Structural Analysis of the Additive Noise
Impact on the α-tree
Baptiste Esteban
1[0000−0002−6477−7666]
, Guillaume
Tochon
1[0000−0003−4617−4922]
, Edwin Carlinet
1[0000−0001−5737−5266]
, and Didier
Verna
1[0000−0002−6315−052X]
EPITA Research Laboratory
14-16 rue Pasteur, Le Kremlin-Bicˆetre, France
baptiste.esteban@lrde.epita.fr
Abstract. Hierarchical representations are very convenient tools when
working with images. Among them, the α-tree is the basis of several
powerful hierarchies used for various applications such as image simplifi-
cation, object detection, or segmentation. However, it has been demon-
strated that these tasks are very sensitive to the noise corrupting the
image. While the quality of some α-tree applications has been studied,
including some with noisy images, the noise impact on the whole struc-
ture has been little investigated. Thus, in this paper, we examine the
structure of α-trees built on images corrupted by some noise with re-
spect to the noise level. We compare its effects on constant and natural
images, with different kinds of content, and we demonstrate the relation
between the noise level and the distribution of every α-tree node depth.
Furthermore, we extend this study to the node persistence under a given
energy criterion, and we propose a novel energy definition that allows
assessing the robustness of a region to the noise.
Keywords: α-tree · noise analysis · persistent hierarchy
1 Introduction
Hierarchical representations are powerful tools for several image processing tasks.
They are divided into two categories [2]: inclusion hierarchies and partitioning
hierarchies. Inclusion hierarchies describe the relation of the connected compo-
nents of an image while partitioning hierarchies stack different image partitions
whose regions are obtained with a given criterion. However, despite their divi-
sion, there exist some links between the different categories [4]. In this article,
we focus on the α-tree [14], a partitioning hierarchy. It is used for tasks such as
segmentation, simplification [10], or attribute profiles [1].
To evaluate the quality of these hierarchies, a set of metrics such as the
quality of regions and contours in the context of horizontal and optimal cut, is
proposed and applied to hierarchical watersheds [11]. However, this evaluation
does not take into account the case where a hierarchy is built on a noisy image.
The impact of the noise on hierarchies applied to attribute profiles is investigated
2 Baptiste Esteban, Guillaume Tochon, Edwin Carlinet, and Didier Verna
4 7 3 2
1 2 5 3
1 1 5 7
(a) An image
1 1 1 2 7 4 7 5 5 3 3 2
0
1
2
3
(b) α-tree of the image (a)
1 1 5 7
1 2 5 3
4 7 3 2
(c) 1-partition of (b)
Fig. 1: Illustration of the α-tree representation.
in [7], where the superiority of inclusion trees and the ω-tree compared to the
α-tree is demonstrated for such applications. Finally, the α-tree structure is
investigated in [15] in order to efficiently build the tree by estimating its size
and only allocating the needed memory space to store the tree. Nonetheless, the
impact of the noise on its structure has not been really studied.
Here, we focus on the evolution of some attributes computed on the structure
of the α-trees built on images corrupted by some noise with respect to its level.
The first attribute is the depth of every node in the tree, particularly their sta-
tistical distribution. The second attribute originates from the scale-set theory [5]
and yields the notion of persistent nodes according to a given energy criterion.
Using these attributes, we highlight the relation between the structure of the
tree and the noise level, and we propose a novel energy criterion, relying on the
values of a region of the tree and the gradient at its contour, in order to assert
the robustness of a region to the noise.
This article is structured as follows: in Section 2, we recall the definition of
the α-tree and explain how to obtain the persistence of a node when constrained
to a particular energy. Then, we study the impact of the noise on the structure of
the tree in Section 3. We extend this study to the node persistence in Section 4.
Finally, we conclude and give the perspectives of this work in Section 5.
2 Hierarchical representations
2.1 The α-tree representation
Let f : Ω → I be an image defined on a domain Ω and whose values be-
long to I. Two points p, q ∈ Ω are α-connected if there exists a path of m
consecutive points (p → q) = (x
0
= p, ..., x
m−1
= q) according to an adjacency
relationship such that for every two consecutive points x
i
and x
i+1
of this path,
w(f(x
i
), f(x
i+1
)) ≤ α, with w a dissimilarity measure between two pixel values.
