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Stratified Dense Matching for Stereopsis
in Complex Scenes
Jana Kostkov´
a and Radim ˇ
S´
ara
Center for Machine Perception, Czech Technical University, Prague, Czech Republic
{kostkova,sara}@cmp.felk.cvut.cz,http://cmp.felk.cvut.cz
Abstract
Local joint image modeling in stereo matching brings more discriminable
and stable matching features. Such features reduce the need for strong prior
models (continuity) and thus algorithms that are less prone to false positive
artefacts in general complex scenes can be applied. One of the principal qual-
ity factors in area-based dense stereo is the matching window shape. As it
cannot be selected without having any initial matching hypothesis we pro-
pose a stratified matching approach. The window adapts to high-correlation
structures in disparity space found in pre-matching which is then followed by
final matching. In a rigorous ground-truth experiment we show that Stratified
Dense Matching is able to increase matching density 3×, matching accuracy
1.8×, and occlusion boundary detection 2×as compared to a fixed-size rect-
angular windows algorithm. Performance on real outdoor complex scenes is
also evaluated.
1 Introduction
The core problem of computational stereopsis is computing disparity map of the scene
which means (1) finding correspondences between binocularly visible points in the input
2-D images and (2) jointly segmenting the images to binocularly visible, half-occluded,
and mutually occluded regions.1There exist two main classes of applications the stereo
matching can be used for: view prediction and 3D structure reconstruction. In our re-
search, we are interested in structure reconstruction which imposes the following require-
ments on stereo matching results: it must not contain incorrect correspondences and the
occluded regions have to be identified accurately. The price paid for these requirements
is lower matching density (mainly in texture-less regions). Nevertheless, as our approach
belongs to the area-based stereo, the results are required to be as dense as possible.
In general, pixels having the most similar neighbourhoods (measured by various statis-
tics: SSD, SAD, NCC) are assigned as the corresponding pairs. In order to produce
accurate results, the matching features have to be as discriminable and stable as possi-
ble. By discriminability we mean the ability to recognize correct correspondences. By
stability we mean independence on distortions introduced by image projection. Hence,
the key-problem is the selection of suitable matching windows over which the statistics
are computed. The simplest approaches use centralized fixed-size rectangular windows.
1To visualize a pair of mutually occluded regions imagine taking a binocular peek through a keyhole. A real
example of this phenomenon is shown in the experiments, Sec. 5.
However, due to geometric distortions, this definition fails at curved surfaces and occlu-
sion boundaries. Image similarity computed over independent rectangular image windows
has been shown inferior to similarity computed over binocularly corresponding windows.
There exist several approaches trying to cope with this problem. Kanade proposed a
method [10], where windows adapt their size: at hypothesized boundaries they are small,
otherwise they remain large. Various versions of windows adapting their shape have also
been introduced [21, 4, 14, 16]. The resulting non-rectangular windows cover the scene
by independent patches (based on the reference image). However, they assume constant
disparity within the window. The specific problem at occlusion boundaries is often solved
by shifting the window away from the boundary to minimize the non-corresponding back-
ground part [2, 8, 15]. Some authors have tried to avoid the occlusion boundary problem
by segmentation [20, 22]: The windows are then defined not to cross a segment boundary.
In [16, 5], a matching process starts by finding the seeds (high-probability matches),
from which the final solution is spread out. Since the seeds determine the matching ac-
curacy, their correctness is crucial. The first stage we see as a kind of pre-matching. We
show this can be done in a much more general way. In [3], a disparity space has been
introduced. First, plausible disparities are computed for each pixel. Based on them, pixels
are assigned to disparity components, which are defined to be of constant disparity. The
final disparity at each pixel is selected by assigning that of its largest component.
