Segmentation Using Superpixels: A Bipartite Graph Partitioning Approach
Zhenguo Li Xiao-Ming Wu Shih-Fu Chang
Dept. of Electrical Engineering, Columbia University, New York
{zgli,xmwu,sfchang}@ee.columbia.edu
Abstract
Grouping cues can affect the performance of segmenta-
tion greatly. In this paper, we show that superpixels (im-
age segments) can provide powerful grouping cues to guide
segmentation, where superpixels can be collected easily by
(over)-segmenting the image using any reasonable existing
segmentation algorithms. Generated by different algorithm-
s with varying parameters, superpixels can capture diverse
and multi-scale visual patterns of a natural image. Suc-
cessful integration of the cues from a large multitude of su-
perpixels presents a promising yet not fully explored direc-
tion. In this paper, we propose a novel segmentation frame-
work based on bipartite graph partitioning, which is able to
aggregate multi-layer superpixels in a principled and very
effective manner. Computationally, it is tailored to unbal-
anced bipartite graph structure and leads to a highly effi-
cient, linear-time spectral algorithm. Our method achieves
significantly better performance on the Berkeley Segmenta-
tion Database compared to state-of-the-art techniques.
1. Introduction
Image segmentation is a fundamental low-level vision
problem with a great potential in applications. While hu-
man can parse an image into coherent regions easily, it is
found rather difficult for automatic vision systems. Despite
a variety of segmentation techniques have been proposed, it
remains challenging for any single method to do segmenta-
tion successfully due to the broad diversity and ambiguity
of visual patterns in a natural image.
Methods developed under different motivations can be-
have quite differently. For example, Comaniciu and Meer’s
Mean Shift [4] seeks the modes of a non-parametric prob-
ability distribution in a feature space, and appears to well
respect object details though it tends to split an object in-
to pieces. Felzenszwalb and Huttenlocher’s graph-based
method (FH) [10] merges regions greedily, and tends to
return gross segmentation. Shi and Malik’s Normalized
Cuts (Ncut) [26] aims to minimize the association between
groups while maximizing the association within groups. It
(a) (b) (c) (d)
(e) (f) (g) (h)
Figure 1. Segmentation using superpixels. (a) Input image. (b–e)
Superpixels generated by over-segmenting the image using Mean
Shift (b–c) and FH (d–e), each with different parameters. (f–g)
Segmentations of Mean Shift and FH, respectively. (h) Segmenta-
tion of our proposed method in aggregating the superpixels.
favors balanced partitions, at the risk of breaking object
boundaries or large uniform regions (e.g. sky and grass)
into chunks due to its “normalization” prior. See Section 4
for examples of the above observations.
Many studies have pointed out the synergistic effects of
fusing complementary information, under ensemble learn-
ing [23], multi-kernel learning [16], etc. In this paper, we
combine different segmentation methods for image segmen-
tation (Fig. 1), which, to our knowledge, has not been suf-
ficiently explored. Our main contributions are summarized
as follows:
1. We propose a novel segmentation framework which
is able to aggregate multi-layer superpixels in a prin-
cipled and effective manner. More generally, it can
incorporate additional group constraints such as those
specifying a set of pixels belong to the same group.
2. We show that spectral clustering [26, 22] can be highly
efficient on unbalanced bipartite graphs, in contrast to
the cases on general graphs. We achieve this by de-
veloping a novel graph partitioning technique that is
tailored to the specific unbalanced graph structure.
3. We achieve highly competitive results on the Berkeley
Segmentation Database with large performance gains
ranging from 0.8146 to 0.8319 in PRI, and 12.21 to
11.29 in BDE, and very close to the best one in VoI
and GCE (Section 4).
The rest of the paper is organized as follows. Related
work is reviewed in Section 2 and our model is presented
in Section 3. Experimental results are reported in section 4
and the paper is concluded in Section 5.
2. Related Work
Over-segmentation occurs when image regions are seg-
mented into smaller regions, each referred to as a “super-
pixel” [24]. Superpixels are usually expected to align with
object boundaries, but this may not hold strictly in practice
due to faint object boundaries and cluttered background.
However, assuming the employed segmentation technique
is reasonable, most pixels, if not all, within a superpixel
can still belong to one coherent region (superpixel cues). In
this sense, a superpixel imposes some “soft” constraints on
“good” segmentations.
