Confidence-based Refinement of Corrupted Depth
Maps
Satoshi Ikehata∗ and Kiyoharu Aizawa∗∗ University of Tokyo, Tokyo, Japan

E-mail: {ikehata, aizawa} @hal.t.u-tokyo.ac.jp

(a)

(b)

(c)

Fig. 1. The depth map refinement results by our method, where (a) Input
images, (b) Initially acquired depth maps (by match propagation [11]) and (c)Depth maps refined by our method.

Abstract—This paper present a practical depth-map refinement system designed for highly corrupted multiple depth
maps. We define a pixel-wise confidence measurement of depth
value and apply the three-steps depth-map refinement scheme(i.e.confidence-based depth-map fusion, confidence-weighted bundle optimization and super-pixel-based planar propagation) tomaximize the whole reliability of depth maps.
Our experimental result shows that our refinement algorithm
can dramatically improve highly corrupted depth maps acquiredby previous approaches.

I. I NTRODUCTION
Depth map is an image where distances between the camera
and the scene are assigned to each pixel. Recently, the depthmap processing has become one of most attractive areas inimage processing and computer vision since many researchers
have reported that depth information helps various visualtasks such as image-based rendering [10], background subtraction [23], object tracking [5] and so on.
As is well known that there are roughly two ways to acquire
depth maps. One is the image-based approach so called stereo
or multi-view stereo (MVS) and using active sensors such as

Time of Flight (TOF) sensors or structured light sensors suchas MicrosoftKinect is the other way. Most existing imagebased depth-map estimation algorithms are further classified
into two categories, one is the feature-based approach, andthe other is the optimization-based approach. In the featurebased approach, each pixel’s depth value is recovered from thelocal matching with the highest stereo confidence score [8].
While effective in a narrow baseline case, this winner-takes-all
strategy is easily disrupted under the large projective distortionand repetitive textures where correct correspondences are
difficult to be found. To avoid incorrect matching, some works
find sparse salient matches around textured areas initially, andthen propagate them into surroundings [11], [9]. Furthermore,various optimization-based algorithms have been proposed,
which introduce spatial constraints of depth maps in their costfunction [19], [20], [13].
Unfortunately, while dramatic developments in the imagebased and the active-sensor-based depth-map acquisition, theaccuracy of depth maps is yet problematic. The active sensors
are still poor around the object boundaries due to the occlusion
problem and sensor satiations. And most existing image-basedalgorithms including both feature-based and optimizationbased approach do not consider any consistency of depth maps
among views (especially when there are more than two views).
The primary contributions of this work are twofold. First,
we present a practical depth-map refinement system designed
for highly corrupted multiple depth maps which enforces theconsistency of depth maps. We define a pixel-wise confidence
measurement of depth value and apply the three-steps depthmap refinement scheme to maximize the whole reliability of
depth maps. Secondly, we propose a confidence-based dense
track extraction algorithm, which is a core part of the bundle
optimization for multiple depth maps. While Li et al. [12] were
the first to introduce the bundle adjustment to merge depth
maps, they extracted dense tracks by simply connecting the
successive two-view correspondences and then optimized eachtrack with re-projection errors, therefore their method cannot
be applied when no explicit correspondence among views isprovided (e.g., only depth maps and camera parameters areprovided). In contrast that, we try to use all of the views
simultaneously to find the optimal tracks by taking advantagesof confidence scores. In our experimental results, we apply our
method to corrupted depth maps acquired by multi-view stereo
algorithms and show that our method improve the accuracy ofdepth maps dramatically.

Initial depth-map acquisition

Confidence-based depth-map refinement
TargetView 1

Low confidence

View 2

Low confidence

View 1
Image-based approach
Image sensor
Depth sensor

S

S
Track

View 3High confidence

View 2

OptimizeSS・・・ S(b) confidence-based bundle-optimization(a) confidence-based depth-map fusion

Active-sensor-based approach

Corrupted depth map

High confidence

Loop untilconvergence(c) super-pixel-based planar propagation

Confidence Evaluation

Fig. 2. System overview.