We claim that in order to produce good results it is essential to assign the matching
over jointly discriminable and stable matching features. By ‘jointly’ we mean that both
images contribute to their definition. This seems to be impossible without first having the
correspondences. Therefore, we have come to the conclusion, that the matching process
should be algorithmically divided into two semi-independent stages, where the first one
(pre-matching) hypothesizes reliable matching features, while the second one resolves
ambiguities and establishes the final matching. This division follows from requirements
on these stages: the pre-matching is to produce dense results (low false-negative error) but
it need not be a one-to-one mapping. The final matching is to produce accurate results
(low mismatch and false-positive errors) and it is to be a one-to-one mapping.
In this paper, we follow this view and propose a straightforward method, the Strat-
ified Dense Matching. We pose the matching problem in disparity space and design a
full-generality pre-matching, which is used to define the windows: They adapt to 3D-
connected structures of high similarity, which we call disparity components. Such win-
dows are of various shapes, non-constant disparity and are not independent (they are
consistent at their overlap). The final matching problem is solved using the similarity
statistics re-computed over these adaptive windows.
2 Disparity Components
In our approach, the windows collectively adapt to the 3D-image of the scene in the
disparity space. It increases the probability that a high similarity implies a correct match.
As a consequence, the discriminability of matching features is improved, which results in
higher matching quality. Due to projective distortions this would be impossible to do in
input images. The window definition in disparity space enables to get not only windows
covering the same scene patches, but also its symmetric left-right and right-left form.
Disparity space Tis the set of all possible matches between two or among more
images. In the rectified binocular case it can be visualized as matching tables computed
separately for each pair of image rows and stacked on top of each other. Each matching
4-neighbourhood: constant disparity 20-neighbourhood: varying disparity
Figure 1: Neighbourhoods of a disparity space point (r,i,j)(empty blue circle): for constant dis-
parity components (left) and for varying disparity components (right).
table consists of similarity values evaluated on the Cartesian product of left and right
image pixels in the equivalent row (epipolar line). Matching table elements are called
(tentative) pairs. A part of the disparity space (matching tables for rows r−1, rand
r+1) is shown in Fig. 1. Matching table rows represent positions iin the left image, the
columns represent positions jin the right image. Since the disparity space corresponds to
all possible pairwise optical ray intersections it is clear that a surface point neighbourhood
in the scene maps on disparity point neighbourhood.
Disparity components are defined in disparity space as connected structures of pairs
with high similarity values. The connectedness within a disparity component is defined
by a neighbourhood relation. Two high-similarity pairs (r,i,j)and (r0,i0,j0), where r,r0
denote corresponding image rows, i,i0columns in left image and j,j0columns in right
image, are neighbours in the disparity space if and only if (1) they are neighbours to each
other in the left or the right image, and (2) the difference of their disparities is smaller or
equal to a predefined value δ. In mathematical terms the neighbourhood relation can be
formulated in the following way:
Definition 1 The pairs (r,i,j)and (r0,i0,j0)in the disparity space are neighbours to each
other if and only if the following three conditions hold:
1. |r−r0| ≤ 1,
2. |i−i0| ≤ 1 or |j−j0| ≤ 1 for r=r0,
i=i0or j=j0for r6=r0,
3. |d(r,i,j)−d(r0,i0,j0)| ≤ δ,
where r,r0are the corresponding image rows, i,i0are the left-image positions, j,j0are the
right-image positions and d(r,i,j) = i−j is the disparity of the point (r,i,j).
Using the neighbourhood relation recursively, the disparity components are traced out.
For each pair (r,i,j), the corresponding disparity component can be identified uniquely (a
single match can be part of at most one component). Based on the component, the shape
of the adaptive window is found (it brings entire image patches into correspondence).
The parameter δin the definition of the neighbourhood relation allows disparity varia-
tions within one disparity component. If we restrict our definition to points with the same
disparity (δ=0), we get constant disparity components with 4-neighbourhood relation,
see Fig. 1 (left). The 4-connected components correspond to planparallelity assumption,
proposed by Boykov [3]. In our approach, the difference of neighbouring pixel disparities
is allowed to be smaller or equal to one (δ=1). Consequently, we get varying-disparity
components with 20-neighbourhood relation, see Fig. 1 (right). This definition corre-
sponds to continuity assumption and allows to capture even small variations in disparity.
