DEPTH INPAINTING WITH TENSOR VOTING USING LOCAL
GEOMETRY
Mandar Kulkarni
1
, A. N. Rajagopalan
1
and Gerhard Rigoll
2
1
IPCV Lab, Electrical Engineering Department, Indian Institute of Technology Madras, Chennai, India
2
Institute for Man-machine Communication, Technical University Munich, Munich, Germany
Keywords:
Tensor Voting, Range Maps, Range Inpainting.
Abstract:
Range images captured from range scanning devices or reconstructed form optical cameras often suffer from
missing regions due to occlusions, reflectivity, limited scanning area, sensor imperfections etc. In this paper,
we propose a fast and simple algorithm for range map inpainting using Tensor Voting (TV) framework. From
a single range image, we gather and analyze geometric information so as to estimate missing depth values.
To deal with large missing regions, TV-based segmentation is initially employed as a cue for a region filling.
Subsequently, we use 3D tensor voting for estimating different plane equations and pass depth estimates from
all possible local planes that pass through a missing region. A final pass of tensor voting is performed to
choose the best depth estimate for each point in the missing region. We demonstrate the effectiveness of our
approach on synthetic as well as real data.
1 INTRODUCTION
Due to decreasing costs of range scanners, range im-
ages of 3D structures are becoming easily available.
Range maps are widely used for applications such as
3D reconstruction, image based rendering (IBR) and
matting. Also, depth estimation from optical images
continues to be an exciting area of research. Many
a time, range maps derived from these modalities of-
ten have missing regions. This could be due to vari-
ous factors such as occlusion, low reflectivity, limited
field of view, sensor imperfections etc. One needs to
fill-in the missing regions for effective use of this 3D
data.
Many approaches exist in the literature for es-
timating small or medium sized missing regions in
range data. In (Stavrou et al., 2006), algorithms for
2D image inpainting are applied to 3D data assum-
ing range maps as images. In (Sharf et al., 2004), an
example-based approach is used for estimating miss-
ing values. The best match from an example dataset
is found and the patch is fitted by aligning it with
the surrounding surface. In (Xu et al., 2006), a sin-
gle range image is used for estimating missing values.
First, the normal directions at the missing values are
estimated based on training data from image patches.
Then a 3D surface is produced based on estimated
normal directions. In (Bhavsar and Rajagopalan,
2010), an intensity image along with the range im-
age is used for estimating missing values. The in-
tensity image is registered with the range map. Us-
ing the segmented intensity image, missing values are
estimated. Techniques based on stereo (Frueh et al.,
2005)(Frueh et al., 2004) and structure-from motion
(Abdelhafiz et al., 2005)(Brunton et al., 2007)(Dias
et al., 2003) also exist and they use multiple intensity
images to estimate 3D geometry.
In this work, we attempt inpainting given a single
range observation derived directly from a range scan-
ner or from optical images using the multi-view stereo
principle. We achieve this objective within a ten-
sor voting framework. Tensor Voting (TV) is a non-
iterative framework originally developed by Medioni
et al. (Guy and Medioni, 1997)(Medioni et al., 2000).
This formalism which is based on tensor calculus for
representing data and on a voting process for commu-
nication, identifies geometrical structures described
by the layout of a sparse and potentially noisy N-D
dataset. It also provides for each point a saliency mea-
sure about the possibility of its belonging to the type
of structure being inferred. In our approach, we first
detect edges in the range map using tensor voting.
We extrapolate edge components inside the missing
region. Segmentation based on edge interconnection
provides a cue for region filling. We model depth vari-
ations based on local geometry and estimate individ-
22
Kulkarni M., N. Rajagopalan A. and Rigoll G..
DEPTH INPAINTING WITH TENSOR VOTING USING LOCAL GEOMETRY.
DOI: 10.5220/0003840100220030
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 22-30
ISBN: 978-989-8565-03-7
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

ual plane equations using 3D tensor voting. Note that
we perform local plane fitting as in (Bhavsar and Ra-
jagopalan, 2010) but we do not use the intensity im-
age for synthesizing missing values. This eliminates
computations required for registering intensity image
and range map. We pass depth estimates, obtained
from different plane equations, inside the missing re-
gion. A final pass of 3D TV is performed to decide
the best depth estimate. As lower depth regions oc-
clude higher depth regions, we perform region filling
starting with the lowest depth region and moving on
to higher depth regions.
