# Multi-scale Image Decomposition
using a Local Statistical Edge Model

Kin-Ming Wong  School of Creative Media
City University of Hong Kong
Hong Kong S.A.R., China
<EMAIL_ADDRESS>
###### Abstract

We present a progressive image decomposition method based on a novel non-
linear filter named Sub-window Variance filter. Our method is specifically
designed for image detail enhancement purpose; this application requires
extraction of image details which are small in terms of both spatial and
variation scales. We propose a local statistical edge model which develops its
edge awareness using spatially defined image statistics. Our decomposition
method is controlled by _two intuitive parameters_ which allow the users to
define what image details to suppress or enhance. By using the summed-area
table acceleration method, our decomposition pipeline is highly parallel. The
proposed filter is gradient preserving and this allows our enhancement results
free from the gradient-reversal artefact. In our evaluations, we compare our
method in various multi-scale image detail manipulation applications with
other mainstream solutions.

###### Index Terms:

Multi-scale image decomposition, Image processing, Edge-preserving filter

##  I Introduction

Many image manipulation applications use the multi-scale image decomposition
method [1] for image detail enhancement. This kind of image decomposition is
often represented by the following equation:

$I=B+\sum_{i=1}^{N}D_{i}$ (1)

where $I$ is the original image. The decomposition process produces a base
layer $B$ and multiple detail layers $D_{i}$. The base layer is often a
piecewise smooth image which maintains the low-frequency information and the
primary image structures. The detail layers $D_{i}$ hold image details of
various scales.

Multi-scale image decomposition relies on coarsening an image with a smoothing
filter to obtain a detail layer which is the difference between the original
and the filtered image. This process repeats with the filter support increased
at each iteration, and stops when the desired number of detail layers have
been obtained. Fig. 1 shows an example of the decomposition process where the
fine details which are small in both spatial and signal variation scales are
separated.

Figure 1: Fine details (in terms of both spatial _and_ signal variation
scales) extraction from the input signal.

The early form of multi-scale image manipulation technique is based on
Laplacian pyramid [2]. However, this non edge-preserving approach often
produces halo artefacts, so edge-awareness becomes an essential requirement of
image decomposition. Anisotropic diffusion [3] is a non-linear filtering
method which uses the heat transfer model to detect edges, and there is an
improved extension [4] which includes the use of robust statistics.

The bilateral filter [5, 6] has become the mainstream edge-preserving
filtering method. Many image decomposition and dynamic range compression
related works [7, 8, 9] rely on the bilateral filter. However, its over-
sharpening characteristic is undesirable for certain image manipulation
purposes, and is discussed in some recent studies [1, 10]. Two essential
properties of an edge-preserving filter suitable for multi-scale image
decomposition application are identified by the Weighted Least Square (WLS)
method article [1]. The authors emphasize that the decomposition filter should
preserve important edge signals and it must be gradient preserving in order to
avoid the over-sharpening effect.

In this paper, we propose a high quality edge-preserving local filter based on
a novel local statistical edge model. Our filter preserves user-defined edges
practically unchanged, and it is gradient preserving. We discuss a quality
factor that has not been studied in previous literatures; it is _the
uniformity of spatial scale and signal variations in the detail layers_. Our
proposed method allows a flexible progressive decomposition without a
constraint of the spatial scale increment as in wavelet based or other
pyramidal schemes.

##  II Related Work

The bilateral filter has many enhanced variants available [11, 12, 8, 9, 13,
14, 15, 16]. However, most of them maintain the over-sharpening characteristic
which often results gradient reversal artefacts in the detail enhanced images.
Many modern edge-preserving filters [17, 18, 19, 20] share this
characteristic. A thorough analysis of artefacts caused by using the bilateral
filter and its variations is available [1]. The guided image filter [10] has
attractive performance but the authors recognize that its edge-preserving
capability cannot eliminate the halo artefacts.

Optimization based filters rely on carefully designed constraints to achieve
edge-preserving filtering using the global optimization approach. The weighted
least square (WLS) framework [1] overcomes the two artefacts which hinder most
local filters. However, we notice that the WLS method cannot differentiate
spatially small-scale image structures from the less important image details
which are small in both spatial and signal variation scales. Other
optimization based filters [21, 22, 23] tend to summarize the piecewise smooth
areas into step edges, i.e. non gradient-preserving. In some cases, spatial
displacement of features are observed [24], and this side effect is not
desirable for the image detail enhancement purpose. Another practical concern
about the optimization based filters is the opaqueness of the filter
parameters. Unlike the ones in kernel based filters, the parameters in the
optimization based filters are often non-intuitive from an end-user’s
perspective.

Wavelet based solution [25] offers good edge-aware decomposition but the rigid
decimation of spatial scale prevents flexible manipulation of image details.
The more recent deep learning approach [26] learns the filtering results from
a selection of curated filtered images. As a result, their effectiveness for
image detail enhancement becomes largely dependent on the training data set.

##  III Sub-window Variance Filter

###  III-A Local Statistical Edge Model

We propose an edge model which uses simple statistical information to identify
edge-alike features on local image patches. Our model comprises three
conditions as follows:

  1. 1.

An edge is formed by two adjacent groups of pixels.

  2. 2.

These two groups have contrasting intensities (global high variance).

  3. 3.

An edge is more distinct if the pixels of either group share similar intensity
(locally low variance in a subregion).

The first condition states the importance of the spatial property required to
define an edge, and the second condition describes the signal strength
difference necessary for an edge to be perceived. Given an image patch which
contains a distinct edge, the global statistical variance of intensity values
should be high as suggested by the second condition. However, a patch which
has a high variance value does not necessarily imply the existence of an edge
due to the lack of spatial information. Fig. 2(a) illustrates an image patch
containing a fuzzy edge which exhibits high variance.

((a)) A fuzzy edge formed by two contrasting groups of pixels exhibiting
overall high variance

((b)) A distinct edge formed by two contrasting groups with one group exhibits
low variance (highlighted)

Figure 2: Presence of a relatively low variance sub-region as an evidence of
the presence of edge(s)

The most important observation we made about the presence of a distinct edge
is stated in the third condition. Fig. 2(b) illustrates such a distinct edge;
it is the presence of a subregion which has a low local variance serves as a
clear background to highlight the adjacency of two groups of pixels. In order
to validate our edge model, we present a simple analysis in next section to
show how the statistical variance values of the local sub-regions, i.e. how
the sub-window variance values relate to the global variance.

###  III-B Properties of Sub-window Variance

By definition, the variance of a discrete signal is the expected value of the
squared deviation from the mean value of the signal. The variance value of any
arbitrary rectangular image patch on an image can be computed quickly by using
a pair of precomputed summed-area tables [27, 28]. Let us consider a simple 1D
signal $X$ as shown in Fig. 3, and we use two equal-sized sub-windows to
divide the signal into two halves namely $X_{A}$ and $X_{B}$. We express the
expected value of $X$ as the mean of the expected values evaluated from the
two sub-windows as expressed in equation (2).

$E[X]=\frac{E[X_{A}]}{2}+\frac{E[X_{B}]}{2}$ (2)

By using equation (2), we explore the relationship between the overall
variance $Var(X)$ and the sub-window variance values $Var(X_{A})$ and
$Var(X_{B})$ through the steps as shown in equation (3).

$\displaystyle\begin{split}Var(X)&=E[X^{2}]-(E[X])^{2}\\\
&=\frac{E[X_{A}^{2}]}{2}+\frac{E[X_{B}^{2}]}{2}-\Big{(}\frac{E[X_{A}]}{2}+\frac{E[X_{B}]}{2}\Big{)}^{2}\\\
&=\frac{E[X_{A}^{2}]-(E[X_{A}])^{2}}{2}+\frac{E[X_{B}^{2}]-(E[X_{B}])^{2}}{2}\\\
&~{}~{}~{}+\Big{(}\frac{E[X_{A}]}{2}-\frac{E[X_{B}]}{2}\Big{)}^{2}\\\
&=\frac{Var(X_{A})}{2}+\frac{Var(X_{B})}{2}+\frac{(E[X_{A}]-E[X_{B}])^{2}}{4}\end{split}$
(3)

Figure 3: A simple 1D signal and the two equal-sized sub-windows A and B

For a given variance value $Var(X)$ of a signal, if both sub-window variance
values $Var(X_{A})$ and $Var(X_{B})$ are small or close to zero, then
according to equation (3), the difference between $E[X_{A}]$ and $E[X_{B}]$ is
maximized and therefore it concludes the presence of a clear step edge
feature. This important property allows us to use sub-window variance values
as a means to detect edge-alike features in a signal.

Furthermore, according to equation (3), when the values of $Var(X)$,
$Var(X_{A})$ and $Var(X_{B})$ are identical, the third term vanishes. For the
third term to become zero, $E[X_{A}]$ must be equal to $E[X_{B}]$ and by
equation (2), it further concludes that all three expected values $E[X]$,
$E[X_{A}]$ and $E[X_{B}]$ must be equal too. One possible interpretation is
the presence of self-similarity such as a constant flat signal or the presence
of regular patterns with its feature wavelength smaller than the filter
support.

###  III-C Sub-window Variance Filter

We formulate the edge-preserving filtering process on an image patch as a
linear blend of the original patch and a filtered version of the patch. For a
given local patch $I_{k}$ centred on pixel $k$, our filtered patch
$I_{k}^{\prime}$ is computed by:

$I_{k}^{\prime}=A_{k}I_{k}+(1-A_{k})F(I_{k}),$ (4)

where $A_{k}\in[0,1]$. $A_{k}$ is called the _per-patch preservation factor_
which governs the degree of contribution by the original patch. $F()$ is a
generic smoothing filter, and we choose the box filter for its simplicity
which evaluates the mean intensity values of the original patch. This simple
linear blend formulation forces the filtered value bound to the original value
and the mean of the whole patch. We rely on this formulation to eliminate the
possibility of over-sharpening. We rewrite equation (4) as follows:

$I_{k}^{\prime}=A_{k}I_{k}+B_{k},$ (5)

where
$B_{k}=(1-A_{k})(\frac{1}{|\omega|}\sum\limits_{i\in{\omega_{k}}}p_{i})$,
$\omega_{k}$ is the local domain defined by the filter support centred at
pixel $k$, and $p_{i}$ is the intensity of the $i^{th}$ pixel.

Per-patch Preservation Factor. Based on our proposed edge model, we divide the
filter window into four equal-sized sub-windows namely A, B, C and D (Fig.
4(c)) in order to evaluate the variance value of each sub-window. This simple
uniform sampling strategy is motivated by the advantage of using summed-area
table [27, 28] for high performance filtering. This subdivision scheme also
resembles the one used in the Kuwahara filter [29].

((a)) Local image patch highlighted

((b)) image patch variance value

((c)) Sub-window variance values

Figure 4: Edge model exploiting locally low variance subregion as an evidence
of presence of an edge

Let $V=\\{\sigma_{A}^{2},\sigma_{B}^{2},\sigma_{C}^{2},\sigma_{D}^{2}\\}$ be
the set of intensity variances evaluated on the four sub-windows, and
$\sigma_{W}^{2}$ be the intensity variance of the whole patch. We define the
value of _per-patch preservation factor_ , $A_{k}$ by:

$A_{k}=min(1,\frac{\sigma_{max}^{2}}{\sigma^{2}_{min}+\epsilon}),$ (6)

where $\sigma_{max}^{2}=max(\\{\sigma_{W}^{2},V\\})$,
$\sigma_{min}^{2}=min(V)$ and $\epsilon$ is a positive valued user parameter.

Based on the second condition of our edge model (the requirement of
contrasting intensities), we use $\sigma_{max}^{2}$ as a potential edge
indicator on the patch. At the same time, we include the minimum sub-window
variance value $\sigma_{min}^{2}$ in the denominator of equation (6) to
promote strong preservation of edges following the third condition of our edge
model.

The user parameter $\epsilon$ has an intuitive meaning in this formulation.
Suppose we have a distinct edge with $\sigma_{min}^{2}=0$ and if
$\sigma_{max}^{2}$ is equal to the value of $\epsilon$, then the whole patch
will be fully preserved with $A_{k}=1$. In short, $\epsilon$ defines the
threshold variance value of a clear edge in a given filter window. In
addition, our simple filter model guarantees two neighbour points on a
gradient shall only have their intensity values become closer to each other
when filtered, i.e. there is no risk of over-sharpening.

