilized in (h). Table 1 shows re-

sults of the quantitative performance evaluations. The Mid-

dlebury website (http://vision.middlebury.edu/stereo/) pro-

vides the stereo image pair, the ground truth data and the

deﬁnitions of the area.

INTEGRATION OF INTENSITY EDGE INFORMATION INTO THE REACTION-DIFFUSION STEREO

ALGORITHM

585

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