The paper is organized as follows. Section 2
describes our TV-based edge detection and Least
Squares (LS) based edge interlinking process to en-
able robust segmentation. In section 3 we discuss re-
gion filling by local plane fitting using 3D TV. Exper-
imental results are provided in Section 4 for purpose
of validation.
2 TV FOR EDGE LINKING
Range maps usually have regions with varying depths.
Some depth maps (such as piecewise planar scenes)
exhibit sharp variations in depth from one plane to an-
other while some display smooth variations. Without
the missing region, we have continuous boundaries
between these regions while a defect renders these
boundaries discontinuous. It is important to recon-
nect these boundaries as they define the extent of each
region in the range map.
Edge detection based on Canny, Sobel or Prewitt
is sensitive to noise and tends to produce weak and
spurious edges when applied to real depth maps. For
robust edge detection, we resort to 2D tensor voting.
The key idea is that true edge pixels form coherent
edges in the range image. We first apply basic Sobel
operator on the image. We set edge strength thresh-
old to a high value so that only strong edge pixels get
selected. These pixels are more likely to be true edge
pixels. We refer to them as Reference edge pixels as
they serve as reference for selecting weaker edge pix-
els subsequently. Next, we reduce the edge strength
threshold by a small value so as to allow selection
of weak edge pixels (some of them might even be
noise). We refer to them as Candidate edge pixels.
We use 2D tensor voting for selecting true edge pix-
els from Candidate edge pixels. Here, Reference and
Candidate edge pixel locations act as tokens for ten-
sor voting. We find the coherency of Candidate edge
pixels with respect to Reference edge pixels. Here,
Curve saliency (obtained from the tensor voting) acts
as a measure of coherency. For 2D tensor voting, a
tensor T can be decomposed as
T = (λ
1
− λ
2
)e
1
e
T
1
+ λ
2
(e
1
e
T
1
+ e
2
e
T
2
) (1)
Here λ
1
and λ
2
are eigenvalues of T correspond-
ing to eigenvectors e
1
and e
2
, respectively. The
Curve saliency is given by
Curve saliency = λ
1
− λ
2
(2)
Thus, among all Candidate edge pixels, only those
with high Curve saliency value are retained. The
threshold for saliency is chosen empirically. The re-
tained Candidate edge pixels now become a part of
Reference edge pixels to be used for the next stage.
We iteratively reduce the edge strength threshold in
small steps and select edge pixels using the above pro-
cedure. As only those edge pixels that form coherent
connections are retained, our edge detection process
is robust to noise.
We demonstrate the performance of our edge de-
tection method on a real range map. Fig.1(a) is a sam-
ple real depth map. Fig.1(b) shows reference edge
pixels for one of the stages and Fig.1(c) shows can-
didate edge pixels obtained with smaller threshold.
It can be seen from Fig.1(d) that our approach se-
lects only those candidate pixels which form coherent
connections with reference edge pixels. The selected
edge pixels (after the first iteration) are shown in red
in Fig.1(d) along with reference edge pixels.
For interconnections, we only consider edges that
are close to the missing region. The approach de-
scribed in (Jia and Tang, 2003) uses 2D tensor vot-
ing for connecting edges. It assumes that there are no
possible edge intersections within a missing region.
In contrast, we use a least squares (LS) approach for
modeling and connecting edges. The TV based ap-
proach, due to its limited voting range, imposes con-
straints on the size of the missing region. Also, it can-
not handle possible edge intersections within a miss-
ing region. We attempt to connect two broken (dis-
continuous) edge components by a smooth curve. To
enforce smoothness, we fit a second order polynomial
to the edge data. A general equation of a second order
polynomial is given by
Y = aX
2
+ bX +c (3)
where (a,b,c) are the parameters of the polynomial
fit. Here, X and Y are the spatial coordinates of a
data point. We consider combinations of two separate
(broken) edge components at a time. We define
Edge(i) = [x
i
,y
i
] i = 1 : N (4)
where [x
i
, y
i
] is the co-ordinate of the i
th
edge pixel
and N is the total number of edge pixels from the two
edge components. We use LS to fit a second order
DEPTH INPAINTING WITH TENSOR VOTING USING LOCAL GEOMETRY
23

(a) (b)
(c) (d)
Figure 1: Robust edge detection. (a) A real depth map. (b) Reference edge pixels. (c) Candidate edge pixels. (d) Retained
candidate edge pixels after the first iteration (shown in red).
polynomial to it and estimate the parameters (a,b,c).