Although our theoretical development focuses on edges, the formulation is not
biased on any geometrical property. We extract two patches from Fig. 5(a), and
while they look seemingly different (Fig. 5(b) and 5(c)), they have
numerically close variance values. Sub-window variance values and per-patch
preservation factors of both patches are computed and tabulated in Tables I
and II. The computed per-patch preservation factors successfully differentiate
these two patches, and conclude that the thin lines patch deserves stronger
preservation.

TABLE I: Sub-window variance values of patches in Fig. 5(b) (Thin lines) and 5(c) (Fuzzy) Patch |  ${\sigma_{A}^{2}}$ |  ${\sigma_{B}^{2}}$ |  ${\sigma_{C}^{2}}$ |  ${\sigma_{D}^{2}}$
---|---|---|---|---
Lines |  0.001352 |  $0.000375^{*}$ |  0.003283 |  0.003011
Fuzzy |  0.002491 |  0.001343 |  $0.001061^{*}$ |  0.001369
TABLE II: Per-patch preservation using equation (6), $\epsilon=0.0028$ Patch |  $\sigma_{min}^{2}$ |  $\sigma_{max}^{2}$ |  $A_{k}$
---|---|---|---
Lines |  0.000375 |  0.003327 |  $\min(1,1.0479)$
Fuzzy |  0.001061 |  0.003342 |  $\min(1,0.8656)$

Per-pixel preservation factor for image filtering. In order to filter a whole
image, we densely evaluate the per-patch preservation factors over the whole
image. Since each pixel receives multiple per-patch preservation estimations
from a fixed number of overlapping patches, the final filtered output is given
by:

$p_{i}^{\prime}=\bar{A_{k}}p_{i}+\bar{B_{k}},$ (7)

where $\bar{A_{k}}=\frac{1}{|\omega|}\sum\limits_{k\in{\omega_{i}}}A_{k}$,
$\bar{B_{k}}=\frac{1}{|\omega|}\sum\limits_{k\in{\omega_{i}}}B_{k}$ and
$\omega_{i}$ is the domain of all overlapping patches that offer estimation to
pixel $i$. $\bar{A_{k}}$ can be understood as a _per-pixel preservation
factor_. This simple averaging strategy guarantees piece-wise continuity, and
is a common practice among modern image filters [30, 21, 10, 31]. Fig. 6
illustrates how the per-pixel preservation factors (blue line) accurately
capture the strong edge locations of a given 1D signal.

((a)) Local image patches highlighted

((b)) Thin lines
$\sigma_{W}^{2}=0.003327$

((c)) Fuzzy
$\sigma_{W}^{2}=0.003342$

Figure 5: Feature sensitivity of sub-window variance filter Figure 6: Per-
patch and per-pixel preservation factors
$filter~{}width=11,~{}\epsilon=0.025$

((a)) Original

((b)) Bilateral filter
$\sigma_{d}=10,~{}\sigma_{r}=0.1$
SSIM = 0.76319

((c)) Domain Transform
$\sigma_{s}=10,~{}\sigma_{r}=0.45^{*}$
SSIM = 0.64674

((d)) ResNet
SSIM = 0.69766

((e)) Guided filter
$radius=10,~{}\epsilon=0.01$
SSIM = 0.76527

((f)) Local Laplacian
$\sigma_{r}=0.28,\alpha=3$
SSIM = 0.81236

((g)) $L_{0}$ smoothing
$\lambda=0.01$
SSIM = 0.71559

((h)) WLS
$~{}\alpha=0.9,\lambda=0.6$
SSIM = 0.70515

((i)) Ours
$radius=10,~{}\epsilon=0.01$
SSIM = 0.82533

Figure 7: Edge-preserving and gradient-preserving performance comparison. All
filter parameters are set according to the approach of matching the bilateral
filter documented in the respective literatures; *Parameters of the Domain
Transform filter were hand tuned for improved smoothing performance.

###  III-D Sub-window variance filter characteristics

Basic smoothing and detail layer properties. We compare our filter with the
leading edge-preserving filters commonly used in image decomposition (Fig. 7),
such as the bilateral filter [6] and the WLS smoothing method [1]. We have
included several modern edge-preserving filters in our comparison. Fig. 7
shows the smoothing results from each filter, and Fig. 9 shows their
respective extracted detail layers.

Fig. 7(b) shows the result using the bilateral filter, and it is the control
that all filters attempt to match. The domain transform (normalized
convolution) filter [18] (Fig. 7(c)) demonstrates satisfactory smoothing
performance but it results over-sharpening on the edges. Fig. 9(b) and 9(c)
show the detail layers extracted by these filters; they contain features are
small in both spatial and variation scales. However, they both exhibit over-
sharpening characteristics on the edge of the pupil as shown in Fig. 8.

The ResNet based filtering [26] result shown in Fig. 7(d) exhibits
characteristics shared by most optimization based filters. There are step
edges and loss of the important gradients especially on the eyelid. Among all
the detail layers produced, the one by ResNet (Fig. 9(d)) exhibits a strong
presence of image structures. The guided filter [10] (Fig. 7(e)) delivers
reasonable smoothing but there is an image-wide contrast reduction due to weak
edge preservation. This will likely lead to halos when the detail layers are
aggressively enhanced. Fig. 9(e) shows that its detail layer carries important
image structures.

Local Laplacian filtering [32, 33] has difficulty to deliver comparable
smoothing without aggressive filtering. The result (Fig. 7(f)) has
considerable amount of medium-scale gradients removed. Its detail layer (Fig.
9(f)) also suggests the presence of strong image structures. We have mentioned
that most structure extraction filters [22, 34, 23] are unsuitable for multi-
scale image decomposition, and we evaluate the $L_{0}$ smoothing filter [22]
in order to illustrate our argument. Fig. 7(g) shows that the result is over-
sharpened, and the banded edges are the natural consequence of its
approximation based formulation. Fig. 9(g) further shows the non-uniform
detail layer which exhibits both image structures and low frequency
information.

WLS filtering approach [1] is proposed specifically for multi-scale image
decomposition. Its smoothing result (Fig. 7(h)) matches the bilateral filter
but there is an obvious contrast reduction on the strong edges. Fig. 9(h)
shows that its respective detail layer holds structural edges, and it will
result considerable halos upon detail enhancement.

Figure 8: Close examination of the 1D slices extracted from the filtering
results presented in Fig.7. The blue dotted line represents the original
signal, and the red ones represent the filtered signals. The over-sharpened
regions are highlighted with red coloured circles.

Fig. 7(i) shows our filtering result, and it matches closely the smoothing
result of the bilateral filter. Our result is free from any over-sharpening or
contrast reduction. Fig. 9(i) further illustrates the strong edge-preservation
capability where the high-contrast area of the iris and the eyelashes are
practically untouched, and the detail layer demonstrates uniform fine scale
details.

This basic comparison helps us to identify the candidates for more in-depth
comparisons in the context of multi-scale image decomposition. We shall now
focus on the bilateral filter and the multi-scale specific WLS approach for
further comparisons.

((a)) Original

((b)) Bilateral filter

((c)) Domain transform

((d)) ResNet

((e)) Guided filter

((f)) Local Laplacian

((g)) $L_{0}$ smoothing

((h)) WLS

((i)) Ours

Figure 9: Detail layers extracted by various filters. Our method (i) does not
extract the eyelashes, and other important image structures into the detail
layer.

((a)) Original

((b)) Small Details Enhancement
($B+2\times$D1 + $4\times$D2 + $2\times$D3)

((c)) Fine Details Enhancement
($B+5\times$D1 + $1.5\times$D2 + $1.25\times$D3)

((d)) 1st detail layer (D1)
Fine-scale Details

((e)) 2nd detail layer (D2)
Small-scale Details

((f)) 3rd detail layer (D3)
Medium-scale Details

Figure 10: multi-scale image decomposition using the sub-window variance
filter
3 iterations with $radius=\\{2,4,8\\}$ and $\epsilon=0.015$

##  IV Multi-scale image decomposition

Applying our edge-preserving filter for multi-scale image decomposition is
straight forward. We follow the practice used in the WLS method [1] which is
modelled after the Laplacian pyramid [2]. The multi-scale decomposition is
achieved by decomposing the base image iteratively. If we write the original
image $I$ as $B_{0}$ with the base and detail layers of the $i^{th}$ level
decomposition as $B_{i}$ and $D_{i}$, we can show that

$B_{k}=B_{k+1}+D_{k+1}$ (8)

and a multi-scale decomposition with N detail layers can be written as

$I=B_{N}+\sum_{i=1}^{N}D_{i}$ (9)

with

$B_{k}=E(B_{k-1})$ (10)

and

$D_{k}=B_{k-1}-B_{k}$ (11)

where $E()$ is an edge-preserving filter.

((a)) WLS base layer
$\alpha=1.2,\lambda=0.1$

((b)) WLS base layer scan-line

((c)) WLS detail layer

((d)) WLS detail layer signal

((e)) Our base layer
$radius=2,\epsilon=0.015$

((f)) Our base layer scan-line

((g)) Our detail layer

((h)) Our Detail layer signal

Figure 11: Decomposition difference between the WLS approach and our method

In each iteration, the filter radius and the edge threshold parameter
$\epsilon$ may be adjusted in a progressive fashion in order to capture the
desired features. We apply our decomposition on an image as shown in Fig.
10(a), it is a four-scale decomposition using a constant $\epsilon=0.015$ and
a doubled filter support in each step with an initial filter radius of 2.
Strong edges are consistently preserved in each iteration, and our detail
layers exhibit uniform similar-scale features and variations. Fig. 10(b) shows
an enhancement where the small-scale details (Fig. 10(e)) is emphasized, and
Fig. 10(c) focuses on fine detail (Fig. 10(d)) enhancement.

###  IV-A Importance of Uniformity in the Detail Layers

The primary goal of multi-scale image decomposition is to enable flexible
manipulation of the details of different scales, and it is especially useful
to consider _signals of similar spatial-scale and variations_ together. An
alternative method [35] proposed to group all small spatial-scale signals
together regardless of their signal amplitudes. The disadvantage of such
method is that it directly reduces the headroom of possible contrast
enhancement for the details which have small amplitude of variations.

Fig. 11 shows a detailed comparison between our approach and the WLS method.
We tried to extract a fine detail only layer using the WLS method but the best
result we obtained still contains details of different scales, and is shown in
Fig. 11(c). This WLS extracted detail layer contains both smooth gradient in
the background, and structural details such as the eye’s bright highlight. It
is further visualized by the scan-line plot in Fig. 11(d).

Figures 11(e) and 11(f) show the base layer obtained using our method. The
eye’s bright highlight is well preserved, and our detail layer (Fig. 11(g))
contains uniformly high frequency details only. Fig. 12 shows a qualitative
comparison between the two decomposition methods. The common goal is to
enhance aggressively (a $10\times$ boost of details) the fur details of a
cat’s picture (original shown in Fig. 10(a)) using the decomposition results
shown in Fig. 11. The WLS enhancement result shown in Fig. 12(a) shows that
the global contrast of the whole picture is altered, and many regions have
reached peak brightness or blackened, and the whiskers become aliased.

Our enhancement result as shown in Fig. 12(b) exhibits a uniform enhancement
of the fur details without affecting the global image contrast.

((a)) $10\times$ detail enhancement using the WLS approach.

((b)) $10\times$ detail enhancement using our approach

Figure 12: Quality comparison between the WLS approach and our method in an
aggressive detail enhancement.

##  V Applications and Discussions

###  V-A Interactive Multi-scale Detail Manipulation

Unlike most previous methods which require pre-computation of the
decomposition; our method allows the decomposition stage to run at interactive
rate. We have implemented a simple GPU application using the GLSL shading
language, and this application runs on a modest nVIDIA GTX760 GPU. It is able
to decompose one single 1024$\times$1024 RGB floating point image in 20ms
using our method, and a three-scale decomposition can run at interactive rate.
We believe that a multi-scale image detail manipulation process is often based
on subjective creative decisions, so the possibility to adjust all the
decomposition parameters interactively should offer a good manipulation
experience.

((a)) Original image

((b)) WLS suppression result
$\alpha=2.0,\lambda=0.4$

((c)) The bilateral filter suppression result
$\sigma_{d}=\\{10,20\\},\sigma_{r}=\sqrt{0.03}$

((d)) Our suppression result
$radius=\\{10,20\\},\epsilon=0.03$

Figure 13: Uniform multi-scale details suppression

###  V-B Uniform Image Details Suppression

The ability to extract uniformly same-scale details by our method becomes an
important feature which allows us to handle some difficult detail extraction
problem as shown in Fig. 13. The image shown in Fig. 13(a) possesses details
of different scales, and our goal is to remove the bokehs and the raindrop
trails while keeping all other elements intact.