II. C ONFIDENCE - BASED S IMULTANEOUS D EPTH - MAPS
R EFINEMENT
In this section, we present the three-steps depth refinement
algorithm which is composed by, namely, (a) confidence-based
depth-map fusion, (b) confidence-based bundle optimizationand (c) super-pixel-based planar propagation (see Figure 2).
We define a confidence metric which evaluates the consistencyof depth maps among views by borrowing the investigationof Hu et al. [8] which quantitatively evaluated thirteen kindsof stereo confidence metrics using both indoor and outdoor
datasets.
Henceforth we rely the following assumptions:(1) There are at least two initial depth maps of a static scene.
(2) Each view has aligned depth and color image.
(3) The position and intrinsic parameters of sensors are known.
When initial depth maps are recovered by the imagebased approach (e.g., multi-view stereo), those assumptions
are mostly satisfied. And fortunately, we can satisfy thoseassumptions when we use active sensors since recent active
depth sensors can also capture color images and camera
parameters could be recovered by the applying structure-frommotion method [17] with them. To align depth and color
images, we can usually use the calibration software provided
by the manufacturer, otherwise other calibration methods suchas [4] are available.
A. Corresponding points among depth maps
Corresponding points of two depth maps are uniquelydetermined if the camera parameters of each viewpoint andone of each depth value are known. It means that even ifeither of depth value is incorrect, we can get the correct match(and corresponding scene 3D point), which is the importantcharacteristics of a depth map.
Now, we assume that xi and xj in image i and j arecorresponding each other, their relationship is represented asfollows,
1
Kj RjT (tj − ti ),di1x̃i = f ji (xj ) ∼ Ki (RiT Rj )Kj−1 x̃j − Ki RiT (ti − tj ),dj(1)x̃j = f ij (xi ) ∼ Kj (RjT Ri )Ki−1 x̃i −where di,j are depth values of each pixel, Ki,j are intrinsic
matrices, Ri,j are rotation matrices and ti,j are translations ofeach camera. Note that in Eq. (1), each point is represented
by a homogeneous coordinate system.
When a correspondence (xi = [ui , vi ]T , xj = [uj , vj ]T )and each camera matrix P i = Ki (RiT | − RiT ti ), P j =Kj (RjT | − RjT tj ) are provided, we can compute the 3D point
X by the triangulation as follows,AX̃ = 0 ii
P11 − P31ui
iivi
 P21 − P31
A= jj
P11 − P31
uj
j
j
P21
− P31
vj

i
i
P12
− P32
ui
i
i
P22 − P32
vi
j
j
P12
− P32
uj
j
j
P22
− P32
vj

i
i
P13
− P33
ui
i
i
P23 − P33
vi
j
j
P13
− P33
uj
j
j
P23
− P33
vj

i
i
P14
− P34
ui
i
i
P24 − P34
vi 
j
j
P14
− P34
uj 
j
j
P24
− P34
vj ,



(2)

where X = [X̃(1) /X̃(4) , X̃(2) /X̃(4) , X̃(3) /X̃(4) ]. Inversely, a
3D point X and camera matrices give the projections into
each viewpoint are as follows,di (xi )[ui , vi , 1]T = P i [X(1) , X(2) , X(3) , 1]T
dj (xj )[uj , vj , 1]T = P j [X(1) , X(2) , X(3) , 1]T .

(3)Here we emphasize that one of each depth value is enoughto find the correspondence between two views and we can reproject the 3D point recovered from the correspondence into
each view with depth values. It means that one depth value
gives estimation of the depth value of corresponding pixels,
which is the core of our confidence-based refinement scheme
described in following sections.