Least squares, if used directly for parameter estima-
tion, can be erroneous as it is sensitive to outliers. We
first perform 2D TV on the edge components and re-
move pixels with curve saliency less than a threshold.
We calculate the goodness of the fit with the data
as
Error
j
=
N
∑
i=1
(y
i
− ax
2
i
− bx
i
− c)
2
(5)
where Error
j
is the polynomial fit error for the j
th
edge combination.
For a given edge segment, we consider all poten-
tial combinations of edge components and find that
edge combination which results in the least value for
Error. We connect these edge segments using the
parametric equation for that combination. The miss-
ing edge points [x,y] are generated as
y − ax
2
− bx − c ≤ Edge threshold (6)
where (a,b,c) are parameters for the best match. The
value of Edge threshold should be close to zero. We
set its value to 0.5 during experimentation.
For each individual edge segment, we carry out
the same procedure and connect it to the best match
derived from the remaining edge components. An
edge segment which does not have a good match with
any of the broken edge segments may result in an in-
tersection within a missing region. We extrapolate
such an edge segment across the missing region us-
ing its own parametric equation. This enables us to
handle even edge intersections within the missing re-
gion. Fig.2(b) shows edge linking result on the broken
edges of Fig.2(a).
(a) (b)
Figure 2: (a) Edge components near defect area. (b) Edge
linking result using our LS approach.
3 REGION FILLING
In this section, we discuss two important steps
namely, missing region segmentation and local plane
fitting.
The edge interconnections, as described in Sec-
tion 2, segment the missing regions and hence act as a
cue for region filling. They define the correspondence
i.e. part of available information that must be used for
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
24

Figure 3: Missing region segmentation. Green color indicates the missing region.
synthesizing a given missing section. Each edge inter-
connection passing through a missing region divides
it in two (left/right) sections. Each missing point co-
ordinate [X,Y ] is locally assigned a region label L/R
as follows.
If (Y − a
i
X
2
− b
i
X − c
i
) ≤ 0, M(X ,Y ) → L
else M(X,Y ) → R (7)
where (a
i
,b
i
,c
i
) is the parametric equation of the i
th
edge interconnection and M indicates missing region
area. Based on the parametric equations of all the
edge interconnections and the relative position of a
missing point with respect to these interconnections,
each point in the missing region is globally assigned a
region label S
i
(i=1:p) where p is the total number of
segments. Using a bigger sub-image around the miss-
ing region, the available information segment with re-
gion label S
i
is chosen to fill in the values for the miss-
ing segment with the same region label S
i
. Fig.3 illus-
trates segmentation and labeling of a missing region
based on the curve equations.
We next move onto assignment of depth values for
each missing point.
3.1 Local Plane Fitting
We assume that points with similar depth values are
part of a local linear plane in 3D. Assuming local pla-
narity, a missing point is likely to belong to any one of
the many planes that pass through the missing region.
The task that remains is to identify the correct plane
from a set of planes.
The general equation of a plane in 3D is given by
Ax + By +Cz + D = 0 (8)
where (A,B,C) is the normal to the plane. We use 3D
tensor voting to estimate the normal direction. For
a missing segment with label S
i
, we collect all the
points with label S
i
from known information. These
points act as tokens for 3D TV. The tokens are formed
as
Data(i) = [x
i
,y
i
,z
i
] i = 1 : N (9)
where [x
i
,y
i
] is the coordinate and z
i
is the depth at
known points. Here, N is the total number of tokens
available to synthesize the missing segment informa-
tion.