The WLS method’s approach makes it difficult to relate its filter’s parameters
with this task. It is almost impossible to isolate the target details from the
main subject. We attempted to apply iterative filtering tactics filter but it
delivered no improved results. Fig. 13(b) shows its best result with our best
effort. The detail layer (the block on the right hand side of Fig. 13(b))
extracted by the WLS approach shows that under the WLS filtering logic, it is
fairly difficult to isolate uniformly the details of similar scales. As a
result, its suppression becomes blurry and abstracted.

We apply the bilateral filter to this task, and we use iterated filtering in
order to obtain the two desired detail layers. The filter support size is
chosen to remove the thin and small features. Unfortunately, the detail layers
extracted (the blocks on the right side of Fig. 13(c)) by the bilateral filter
contain details of both scales, and we believe it is a consequence of the
filter’s edge-preservation strategy. The suppression result (Fig. 13(c)) by
the bilateral filter looks abstracted as expected.

Our filter delivers satisfactory results. The two detail layers (the blocks on
the right side of Fig. 13(d))) extracted possess uniform variations and share
similar spatial scales. The final base layer (Fig. 13(d)) also fulfils the
goal, and the result is significantly better than the other two leading
methods. The high sensitivity of structure by the sub-window variance filter
enables a lot of fine edge-like features to be well preserved on the base
layer. This uniform extraction capability makes our proposed method
particularly suitable for refined detail manipulation tasks which demand
separation of details.

###  V-C Limitations

There exist certain situations where the users may want to extract or suppress
image details of a limited range of tone only. Our current formulation
preserves edge-alike features based on a constant threshold variance value,
i.e. the parameter $\epsilon$. This tone selective suppression may become
challenging to our method, one possibility is to make the parameter $\epsilon$
tone adaptive, such as to extend this parameter into a function of mean pixel
intensity of the filter window.

##  VI Conclusion

We have presented a multi-scale image decomposition approach based on a novel
local statistical edge model which delivers interactive performance. Our
method emphasizes the quality of edge preservation, and the importance of
uniformity of details in terms of both spatial scale and variations during
decomposition. The simplicity of our intuitive method is based on an important
observation of edge properties from the perspective of perception and simple
image statistics.

## References

  * [1] Z. Farbman, R. Fattal, D. Lischinski, and R. Szeliski, “Edge-preserving decompositions for multi-scale tone and detail manipulation,” in _ACM Transactions on Graphics (TOG)_ , vol. 27, no. 3. ACM, 2008, p. 67.
  * [2] P. J. Burt and E. H. Adelson, “The laplacian pyramid as a compact image code,” _Communications, IEEE Transactions on_ , vol. 31, no. 4, pp. 532–540, 1983.
  * [3] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” _Pattern Analysis and Machine Intelligence, IEEE Transactions on_ , vol. 12, no. 7, pp. 629–639, 1990.
  * [4] M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” _Image Processing, IEEE Transactions on_ , vol. 7, no. 3, pp. 421–432, 1998.
  * [5] V. Aurich and J. Weule, “Non-linear gaussian filters performing edge preserving diffusion,” in _Mustererkennung 1995_. Springer, 1995, pp. 538–545.
  * [6] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in _Computer Vision, 1998. Sixth International Conference on_. IEEE, 1998, pp. 839–846.
  * [7] S. Bae, S. Paris, and F. Durand, “Two-scale tone management for photographic look,” in _ACM Transactions on Graphics (TOG)_ , vol. 25, no. 3. ACM, 2006, pp. 637–645.
  * [8] F. Durand and J. Dorsey, “Fast bilateral filtering for the display of high-dynamic-range images,” in _Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques_ , ser. SIGGRAPH ’02. New York, NY, USA: ACM, 2002, pp. 257–266. [Online]. Available: http://doi.acm.org/10.1145/566570.566574
  * [9] R. Fattal, M. Agrawala, and S. Rusinkiewicz, “Multiscale shape and detail enhancement from multi-light image collections,” _ACM Trans. Graph._ , vol. 26, no. 3, p. 51, 2007.
  * [10] K. He, J. Sun, and X. Tang, “Guided image filtering,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 35, no. 6, pp. 1397–1409, 2013.
  * [11] J. Chen, S. Paris, and F. Durand, “Real-time edge-aware image processing with the bilateral grid,” in _ACM SIGGRAPH 2007 Papers_ , ser. SIGGRAPH ’07. New York, NY, USA: ACM, 2007. [Online]. Available: http://doi.acm.org/10.1145/1275808.1276506
  * [12] H. Cho, H. Lee, H. Kang, and S. Lee, “Bilateral texture filtering,” _ACM Transactions on Graphics (TOG)_ , vol. 33, no. 4, p. 128, 2014.
  * [13] R. G. Gavaskar and K. N. Chaudhury, “Fast adaptive bilateral filtering of color images,” in _2019 IEEE International Conference on Image Processing (ICIP)_. IEEE, 2019, pp. 180–184.
  * [14] S. Paris and F. Durand, “A fast approximation of the bilateral filter using a signal processing approach,” in _Computer Vision–ECCV 2006_. Springer, 2006, pp. 568–580.
  * [15] P. Xu and W. Wang, “Improved bilateral texture filtering with edge-aware measurement,” _IEEE Transactions on Image Processing_ , vol. 27, no. 7, pp. 3621–3630, 2018.
  * [16] Q. Yang, K.-H. Tan, and N. Ahuja, “Real-time o (1) bilateral filtering,” in _Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on_. IEEE, 2009, pp. 557–564.
  * [17] J.-H. R. Chang and Y.-C. F. Wang, “Propagated image filtering,” in _2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_. IEEE, 2015, pp. 10–18.
  * [18] E. S. Gastal and M. M. Oliveira, “Domain transform for edge-aware image and video processing,” in _ACM Transactions on Graphics (TOG)_ , vol. 30, no. 4. ACM, 2011, p. 69.
  * [19] H. Lee, J. Jeon, J. Kim, and S. Lee, “Structure-texture decomposition of images with interval gradient,” in _Computer graphics forum_ , vol. 36, no. 6. Wiley Online Library, 2017, pp. 262–274.
  * [20] Q. Zhang, X. Shen, L. Xu, and J. Jia, “Rolling guidance filter,” in _European Conference on Computer Vision_. Springer, 2014, pp. 815–830.
  * [21] B. Gu, W. Li, M. Zhu, and M. Wang, “Local edge-preserving multiscale decomposition for high dynamic range image tone mapping,” _Image Processing, IEEE Transactions on_ , vol. 22, no. 1, pp. 70–79, 2013.
  * [22] L. Xu, C. Lu, Y. Xu, and J. Jia, “Image smoothing via l 0 gradient minimization,” in _ACM Transactions on Graphics (TOG)_ , vol. 30, no. 6. ACM, 2011, p. 174.
  * [23] L. Xu, Q. Yan, Y. Xia, and J. Jia, “Structure extraction from texture via relative total variation,” _ACM Transactions on Graphics (TOG)_ , vol. 31, no. 6, p. 139, 2012.
  * [24] M. Hua, X. Bie, M. Zhang, and W. Wang, “Edge-aware gradient domain optimization framework for image filtering by local propagation,” in _2014 IEEE Conference on Computer Vision and Pattern Recognition_. IEEE, 2014, pp. 2838–2845.
  * [25] R. Fattal, “Edge-avoiding wavelets and their applications,” in _ACM Transactions on Graphics (TOG)_ , vol. 28, no. 3. ACM, 2009, p. 22.
  * [26] F. Zhu, Z. Liang, X. Jia, L. Zhang, and Y. Yu, “A benchmark for edge-preserving image smoothing,” _IEEE Transactions on Image Processing_ , vol. 28, no. 7, pp. 3556–3570, 2019.
  * [27] W. Burger and M. J. Burge, _Digital image processing: an algorithmic introduction using Java_. Springer, 2016\.
  * [28] F. C. Crow, “Summed-area tables for texture mapping,” in _Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques_ , ser. SIGGRAPH ’84. New York, NY, USA: ACM, 1984, pp. 207–212. [Online]. Available: http://doi.acm.org/10.1145/800031.808600
  * [29] M. Kuwahara, K. Hachimura, S. Eiho, and M. Kinoshita, “Processing of ri-angiocardiographic images,” in _Digital processing of biomedical images_. Springer, 1976, pp. 187–202.
  * [30] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” _IEEE Transactions on image processing_ , vol. 16, no. 8, pp. 2080–2095, 2007.
  * [31] V. Katkovnik, A. Foi, K. Egiazarian, and J. Astola, “From local kernel to nonlocal multiple-model image denoising,” _International journal of computer vision_ , vol. 86, no. 1, pp. 1–32, 2010.
  * [32] M. Aubry, S. Paris, S. W. Hasinoff, J. Kautz, and F. Durand, “Fast local laplacian filters: Theory and applications,” _ACM Transactions on Graphics (TOG)_ , vol. 33, no. 5, p. 167, 2014.
  * [33] S. Paris, S. W. Hasinoff, and J. Kautz, “Local laplacian filters: edge-aware image processing with a laplacian pyramid.” _ACM Trans. Graph._ , vol. 30, no. 4, p. 68, 2011.
  * [34] L. Xu, J. S. Ren, Q. Yan, R. Liao, and J. Jia, “Deep edge-aware filters,” in _Proceedings of the 32nd International Conference on Machine Learning (ICML-15)_ , 2015, pp. 1669–1678.
  * [35] K. Subr, C. Soler, and F. Durand, “Edge-preserving multiscale image decomposition based on local extrema,” in _ACM Transactions on Graphics (TOG)_ , vol. 28, no. 5. ACM, 2009, p. 147\.


# Generating Connected, Simple, and Realistic Cyber Graphs for Smart Grids
††thanks: This work was supported by NSF under Award Number 1808064.

Osman Boyaci  Electrical Engineering
Texas A&M University
College Station, TX, 77843
<EMAIL_ADDRESS>M. Rasoul Narimani  College of Engineering
Arkansas State University
Jonesboro, AR, 72404
<EMAIL_ADDRESS>Katherine Davis  Electrical Engineering
Texas A&M University
College Station, TX, 77843
<EMAIL_ADDRESS>Erchin Serpedin  Electrical Engineering
Texas A&M University
College Station, TX, 77843
<EMAIL_ADDRESS>
###### Abstract

Smart grids integrate communication systems with power networks to enable
power grids operation and command through real-time data collection and
control signals. Designing, analyzing, and simulating smart grid
infrastructures as well as predicting the impact of power network failures
strongly depend on the topologies of the underlying power network and
communication system. Despite the substantial impact that the communication
systems bring to smart grid operation, the topology of communication systems
employed in smart grids was less studied. The power community lacks realistic
generative communication system models that can be calibrated to match real-
world data. To address this issue, this paper proposes a framework to generate
the underlying topological graphs for the communication systems deployed in
smart grids by mimicking the topology of real-world smart grids. In this
regard, we have updated the Chung-Lu algorithm to guarantee the communication
network connectivity and to match the degree distribution of a real-world
smart grid rather than following an expected degree distribution. In addition,
key characteristics of communication systems such as diameter, average
shortest paths, clustering coefficients, assortativity, and spectral gap were
taken into consideration to generate the most similar real-world communication
network for smart grid studies. We believe that the proposed algorithm to
generate realistic cyber graphs for smart grid studies will benefit the power
community.