3 Stratified Dense Matching
In this section we overview the Stratified Dense Matching Algorithm. The input to this
algorithm is a pair of left and right rectified images and the output is the disparity map of
a scene. The algorithm consists of four steps: (1) pre-matching, (2) disparity component
tracing, (3) similarity value re-computation, and (4) final matching.
In the first step, an algorithm which is able to produce multivalued disparity maps
is run. These pre-matches segment the disparity space to a set of connected subsets by
eliminating the least number of low-similarity pairs. The requirements on this step are as
follows: (1) multivalued disparity map (to obtain all prospective matching hypotheses),
and (2) dense results (without unmatched rows or columns in the matching table). The
pre-matching step can be based both on global energy minimizations [9, 17, 11] or local
correlation methods [10, 18]. We have applied a local correlation method [18].
In the second step, connected disparity components are traced out on pre-matches
resulting from the first step. The tracing is based on applying recursively the above-
defined 20-neighbourhood relation. For each pre-match, the unique disparity component
is identified. Based on the disparity component, the pair of equivalent matching windows
is found in the input images. Note that it is not necessary to trace out the entire disparity
component, all we need is a set of n-th order neighbours for every pre-match (to get the
tracing computationally efficient).
In the third step, for each pre-match, the similarity statistics is recomputed using
the pair of equivalent matching windows resulting from the second step. In order to
get similarity values comparable in their statistical properties, for each pre-match only
a fixed-size disparity component neighbourhood is used to jointly define the windows.
The match similarity is re-computed only if the corresponding disparity component is
large enough (the minimal size is a parameter), otherwise the pre-match is discarded (to
suppress mismatches caused by noise or weak textures).
In the fourth step, the final univalued disparity map is computed using the re-com-
puted similarities. The most important requirement is a low error rate. Full density is not
strictly required. The results are desired to be as dense as possible, however. In principle,
various stereo matching algorithms can be used to compute the final matching. We have
selected Confidently Stable Matching [18] for its accuracy: it produces disparity maps
that are not necessarily dense but have very low error rates [13].
4 Implementation
Confidently Stable Matching (CSM) solves an optimization task which is defined on mu-
tual occlusion graph G= (T,E)in which the vertex set Tis the set of all tentative matches
(pairs) and (t,s)is an edge in Eif pairs tand sare mutually exclusive, i.e. cannot be both
elements of the same matching due to occlusion. We use uniqueness and ordering con-
straints as the occlusion model. Every pair tin Tthus has a set of competitors (neighbours
in G)N(t)which we call the inhibition zone of t.2Every pair t∈Thas associated the
2Inhibition zone for matchings is as follows: if t= (i,j)then N(t) = {(k,l)|k=ior l=j,(k,l)6= (i,j)}.
value c(t)of match similarity statistics. We say a pair t∈Tis confidently excluded by
another pair e∈Tif (t,e)∈Eand c(t)≤c(e)−∆(t,e). The value of ∆(t,e)is related to
the confidence interval widths of c(t)and c(e). Confidently stable subset Mis the largest
subset of Tsuch that every pair not in Mhas either no unambiguous competitors in Mor
is confidently excluded by a competitor in M. Simply and somewhat imprecisely, all pairs
not in Mare either ambiguous in c(·)or confidently occluded by some strongly better
match in M. If exclusion takes into account uniqueness the stable subset is a (univalued)
matching but multivalued ‘matchings’ are also possible when the inhibition zone has a
finite extent as in Fig. 2. For precise definitions, existence and uniqueness theorems, and
the algorithm see [18]. The advantage of CSM is that it does not need a prior detection of
matchable image features: it automatically recovers them in the matching process.