What roles can superpixels play in segmentation? First,
superpixel cues enforce local smoothness since superpixels
generally occupy consecutive image areas in which pixels
are likely to be grouped together. Second, large elongated
superpixels (e.g. the ones from FH in Fig. 1(e)) incorpo-
rate long-range grouping cues, which has shown to improve
segmentation substantially [25, 2, 5]. Third, and more im-
portantly, superpixels generated by different methods with
varying parameters can capture diverse and multi-scale vi-
sual contents of a natural image. Enforcing such a large col-
lection of superpixels simultaneously is expected to achieve
synergistic effects. Other benefits include robust feature
representation due to larger supports as well as the efficien-
cy brought by the relatively small number of superpixels.
Superpixels have been exploited to aid segmentation in
several different guises. In most cases, they are used to ini-
tialize segmentation [29, 28, 1, 8, 21]. However, an un-
satisfactory over-segmentation often degrades performance
substantially. In this paper, we tackle this by using multiple
over-segmentations. Another approach works with multiple
binary segmentations [20 , 9, 14], relying on a strong as-
sumption that each superpixel corresponds to one whole co-
herent region, which seems hard to achieve for real images.
In [15], superpixels are incorporated in semi-supervised
learning to derive a dense affinity matrix over pixels for
spectral clustering, which can be computationally intensive.
In contrast, our method is much more efficient by using un-
balanced bipartite graphs. Also related is the work in [11]
from machine learning, where bipartite graph partitioning
is used for cluster ensemble. Our work differs in the way
of constructing and partitioning the bipartite graph, and the
new image segmentation application.
3. Superpixel Aggregation
In this section, we propose a novel graph-based segmen-
tation framework which is able to efficiently integrate cues
from multi-layer superpixels simultaneously. We rely on
two simple observations, 1) pixels within one superpixels
tend to belong to one coherent region (superpixel cues); and
2) neighboring pixels which are close in feature space tend
to belong to one coherent region (smoothness cues). We
show that both cues can be effectively encoded using one
bipartite graph. We further develop an efficient algorithm
for unbalanced bipartite graph partitioning.
3.1. Bipartite Graph Construction
We construct a bipartite graph over pixels and superpix-
els collectively, as shown in Fig. 2. To enforce superpixel
cues, we connect a pixel to a superpixel if the pixel is in-
cluded in that superpixel. To enforce smoothness cues, we
could simply connect neighboring pixels weighted by simi-
larity, but this would end up with redundancy because the s-
moothness regarding neighboring pixels within superpixels
were incorporated when enforcing superpixel cues. It may
also incur complex graph partitioning due to denser con-
nections on the graph. To compensate for the smoothness
on the neighboring pixels across superpixels, we connect
neighboring superpixels that are close in feature space.
Formally, denote S as a set of (multi-layer) superpixels
over an image I, and let G = {X, Y, B} be a bipartite graph
with node set X ∪ Y , where X := I ∪ S = {x
i
}
N
X
i=1
and
Y := S = {y
j
}
N
Y
j=1
with N
X
= |I| + |S| and N
Y
=
|S|, the numbers of nodes in X and Y , respectively. The
across-affinity matrix B = (b
ij
)
N
X
×N
Y
between X and Y
is constructed as follows:
b
ij
= α, if x
i
∈ y
j
, x
i
∈ I, y
j
∈ S; (1)
b
ij
= e
−βd
ij
, if x
i
∼ y
j
, x
i
∈ S, y
j
∈ S; (2)
b
ij
= 0, otherwise, (3)
where d
ij
denotes the distance
1
between the features of su-
perpixels x
i
and y
j
, ∼ denotes a certain neighborhood be-
tween superpixels
2
, and α > 0 and β > 0 are free param-
eters controlling the balance between the superpixel cues
and the smoothness cues, respectively. By this construc-
tion, a pixel and the superpixels containing it are likely to
be grouped together due to the connections between them.
Two neighboring (defined by ∼) superpixels are more like-
ly to be grouped together if they are closer in feature space.
Particularly, superpixels are included in both parts of the
graph to enforce the smoothness over superpixels.
Compared to the multi-layer and multi-scale graph mod-
els respectively presented in [15] and [5], our graph model
1
For example, if color space is used as the feature space, and a su-
perpixel x
i
(y
j
) is represented by the average color c
i
(c
j
) of the pixels
within it, then d
ij
= ∥c
i
− c
j
∥
2
. We use this setting in this paper, but
note that our method applies to other features as well.