## Nomenclature

$\mathcal{G},\mathcal{V},\mathcal{E}$

Graph, set of nodes, set of edges

$n,\ m\in\mathbb{Z}$

$|\mathcal{V}|,|\mathcal{E}|$

$u.d\in\mathbb{R}$

Degree of a node $u$

$S\in\mathbb{Z}^{n}$

Degree sequence

$\overline{d}\in\mathbb{Z}$

Max. degree

$\widehat{(\cdot)}$

Normalization operation s.t. $\sum_{1}^{n}\widehat{(\cdot)}=1$

$D\in\mathbb{Z}^{\overline{d}}$

Degree vector $D=[1,\dots,\overline{d}]$

$\widehat{K}\in\mathbb{R}^{\overline{d}}$

Normalized frequency vector

$\mathcal{G}.g\in\\{0,1\\}$

$\mathcal{G}$ is graphical (no self loops and parallel edges)

$\mathcal{G}.c\in\\{0,1\\}$

$\mathcal{G}$ is connected

$\rho\in\mathbb{R}$

Density as ${m}/{n}$

$\diameter\in\mathbb{R}$

Diameter as the longest shortest path

$\widetilde{sp}\in\mathbb{R}$

Average shortest path

$cc\in\mathbb{R}$

Clustering coefficient

$a\in\mathbb{R}$

Assortativity as the Pearson correlation coefficient of degrees between pairs
of linked nodes

$\lambda\in\mathbb{R}$

Spectral gap as the minimum non-zero eigenvalue of the normalized Laplacian

##  I Introduction

Communication systems play a major role in the deployment of smart grids
empowering them to be more resistant, secure, reliable and manageable and
ensuring connectivity of the grid components. The backbone of communication
systems in smart grids is represented by the information and communication
technologies that allow two-way communication and automated control.
Communication systems improve the efficiency and reliability of smart grids by
gathering and transmitting a wide variety of data for grid control and
decision-making purposes. The integration of cyber communications and control
systems into the power distribution infrastructure has a profound impact on
the operation, reliability, and efficiency of the power grid. The power and
communication systems in modern power grids are highly intertwined. Analyzing,
simulating, designing, and predicting the impact of network failures strongly
rely on the knowledge of a communication network topology [1]. Thus, studying
the underlying communication network topology is essential for the smart grid
operation and control [2].

In spite of the many models proposed for electrical power systems [3], the
problem of modeling the underlying communication network in smart grids was
less studied. In fact, despite the huge efforts deployed for studying smart
grids operation and control, modeling smart grids is still at its infancy.
There is not enough realistic and practical information about the topology of
the underlying communication network in smart grids. So far, various efforts
have focused on developing cyber-physical test models for general use by the
power system community [4, 5, 6, 7]. These studies consider different
characteristics of communication systems including vulnerabilities of
communication devices, attack paths, etc., to design a practical cyber layer
for cyber-physical power systems. However, taking all these characteristics
into consideration makes these approaches computationally intractable for
larger cyber graphs as the number of attack paths increases exponentially with
the number of nodes.

To analyze the impact of cyber-graphs on power networks operation, e.g.,
cascading failure analysis, we first need a fast and reliable framework to
generate realistic cyber graphs for power test cases irrespective of their
size. A few efforts were conducted for generating realistic cyber graphs for
power test cases. A graph generator based on the characteristics of Luxembourg
smart grid, which is a power-line communication (PLC) system [8], was
presented in [1] to create random but realistic smart grid communication
topologies. Different characteristics of power grid including nodal degree
distribution, graph spectrum and connectivity scaling property, etc., were
analyzed in [9] for designing efficient communication schemes for power test
cases. Heuristic algorithms were employed in [10] to improve the communication
reliability for smart grids at the transmission level. However, many generic
graph generation algorithms such as configuration model [11], Havel-Hakimi
algorithm [12], and Chung-Lu algorithm do not guarantee the graphical and
connected graphs properties, and thus are not fit for designing communication
systems for smart grids. Horvát-Modes model, on the contrary, yields both
connected and graphical outputs. Due to its edge connection mechanism it
produces graphs with large diameter and low assortativity, which are not
realistic for communication systems. Thus, it is important to propose
generative graph algorithms that take into account the real-world
characteristics of communication systems in the design process.

Power system statistics were leveraged to design optimal communication systems
for smart grids in [9]. Similarly, the communication system statistics can be
leveraged for designing a realistic communication system for smart grids which
is the centerpiece of this paper. Along this parallel, we first derive the
statistical metrics of a real-world smart grid’s communication system and then
propose a graph generator based on the statistical information of the smart
grid’s communication system. Different graph attributes of a real-world smart
grid communication graph such as diameter, assortativity, etc., are taken into
consideration in designing the communication system for power test cases.
Moreover, we adapt the Chung-Lu algorithm to preserve the connectivity of the
graph since the connectivity is a key characteristic of communication systems.
In addition, edge switching operation is employed to prevent self loops and
parallel paths in the communication graph.

The contributions of this work are outlined as follows: (1) To generate
connected, simple, and realistic cyber graphs for smart grids, we propose a
simple and elegant framework by updating the Chung-Lu algorithm. (2) To
satisfy the required degree distribution, we propose an adaptive remaining
degree approach instead of the fixed expected degrees. (3) To minimize the
length of cross edges between power and cyber graphs, we employ the Hungarian
algorithm for optimal matching between cyber and power nodes. (4) To compare
the proposed method with the currently available approaches, we implement
other graph generation methods in the literature such as configuration model,
Havel-Hakimi, Horvát-Modes, and Chung-Lu algorithms using the same degree
sequence and analyze the global characteristics of the output graphs.

The remainder of this paper is divided into four sections. Section II proposes
the cyber graph generation framework. Section III presents the generated
graphs and discusses their global characteristics. Finally, Section IV
concludes the paper.

##  II Cyber-Graph Generation

###  II-A Analyzing a real-world communication system

To be able to generate realistic cyber graphs, we first analyze a real
communication system of a smart grid given in [2, Table (3.2)]. Its degree
count vector $K^{*}$ is extracted as $K^{*}=[162,101,30,25,11,4,3,4,2,1]$ and
its global graph characteristics are tabulated in Table I.

Table I: Global graph characteristics of the reference system [2, Table (3.1)] $|V|$ | $|E|$ | $\rho$ | $\diameter$ | $\widetilde{sp}$ | $cc$ | $a$ | $\lambda$
---|---|---|---|---|---|---|---
343 | 357 | 1.04 | 28 | 11.47 | 0.05 | -0.22 | $1.16e^{-3}$

It is clear from the distribution of $K^{*}$ that the reference graph shows
scale free topology since $K^{*}$ values tend to diminish for larger degrees
($K^{*}_{1}=362$ $\gg$ $K^{*}_{10}=1$). In addition, its global
characteristics indicate that it is a sparse ($\rho=1.04$) and tree like graph
(relatively high $\diameter$ and $\widetilde{sp}$). Moreover, it has only a
few cyclic structures ($cc=0.05$), tends to connect higher degree nodes with a
lower degree ones ($a=-0.22$) and presents some bottleneck edges
($\lambda=1.16e^{-3}$) that removal of them may split the graph into different
components[2].

###  II-B Selecting the degree distribution function

To select an appropriate degree distribution function that can generate the
required degree distribution, we consider and optimize the parameters of three
main distribution: generalized lognormal distribution [13], power law
distribution [14] and zipf distribution [14]. Specifically, we first obtain
the frequency vector $\widehat{K^{*}}$ of the reference system and optimize
the parameters $\theta$ of each distribution function $f$ by:

$\min_{\theta}||\widehat{K^{f}}-\widehat{K^{*}}||_{2},$ (1)

where $\widehat{K^{*}}$ is the normalized frequency vector of the reference
system and $\widehat{K_{f}}$ denotes the frequency vector of $K^{f}$ with
$K^{f}_{d}=f_{\theta}(d)$. As can be seen from Fig. 1 while powerlaw and zipf
distributions underestimate $\widehat{K^{*}}$ when $D<4$ and overestimates it
when $D>4$, lognormal distribution better approximates the reference
distribution for all $D$ values. Formulation of the distributions, their
optimal parameters, and mean square errors when estimating $\widehat{K^{*}}$
is given in Table II. Note that since the lognormal distribution provides the
best approximation, we use the lognormal distribution
$f^{*}(x)=f_{\alpha=1.371,\beta=1.986}(x)$ to generate the degree sequences
for the rest of the paper.

Figure 1: Normalized frequencies of the optimized degree distribution functions to approximate the reference distribution Table II: Optimized degree distribution functions distribution | formula | $\bm{\alpha}$ | $\bm{\beta}$ | MSE
---|---|---|---|---
lognormal  | $f_{\alpha,\beta}(x)=\textrm{exp}\Big{(}\shortminus\big{(}\frac{log(x)}{\alpha}\big{)}^{\beta}\Big{)}$ | 1.371 | 1.986 | $2e^{-4}$
powerlaw  | $g_{\alpha,\beta}(x)=\beta x^{-\alpha}$ | 1.440 | 3.745 | $1.6e^{-3}$
zipf  | $h_{\alpha}(x)=x^{-\alpha}/\zeta(\alpha)$ | 1.440 |  | $1.6e^{-3}$

###  II-C Generating valid degree sequence

The first step for generating a valid degree sequence for a realistic cyber
graph is to specify the number of nodes, $n$, number of edges, $m$, and
maximum degree for each node, $\overline{d}$, [2]. Then, we can generate the
degree sequence $S$ by randomly drawing $n$ samples from the degree vector
$D=[1,\dots,\overline{d}]$ with the corresponding probabilities
$P(d=\widehat{F}_{d})$ where $F=[f^{*}(1),\dots,f^{*}(\overline{d})]$. Note
that not every $S$ is realizable since it should satisfy
$\sum_{i=1}^{n}S_{i}=2m$ for an undirected graph. In addition, to be able to
generate a simple graph without any self-loops or parallel edges, $S$ should
be graphical. For instance, $S1=[1,1,4]$ and $S2=[2,2]$ are not graphical due
to a self-loop and a parallel edge, respectively. To test whether a given
degree sequence $S$ is graphical or not, we utilize the well-known Havel
Hakimi algorithm [12] given in Algorithm 1. If $S$ is not graphical, a new
degree sequence should be generated.

1 function is_simple_graph($S$)

2 while True do

3 sort S ascending

4 $n\leftarrow|S|$

5 if $S_{1}=0$ and $S_{n}=0$ then return True

6 $v\leftarrow\ $popleft($S$)

7 if $v>n-1$ then return False

8 for $i\leftarrow 1\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}to}}\
v$ do

9 decrement $S_{i}$ if $S_{i}<0$ then return False

10

11 end for

12

13 end while

14

15

Algorithm 1 Check if the degree sequence is graphical

###  II-D Generating simple connected cyber graph

The main algorithm outlined in Algorithm 2, for generating a simple graph
$\mathcal{G}$ from a degree sequence $S$, presents five basic steps.

1 function main($S$)

2 $n,m\leftarrow|S|,sum($S$)/2$

3 create nodes ${v_{1},v_{2},\ldots,v_{n}}$ s.t. $v_{i}.d\leftarrow S[i]$

4 $I,J\leftarrow\\{\ \\},\\{v_{1},v_{2},\ldots,v_{n}\\}$

5 create $\mathcal{G}$ s.t. $\mathcal{G}.\mathcal{V}=\\{\
\\},\mathcal{G}.\mathcal{E}=[\ ]$

6 $v_{max}\leftarrow$ max. degree node

7 add $v_{max}$ to $\mathcal{G}.\mathcal{V}$ as the first node

8 remove $v_{max}$ from $J$ and add it to $I$

9 generate_tree($\mathcal{G},I,J,n$)

10 $E_{s},E_{p}\leftarrow$ add_remaining_edges($\mathcal{G},I,n,m$)

11 remove_self_loops($\mathcal{G},E_{s}$)

12 remove_parallel_edges($\mathcal{G},E_{p}$)

13 return $\mathcal{G}$

14

Algorithm 2 Generate a simple connected graph (main)

The first phase in Algorithm 2 is the initialization phase (lines 1 to 9) in
which $n$ and $m$ are determined, nodes are created with their required
degrees, visited and unvisited sets $I$ and $J$ are created, and $\mathcal{G}$
which includes the maximum degree node is created. The second phase is the
tree generation, and third phase is adding the remaining degrees. The fourth
and fifth phases are removing the sell loops and parallel edges in order to
make $\mathcal{G}$ simple.

Dynamic adaptive sampling outlined in Algorithm 3 is the backbone of the
proposed algorithm to satisfy the required degree sequence. It draws a sample
$v$ from the input set $I$ by a probability proportional to its nodes’
remaining degrees and decrements the degree of $v$ for further samplings.

1 function sample($I$)

2 draw random $v$ from set $I$ w.r.t. node degrees

3 decrement $v.d$

4 return $v$

5

6

Algorithm 3 Draw a sample from a set

Algorithm 4 outlines the tree generation algorithm in which at each iteration
a random node pair sampled from the visited set $I$ and the unvisited set $J$
are connected. Sampling random nodes from $I$ and $J$ guarantees the
connectivity of the output graph $\mathcal{G}$ in which any two nodes are
connected by exactly one path, which brings the tree property.