B. Confidence metric for evaluating consistency of depth maps
Within the field of the stereo matching, there have beenmuch researches about the confidence metric to accurately
estimate how trustworthy the correspondences are (e.g., [6],[8]). However, unfortunately, there is little research about theconfidence of depth [19], [16] or limited in the domain of
active sensors [15]. Therefore, we design a new heuristic
confidence metric for our algorithm by importing ideas fromstereo confidence metrics [8].
Inspired by [19], we define the confidence of the depth as
the maximum of the confidence of two-view correspondences
given by Eq. (1),Ci (xi ) , max cij (xi ),j(4)where xi is a pixel in the i-th view and cij is a confidence
metric for each two-view correspondence from xi to the j-th
view. Note that maximizing cij can implicitly compensate forthe case where an accurate depth value wrongly gives small
cij because of the occlusion.
Requirements for our confidence metric in our algorithms
are threefold:(1) Our confidence metric can estimate how trustworthy theprovided depth values are.
(2) It leads as small as possible false positive errors.
(3) Our confidence metric should give a high score for highconsistency among views.
Under those requirements, we defined cij as follows,cij (xi ) ,Confgeo ,Confphoto ,Confgeo ∗ Confphoto ≤ 1,
(5)
1
,
(6)
|xi − f ji (xj )|
1+
ηa
1
, (7)
|F (xi , xj ) − minpi ∈Nxi F (pi , xj )|1+ηbwhere xj = f ij (xi ) and ηa and ηb are weights to regularize
each confidence to 1, and the function F evaluates the photoconsistency of two views locally (we will discuss about thisfunction later). pi are pixels in the i-th view collected fromthe epipolar line of xj . cij is composed of two terms, one
evaluates the geometry-consistency and the other evaluates thephotometry-consistency. Note that we observed that introducing both of them decreased the false positive errors rather thanintroducing either of them.
We simply use the well known metric called forwardbackward error [19] as the geometry-consistency term which
gives high score when the correspondence is bidirectional. As
for the photometry-consistency term, we are inspired by the
Hu et al. [8]’s LRD (Left-Right Difference) which gives a highscore if the photoconsistency of the correspondence is locallyminimum since otherwise, it means that there exists a better
correspondence.
There are many options for the photoconsistency function
F . The most commonly used function in the stereo case may
be a color distance or block-based similarity measures such

as SSD or NCC. However in case viewpoints are widely
separated, we should choose the metric which have a highreliability under large perspective distortions. Here we use theDAISY descriptor [21], which are robust to distortions yet lesscomputational cost comparing to well known local descriptors
such as SURF [2].
C. Confidence-based depth-map fusion
Our confidence-based depth-map fusion scheme relies thefact that each of depth value in corresponding points gives
another depth value (see section II-A). It means that reliable
pixels which correspond to unreliable pixels can give better
estimation of depth of those unreliable pixels, which is thecore concept of our fusion scheme.
The algorithm is described in Algorithm. 1. Assume wehave calculated confidence values for each pixel as describedin II-B, and we try to refine the unreliable depth d(xi ) of xi
in i-th view. Firstly, we search for all pixels xj from otherviews which can project onto xi by using Eq. (1). While
we may find many correspondences, we try to find the best
correspondence between xi and the pixel with most reliable
depth value since an inaccurate correspondence can rather
disrupt d(xi ). Therefore we choose the pixel with the highest
confidence score as follows,
xmax , argmaxxj Cj (xj ).