An α-connected component is a connected component composed of α-connected
points. Thus, a 0-connected component is a flat zone. An α-partition α-P is a par-
tition composed of disjoint α-connected components whose union is Ω. An α-tree
T
α
is the tree representation of the hierarchy H
α
= (0-P, ..., (m − 1)-P) composed
of m α-partitions. Each node of T
α
represents an α-connected component and
its parent represents the fusion of this node with all its siblings. Finally, a cut ζ
Structural Analysis of the Additive Noise Impact on the α-tree 3
is a set of disjoint regions (R
α
)
i
represented by the nodes of T
α
whose union
is Ω. A particular case of cut is the horizontal cut at a given level t which results
in an α-partition with α = t.
By applying these notions to graphs, and by the links between different hi-
erarchical representations on edge-weighted graphs [4], the α-tree is the min-
tree [13] of the minimum spanning tree of a graph, such as an adjacency graph
of an image. This link leads to an efficient construction procedure based on the
Kruskal algorithm [9]. Furthermore, there exist more efficient algorithms such as
one based on flooding [15] or a parallel version of the α-tree construction [6].
An example of α-tree is illustrated in Fig. 1. It is built on the image in Fig. 1a
and displayed in Fig. 1b as a dendrogram. In this representation, each pixel is
represented by a leaf of the tree and each inner node represents the fusion of
different sets of pixels. Finally, a partition of the α-tree is given in Fig. 1c.
2.2 Persistent hierarchies
Each region R of a partitioning hierarchy appears in the tree for a given continu-
ous set of scale values associated with the hierarchy. This set is called intervale of
persistence and is defined by Λ(R) = [λ
+
(R), λ
−
(R)[, where λ
+
(R) is the scale
of appearance of R and λ
−
(R) is its scale of disappearance. Thus, for each region
R
α
represented by a node r
α
of an α-tree T
α
, λ
+
(R
α
) is the value α associated
with r
α
and λ
−
(R
α
) is the value α of the parent of r
α
. The scale of disappearance
of the root is a particular case where λ
−
(Ω) = +∞.
There exist several image processing approaches relying on energy minimiza-
tion for different tasks such as segmentation or denoising. Guigues et al. [5]
propose to apply energy minimization to hierarchical representations to obtain
a cut ζ
∗
, which is optimal according to a separable energy of the form:
E
λ
(ζ
∗
) =
X
R
i
∈ζ
∗
D(R
i
) + λ
X
R
i
∈ζ
∗
C(R
i
)
with D(R
i
) a data-fidelity term to R
i
, C(R
i
) a regularization term and λ a
parameter of this energy. When λ is varying from low value to high value, this
produces different cuts whose regions are evolving from fine to coarse. Therefore,
this parameter may be seen as a scale parameter, and it is possible to obtain an
interval of persistence using a functional dynamic programming problem [5] by
subjecting an energy of the form E
r
= D(r) + λC(r) to a node r. This reveals
some non-persistent nodes, with λ
−
(r) ≤ λ
+
(r), which are removed from the
hierarchy, leading to a persistent hierarchy.
3 Noise impact on the tree structure
In the following, a noisy image is defined by f
σ
= f + n
σ
with n
σ
a sample of
values drawn from a normal law N (0, σ
2
). A particular case of f is the constant
image f
c
such that ∀p, f
c
(p) = c, and its noisy version is denoted by f
c,σ
. For all
the experiments performed in this paper, the set of pixel values I is included or
equal to J0 − 255K.
4 Baptiste Esteban, Guillaume Tochon, Edwin Carlinet, and Didier Verna
0
1 1
2 2
3 3 2 1 2 3 3 3
(a) Depth P
T
of the nodes of a tree T
0 25 50 75 100 125 150 175 200
Depth
0
5000
10000
15000
20000
25000
30000
35000
Amount of nodes
20
40
60
80
100
120
140
Standard deviation
(b) Distributions h
c,σ
Fig. 2: The depth attribute and its representation as histograms for f
c,σ
.
3.1 Study on a noisy constant image
We propose here to evaluate the impact of the noise on a tree T by studying the
depth P
T
(r) of each node r, defined by
P
T
(r) =
(
0 if r is the root of T
P(r
p
) + 1 else
with r
p
the parent node of r in T . This attribute is illustrated in Fig. 2a, rep-
resenting a tree whose labels in blue are the depth value of each node. The
depth distribution of T is studied by observing its histogram h, whose values
are defined by h(d) = |{r ∈ T | P
T
(r) = d}| for a particular d ∈ P
T
. The mode
m(h) of the distribution h and its empirical mean µ(h) are used throughout
this paper. They are respectively obtained by m(h) = argmax
d∈P
T
h(d) and
µ(h) =
1
|h|
P
d∈P
T
h(d). The depth distributions obtained from the α-trees built
on f
σ
and f
c,σ
are respectively denoted by h
σ
and h
c,σ
.