1 function generate_tree($\mathcal{G},I,J,n$)

2 for $i\leftarrow 1\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}to}}\
n-1$ do

3 $u\leftarrow$ sample($I$)

4 $v\leftarrow$ sample($J$)

5 remove $v$ from $J$ and add it to $I$

6 add $v$ to $\mathcal{G}.\mathcal{V}$

7 add $(u,v)$ to $\mathcal{G}.\mathcal{E}$

8

9 end for

10

11

Algorithm 4 Generate the tree.

If there is any remaining edge to be added to $\mathcal{G}$, in other words,
if there is any positive degree node pairs in $\mathcal{G}$, it is simply
added between these two randomly sampled nodes as summarized in Algorithm 5.
Note that self loops and parallel edges are saved in $E_{s}$ and $E_{p}$ for
later removal.

1 function add_remaining_edges($\mathcal{G},I,n,m$)

2 $E_{s},E_{p}\leftarrow[\ ],[\ ]$

3 for $i\leftarrow n\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}to}}\
m$ do

4 $u\leftarrow$ sample($I$)

5 $v\leftarrow$ sample($I$)

6 if $u=v$ then append $u$ to $E_{s}$

7 else if $(u,v)\in\mathcal{G}.\mathcal{E}$ then append $(u,v)$ to $E_{p}$

8 add $(u,v)$ to $\mathcal{G}.\mathcal{E}$

9

10 end for

11 return $E_{s},E_{p}$

12

Algorithm 5 Add remaining edges.

To remove the self-loops, we implement the edge switching strategy in
Algorithm 6. Assume that node $u$ has a self loop and other nodes $s$ and $t$
present an edge. We first remove the edges $(u,u)$ and $(s,t)$ and then add
edges $(u,s)$ and $(u,t)$ to remove the self loop at $u$. Note that remaining
node degrees do not change after this operation as indicated by the numbers
beneath each node in Fig. 2 which show the number of edges that need to be
added to each node.

Figure 2: Edge switching operations to remove a self loop

1 function remove_self_loops($\mathcal{G},E_{s}$)

2 while $|E_{s}|>0$ do

3 draw sample $u\in E_{s}$

4 draw sample $(s,t)\in\mathcal{G}$

5 if $|\\{u,s,t\\}|=3$ then

6 if $(u,s)\not\in G.\mathcal{E}\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}and}}\
(u,t)\not\in G.\mathcal{E}$ then

7 remove $u$ from $E_{s}$

8 remove $(u,u)$ and $(s,t)$ from $\mathcal{G}.\mathcal{E}$

9 add $(u,s)$ and $(u,t)$ to $\mathcal{G}.\mathcal{E}$

10

11 end if

12

13 end if

14

15 end while

16

17

Algorithm 6 Remove self-loop by edge switching

Parallel edges are removed similarly as shown in Algorithm 7. Assume
$\mathcal{G}$ has parallel edges $(u,v)$ and other nodes $s$ and $t$ present
an edge. To remove one of the parallel edges between $u$ and $v$, we first
remove one of them, i.e., $(u,v)$ and $(s,t)$. Next, we add edges $(u,s)$ and
$(v,t)$ to the graph. The process of eliminating parallel edges is illustrated
in Fig. 2 where the required node degrees are shown by the numbers beneath
each node.

Figure 3: Edge switching operations to remove a parallel edge

1 function remove_parallel_edges($\mathcal{G},E_{p}$)

2 while $|E_{p}|>0$ do

3 sample $(x,y)\in\mathcal{G}$

4 if $|\\{u,v,x,y\\}|=4$ then

5 if $(u,x)\not\in\mathcal{G}.\mathcal{E}\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}and}}\
(v,y)\not\in\mathcal{G}.\mathcal{E}$ then

6 remove $(u,v)$ from $E_{p}$

7 remove $(u,v)$ and $(x,y)$ from $\mathcal{G}.\mathcal{E}$

8 add $(u,x)$ and $(v,y)$ to $\mathcal{G}.\mathcal{E}$

9

10 end if

11 else if $(u,y)\not\in\mathcal{G}.\mathcal{E}\
\textnormal{{\color[rgb]{0,0,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.9}and}}\
(v,x)\not\in\mathcal{G}.\mathcal{E}$ then

12 remove $(u,v)$ from $E_{p}$

13 remove $(u,v)$ and $(x,y)$ from $\mathcal{G}.\mathcal{E}$

14 add $(u,y)$ and $(v,x)$ to $\mathcal{G}.\mathcal{E}$

15

16 end if

17

18 end if

19

20 end while

21

22

Algorithm 7 Remove parallel edges by edge switching

###  II-E Relabeling the output graph’s nodes

Since the proposed algorithm randomly generates the cyber graph, the nodes’
position can be imperfect to match their corresponding nodes in the power
graph. For instance, a cyber node can be placed far away from the power node
that it controls as demonstrated by the cyber node 6 given in Fig.4a and the
power node 6 given in Fig.4b.

(a)  (b)  (c)

Figure 4: Relabeling the generated cyber graph to minimize the cross edge
distances. (a) Generated cyber graph. (b) Given power graph. (c) The cyber
graph after relabeling. After relabelling, the distance between a power node
in (b) and its corresponding cyber node in (c) with the same label is
minimized.

To address this issue, we keep the node positions fixed and relabel the cyber
graph labels to minimize the cross links’ distances. Specifically, we
formalize this as an optimization problem (2) based on the Hungarian algorithm
[15] to find the optimal matching between power and cyber nodes:

$\min_{L,R},\textrm{Tr}(LCR)$ (2)

where $L$ and $R\in\\{0,1\\}^{n\times n}$ are the permutation matrices and
$C\in R^{n\times n}$ is the cost matrix whose elements are defined by
Euclidean distances $C_{u,v}=||u-v||_{2}$ for each $u,v$ pairs from power and
cyber graphs, respectively. As shown in Figure 4 for a 7-bus test case, the
node “4” in the generated cyber graph matches with power node “6” which is not
in its vicinity before applying the relabeling process. After applying the
relabeling process, the node “6” in the generated cyber graph matches with
node “6” in power graph that minimizes crossing edge distance between two
graphs.

##  III Results and Discussion

For a fair comparison, we implemented the existing approaches in the
literature such as configuration model (CM), Havel-Hakimi (HH), Chung-Lu (CL),
and Horvát-Modes (HM) algorithms. Then, we generate random graphs for 30-,
118-, and 300-buses IEEE test systems using the same degree sequence.
Generated graphs are visualized in Fig. 5 where nodes’ color and sizes
indicate their degrees. In addition, the global characteristics of these
graphs are tabulated in Table III.

Model |  30-nodes |  118-nodes |  300-nodes
---|---|---|---
Configuration model |  |  |
Havel-Hakimi |  |  |
Chung-Lu |  |  |
Horvát-Modes |  |  |
Proposed work |  |  |

Figure 5: Generated graphs for each model (rows) and system (columns). Node’s colors and sizes are given according to their degrees for better visualization Table III: Global graph characteristics for each model and system (CM:Config. model, HH: Havel-Hakimi, CL:Chung-Lu HM:Horvát-Modes, PW:Proposed work n | model | $\mathcal{G}.g$ | $\mathcal{G}.c$ | $\rho$ | $\diameter$ | $\widetilde{sp}$ | $cc$ | $a$ | $\lambda$
---|---|---|---|---|---|---|---|---|---
30 | CM | ✗ | ✗ | 1.167 | – | – | – | -0.128 | $9.62e^{-2}$
| HH | ✓ | ✗ | 1.167 | – | – | 0.303 | 0.776 | $2.19e^{-1}$
| CL | ✓ | ✗ | 0.933 | – | – | 0.098 | -0.132 | $9.80e^{-2}$
| HM | ✓ | ✓ | 1.167 | 8 | 3.84 | 0.050 | -0.676 | $4.74e^{-2}$
| PW | ✓ | ✓ | 1.167 | 7 | 3.44 | 0.048 | -0.220 | $8.62e^{-2}$
118 | CM | ✗ | ✗ | 1.102 | – | – | – | -0.010 | $3.13e^{-2}$
| HH | ✓ | ✗ | 1.102 | – | – | 0.417 | 0.880 | $1.22e^{-1}$
| CL | ✓ | ✗ | 1.220 | – | – | 0.058 | -0.002 | $8.86e^{-2}$
| HM | ✓ | ✓ | 1.102 | 17 | 7.88 | 0.000 | -0.737 | $7.68e^{-3}$
| PW | ✓ | ✓ | 1.102 | 13 | 5.64 | 0.026 | -0.220 | $1.68e^{-2}$
300 | CM | ✗ | ✗ | 1.040 | – | – | – | -0.059 | $7.77e^{-3}$
| HH | ✓ | ✗ | 1.040 | – | – | 0.365 | 0.927 | $1.79e^{-3}$
| CL | ✓ | ✗ | 0.943 | – | – | 0.008 | -0.029 | $4.50e^{-2}$
| HM | ✓ | ✓ | 1.040 | 54 | 21.22 | 0.000 | -0.606 | $7.62e^{-4}$
| PW | ✓ | ✓ | 1.040 | 22 | 9.22 | 0.007 | -0.226 | $2.86e^{-3}$

As can be seen from the Fig. 5 and Table III, although these graphs are
generated by the same degree distribution, their configurations are totally
different. The graph generated by “Configuration Model” is neither graphical,
nor connected. Havel-Hakimi model, in contrast, generates graphical outputs.
However, this model does not guarantee connectivity of the graph and also it
has a high clustering coefficient $(cc>0.3)$ and assortativity $(a>0.7)$.
These features makes Havel-Hakimi model ineligible for generating realistic
cyber graphs. Similar to the Havel-Hakimi model, Chung-Lu model generates
graphical but not connected graphs. However, since it utilizes the expected
degree distribution instead of the exact one, its generated graphs may show
different characteristics from the graph that generated by the exact degree
distribution.

For instance, the graphs generated by the Chung-Lu model for IEEE 30- and
300-bus test systems present less edges $(\rho<0.95)$ than the required edges.
Conversely, the graph generated by the Chung-Lu model for the 118-bus test
system has more edges $(\rho>1.2)$ compared to the required one. Horvát-Modes
model, contrary to the previously mentioned models, generates both graphical
and connected outputs. Yet, its highly low assortativity $(a<-0.6)$ is not
compatible with the real-world cyber graph. In addition, the graph generated
for 300-bus test system presents high diameter $(\diameter=54)$ and high
average shortest path ($\widetilde{sp}>21$) which indicate its “bias” induced
to preserve the connectivity. In contrast, our proposed method generates tree-
like connected structures that have the exact degree distribution and similar
graph attributes to the reference system. Moreover, it generates a graph which
has a diameter and an average shortest paths proportional to its node size, an
appropriate clustering coefficient, a spectral gap decreasing with its node
size, and almost the same assortativity $(a=-0.22)$ with the reference system.

Another finding of our experiments is that although the graphs generated by
Havel-Hakimi, Horvát-Modes, and the proposed work exhibit the same
distribution, they produce totally different topologies, especially for the
300-bus test system in Fig. 5. For instance, Havel-Hakimi produces many
2-vertex or 3-vertex disconnected cliques. Horvát-Modes, on the contrary,
tends to create graphs with higher assortativity to preserve the connectivity.
In contrast, the proposed model generates tree like structures to better model
a real-world cyber graph. Aside from the global graph characteristics given in
Table III, this situation corroborates the supremacy of the proposed algorithm
over the existing ones.

##  IV Conclusion

In this paper, we propose a simple and elegant framework by modifying the
Chung-Lu algorithm to generate connected, simple, and realistic cyber graphs
for power systems. For generating graphs which have the exact required node
degree distribution, we propose an adaptive remaining degree approach instead
of the fixed expected degree in the Chung-Lu method. In addition, we utilize
the Hungarian algorithm to minimize the length of cross-links between the
power graph and the generated cyber graph. We implement other graph generation
methods to compare the suitability of the proposed algorithm with those in
literature including configuration model, Havel-Hakimi, Chung-Lu, and Horvát-
Modes algorithms and generate cyber graphs using the same degree sequence for
each model. Generated cyber graphs for IEEE 30-, 118-, and 300-bus test
systems demonstrate that the proposed model yields better results compared to
the existing approaches in terms of global characteristics, connectivity, and
graphicality to mimic a real-world communication system. The proposed
framework can be utilized by power system community to generate realistic
cyber graphs for cyber-physical power system studies.