(8)Note that we use Cj (xj ) instead of cji (xj ) since the priorityof the consistency in cji (xj ) gives a low confidence score
if di (xi ) is inaccurate. In addition, we remove xj whose
corresponding point is occluded in i-th view by a simple
photo-consistency check (i.e., we only collect pixels wherethe `2 color distance from xi is less than 10).
Although the correspondence of xi and xmax gives anotherdi (xi ) by Eq. (3), we should prevent the accuracy of d(xi )from decreasing by this fusion. Therefore, we only update
d(xi ) when these two confidence checks between C(xi ) andC(xmax ) are satisfied,C(xmax ) > εa , C(xmax ) − Ci (xi ) > εb(9)where εa and εb are thresholds set to 0.5 and 0.2, respectively,in our implementation (note that confidence values are from0 to 1). The first condition in Eq. (9) is not necessary but
important to avoid any inefficient propagation.
While our depth-map fusion scheme refines low-confidence
pixels, especially where depth values are not assigned initially,
there are two limitations. First, only low-confidence pixels arerefined since the direction of our propagation is from highconfidence pixels to low-confidence pixels. Secondly, refinement is based on per-pixel, two-view correspondences without
considering about sub-pixel correspondences and multi-viewgeometry. Thus, after this step we apply bundle optimization
described in the next section for further refinement.
D. Confidence-based bundle optimization
Our confidence-based bundle optimization algorithm mainly
builds on the standard bundle adjustment scheme which isAlgorithm 1 Confidence-based depth-map fusion
Cmax ← 0, C and c are already calculated
for j = 1 to n − 1 dofor i = j + 1 to n do
for all xj in j-th view do
xi ← f ji (xj )if Cj (xj ) > Cmax (xi ), Cj (xj ) − Ci (xi ) > εb andCj (xj ) > εa thenCmax (xi ) ← Cj (xj ), xmax ← dnew
end ifend forfor all xi in i-th view doxj ← f ij (xi )if Ci (xi ) > Cmax (xj ), Ci (xi ) − Cj (xj ) > εb andCi (xi ) > εa thenCmax (xj ) ← Ci (xi ), dj (xj ) ← dnew
end ifend forend forend formainly applied in the structure-from-motion framework [17]to simultaneously refine 3D points and camera parameters byminimizing the sum of re-projection errors (i.e., `2 distances
between projections of original and refined 3D points). Since
bundle adjustment builds on an assumption that re-projection
errors obey the zero-mean Gaussian distribution [1], outliersin the input sequence of points easily disrupt the optimization,
which is a critical issue when we apply the bundle adjustment
to correspondences extracted from highly corrupted depth
maps. Unlike Li et al. [12] which simply connected matchingpixels of stereo pairs to build a track and then filtered out
the pixels as outliers whose re-projection error is less than acertain threshold, we use the confidence score to intelligently
extract the tracks from highly corrupted depth maps and apply
confidence-weighed bundle adjustment to refine whole track.
1) Dense Track Extraction: We define a track as a sequenceof pixels over views which assumed to be pointing at the same
3D point. tk = {xk1 , xk2 , ....xkN (t) }.
We illustrate the algorithm in Table I. Track extraction
process is described by two steps, i.e., the selection of thestarting point and the order of next viewpoints. The starting
points of each track are critically important since inaccurate
depth values can lead unreliable tracks. Therefore we firstly
set a pixel with the highest depth-confidence score (Ci ), as astart point of each track. Then we collect correspondences inthe track considering two factors for selecting the next view;
first is the reliability of the correspondence between current
view and the next view (cij ), and second is the possibility of acorrespondence between the next view and the view after the
next (Cj ) because we would like to build the longer tracks.
We finish the track extraction if the confidence of current pixel(Ci ) is under a threshold (0.5 in our implementation).
The important property of our track extraction algorithm
is twofold. First, our method intelligently chooses the first

viewpoint from all of viewpoints which is practical when using
depth maps where there is no explicit first view (e.g., capturedby active sensors). Secondly, when tracks are extracted, we
can also acquire the confidence of each pixel, which could be
used as the weight for the bundle optimization described in
the next section.
2) Confidence-weighted Bundle Adjustment: When tracks
are built, we firstly reconstruct a 3D point for each track by
applying triangulation in a similar manner with Eq. (2). Then
we apply the weighted bundle adjustment for each track asfollows,
Nt
X
min E =
wi2 f (xi , Pi X),
(10)
X

i=1

wi = C1
Qi
wi = C1 k=2 c(k−1,k)(i = 1),(else).