To simplify the implementation we use stable matching
i
j
Figure 2: Inhibition zone
for pre-matching (black
circles) for a pair (i,j).
for the pre-matching step. It is defined as confidently stable
matching with ∆(·,·)≡0. The corresponding algorithm is
considerably simpler. Multivalued disparity is achieved by
using the smallest inhibition zone possible, shown in Fig. 2.
For the component tracing we used a brute-force method:
For each pre-match, the neighbours are found by directly test-
ing all their possible positions (based on the defined neigh-
bourhood) without any optimization, which is very time con-
suming, but we suppose it can be speeded up about 100-fold.
In the third step, we did not recompute the similarities c(t)from image data, we
only averaged the values computed in the first step over neighbourhoods in the traced-out
disparity component.
There are a few procedure parameters which can be adjusted: disparity search range,
initial matching window size, minimal disparity component size, and the confidence level.
However, none of them (except for the search range) are critical for the matching process.
The default values are: 5×5 initial matching window, and minimal component size of 25
pixels. The confidence level, parametrized by two constants αand β[18], determines the
quality of the results. Its choice is left to the user (see the experiments).
On the Middlebury dataset [19] the running time of our current implementation av-
erages 5.2min, of which pre-matching is 2.1 sec, correlation computation 0.63 sec, final
matching 0.64sec and the rest is spent on component tracing.
5 Experiments
We demonstrate the disparity map improvement due to our adaptive windows over stan-
dard rectangular windows. Therefore, in the experiments, we compare the results of Strat-
ified Dense Matching (SDM) with the plain Confidently Stable Matching (CSM) with
5×5 rectangular windows.
We divide the experiments reported here into two groups: the first one is based on a
rigorous ground-truth evaluation and focuses on matching failure mechanisms related to
insufficient image feature discriminability. The second one demonstrates the results on
complex outdoor scenes. For a comparison based on the Middlebury dataset (which is
omitted here due to space limitations) see [12].
Ground-Truth Evaluation
This evaluation method [13] is based on a designed artificial scene with known ground-
truth, shown in Fig. 3. The scene consists of five thin textured stripes (foreground) in front
Ground Truth
lowest contrast medium contrast highest contrast ground-truth
Figure 3: A selection of tested scene: contrasts of 1, 13 and 20, and the ground-truth disparity map.
The rightmost bar shows disparity map colour coding: low disparities are dark blue, high disparities
are red, half-occluded regions are gray.
1 13 20
10−3
10−2
10−1
100Mismatch Rate (δ>1)
MIR
texture contrast
SDM
CSM
1 13 20
10−2
10−1
100False Negative Rate
FNR
texture contrast
SDM
CSM
1 13 20
10−5
10−4
10−3
10−2
10−1
100False Positive Rate
FPR
texture contrast
SDM
CSM
1 13 20
10−2
10−1
100Failure Rate
FR
texture contrast
SDM
CSM
1 13 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Occlusion Boundary Inaccuracy
OBI
texture contrast
SDM
CSM
1 13 20
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05 Bias
B
texture contrast
SDM
CSM
Figure 4: Matching error results. The CSM algorithm: red solid, the SDM algorithm: blue dashed.
of a textured plane (background). Twenty stereo images of the scene have been captured
under 20 different texture contrast values, which emulate varying signal-to-noise ratio.
Three images of those are shown in Fig. 3 (the dark stripes are shadows). The confidence
level parameters were set to α=20σ2/1000 and β=0.05, where σ2is the image inten-
sity variation. The goal of this test set is not to provide a complete cover of all possible
stereo data but to expose weaknesses related to image feature discriminability.
Types of Error In the experiment, the following six types of error were distinguished:
Mismatch Rate (MIR) measures the accuracy of matching, False Negative Rate (FNR)
measures the disparity map sparsity, False Positive Rate (FPR) measures the quality of
occluded regions detection, Failure Rate (FR) measures the overall disparity map quality
(for view prediction), Occlusion Boundary Inaccuracy (OBI) measures the precision of
occlusion boundary detection, Bias (B) measures the algorithm’s bias to large objects.