2
For example, x ∼ y, x ∈ S, y ∈ S, if x = y, or y is adjacent to
x and is most similar to x in terms of (average) color. This neighborhood
relationship is adopted in this paper.
X
Y
Image
Over-segmentation 1
Over-segmentation K
Over-segmentation KOver-segmentation 1
Figure 2. The proposed bipartite graph model with K over-
segmentations of an image. A black dot denotes a pixel while a
red square denotes a superpixel.
has a distinct bipartite structure, which allows highly effi-
cient graph partitioning as shown later. Besides, our graph
model is much sparser because pixels are only connected
to superpixels, while in [15] and [5], neighboring pixels are
also connected to each other. Common to the three methods
is that they all incorporate long-range grouping cues.
3.2. Bipartite Graph Partitioning
Given the above bipartite graph G = {X, Y, B}, the task
is to partition it into k groups, where k is manually speci-
fied. Each group includes a subset of pixels and/or super-
pixels, and the pixels from one group form a segment. Var-
ious techniques can be employed for such a task. Here we
use spectral clustering [26, 22] since it has been success-
fully applied to a variety of applications including image
segmentation [26].
Spectral clustering captures essential cluster structure of
a graph using the spectrum of graph Laplacian. Mathemat-
ically, it solves the generalized eigen-problem [26]:
Lf = γDf , (4)
where L := D − W is the graph Laplacian and D :=
diag(W 1) is the degree matrix with W denoting the affini-
ty matrix of the graph and 1 a vector of ones of appropriate
size. For a k-way partition, the bottom
3
k eigenvectors are
computed. Clustering is performed by applying k-means
[22] or certain discretization technique [30].
To solve (4), one can simply treat the bipartite graph
constructed above as a general sparse
4
graph and apply the
Lanczos method [13] to
¯
W := D
−1/2
W D
−1/2
, the nor-
malized affinity matrix, which takes O(k(N
X
+ N
Y
)
3/2
)
running time empirically [26].
Alternatively, it was shown, in the literature of document
clustering, that the bottom k eigenvectors of (4) can be de-
rived from the top k left and right singular vectors of the
3
Bottom (top) eigenvectors refer to the ones with smallest (largest)
eigenvalues. Similar arguments apply to singular vectors.
4
In our bipartite graph model, a pixel is connected to only K superpix-
els for K over-segmentations of an image. Given K ≪ (N
X
+ N
Y
), the
graph {V, W } is actually highly sparse. In our experiments in Section 4,
K = 5 or 6, and N
X
+ N
Y
> 481 × 321 = 154401.
normalized across-affinity matrix
¯
B := D
−1/2
X
BD
−1/2
Y
,
where D
X
= diag(B1) and D
Y
= diag(B
⊤
1) are the de-
gree matrices of X and Y , respectively [7, 31]. This par-
tial SVD, done typically by an iterative process alternating
between matrix-vector multiplications
¯
Bv and
¯
B
⊤
u, is es-
sentially equivalent to the Lanczos method on
¯
W [13]. It
does not bring substantial benefit on solving (4) and is still
subject to a complexity of O(k(N
X
+ N
Y
)
3/2
) (Fig. 3).
3.3. Transfer Cuts
The above methods do not exploit the structure that the
bipartite graph can be unbalanced, i.e., N
X
= N
Y
. In our
case, N
X
= N
Y
+ |I|, and |I| ≫ N
Y
in general. Thus
we have N
X
≫ N
Y
. This unbalance can be translated into
the efficiency of partial SVDs without losing accuracy. One
way is by using the dual property of SVD that a left sin-
gular vector can be derived from its right counterpart, and
vice versa
5
. In this paper, we pursue a “sophisticated” path
which not only exploits such unbalance but also sheds light
on spectral clustering when operated on bipartite graphs.
Specifically, we reveal an essential equivalence between
the eigen-problem (4) on the original bipartite graph and the
one on a much smaller graph over superpixels only:
L
Y
v = λD
Y
v, (5)
where L
Y
= D
Y
− W
Y
, D
Y
= diag(B
⊤
1), and W
Y
=
B
⊤
D
−1
X
B. L
Y
is exactly the Laplacian of the graph G
Y
=
{Y, W
Y
} because D
Y
= diag(B
⊤
1) = diag(W
Y
1). It
should be noted that our analysis in this section applies to
spectral clustering on any bipartite graph.