Each token has a 3x3 symmetric, positive semi-
definite tensor matrix associated with it. At start, each
tensor matrix is initialized to a 3x3 identity matrix.
Following this, ball and stick voting are performed in
the feature space. With the initialized shape and size,
the tensors gradually deform due to the accumulation
of votes cast from neighboring tokens within a cer-
tain range. The scale σ of the voting field controls the
size of the voting neighborhood and the strength of
the votes. A large value of σ corresponds to a higher
voting range. The votes received contain both mag-
nitude and orientation information. A generic tensor
T with eigenvectors and eigenvalues can be decom-
posed into stick and ball components as
T = (λ
1
− λ
2
)e
1
e
T
1
+ (λ
2
− λ
3
)(e
1
e
T
1
+ e
2
e
T
2
)
+λ
3
(e
1
e
T
1
+ e
2
e
T
2
+ e
3
e
T
3
) (10)
Here λ
1
, λ
2
and λ
3
are eigenvalues of T corresponding
to the eigenvectors e
1
, e
2
and e
3
, respectively.
We use the method proposed in (Lee and Medioni,
1997) to find the normal direction. We perform ball
voting and stick voting on the variable Data. We ob-
tain a 3x3 tensor matrix at each token. The eigen-
vector with the highest eigenvalue shows the normal
direction at the token. We take normal directions at all
token points and find the co-variance matrix. As each
point on the same plane should have a unique normal
direction, the eigenvector of the co-variance matrix
corresponding to the highest eigenvalue is treated to
be the normal direction to the plane. Thus, we can
find D in Eq.(8) using one of the tokens from Data.
We choose a token whose normal direction is most
consistent with the estimated normal direction. Thus,
we obtain the parameters (A,B,C,D) of the plane equa-
tion.
Note that all the tokens in the Data can be part of
a single linear plane (e.g. for a planar scene) or there
DEPTH INPAINTING WITH TENSOR VOTING USING LOCAL GEOMETRY
25

can exist multiple local planes (e.g for a curved sur-
face) that pass through the missing region. To find
equations for all possible planes, we first choose a set
of tokens from Data which are clustered in space and
similar in depth values as well (as these tokens are
more likely to form a local plane). Using only those
tokens, we estimate the equation of the plane as de-
scribed earlier. We evaluate all other tokens against
the estimated plane equation. A token [x,y,z] is a part
of the k
th
local plane if
A
k
.x + B
k
.y +C
k
.z + D
k
≤ Plane threshold (11)
where (A
k
,B
k
,C
k
,D
k
) are the parameters of the k
th
lo-
cal plane. The value of Plane threshold should be
close to zero. We set its value to 0.3 during experi-
mentation. We eliminate tokens from Data which sat-
isfy Eq.11 as these are redundant. On the remaining
tokens, we iteratively perform the above procedure
until all the tokens are modeled by their local planes.
Thus, we get a set of local plane equations which fa-
cilitate filling of missing information.
We compute K number of depth estimates(z
k
),
corresponding to K number of plane equations, at
each missing point co-ordinate [x,y] as
z
k
=
−A
k
.x − B
k
.y − D
k
C
k
k = 1 : K (12)
Thus, the number of depth estimates at a missing
point is equal to the number of estimated local planes.
The points from the available segment as well as miss-
ing point co-ordinates along with their depth esti-
mates (from Eq.(12)) act as tokens for 3D TV. The
surface saliency at each token is computed as
Surface saliency = λ
1
− λ
2
(13)
Out of the K depth estimates, the token [x,y,z
k
] with
the highest surface saliency is selected and that z
k
is
considered to be the true depth value for the miss-
ing point [x,y]. It is intuitive that lower depth regions
will occlude higher depth regions. Hence, we fill re-
gions starting from lowest depth segments and pro-
ceed to higher values. For each available informa-
tion segment, we find its average depth value. Us-
ing an available segment with lowest average depth
value, the corresponding missing segment is synthe-
sized first. Other segments are filled subsequently (in
the order from lower to higher depth).
Note that σ is the only tuning factor in the TV
framework and it controls the voting range. For large
missing regions, we need a high value of σ. To auto-
mate the process, we adapt the value of σ to the size
of the missing region.