The range of all errors is [0,1], only for bias it is [−1,1].
Evaluation The results are shown in Fig. 4. Texture contrast (horizontal axis in all
plots) is directly related to signal-to-noise ratio. The respective error rates are shown on
vertical axes. Note that both axes have logarithmic scale (except in Band OBI plots).
Both algorithms reach a low level of MIR very fast and then they stay constant.3The
MIR is normalized by the matching density, we can therefore conclude that the accuracy
of both results is better than 99%, while the SDM results improve about 1.8×. The False
Negative Rate in CSM is about 25%, in SDM about 9%, i.e. the matching density is im-
proved about 3×. The FPR vanishes in both CSM and SDM except for the worst two
contrasts in CSM (where the SDM results are better by about one order of magnitude),
which demonstrates the ability of both algorithms to detect occlusions correctly. The FR
error was improved about 3.3×by applying the SDM approach. Even the OBI results are
improved: the occlusion boundary is detected about 2×more precisely. None of the algo-
rithms exhibits bias Btowards large objects. The CSM slightly “prefers” the background
(larger object), unlike the SDM, which appears to prefer the foreground (smaller object).
However, the bias in the SDM is about 2×smaller than in the CSM.
To conclude, we can say the SDM preserves the good property of CSM (low false
positive rate), while the density of the results is improved 3×, the occlusion boundary
accuracy 2×, the matching accuracy 1.8×and the overall error 3.3×.
Real Outdoor Scenes
The purpose of this section is to demonstrate the ability of our method to cope even with
complex outdoor scenes. We have selected scenes of wide disparity range, thin objects
(obstacles) at the foreground, and slanted ground plane. We show our approach is able to
correctly detect all the objects in the scene without false positive artefacts (illusions). The
confidence level parameters were fixed to the values of α=20 and β=0.05 in both SDM
and CSM. For comparison, we show the results of the state-of-the-art algorithms: MAP
matching via dynamic programming DP (our reimplementation of Cox’s algorithm [7]),
and MAP matching via graph cuts GC (the authors’ implementation [11]). The parameters
of the two MAP algorithms (the occlusion penalty λof DP and λand penalty0of GC)
were manually adjusted to give the visually best results on each of the tested datasets.
The first selected image pair is a photo of a meadow with an apple tree at foreground,
a shrub at midrange, and a forest at background. The sky above the scene is completely
featureless. The input images together with the results are shown in Fig. 5. The main
difference between SDM and CSM is the density, which is about 1.4×greater in SDM
(35% of pixels is matched in CSM and 50% in SDM), while the accuracy is preserved.
The SDM detects correctly even very fine features in the scene, e.g. the tall grasses at
the road side on the right of the tree. The results on the sky and on the tree trunk have
been also improved. The DP exhibits the typical “streaking” artefacts around objects (tree
crown, shrub), and the tree trunk is disconnected. However, the meadow and the forest are
detected correctly. The GC disparity map consists of piecewise constant disparity regions,
which do not correspond to the scene at all. The top of the tree crown has been “cut” and
the trunk of the tree has not been detected precisely: a higher disparity is assigned to
it and this disparity is propagated to the neighbouring meadow, which means GC sees
a constant disparity “wall” standing in place of the meadow. A similar frontoparallel
wall is hallucinated in place of the meadow on the left of the tree. The forest at the
background has not been detected correctly either. Only the sky is correct with unassigned
correspondences, but it partially cuts the objects.
The second image is a photo of a larch grove, shown in Fig. 6. In this image pair, we
can see a mutually occluded region: the area in between the two leftmost trees corresponds
3Which is a remarkable property confirmed in real scenes as well.