Our main result states that the bottom k eigenvectors of
(4) can be obtained from the bottom k eigenvectors of (5),
which can be computed efficiently given the much smaller
scale of (5). Formally, we have the following Theorem 1.
Theorem 1. Let {(λ
i
, v
i
)}
k
i=1
be the bottom k eigen-
pairs of the eigen-problem (5) on the superpixel graph
G
Y
= {Y, B
⊤
D
−1
X
B}, 0 = λ
1
≤ · · · ≤ λ
k
< 1.
Then {(γ
i
, f
i
)}
k
i=1
are the bottom k eigenpairs of the eigen-
problem (4) on the bipartite graph G = {X, Y, B}, where
0 ≤ γ
i
< 1, γ
i
(2 − γ
i
) = λ
i
, and f
i
= (u
⊤
i
, v
⊤
i
)
⊤
with
u
i
=
1
1−γ
i
P v
i
. Here P := D
−1
X
B is the transition proba-
bility matrix from X to Y .
Proof. See the Appendix.
It is interesting to note that, if B is by construction a
transition probability matrix from Y to X (i.e. non-negative
5
Let v
i
and u
i
be the i-th right and left singular vector of
¯
B with
singular value λ
i
. Then
¯
Bv
i
= λ
i
u
i
The top k right singular vectors
v
1
, · · · , v
k
are exactly the top k eigenvectors of
¯
B
⊤
¯
B, which can be
computed efficiently by the Lancsoz method or even a full eigenvalue de-
composition given the much smaller size (N
Y
× N
Y
) of
¯
B
⊤
¯
B.
Table 1. Complexities of eigen-solvers on bipartite graphs
Algorithm Complexity
Lanczos [26] O (k(N
X
+ N
Y
)
3/2
)
SVD [31] O (k(N
X
+ N
Y
)
3/2
)
Ours 2k(1 + d
X
)N
X
operations + O(kN
3/2
Y
)
and the sum of the entries in each column is 1.), then the in-
duced affinity matrix W
Y
over Y , W
Y
:= B
⊤
D
−1
X
B, is ex-
actly the one-step transition probability matrix on the graph
G
Y
= {Y, B
⊤
D
−1
X
B}, or the two-step transition probabil-
ity matrix on the original bipartite graph G = {X, Y, B}
following Y → X → Y [17].
By Theorem 1, computing f
i
from v
i
needs
2N
X
d
X
+ 2N
X
operations, following the execution
order
1
1−γ
i
(D
−1
X
(Bv
i
)), where d
X
is the average num-
ber
6
of edges connected to each node in X. So it takes
2k(1 + d
X
)N
X
operations for computing f
1
, . . . , f
k
from v
1
, . . . , v
k
, plus a cost of O(kN
3/2
Y
) for deriving
v
1
, . . . , v
k
with the Lanczos method [26].
So far we have presented three methods to solve (4) for
the bottom k eigenvectors, whose complexities are listed in
Table 1, where we can see that only our method comes with
a linear time (w.r.t. N
X
) with a small constant. To compare
their performance in practice, we test on a series of bipartite
graphs G = {X, Y, B} of different sizes. The results are
shown in Fig. 3. For Fig. 3(a), N
Y
is fixed to 200 while N
X
ranges from 10, 000 to 90, 000. For Fig. 3(b), N
X
is fixed
to 100, 000 while N
Y
ranges from 50 to 2000. For each
pair {N
X
, N
Y
}, B is randomly generated in MATLAB with
B = rand(N
X
, N
Y
), and entries other than the 5 largest
per row are set to zeros (i.e., each node in X is connected
to 5 nodes in Y ). For each graph, 10 eigenvectors of (4)
are computed. The results reported are averaged over 10
graphs. For the Lanczos method and SVD, we use eigs and
svds in MATLAB; for our method we also use the Lanczos
method (i.e. eigs) to solve (5). For all the three methods,
the tolerance for convergence is set to 1e − 10. From Fig. 3,
we can see that our cost is much less than those of SVD and
the Lanczos method in both cases. The costs of SVD and
the Lanczos are quite close to each other. The small slope
of our method in Fig. 3(a) confirms that the constant factor
of our linear complexity is quite small.