(11)

Here, Nt is the number of points included in a track and P
is a projective matrix for each viewpoint. xi is the i-th points
in the track and X is the 3D point corresponding to the track
initialized by the triangulation result. f is the `2 re-projection
error and wi is a penalty weight reflecting confidence scores(we discuss about the weight later). The objective is minimized
with the Levenberg-Marquardt algorithm. When the optimization is finished, we can get a set of refined 3D point cloud.
Unlike Li et al. [12] which used current re-projection
errors as the penalty weight, we relies confidence of eachcorrespondence which has higher score the former in the
track. Therefore, our method not only filters out outliers,
but also refine unreliable correspondences by propagating
reliable information of former part of the tracks. It means thateven if the depth value of the last pixel is not accurate, theoptimization is not disrupted by the pixel since the weight isvery small, in spite that, the inaccurate pixel could receive
the benefit from the optimization. One problem is that we
acquire the refined quasi-dense 3D points as the output ofour bundle optimization. To get refined dense depth maps, wefinally apply super-pixel based planar fitting described in thenext section.
E. Super-pixel based planar propagation
While refined 3D points are not dense, we can use themas the seeds of segmentation based approach [9], [22] to get
dense depth maps. This approach assumes that the region withthe similar observation has a loose change of structure andapproximate the underlying local structure as a simple plane.
Although this approach is usually used for the last step ofstereo algorithms to remove outliers, it could be also applied
to estimate depth maps.
Firstly we divide images into super-pixels by mean-shift
color segmentation [3]. We assume that the depth map of eachsuper pixel is approximated by a plane and depth values inside
the super pixel is represented as follows,d(xi ) = aui + bvi + c.

(12)

where d is the depth value provided by the projection of arefined 3D point, xi = [ui , vi , 1]t is the i-th point in theTABLE I
A LGORITHM OF CONFIDENCE - BASED TRACK EXTRACTION
1.
2.
2.
3.
4.

5.
6.
7.
8.

Create List, M atch and T rack. List and M atch arestacks of pixels and T rack is the stack of M atch.
Put all pixels into List with confidence scores Ci and cij
Sort List in a descending order of Ci .
Pull the top of List and set it on xi and put xi into M atch
if it has not been included in T rack.
Put xj in j-th iamge into M atch, which satisfies following
conditions.
(1) xj is a correspondence of xi .
(2) j = argmaxk cik Ck .
(3) j-th view has not been included in M atch.
If all views have been included in M atch or cij Cj is lessthan a certain threshold, go to 4. Otherwise set xj on xi theniterate 3.
If the size of M atch is less than three, then go to 7.
Otherwise, go to 4.
Put M atch into T rack
If List is not empty, clear M atch and go back to 2.
Get a set of reliable tracks in T rack.

super-pixel and a, b, c are the plane parameters. While threeprojections is enough to estimate the parameters, we use all
projections to calculate the plane parameters using the voting
approach [22] for a robust estimation. After plane parameters
for each super pixel are estimated, we can recover depth
values of regions with no projection from refined 3D points
by Eq. (12), and finally get refined dense depth maps.
We note that our three-step refinement scheme works wellwhen it is applied for several times (at least twice). The
computational cost is less problematic since there is no large
scale optimization in our scheme and each bundle optimization
could be processed in parallel because it is performed on tracklevel.

used five of eleven images. Because of the limited memory
space, we down sampled the images to 768 x 512 from theoriginal resolution of 3,072 x 2,048. For evaluating depth map
quantitatively, we introduced relative depth error histogramsas the measurement of accuracy which proposed by Strecha
et al [18] which is defined as follows.
Xhk ∝δk (|Dlij − Dsij |0.01Dσj )(13)ij

Dlij and Dsij are the ground truth depth and the estimateddepth value of camera j and position i. Dσj is the varianceof the correct depth values in image j given by ground truth
data. δk () returns 1 if |Dlij − Dsij | is in [k × 0.01Dσj , (k + 1) ×
0.01Dσj ], otherwise returns 0. Then we construct histogramsof the relative errors, which regularized by the total numberof the pixels which are seen at least from two views. If theerror is larger than the range of k > 20, we include it in thehistogram corresponding to k = 20.
We illustrate the experimental result in Figure 3, Figure 4
and Figure 5. In Figure 4, we construct error maps whosecolors are decided by the error level. When k is increased,
the color changes from blue (k = 0) to red (k = 20). And
green areas are not used in the evaluation because they don’t
have ground truth values or they could not be seen from morethan two views. The experimental results show that depthfusion step reduced high-level errors (k = 20) of input depth
maps, in contrast our confidence-weighted bundle optimizationand super-pixel-based planar propagation dramatically reduced
errors of depth maps especially with low-level errors (k < 3).
These results show our three-step scheme worked as we had
intended. One example of refined depth map is presented in
Figure 5.