In this part, the studied depth distributions are obtained from α-trees built
on images containing only noise, without any texture information, in order to
observe the evolution of the tree structure in the presence of noise with respect
to its level. To this aim, the distributions h
c,σ
are built from α-trees constructed
on constant images f
c,σ
, with c = 127, which have been corrupted with noise
whose level σ is varying from 1 to 150. The resulting distributions are displayed
in Fig. 2b. The different distributions h
c,σ
are represented by plots whose color
corresponds to the noise level σ of the image on which the α-tree is built and
depicted by the color bar.
By examining these distributions, the evolution of the tree structure related
to the noise level corrupting the image is studied. First, while the noise level
increases, the depth distribution becomes a tailed distribution for nodes with a
low depth. These nodes are α-connected components resulting from the fusion of
another component and a small region, usually of size 1, which have an intensity
significantly different from its surrounding pixel values. Then, the mode of the
distributions increases while the noise level grows up to some high noise level,
beyond which this mode decreases slowly. This is due to the clipping of values to
the limits of I during the noising process of the image, creating new flat zones.
Structural Analysis of the Additive Noise Impact on the α-tree 5
(a) Textured image
0 50 100 150 200 250
Depth
0
5000
10000
15000
20000
Amount of nodes
20
40
60
80
100
120
140
Standard deviation
(b) Distributions h
σ
for the α-tree of (a)
(c) Low brightness image
0 50 100 150 200 250
Depth
0
5000
10000
15000
20000
25000
Amount of nodes
20
40
60
80
100
120
140
Standard deviation
(d) Distributions h
σ
for the α-tree of (c)
Fig. 3: Distributions h
σ
on different kind of images.
3.2 Comparison with natural images
In this section, natural images from the database of Laurent Condat
1
are used
to take into account different characteristics likely to be impacted by the noise
such as a high texture or a low brightness. Examples of images are displayed in
Fig. 3a and 3c, with their respective depth distributions h
σ
in Fig. 3b and 3d.
These distributions are obtained by the same process as previously described
and using the same noise level ranges.
These distributions have similar behaviors as the ones in Fig. 2b. Their mode
increases up to some noise level, and then decreases slowly. Then, the variance
of each distribution is increasing as the noise level becomes high. However, there
also are several differences between the distributions of tree depth obtained from
f
c,σ
and f
σ
, and between the natural images. First, the distribution modes, at
low noise levels, are higher for h
σ
than for h
c,σ
. This is due to the content of
the natural images which, conversely to the constant image, has some texture.
Consequently, for very small σ values, the image content is still prevailing. Fi-
nally, the distributions h
σ
in Fig. 3d with high σ values have a higher variance
than the distributions in Fig. 3b. This demonstrates the impact of the noise on
α-trees built on images with low brightness.
The analysis of the noise impact is then extended to the whole image
database. For this purpose, an α-tree is built on each image and the empiri-
cal mean µ(h
σ
) of the depth distribution h
σ
is computed. This is performed
1
https://lcondat.github.io/imagebase.html
6 Baptiste Esteban, Guillaume Tochon, Edwin Carlinet, and Didier Verna
20 40 60 80 100 120 140
0
50
100
150
200
Standard deviation σ
Mean M
M(σ) for f
σ
M(σ) for f
c,σ
Fig. 4: Comparison between h
σ
from all the images from the base and h
c,σ
N = 10 times to obtain the average M(σ) =
1
N
P
N−1
i=0
µ((h
σ
)
i
), with (h
σ
)
i
the
i
th
depth distribution. This process is carried out for several noise levels varying
between 1 and 150, resulting in the plots in Fig. 4. Each dashed plot is related
to a particular image, leading to a total of 150 dashed plots. Furthermore, the
red plot results from the same experiment, but with f
c,σ
, to compare the noise
impact on the structure of an α-tree built on a pure noisy image and α-trees
built from images with content.
The average M(σ), at low noise levels σ, is much higher for f
σ
than f
c,σ
.