## References

  * [1] Z. Wang, A. Scaglione, and R. J. Thomas, “Generating statistically correct random topologies for testing smart grid communication and control networks,” _IEEE Transactions on Smart Grid_ , vol. 1, no. 1, pp. 28–39, 2010.
  * [2] X. Fan, D. Wang, S. Aksoy, A. Tbaileh, Q. H. andT. Fu, and J. Ogle, “Coordination of transmission, distribution and communication systems for prompt power system recovery after disasters,” _PNNL-28598_ , 2019.
  * [3] IEEE PES Task Force on Benchmarks for Validation of Emerging Power System Algorithms, “The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms,” _arXiv:1908.02788_ , Aug. 2019. [Online]. Available: https://github.com/power-grid-lib/pglib-opf
  * [4] K. R. Davis, C. M. Davis, S. A. Zonouz, R. B. Bobba, R. Berthier, L. Garcia, and P. W. Sauer, “A cyber-physical modeling and assessment framework for power grid infrastructures,” _IEEE Transactions on Smart Grid_ , vol. 6, no. 5, pp. 2464–2475, 2015.
  * [5] G. A. Weaver, K. Davis, C. M. Davis, E. J. Rogers, R. B. Bobba, S. Zonouz, R. Berthier, P. W. Sauer, and D. M. Nicol, “Cyber-physical models for power grid security analysis: 8-substation case,” in _2016 IEEE International Conference on Smart Grid Communications (SmartGridComm)_ , 2016, pp. 140–146.
  * [6] P. Wlazlo, K. Price, C. Veloz, A. Sahu, H. Huang, A. Goulart, K. Davis, and S. Zounouz, “A cyber topology model for the texas 2000 synthetic electric power grid,” in _2019 Principles, Systems and Applications of IP Telecommunications (IPTComm)_ , 2019, pp. 1–8.
  * [7] A. Sahu, H. Huang, K. Davis, and S. Zonouz, “A framework for cyber-physical model creation and evaluation,” in _2019 20th International Conference on Intelligent System Application to Power Systems (ISAP)_ , 2019, pp. 1–8.
  * [8] S. Galli, A. Scaglione, and Z. Wang, “For the grid and through the grid: The role of power line communications in the smart grid,” _Proceedings of the IEEE_ , vol. 99, no. 6, pp. 998–1027, 2011.
  * [9] Z. Wang, A. Scaglione, and R. J. Thomas, “Generating statistically correct random topologies for testing smart grid communication and control networks,” _IEEE Transactions on Smart Grid_ , vol. 1, no. 1, pp. 28–39, 2010.
  * [10] K. Velin, L. Martin, T. David, and G. Teresa, “Reliable communication networks for smart grid transmission systems,” _Journal of Network and Systems Management_ , vol. 24, pp. 629–652, 2016.
  * [11] M. E. Newman, “The structure and function of complex networks,” _SIAM review_ , vol. 45, no. 2, pp. 167–256, 2003.
  * [12] S. L. Hakimi, “On realizability of a set of integers as degrees of the vertices of a linear graph. i,” _Journal of the Society for Industrial and Applied Mathematics_ , vol. 10, no. 3, pp. 496–506, 1962.
  * [13] T. G. Kolda, A. Pinar, T. Plantenga, and C. Seshadhri, “A scalable generative graph model with community structure,” _SIAM Journal on Scientific Computing_ , vol. 36, no. 5, pp. C424–C452, 2014.
  * [14] M. Cristelli, M. Batty, and L. Pietronero, “There is more than a power law in zipf,” _Scientific reports_ , vol. 2, no. 1, pp. 1–7, 2012.
  * [15] H. W. Kuhn, “The hungarian method for the assignment problem,” _Naval research logistics quarterly_ , vol. 2, no. 1-2, pp. 83–97, 1955.


# On singular variants of the Singer-Hopf Conjecture

Laurenţiu Maxim  Department of Mathematics, University of Wisconsin-Madison,
480 Lincoln Drive, Madison WI 53706-1388, USA<EMAIL_ADDRESS>Dedicated
to the memory of Prof. Jean-Pierre Demailly

###### Abstract.

We propose singular variants of the Singer-Hopf conjecture, formulated in
terms of the Euler-Mather characteristic, intersection homology Euler
characteristic and, resp., virtual Euler characteristic of a closed
irreducible subvariety of an aspherical complex projective manifold. We prove
the conjecture under the assumption that the cotangent bundle of the ambient
variety is numerically effective (nef), or, more generally, when the ambient
manifold admits a finite morphism to a complex projective manifold with a nef
cotangent bundle.

###### Key words and phrases:

Euler characteristic, aspherical manifold, Singer-Hopf conjecture,
constructible function, characteristic cycle, nef vector bundle

###### 2010 Mathematics Subject Classification:

14F35, 14F45, 14C17, 32S60, 58K30

##  1\. Introduction

The main purpose of this note is to overview and enhance some of the recent
developments [2, 21] around the Singer-Hopf conjecture in the complex
algebraic context. In order to reach a wider audience, results are formulated
in the convenient language of constructible functions. Along the way, we
generalize a result of [1] on the non-negativity of Euler characteristics of
an important class of constructible functions.

Recall that a connected CW complex is aspherical if its universal cover is
contractible. Closed Riemannian manifolds with non-positive sectional
curvature are aspherical. The following conjecture on the Euler characteristic
of a closed aspherical manifold was made by Hopf (and later on strengthened by
Singer):

######  Conjecture 1.1 (Singer-Hopf).

If $X$ is a closed aspherical manifold of real dimension $2n$, then

(1) $(-1)^{n}\cdot\chi(X)\geq 0.$

The conjecture is clearly true for $n=1$ (i.e., real dimension $2$). For
closed Riemannian manifolds with non-positive sectional curvature, the
conjecture is due to Hopf and Chern, and it was mentioned in the list of
problems [30] compiled by Yau in 1982; it it true in this case for $n=2$ since
the Gauss-Bonnet integrand has the desired sign, cf. [6, Theorem 5] where the
proof is attributed to Milnor. To our knowledge, Conjecture 1.1 is not known
for all closed aspherical $4$-manifolds, and it is open for $n\geq 3$.

In [14], Gromov introduced the notion of Kähler hyperbolicity, including
compact Kähler manifolds with negative sectional curvature, and he verified
Conjecture 1.1 for Kähler hyperbolic manifolds. Cao-Xavier [5] and Jost-Zuo
[15] independently introduced the concept of Kähler nonellipticity, including
compact Kähler manifolds with non-positive sectional curvature, and settled
the Singer-Hopf conjecture in this case (cf. also [10]). All these works
proved a corresponding version of Conjecture 1.1 by means of vanishing
$L^{2}$-cohomology. More recently, Liu-Maxim-Wang [21] proved the complex
projective version of Conjecture 1.1 under the additional assumption that the
(holomorphic) cotangent bundle $T^{*}X$ of $X$ is numerically effective (nef,
for short), and they conjectured that aspherical complex projective manifolds
have nef cotangent bundles. Finally, Arapura-Wang gave in [2] a new proof of
Conjecture 1.1 for compact Kähler manifolds with non-positive sectional
curvature, using the fact that the cotangent bundle of such a manifold is nef.

In this note we propose the following singular variants of the Singer-Hopf
Conjecture 1.1 in the complex projective context:

######  Conjecture 1.2.

If $Z\subseteq X$ is a closed irreducible subvariety of an aspherical complex
projective manifold $X$, then:

  * (i)

$(-1)^{\dim_{\mathbb{C}}Z}\cdot\chi(Z,Eu_{Z})\geq 0$, where $Eu_{Z}$ is the
local Euler obstruction function of MacPherson and $\chi(Z,Eu_{Z})$ is the
corresponding Euler-Mather characteristic.

  * (ii)

$(-1)^{\dim_{\mathbb{C}}Z}\cdot\chi^{IH}(Z)=\chi(Z,ic_{Z})\geq 0$, where
$\chi^{IH}(Z)$ denotes the intersection cohomology Euler characteristic of
$Z$, with $ic_{Z}$ defined by taking the stalkwise Euler characteristic of the
$IC$-complex $IC_{Z}$ on $Z$.

  * (iii)

$\chi_{vir}(Z):=\chi(Z,\nu_{Z})\geq 0$, where $\nu_{Z}$ is Behrend's
constructible function of $Z$ and $\chi_{vir}(Z)$ is the corresponding
Donaldson-Thomas invariant.

If $Z=X$, then all statements in Conjecture 1.2 reduce to the complex
projective version of the Singer-Hopf Conjecture 1.1. More generally, if $Z$ a
smooth closed irreducible subvariety of the aspherical complex projective
manifold $X$, all statements in Conjecture 1.2 become
$(-1)^{\dim_{\mathbb{C}}Z}\cdot\chi(Z)\geq 0$.

In fact, we make the following more uniform conjecture (which, as explained in
Proposition 2.2, turns out to be equivalent to Conjecture 1.2(i)).

######  Conjecture 1.3.

Let $X$ be an aspherical complex projective manifold and let $\varphi$ be a
constructible function on $X$ with effective characteristic cycle. Then the
Euler characteristic of $\varphi$ is non-negative, that is,
$\chi(X,\varphi)\geq 0$.

If $X$ is an abelian variety, Conjecture 1.3 can be deduced from [12, Theorem
1.3], see also [1] and [9]. When $\varphi$ is obtained (by taking the
stalkwise Euler characteristic) from a perverse sheaf, Conjecture 1.3 reduces
to [21, Conjecture 6.2].

In this note, we prove Conjecture 1.3 under the additional assumption that the
cotangent bundle $T^{*}X$ of $X$ is nef (e.g., globally generated), or, more
generally, if $X$ admits a finite morphism to a complex projective manifold
with nef cotangent bundle. Moreover, the inequalities become strict if ``nef''
is replaced by ``ample''.

######  Theorem 1.4.

Let $X$ be a complex projective manifold and let $\varphi$ be a constructible
function on $X$ with effective characteristic cycle. Assume $X$ admits a
finite morphism $f:X\to Y$ to a complex projective manifold $Y$ with nef
cotangent bundle (e.g., $Y$ has non-positive sectional curvature). Then
$\chi(X,\varphi)\geq 0$. Moreover, the inequality is strict if $T^{*}Y$ is
ample (e.g., $Y$ has negative sectional curvature).

A weaker version of Conjecture 1.3, for $\varphi$ coming from a perverse sheaf
on $X$ and $T^{*}X$ nef, was proved in [21, Proposition 3.6]. While the proof
of Theorem 1.4 follows the same lines as that of loc.cit. (see also Theorem
3.5 in Section 3 for a more general statement), what we want to emphasize here
are the various facets of Conjecture 1.3, reflected in the statement of
Conjecture 1.2, if more general coefficients are allowed. Note that asking for
the characteristic cycle of a constructible function to be effective is much
weaker that asking for that function to come from a perverse sheaf. Conjecture
1.2 is obtained directly from Conjecture 1.3 by letting $\varphi$ be one of
the following constructible functions supported on $Z$:
$(-1)^{\dim_{\mathbb{C}}Z}Eu_{Z}$, $ic_{Z}$ and $\nu_{Z}$, respectively, all
of which are known to have effective characteristic cycles, e.g., see the
discussion in [1, Section 7]. Note that, in the notations of Section 2, only
$ic_{Z}$ comes from a perverse sheaf.

We prove Theorem 1.4 (see also Theorem 3.5) following the same approach as in
[21] (see also [2]), based on Kashiwara's index theorem for constructible
complexes of sheaves (cf. [16]), together with (semi-)positivity results for
nef (resp., ample) bundles (cf. [7, 11]). It is well-known that if $X$ has
non-positive (resp., negative) sectional curvature, then $T^{*}X$ is nef
(resp., ample); see, e.g., [7] or [2, Lemmas 4.1, 4.2]. And, as already
mentioned, it was conjectured in [21] that aspherical complex projective
manifolds have nef cotangent bundles.

It was also proved in [2] that Conjecture 1.3 is true if $X$ is an aspherical
complex projective manifold (or, more generally, if $X$ has a large
fundamental group [18]) which admits a cohomologically rigid almost faithful
semi-simple representation, provided that $\varphi$ comes from a perverse
sheaf on $X$. In Section 3, we note that the proof of this result in loc.cit
extends to all constructible functions with effective characteristic cycles,
since the only operations involved in the proof preserve the effectivity of
characteristic cycles of constructible functions.