Left image Right image Stratified Dense Matching
Confidently Stable Matching Dynamic programming Graph cuts
Figure 5: The Apple Tree image pair: the input images are shown together with the results of
selected methods. The disparity map colour coding has been described in Fig. 3.
Left image Right image Stratified Dense Matching
Confidently Stable Matching Dynamic programming Graph cuts
Figure 6: The Larch Grove image pair: the results of the selected methods. The gaps in some of the
tree trunks appear due to ordering constraint violation in the scene. For colour coding see Fig. 3.
to different parts of the scene background, thus no correspondences can be found there.
This scene violates the ordering constraint, which results in discontinuities in some of the
trees. The SDM results are 2.5×denser (CSM matched 19% of all pixels, while SDM
48%), we can find all the trees, ferns on the ground, and the mutually occluded region
detected correctly, although there are a few more mismatches there than in the CSM. The
DP is not able to cope with occlusions and mutually occluded regions at all. The strong
continuity prior causes interpolation artefacts over low-texture (or ambiguous) regions.
In GC the parameter tuning was rather difficult. The disparity map shown here has been
selected from more than 100 results as visually the best one. The disparity map exhibits
similar artefacts as in the previous scene: Neither mutually occluded nor half-occluded
regions have been identified, and many important structural details are missing altogether.
6 Discussion and Conclusions
In this paper, we haveproposed a new method, Stratified Dense Matching. Our approach
improves the discriminability of matching features by selecting a suitable matching win-
dow shape. The windows are defined to adapt to high-correlation structures in disparity
space that represent all matching hypotheses. Non-constant disparity within one disparity
component is allowed. The window definition in disparity space allows the matching to
adapt to slanted and curved surfaces, scale differences, and discontinuities. The stratified
matching approach is independent on the selection of the matching algorithm, although it
is desirable to use a pre-matching algorithm that guarantees low false negative error and
a final matching algorithm that guarantees low false positive error and mismatch error.
We have demonstrated in a quantitative experiment that not only the quality but also
the density of disparity map is considerably determined by discriminable joint image
features. Further improvement might be achieved by proper local image modeling [6, 1].
Why is the density difference between CSM and SDM so large? The CSM prefers
to reject a tentative match if competitors of similar correlation exist. This happens when
there are broad correlation value maxima in disparity space due to statistical dependence
between neighbouring image pixels, e.g. due to image blur or long correlation length
of the visual texture in the scene. In SDM the pre-matching step traces-out the exact
positions of the maxima and removes their close competitors. So this step in effect breaks
the dependencies by removing the competing matching hypotheses that disturb CSM.
The improvement observed in accuracy (MIR and OBI) results from the adaptation of the
matching window shape, which improves the correlation discriminability.
Note that the SDM lies somewhere in between the algorithms that solve one optimiza-
tion problem per rectified image row (like DP [7]) on the one hand and the algorithms that
solve a global optimization problem on the entire disparity space (like GC [11]) on the
other hand. The component tracing step is semi-local in its nature but still able to tie
together solutions on neighbouring epipolar lines. Its semi-locality brings a great algo-
rithmic advantage over the global optimization approaches.
Our work differs from the work of others in:(1) the definition of disparity components
that represent a piecewise contiguous manifold in disparity space and so (2) avoiding
the definition of area-based matching over a collection of independent patches, and (3)
avoiding explicit local surface model (e.g. planar) by (4) the determination of the window
shape directly by the hypothesized solution (as opposed to just selecting the best one from
a group of pre-defined windows), (5) in splitting the matching problem into two semi-
independent stages with different requirements, (6) re-computing similarity statistics from
disparity components, and (7) joining the segmentation of reliable matching features and
the correspondence search into a single procedure.
Acknowledgments
This work has been supported by the Grant Agency of the Czech Republic under project
GACR 102/01/1371, by the Grant Agency of the Czech Technical University under project
CTU 8306413, and by the Czech Ministry of Education under project MSM 212300013.
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