4 RESULTS
We begin this section with demonstration of inpaint-
ing of a simple linear plane range image to depict
the effectiveness of our plane equation estimation.
Fig.4(a) shows a range image of a linear plane with
missing values. We use points around the missing re-
gion for estimating the plane equation. In Fig.5, the
red colored points indicate the tokens used for 3D ten-
sor voting for plane normal estimation. The green ar-
row shows the estimated plane normal direction. Us-
ing the estimated normal vector, we estimate the com-
plete plane equation from Eq.(8). As there exists only
one plane, we have a single estimate for each miss-
ing value and we synthesize these values within the
missing region using Eq.(12).
(a) (b)
Figure 4: Linear plane inpainting. (a) Defective range map,
(b) inpainted range map.
Figure 5: Plane normal direction: Red points indicate the
tokens used for 3D TV. Estimated normal direction is indi-
cated by green arrow.
Figs.6 and 7, we show the results of our approach
on relatively large missing regions for scenes contain-
ing several planar segments. The range maps were
taken from the Middlebury dataset (Scharstein and
Szeliski., 2002). Once edge interlinking is performed,
due to the planar nature of these scenes, there will be
only one estimate for each segment as in the previ-
ous case. Thus, there are fewer number of tokens for
final pass of 3D TV resulting in reduced execution
time. Due to robust plane parameter estimation, miss-
ing values are indeed synthesized correctly (Fig.6(d)
and Fig.7(b)(d)). In Fig.6(b,c,d), we also illustrate the
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
26

order in which the depth estimates are filled-in by the
proposed approach.
(a) (b)
(c) (d)
Figure 6: Planar range map inpainting. (a) Defective range
maps. (b,c,d) order of filling-lower to higher depth.
(a) (b)
(c) (d)
Figure 7: Planar range map inpainting. (a,c) Defective
range maps. (b,d) inpainted range maps.
Fig.8 demonstrates the performance of our ap-
proach on depth maps from the Middlebury dataset
(Scharstein and Pal., 2007)(Scharstein and Szeliski.,
2002). One needs more number of local planes to
model such a smooth surface. Using 3D tensor vot-
ing, we evaluate depth estimates obtained from lo-
cal plane equations. We choose the estimate with
the highest surface saliency value. As there are more
number of tokens than in the planar case, the execu-
tion time goes up. It can be seen that our approach is
quite effective in synthesizing even such smooth re-
gions (Fig.8(b)(d)).
(a) (b)
(c) (d)
Figure 8: Smooth surface inpainting.(a,c) Defective range
maps, (b,d) inpainted range maps.
Next, we take a scenario where there is a possible
edge intersection within the missing region. Fig.9(a)
shows the edge components existing near the missing
region of Fig.9(c). It can be observed that the verti-
cal edge segment in Fig.9(a) cannot be connected to
other edge components with a smooth curve. There
is a possibility of edge interlinking within the miss-
ing region. We extrapolate the non-matching segment
in the missing region as shown in Fig.9(b). Then re-
gion filling is performed from lower to higher depths
(Fig.9(d)(e)(f)) (in that order) as described earlier.
We also tested our method on range maps of hu-
man faces. Fig.10(a) shows a missing region in the
smooth curved portion of a face. The proposed multi-
ple local plane approach enables us to capture and fill
this surface (Fig.10(b)). Fig.10(c) shows another case
of missing region but now in the nose area. Our ap-
proach works yet again as is evident from Fig.10(d).
We also tested our approach on real range maps
reconstructed using optical images. Interestingly, her-
itage sites provide examples of damaged structures
due to weather, aging etc. Range maps of such struc-
tures show missing regions which we attempt to in-
paint using the proposed method. We visited Maha-
balipuram in Indai which is a UNESCO heritage site
and captured images of monuments using an off-the-
shelf digital camera. Multiple images from different
view points were captured to reconstruct the shape.