The paper is organized as follows. In Section 2, we review the relevant
background about constructible complexes, constructible functions,
characteristic cycles, and prove (in Proposition 2.2) that Conjecture 1.3 is
equivalent to Conjecture 1.2(i). In Section 3, we review (semi-)positivity
results for nef and, resp., ample vector bundles on complex projective
manifolds, and prove Theorems 1.4 (as a consequence of the more general
statement of Theorem 3.5). We conclude with a discussion around another
conjecture of Hopf, whose proof in the projective/Kähler context is known to
follow from classical results in complex algebraic geometry.

Acknowledgments. The author thanks Botong Wang and Jörg Schürmann for useful
discussions. Partial support was provided by the Simons Foundation and by the
Romanian Ministry of National Education.

##  2\. Preliminaries: constructible complexes and characteristic cycles

Let $X$ be a complex algebraic manifold. We denote by $D^{b}_{c}(X)$ the
bounded derived category of $\mathbb{C}$-constructible complexes on $X$.
Consider the functor

$CC:K_{0}(D^{b}_{c}(X))\longrightarrow LCZ(T^{*}X)$

which associates characteristic cycles to (Grothendieck classes of)
constructible complexes on $X$ (e.g., see [8, Definition 4.3.19] or [17,
Chapter IX]). Here, we let $LCZ(T^{*}X)$ denote the free abelian group spanned
by the irreducible conic Lagrangian cycles in the cotangent bundle $T^{*}X$.
Its elements are of the form $\sum_{Z}n_{Z}\cdot T^{*}_{Z}X$, for some
$n_{Z}\in\mathbb{Z}$ and $Z$ closed irreducible subvarieties of $X$. Recall
that, if $Z$ is a closed irreducible subvariety of $X$ with smooth locus
$Z_{\operatorname{reg}}$, its conormal bundle $T^{*}_{Z}X$ is the closure in
$T^{*}X$ of

$T^{*}_{Z_{\operatorname{reg}}}X:=\\{(z,\xi)\in T^{*}X\mid z\in
Z_{\operatorname{reg}},\ \xi\in T^{*}_{z}X,\ \xi|_{T_{z}Z_{\rm reg}}=0\\}.$

One then has a group isomorphism

$T:LCZ(T^{*}X)\longrightarrow Z(X)$

to the group $Z(X)$ of algebraic cycles on $X$, defined on generators by:
$T^{*}_{Z}X\mapsto(-1)^{\dim_{\mathbb{C}}Z}Z.$

A function $\varphi:Z\to\mathbb{Z}$ on a complex algebraic variety (e.g., a
closed subvariety of $X$) is said to be constructible if there is a Whitney
stratification $\mathcal{S}$ of $Z$ so that $\varphi$ is constant on each
(connected) stratum $S\in\mathcal{S}$. The Euler characteristic of such a
constructible function $\varphi$ is defined by

$\chi(Z,\varphi):=\sum_{S\in\mathcal{S}}\chi(S)\cdot\varphi_{|_{S}}.$

Let $F(Z)$ be the group of constructible functions on $Z$.

To any bounded constructible complex ${\mathcal{F}}^{\centerdot}\in
D^{b}_{c}(Z)$ on a complex algebraic variety $Z$, one associates a
constructible function $\chi_{st}({\mathcal{F}}^{\centerdot})\in F(Z)$ by
taking the stalkwise Euler characteristic, i.e.,

$\chi_{st}({\mathcal{F}}^{\centerdot})(z):=\chi({\mathcal{F}}^{\centerdot}_{z})$

for any $z\in Z$. For example, $\chi_{st}(\mathbb{C}_{Z})=1_{Z}$, the
indicator function of $Z$, and if $Z$ is pure-dimensional we let

$ic_{Z}:=\chi_{st}(IC_{Z}),$

where $IC_{Z}$ is the intersection cohomology ($IC$-)complex of $Z$. Note that
if $\varphi=\chi_{st}(\operatorname{{\mathcal{F}}^{\centerdot}})$, then

$\chi(Z,\varphi)=\chi(Z,\operatorname{{\mathcal{F}}^{\centerdot}}).$

For a closed subvariety $Z$ in $X$, a constructible function (or complex) on
$Z$ can be regarded as a constructible function (or complex) on $X$ by
extending it by $0$ on $X\setminus Z$, hence for $\varphi\in F(Z)$ we have
$\chi(Z,\varphi)=\chi(X,\varphi)$. For the purpose of this note we may,
without any loss of generality, work with constructible functions on $X$ (with
support in a closed subvariety) and their corresponding Euler characteristics.

The Euler characteristic induces by additivity an epimorphism

$\chi_{st}:K_{0}(D^{b}_{c}(X))\longrightarrow F(X).$

Moreover, since the class map $D^{b}_{c}(X)\to K_{0}(D^{b}_{c}(X))$ is onto,
$\chi_{st}$ is already an epimorphism on $D^{b}_{c}(X)$. The usual functors in
sheaf theory, which respect the corresponding category of bounded
constructible complexes, induce via $\chi_{st}$ well-defined group
homomorphisms on the level of constructible functions (see, e.g., [26, Section
2.3]).

Another important example of a constructible function on a complex algebraic
variety $Z$ is the MacPherson local Euler obstruction function $Eu_{Z}$, see
[22]. When $Z$ is a closed irreducible subvariety $Z$ of $X$, $Eu_{Z}$ can be
seen as a function defined on all of $X$ by setting $Eu_{Z}(x)=0$ for $x\in
X\setminus Z$. In particular, one may consider the group homomorphism

(2) $Eu:Z(X)\longrightarrow F(X)$

defined on an irreducible cycle $Z$ by the assignment $Z\mapsto Eu_{Z}$, and
then extended by $\mathbb{Z}$-linearity. It is well known (e.g., see [8,
Theorem 4.1.38] and the references therein), that the homomorphism $Eu:Z(X)\to
F(X)$ is an isomorphism.

The local Euler obstruction function appears in the formulation of the local
index theorem, which in the above notations asserts the existence of the
following commutative diagram (e.g., see [26, Section 5.0.3] and the
references therein):

(3)
$\textstyle{K_{0}(D^{b}_{c}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{CC}$$\scriptstyle{\chi_{st}}$$\textstyle{F(X)}$$\textstyle{LCZ(T^{*}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\cong}$$\textstyle{Z(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Eu}$$\scriptstyle{\cong}$

In particular, one can associate a characteristic cycle to any constructible
function. For example, if $Z$ is a closed irreducible subvariety of $X$, one
has:

(4) $CC(Eu_{Z})=(-1)^{\dim_{\mathbb{C}}Z}\cdot T^{*}_{Z}X.$

Note also that

$CC({\mathcal{F}}^{\centerdot})=CC(\chi_{st}({\mathcal{F}}^{\centerdot}))$

for any constructible complex ${\mathcal{F}}^{\centerdot}\in D^{b}_{c}(X)$.

Kashiwara's global index theorem [16] computes the Euler characteristic of any
bounded constructible complex $\operatorname{{\mathcal{F}}^{\centerdot}}$ on
$X$ with $supp(\operatorname{{\mathcal{F}}^{\centerdot}})$ compact, or,
equivalently, that of the constructible function
$\varphi=\chi_{st}(\operatorname{{\mathcal{F}}^{\centerdot}})\in F(X)$, by the
formula:

(5)
$\chi(X,\operatorname{{\mathcal{F}}^{\centerdot}})=\chi(X,\varphi)=CC(\varphi)\cdot
T^{*}_{X}X,$

that is, the intersection index in the complex manifold $T^{*}X$, of the
characteristic cycle of $\varphi$ with the zero section of $T^{*}X$.

######  Definition 2.1.

If $\varphi\neq 0\in F(X)$ with $CC(\varphi)=\sum_{Z}n_{Z}\cdot T^{*}_{Z}X$,
we say that $CC(\varphi)$ is effective if all coefficients $n_{Z}$ are
positive.

For instance, (4) implies that
$CC\left((-1)^{\dim_{\mathbb{C}}Z}Eu_{Z}\right)$ is effective. It is also well
known that if
$0\neq\varphi=\chi_{st}(\operatorname{{\mathcal{F}}^{\centerdot}})$ is a
nontrivial constructible function associated to a peverse sheaf
$\operatorname{{\mathcal{F}}^{\centerdot}}$ on $X$, then $CC(\varphi)$ is
effective (e.g., see [23, Corollary 4.7]). In particular, $CC(ic_{Z})$ is
effective. Finally, the Behrend function $\nu_{Z}$ has an effective
characteristic cycle (see [1, 3]).

We can now prove the following.

######  Proposition 2.2.

Conjecture 1.2(i) is equivalent to Conjecture 1.3.

###### Proof.

Conjecture 1.3 implies Conjecture 1.2(i) by choosing
$\varphi=(-1)^{\dim_{\mathbb{C}}Z}Eu_{Z}$ (extended by $0$ to all of $X$).

Conversely, assume Conjecture 1.2(i) holds for all closed irreducible
subvarieties $Z$ of $X$. Let $\varphi\in F(X)$ with $CC(\varphi)$ effective.
Then

(6) $CC(\varphi)=\sum_{Z\subseteq X}n_{Z}\cdot T^{*}_{Z}X$

for uniquely determined closed irreducible subvarieties $Z$ of $X$ and
positive integers $n_{Z}$. By (4), the coefficients $n_{Z}$ are determined by
the following equality of constructible functions

(7) $0\neq\varphi=\sum_{Z\subseteq X}n_{Z}\cdot(-1)^{\dim_{\mathbb{C}}Z}\cdot
Eu_{Z}.$

Hence

$\chi(X,\varphi)=\sum_{Z\subseteq
X}n_{Z}\cdot(-1)^{\dim_{\mathbb{C}}Z}\cdot\chi(Z,Eu_{Z})\geq 0,$

with the last inequality following by applying Conjecture 1.2(i) to each
subvariety $Z$ in the support of $CC(\varphi)$. ∎

In [1, Proposition 7.2], the authors list some basic operations of
constructible functions which preserve the property of having an effective
characteristic cycle. In particular, one has the following.

######  Proposition 2.3.

[1, Proposition 7.2(2)] Let $Z$ be a closed reduced subscheme of a smooth
complex algebraic variety $X$, and assume that $\varphi$ is a constructible
function on $Z$ with $CC(\varphi)$ effective.

  * (i)

Let $f:Z\to Z^{\prime}$ be a finite morphism, with $Z^{\prime}$ a closed
reduced subscheme of a smooth complex algebraic variety $X^{\prime}$. Then
$f_{*}(\varphi)$ is a constructible function on $Z^{\prime}$ with
$CC(f_{*}(\varphi))$ effective. Here, $f_{*}(\varphi)$ is the constructible
function on $Z^{\prime}$ defined by:

$f_{*}(\varphi)(z^{\prime}):=\sum_{z\in f^{-1}(z^{\prime})}\varphi(z).$

  * (ii)

Let $f:X^{\prime}\to X$ be a morphism of smooth complex algebraic varieties
such that $f\colon Z^{\prime}:=f^{-1}(Z)\to Z$ is a smooth morphism of
relative dimension $d$. Then $(-1)^{d}f^{*}(\varphi)=(-1)^{d}\varphi\circ f$
is a constructible function on $Z^{\prime}$ with effective characteristic
cycle.

As already mentioned, the above functors of constructible functions coincide
with those induces via $\chi_{st}$ from the corresponding functors of
constructible complexes of sheaves.

##  3\. Positivity results for nef bundles. Applications

In this section, we recall (semi-)positivity results for ample (resp., nef)
vector bundles on complex projective manifolds. We use such results to deduce
(semi-)positivity statements for the Euler characteristics of constructible
functions with effective characteristic cycles, thus proving Theorem 1.4.

######  Definition 3.1.

If $\mathcal{E}$ is a vector bundle on a complex projective manifold $X$,
denote by $\mathbf{P}(\mathcal{E})$ the projective bundle of hyperplanes in
the fibers of $\mathcal{E}$. A vector bundle $\mathcal{E}$ on $X$ is called
ample (resp. nef) if the line bundle $\mathcal{O}(1)$ on
$\mathbf{P}(\mathcal{E})$ is ample (resp. nef) in the classical sense.