We summarize our approach to bipartite graph partition-
ing in Algorithm 1, which we call Transfer Cuts (Tcut) s-
ince it transfers the eigen-operation from the original graph
to a smaller one. In step 5, one may employ the discretiza-
tion technique in [30] which is tailored to Ncut, or apply
k-means to the rows of the matrix F := (f
1
, . . . , f
k
) after
each row is being normalized to unit length, which is justi-
fied by stability analysis [22]. In our experiments, we found
6
In our bipartite graph, d
X
is approximately equal to the number of
over-segmentations or layers.
10,000 30,000 50,000 70,000 90,000
0
2
4
6
8
10
Number of rows
Second
Ours
SVD
Lanczos
50 200 500 1000 2000
0
5
10
15
20
25
Number of columns
Second
Ours
SVD
Lanczos
(a) (b)
Figure 3. Cost vs. Solver w.r.t. (4).
the performances were comparable in both cases. So we on-
ly report results on the latter case since it is more efficient.
Algorithm 1 Transfer Cuts
Input: A bipartite graph G = {X, Y, B} and a number k.
Output: A k-way partition of G.
1: Form D
X
= diag(B1), D
Y
= diag(B
⊤
1), W
Y
=
B
⊤
D
−1
X
B, and L
Y
= D
Y
− W
Y
.
2: Compute the bottom k eigenpairs {(λ
i
, v
i
)}
k
i=1
of
L
Y
v = λD
Y
v.
3: Obtain γ
i
such that 0 ≤ γ
i
< 1 and γ
i
(2 − γ
i
) = λ
i
,
i = 1, . . . , k.
4: Compute f
i
= (u
⊤
i
, v
⊤
i
)
⊤
, with u
i
=
1
1−γ
i
D
−1
X
Bv
i
,
i = 1, . . . , k.
5: Derive k groups of X ∪ Y from f
1
, . . . , f
k
.
3.4. Proposed Image Segmentation Algorithm
Our segmentation procedures are listed in Algorithm 2,
which we call Segmentation by Aggregating Superpixels
(SAS). The main cost of SAS is in collecting superpixels
(step 1) and bipartite graph partitioning (step 3). The cost
of step 3 is linear in the number of pixels in the image with a
small constant (Section 3.3), and is negligible compared to
that of step 1 (Section 4). Potentially, there may be a group
consisting of superpixels only. In such cases, the returned
number of segments will decrease by 1. However, we never
encounter such cases in our experiments probably because
of the relatively strong connections between pixels and the
superpixels containing them.
Algorithm 2 Segmentation by Aggregating Superpixels
Input: An image I and the number of segments k.
Output: A k-way segmentation of I.
1: Collect a bag of superpixels S for I.
2: Construct a bipartite graph G = {X, Y, B} with X =
I ∪ S, Y = S, and B defined in (1-3).
3: Apply Tcut in Algorithm 1 to derive k groups of G.
4: Treat pixels from the same group as a segment.
4. Experimental Results
In this section, we evaluate the proposed image segmen-
tation algorithm SAS on a benchmark image database, and
compare it with state-of-the-art methods.
SAS requires a bag of superpixels, which are generated
by Mean Shift [4] and FH [10], though other choices are
also possible. The main reason of choosing them is that
they are complementary and practically efficient. For Mean
Shift, we generate three layers of superpixels with param-
eters (h
s
, h
r
, M) ∈ {(7, 7, 100), (7, 9, 100), (7, 11, 100)}
(denoted as MS), where h
s
and h
r
are bandwidth pa-
rameters in the spatial and range domains, and M is
the minimum size of each segment. For FH, we gen-
erate two or three layers of superpixels with parameter-
s (σ, c, M ) ∈ {(0.5, 100, 50), (0.8, 200, 100)} (denoted
as FH1) or (σ, c, M ) ∈ {(0.8, 150 , 50), (0.8, 200, 100),
(0.8, 300, 100)} (denoted as FH2), respectively, where σ
and c are the smoothing and scale parameters, and M is
the minimum size of each segment. FH1 or FH2 is selected
automatically according to the image variance in the LAB
color space (feature space) using a threshold. We also report
results on different combinations of superpixels.