This observation is true for a large majority of images in the database on every
noise level. Thus, we deduce that, in spite of the noise corrupting the image,
the image content has still an impact on the depth distribution of the nodes in
the α-tree, as it has been observed previously. Furthermore, for every image, the
values of M(σ) is decreasing starting from a given high noise level. This may
come from the clipping of image values during the noising process, as previously
noted for all kind of images.
4 Impact of the noise on nodes persistence
In this section, the α-trees constructed from noisy images are transformed into
persistent hierarchies using a particular energy criterion. Two different ener-
gies are utilized for this purpose. First, the piecewise constant Mumford-Shah
functional [8] is employed. It is defined by
E
ms,r
α
(λ) =
X
p∈R
α
(f(p) −
˜
f(p))
2
+ λ |∂R
α
|
where
˜
f is the average intensity of the values in the region R
α
and ∂R
α
is the
set of elements in the contours of R
α
. We propose to modify the Mumford-Shah
functional to use the sum of gradient values in the contour of R
α
instead of the
Structural Analysis of the Additive Noise Impact on the α-tree 7
20 40 60 80 100 120 140
10
20
30
40
50
60
70
80
Standard-deviation σ
% of non-persistent nodes
Non-persistent nodes percentage with E
ms,r
α
Non-persistent nodes percentage with E
cs,r
α
Fig. 5: Non-persistent nodes percentage related to the noise level σ
length of its contour. This functional, denoted by E
cs,r
α
, is defined by
E
cs,r
α
(λ) =
X
p∈R
α
(f(p) −
˜
f(p))
2
+ λ
X
p∈∂R
α
g(p)
with g the set of contour values computed using the dissimilarity function w
between two adjacent pixel values of the image from which the α-tree is built.
This change of regularization term is proposed because a region with a small
variance and a high gradient along its contour is most likely to be contrasted
relatively to its adjacent regions, and therefore prone to be less affected by the
noise in the image on which the α-tree is built.
To compare the usage of these functionals, but also to evaluate the impact of
the noise on the persistence of the nodes, the percentages of non-persistent nodes
using these energies E
ms,r
α
and E
cs,r
α
are computed on an α-tree built on the
image in Fig. 3a and are displayed on Fig. 5. These plots have a similar behavior:
when the noise level increases, the amount of non-persistent nodes is growing.
Additionally, a greater amount of non-persistent nodes is observed when E
cs,r
α
is used as an energy criterion than E
ms,r
α
, and this difference is twice as large
for E
cs,r
α
at high noise levels.
The evolution of the plots of non-persistent nodes percentage is confirmed
on all the images from the database with E
ms,r
α
and E
cs,r
α
in Fig. 6a and 6b
respectively. In this figure, the average percentage of non-persistent nodes is
displayed as the blue plot. Furthermore, the red line represents the percentage
of non-persistent nodes on the constant image f
c,σ
. On the two figures, the red
plot has a different behavior: with E
ms,r
α
, the amount of non-persistent nodes is
close to 0%, whereas with E
cs,r
α
, it is near 80%. This observation suggests that
using E
cs,r
α
as an energy criterion instead of E
ms,r
α
in the presence of noise
is more relevant. Furthermore, this is enforced due to the fact that when σ is
increasing, the amount of non-persistent nodes on a tree built on f
c,σ
gets closer
to the average percentage when E
cs,r
α
is used.
8 Baptiste Esteban, Guillaume Tochon, Edwin Carlinet, and Didier Verna
(a) With E
ms,r
α
.
(b) With E
cs,r
α
.
Fig. 6: Evolution of the amount of non-persistent nodes with respect to the noise
level.
5 Conclusion and perspectives
To conclude this article, we have shown that there exists a relationship between
the noise level degrading the image and the distribution of the depth attribute
computed from the α-tree built on the noisy image. Furthermore, on natural
images, we observed that the content of the image has an effect on the depth
distributions: for low noise levels, the impact of noise on the α-tree is negligible,
and at a high noise level, the brightness impacts the variance of the distribution.
Finally, we have noticed that the choice of the functional used to obtain persistent
nodes affects the amount of non-persistent nodes in the hierarchy.
We plan to extend this work to other kinds of noise, but also to generalize
our study to different partitioning hierarchies such as the ω-tree [14], the binary
partition trees [12], or the hierarchical watersheds [3], but also to inclusion hi-
erarchies. Finally, we will apply the results obtained in this paper in order to
evaluate the quality of hierarchies built from noisy images.
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