Figs.11(a)(b) show optical images of a stone stair-
case captured from two different view points. This
is an interesting case for inpainting as there are
sharp edges and significant depth variations within
the depth map. Using these images, we obtained the
DEPTH INPAINTING WITH TENSOR VOTING USING LOCAL GEOMETRY
27

(a) (b)
(c) (d)
(e) (f)
Figure 9: Possible edge interconnection in missing region:
Edges are shown thick for demonstration. (a) Broken edge
components around the missing region shown in (c). (b)
Extrapolated edge within a missing region and (d,e,f) order
of filling-lower to higher depth values.
(a) (b)
(c) (d)
Figure 10: Results on face data. (a,c) Defective face range
map (b,d) inpainted face range map.
disparity map. We manually marked a region for in-
painting in this real depth map as shown in Fig.11(c).
Since we perform TV based edge detection, we are
able to locate true pixels accurately even for this noisy
(real) depth map. We then interconnect broken edge
components and synthesize missing values within the
missing region. as shown in Fig.11(d).
(a) (b)
(c) (d)
Figure 11: Results on real data. (a,b) Optical images of a
staircase, (c) defective range map, and (d) inpainted range
map.
Fig.12 depicts yet another example of a real depth
map. A lion face sculpted on one of the pillars in
the above mentioned heritage site was captured from
different view points. Figs.12(a)(b) show the opti-
cal images of the structure taken from two different
view points. Fig.12(c) shows the damaged depth map
where the region to be inpainted has been marked.
Note that there can be smooth variations of depth
within the missing region along with possible edge
interconnection. As TV based edge detection and re-
gion filling is robust to noise, our approach is able to
synthesize the missing values accurately even in this
situation as shown in Fig.12(d).
Fig.13(a) shows the image of a sculpted elephant.
Fig.13(b) shows recovered depth map using multiple
images of this object. Note that the depth map has
missing regions at various places and these are in-
dicated in red (Fig.13(c)). This is a commonly en-
countered scenario where the stereo algorithm fail to
provide reliable depth values for all the scene points.
Using our method, we synthesized the depth values in
the missing regions one at a time. Fig.13(d) shows the
final result of inpainting which is quite good.
We also performed a quantitative analysis of our
results which is given in Table 1. To measure the error
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
28

(a) (b)
(c) (d)
Figure 12: Results on real data. (a,b) Optical images of a
lion face, (c) defective range map, and (d) inpainted range
map.
(a) (b)
(c) (d)
Figure 13: Results on real data. (a) Optical image of a
sculpted elephant,(b) damaged range map,(c) missing re-
gions to be inpainted shown in red, and (d) inpainted range
map.
between the estimated depth values and the ground
truth, we used the following error measure (Favaro
et al., 2008).
Error =
s
Avg.

zb
z
− 1

2
(14)
Here, z is the original and zbis the estimated depth
map. Averaging is performed over missing regions.
From the table, it is clear that our results are close to
the ground truth.
For a medium sized missing region in a planar
scene, an unoptimized Matlab code takes around 5
seconds to execute while it takes around 15 seconds
for a missing region of same size but within a smooth
surface.
Table 1: Quantitative analysis.
Result Error Result Error
Fig.6(d) 0.0173 Fig.7(b) 0.0124
Fig.7(d) 0.0189 Fig.8(b) 0.0049
Fig.8(d) 0.0063 Fig.9(f) 0.0224
Fig.10(b) 0.0075 Fig.10(d) 0.0077
Fig.11(d) 0.0127 Fig.12(d) 0.0062
5 CONCLUSIONS
We proposed a fast and reliable range inpainting
approach for filling large regions given a single
range/depth map. We used robust 2D tensor voting
for edge detection. A least squares based approach
was followed for modeling and interconnecting edge
components around the missing region. 2D tensor
voting was also used for refining and making edge in-
terconnection robust. Edge interconnection enabled
us to segment missing regions. This was followed by
3D TV which was employed to estimate plane equa-
tions using local geometry. Depth estimates obtained
from different local planes were then passed inside
the missing region and the best estimate was chosen
based on surface saliency computed from a final pass
of 3D TV. Results (both synthetic and real) reveal that
our approach is quite effective.
ACKNOWLEDGEMENTS
The second author is grateful to the Humboldt Foun-
dation, Germany for supporting this work.
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