For the graph, each pixel is connected to the superpixels
containing it and each superpixel is connected to itself and
its nearest neighbor in the feature space among its spatially
adjacent superpixels (Fig. 2). Each superpixel is represent-
ed by the average LAB color of the pixels within it. The
inclusion and smoothness parameters α and β are set em-
pirically to α = 10
−3
and β = 20. We also test the sensi-
tivity of SAS w.r.t. the variation of these parameters. Like
other graph partitioning methods, the number of segments
is manually set for each image (e.g. [15]). All the results
of SAS reported in this section use the same parameters as
above, including α, β, and the parameters for Mean Shift
and FH in generating superpixels, unless otherwise stated.
4.1. Berkeley Database
We report results on the Berkeley Segmentation
Database [18], which consists of 300 natural images of di-
verse scene categories. Each image is manually segmented
by a number of different human subjects, and on average,
five ground truths are available per image.
To quantify the segmentation results, we follow common
practice (e.g. [15, 9, 28]) to use the four criteria: 1) Proba-
bilistic Rand Index (PRI) [27], measuring the likelihood of
a pair of pixels being grouped consistently in two segmen-
tations; 2) Variation of Information (VoI) [19], computing
the amount of information of one result not contained in
the other; 3) Global Consistency Error (GCE) [18], measur-
ing the extent to which one segmentation is a refinement of
the other; and 4) Boundary Displacement Error (BDE) [12],
computing the average displacement between the bound-
aries of two segmentations. A segmentation is better if PRI
Table 2. Performance evaluation of the proposed method (SAS)
against other methods over the Berkeley Segmentation Database
Methods PRI VoI GCE BDE
Ncut 0.7242 2.9061 0.2232 17.15
Mean Shift 0.7958 1.9725 0.1888 14.41
FH 0.7139 3.3949 0.1746 16.67
JSEG 0.7756 2.3217 0.1989 14.40
MNcut 0.7559 2.4701 0.1925 15.10
NTP 0.7521 2.4954 0.2373 16.30
SDTV 0.7758 1.8165 0.1768 16.24
TBES 0.80 1.76 N/A N/A
UCM 0.81 1.68 N/A N/A
MLSS 0.8146 1.8545 0.1809 12.21
SAS 0.8319 1.6849 0.1779 11.29
SAS(MS) 0.7991 1.9320 0.2222 15.37
SAS(FH1) 0.8070 1.8690 0.2167 14.28
SAS(FH2) 0.8007 1.7998 0.2105 17.17
SAS(MS+FH1) 0.8266 1.7396 0.1868 11.83
SAS(MS+FH2) 0.8246 1.7144 0.1904 12.63
is larger and the other three are smaller, when compared to
the ground truths.
We compare the average scores of SAS and the ten
benchmark algorithms, Ncut [26], Mean Shift [4], FH [10],
JSEG [6], Multi-scale Ncut (MNcut) [5], Normalized Tree
Partitioning (NTP) [28], Saliency Driven Total Variation (S-
DTV) [9], Texture and Boundary Encoding-based Segmen-
tation (TBES) [21], Ultrametric Contour Maps (UCM) [1],
and Multi-Layer Spectral Segmentation (MLSS) [15]. The
scores of these algorithms are collected from [15, 1, 9, 28].
To see how SAS affected by the superpixels used, we al-
so report the scores of SAS with different combinations of
superpixels, where SAS(MS) represents SAS using the su-
perpixels generated by Mean Shift alone, with parameters
MS; SAS(MS+FH1) denotes SAS with the superpixels gen-
erated collectively by Mean Shift and FH, with parameters
MS and FH1, respectively; and similar arguments hold for
SAS(FH1), SAS(FH2), and SAS(MS+FH2). All our meth-
ods use the same parameters, including the number of seg-
ments for each image.
The scores are shown in Table 2, with the three best re-
sults highlighted in bold for each criterion. We see that SAS
ranks first in PRI and BDE by a large margin compared
to previous methods, and second in VoI and third in GCE
with performance quite close to the best one. The scores of
SAS with superpixels from a single algorithm (SAS(MS),
SAS(FH1), SAS(FH2)) are not comparable to those from
both, suggesting that complementary superpixels do im-
prove SAS significantly (see Fig. 6). Although SAS with
a fixed set of complementary superpixels (SAS(MS+FH1),
SAS(MS+FH2)) is already highly competitive, a simple
adaptive selection of superpixels based on the image vari-
ance can make it even better (SAS) (see Fig. 4). Consider-
ing all four criteria, SAS appears to work best overall.