III. E XPERIMENTAL R ESULTS
A. Implementation and Materials
We implemented the proposed approach in C/C++ platform.
We used DAISY library provided by Tola et al. [21] andlevmar [14]; the library for Levenberg-Marquartdt algorithm.
All experiments are performed on a 3.06 GHz Intel Core2
Duo CPU and 4GB RAM. The main computational costs are
the calculation of confidence metrics (each needs about 1
minute per image). For evaluating our algorithm, we test our
method on two kinds of standard datasets; Strecha et al.’s [18](outdoor datasets) and Middlebury [7] (indoor datasets). First,
by using outdoor datasets, we quantitatively evaluate how eachof our step refine the depth maps. Then we show the 3D
reconstruction result from indoor datasets.
B. Evaluation of contribution for each refinement step
We applied our method to the output of Strecha et al. [19]’salgorithm, which indicates middle-level performance in thecomparison in Middlebury online evaluation (many regionsof recovered depth maps are corrupted). In this experiment,
we use the fountain-P11 datasets since the ground truth data
is provided by [21] which are captured by a laser scanner.
Those examples contained eleven images, however we only

C. Evaluation with Middlebury datasets
In this experiment, we demonstrate the effectiveness of
our method by Middlebury dinoSparseRing dataset [7] which
includes 16 images captured from largely different viewpoints
and have little textures. In this experiment, the input depth
maps are recovered from one of local feature-based approaches
called match propagation [11] using the SURF descriptors [2]as seeds. Experimental results are illustrated in Figure 1. Since
the dataset has less texture, the initially recovered depth maps
have large amount of holes. However, by applying our threestep refinement scheme (in this case five times), we can recoverrefined depth maps where most of holes are filled in.
Finally, we illustrate the qualitative comparison with Hu
et al. [8] in Figure 6. To demonstrate the performance of
our track extraction algorithm and confidence-weighted bundle
optimization, we applied the bundle optimization algorithm
which is similar manner with Hu et al. [8](i.e., extract tracks
from two successive image pairs and set a penalty weightby re-projection errors). Note that the algorithm is not completely same with them since original work extracts tracks
based on dense two-view matches not from depth maps. The
experimental result show that our intelligent system of bundle

50

Strecha et al. (Input)
depth-map fusion
confidence-weighted bundle optimization + super-pixel-based planar propagation

Occupancy %

45
40
35
30
25

20
15
10
5
0
1

Accurate

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

k (0.01×σ)

18

19

20

Inaccurate

Fig. 3. Error histograms of fountain-P11 datasets. We averaged results of {3,4,5,6,7 }-th images.
(a)

(b)

Fig. 6. Comparison of refined depth maps by bundle optimization algorithms,(a) the similar manner with Hu et al. [8], (b) our method.

Strecha et al. (Input)

depth-fusion

confidence-weighted bundle optimization + super-pixel-based planar propagation

Fig. 4. Examples of error maps. Here, we show only the error maps
corresponding to 5-th image of fountain-P11 datasets.

without refinement

refined by our three-step methods

Fig. 5. Examples of the depth map refinement. The left one is the input depth
map and the right one is refined by our three-step scheme (our refinement
scheme is applied once).

optimization works better than simple scheme of Hu et al. [8].
IV. C ONCLUSION
In this paper, we proposed a novel framework of theconfidence-based depth-map refinement scheme. Our experimental results showed that our algorithm can estimate depth
maps in detail. Future work will focus on tuning our parameters empirically by training on some data, and applying ouralgorithms to images captured by active sensors.
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