The nef condition is a degenerate ampleness condition. Properties of nef/ample
bundles are studied, e.g., in [7, 11, 20].

The following semi-positivity result was proved by Fulton-Lazarsfeld [11] for
ample bundles, and extended to nef bundles by Demailly-Peternell-Schneider [7]
(cf., e.g., [20, Section 8.1.B] for the definition of the intersection
number).

######  Theorem 3.2.

([7, Proposition 2.3], [11, Theorem II])) Let $X$ be a complex projective
manifold and let $E$ be a rank $r$ nef (resp., ample) vector bundle on $X$.
For any $r$-dimensional conic subvariety $C$ of $E$, one has

$C\cdot Z_{E}\geq 0\ \ \ ({\rm resp.},\ >0),$

where $Z_{E}$ is the zero section of $E$, and $C\cdot Z_{E}$ denotes the
intersection number of cycles in $E$.

######  Remark 3.3.

The notion of nef vector bundle can be extended to the Kähler manifold
context, with nefness of a line bundle understood in the sense of [7,
Definition 1.2]; this coincides with the usual definition in the projective
case. Then the above theorem holds more generally, for nef vector bundles on
compact Kähler manifolds (cf. [7, Proposition 2.3]).

The first result of this section involves nef/ample cotangent bundles. As
already mentioned in the Introduction, complex projective manifolds with non-
positive (resp., negative) sectional curvature have nef (resp., ample)
cotangent bundle.

######  Example 3.4.

The class of complex projective manifolds whose cotangent bundles are nef is
closed under taking finite (unramified) covers, products, and subvarieties,
and it includes smooth subvarieties of abelian varieties. However, if $A$ is
an abelian variety of dimension $n$ and $X$ is a smooth subvariety of $A$ of
dimension $d$ and codimension $n-d<d$, then the cotangent bundle of $X$ is not
ample (e.g., see [20, Example 7.2.3]). On the other hand, for an arbitrary
$m$-dimensional complex projective manifold $M$ and each $n\leq m/2$, there
are plenty of smooth $n$-dimensional subvarieties $X$ of $M$ with ample
cotangent bundle (e.g., complete intersections of sections of $M$ by general
hypersurfaces of sufficiently high degrees in the ambient projective space,
see [4, 29]). Finally, Kratz [19, Theorem 2] showed that if $X$ is a complex
projective manifold whose universal cover is a bounded domain in
$\mathbb{C}^{n}$ or in a Stein manifold, then $T^{*}X$ is nef. This result
prompted Liu-Maxim-Wang [21] to conjecture that the nefness of the cotangent
bundle should hold more generally, if the universal cover is Stein, a claim
refuted recently in [28].

We can now prove the following result.

######  Theorem 3.5.

Let $X$ be a complex algebraic variety, and let $f\colon X\to Y$ be a morphism
to a complex projective manifold $Y$ with $T^{*}Y$ nef. Let $\varphi\in F(X)$
be a constructible function on $X$ such that the characteristic cycle of the
constructible function $f_{*}(\varphi)\in F(Y)$ is effective. Then
$\chi(X,\varphi)\geq 0$, and the inequality is strict if $T^{*}Y$ is ample.

###### Proof.

First note that

(8) $\chi(X,\varphi)=\chi(Y,f_{*}(\varphi)),$

so it suffices to show that $\chi(Y,f_{*}(\varphi))\geq 0$, with strict
inequality if $T^{*}Y$ is ample. Let

$CC(f_{*}(\varphi))=\sum_{Z\subseteq Y}n_{Z}\cdot T^{*}_{Z}Y,$

for uniquely determined closed irreducible subvarieties $Z$ of $Y$, and
$n_{Z}>0$ by the effectivity assumption. Since $T^{*}Y$ is nef, one gets by
Theorem 3.2 that

$T^{*}_{Z}Y\cdot T^{*}_{Y}Y\geq 0,$

with strict inequality if $T^{*}Y$ is ample. Kashiwara's global index formula
(5) then yields:

(9) $\chi(Y,f_{*}(\varphi))=\sum_{Z\subseteq Y}n_{Z}\left(T^{*}_{Z}Y\cdot
T^{*}_{Y}Y\right)\geq 0,$

with strict inequality if $T^{*}Y$ is ample. The assertion of the theorem
follows now by combining (8) and (9). ∎

Theorem 1.4 is an immediate consequence of Theorem 3.5, as we now show.

###### Proof of Theorem 1.4.

Let $f:X\to Y$ be a finite morphism between complex projective manifolds, with
$T^{*}Y$ nef. Since $\varphi\in F(X)$ has, by assumption, an effective
characteristic cycle, it follows from Proposition 2.3(i) that
$CC(f_{*}(\varphi))$ is effective in $T^{*}Y$. The assertion follows now from
Theorem 3.5. ∎

######  Example 3.6.

Besides the situation considered in Theorem 1.4, there are other interesting
classes of morphisms $f:X\to Y$ (and constructible functions $\varphi\in F(X)$
defined on their domain) satisfying the assumptions of Theorem 3.5. For
instance, if $\varphi\in F(X)$ comes from a perverse sheaf, let $f:X\to Y$ be
a morphism such that $Rf_{*}$ preserves perverse sheaves (e.g., a closed
embedding or a quasi-finite affine morphism); then $f_{*}(\varphi)\in F(Y)$
also comes from a perverse sheaf and hence, since $Y$ is assumed smooth,
$CC(f_{*}(\varphi))$ is effective in $T^{*}Y$. Another example is provided by
the projection $f:X\to Y$ from a semi-abelian variety $X$ onto its abelian
part $Y$. In this case, it was shown in [1, Section 8] that if $\varphi\in
F(X)$ comes from a perverse sheaf, then $f_{*}(\varphi)\in F(Y)$ also comes
from a perverse sheaf. Hence $CC(f_{*}(\varphi))$ is effective, and Theorem
3.5 applies to give $\chi(X,\varphi)\geq 0$, an inequality initially proved in
[12, Corollary 1.4]. For more examples in this direction, the interested
reader may consult [1, Proposition 8.4, Example 8.5].

We next discuss the following extension of the main result of [2] to the case
of effective characteristic cycle.

######  Theorem 3.7.

Let $X$ be a smooth complex projective variety with large algebraic
fundamental group (e.g., $X$ is aspherical). Suppose that there exists a
cohomologically rigid almost faithful semi-simple representation
$\rho:\pi_{1}(X)\to GL(r,\mathbb{C})$. Let $\varphi$ be a constructible
function on $X$ with $CC(\varphi)$ effective. Then $\chi(X,\varphi)\geq 0$.

###### Proof.

The proof of this result in [2] for perverse sheaves coefficients can be
readily extended to the more general setup of effective characteristic cycles,
since the only operations involved in the proof preserve the effectivity of
characteristic cycles of constructible functions. For the benefit of the
reader, we summarize the main steps in the proof, by adapting [2, Theorems 1.8
and 1.6] to the setup of our paper.

Let $L_{\rho}$ be the local system on $X$ corresponding to the representation
$\rho$. As noted in [2], $L_{\rho}$ underlies a complex variation of Hodge
structure (CVHS) $V$ with discrete monodromy.

Step 1. Upon passing to a finite unramified cover $\pi:X^{\prime}\to X$, one
can moreover assume that the monodromy group $\Gamma=\rho(\pi_{1}(X))$ of
$L_{\rho}$ is torsion free (cf. [2, Lemma 5.1]). After performing this step,
the Euler characteristic $\chi(X,\varphi)$ gets multiplied by the degree of
the cover $\pi$ and hence it preserves its sign, and the characteristic cycle
of $\pi^{*}(\varphi)$ is still effective (cf. Proposition 2.3(ii)).

The effect of performing Step 1 is that the quotient $M=\Gamma\backslash D$ of
the Griffiths period domain by the monodromy group $\Gamma$ is now a manifold,
and the period map of the CVHS $V$ induces a horizontal map $\alpha:X\to M$
(i.e., its derivative lies in the subbundle $T^{h}M\subset TM$ induced from
the horizontal subbundle of $D$).

Step 2. Since $\chi(X,\varphi)=\chi(M,\alpha_{*}(\varphi))$, it thus suffices
to show that $\chi(M,\alpha_{*}(\varphi))\geq 0$. Conditions of the theorem
force $\alpha:X\to M$ to be quasi-finite, and hence a finite morphism (since
$X$ is compact). Hence, by Proposition 2.3(i), the characteristic cycle of the
constructible function $\alpha_{*}(\varphi)$ (supported on the compact
subvariety $\alpha(X)$) is effective in $T^{*}M$. So if

$CC(\alpha_{*}(\varphi))=\sum_{Z}n_{Z}\cdot T^{*}_{Z}M,$

it suffices to show that for any subvariety $Z$ appearing in the sum one has

(10) $T^{*}_{Z}M\cdot T^{*}_{M}M\geq 0.$

Griffiths transversality implies that $\alpha(X)$ is a horizontal subvariety
of $M$ (in the sense of [2, Definition 5.2]), and hence by [2, Lemma 5.3], so
is any irreducible subvariety appearing in $CC(\alpha_{*}(\varphi))$. The
desired inequality (10) follows now from [2, Proposition 6.1]. ∎

We conclude this note with the following discussion about complex projective
(or compact Kähler) manifolds with nef tangent bundles. Let $X$ be a complex
projective manifold (or even a Kähler manifold) whose tangent bundle is nef.
Examples include rational homogeneous manifolds (e.g., complex projective
spaces, flag manifolds, or quotients $G/P$ of a simply connected complex Lie
group $G$ by a parabolic subgroup), abelian varieties (or complex tori), etc.
(cf. [7, Section 3.A]). Moreover, as we will argue below, complex projective
(or compact Kähler) manifolds with non-negative sectional curvature have nef
tangent bundles. By applying Theorem 3.2 to the tangent bundle $TX$, one gets
via the Gauss-Bonnet formula that

(11) $\chi(X)=\int_{X}c_{n}(TX)=T_{X}X\cdot T_{X}X\geq 0,$

where $T_{X}X$ is the zero section of $TX$. Hence one gets the following
result from [7].

######  Proposition 3.8.

If $X$ is a complex projective (or compact Kähler) manifold with nef tangent
bundle, then $\chi(X)\geq 0$.

Note that Theorem 3.2 also shows that the inequality in (11) is strict if $TX$
is ample. In fact, Mori [25] proved that a complex projective manifold with
ample tangent bundle is isomorphic to a complex projective space.

Proposition 3.8 is a generalization in the complex projective (or Kähler)
context of another conjecture of Hopf (also appearing on Yau's problem list
[30]):

######  Conjecture 3.9 (Hopf).

A compact, even-dimensional Riemannian manifold with positive sectional
curvature has positive Euler characteristic. A compact, even-dimensional
Riemannian manifold with non-negative sectional curvature has non-negative
Euler characteristic.

Indeed, let $X$ be a complex projective or compact Kähler manifold with non-
negative (resp., positive) sectional curvature. The bisectional curvature can
be written as a positive linear combination of two sectional curvatures (e.g.,
see [31]), hence the bisectional curvature of $X$ is non-negative (resp.,
positive). This means (by definition) that the tangent bundle $TX$ is
Griffiths semipositive (resp., positive). Furthermore, by [7] (resp., [13]),
Griffiths semipositive (resp., positive) bundles are nef (resp., ample).

Let us finally note that in the Kähler context, Siu-Yau [27] showed that if
$X$ has positive bisectional curvature then $X$ is biholomorphic to a complex
projective space. Furthermore, a classification of all compact Kähler
manifolds with non-negative bisectional curvature was obtained by Mok [24],
and this can be used along with results of [7] to give a direct proof of
Conjecture 3.9 in this case.

## References

  * [1] Aluffi, P., Mihalcea, L., Schürmann, J., Su, C., Positivity of Segre-MacPherson classes, arXiv:1902.00762.
  * [2] Arapura, D., Wang, B., Perverse sheaves on varieties with large fundamental groups, arXiv:2109.07887.
  * [3] Behrend, K., Donaldson-Thomas type invariants via microlocal geometry, Ann. Math. (2) 170 (2009), 1307–1338.
  * [4] Brotbek, D., Darondeau, L., Complete intersection varieties with ample cotangent bundles, Invent. Math. 212 (2018), no. 3, 913–940.
  * [5] Cao, J., Xavier, F., Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature, Math. Ann. 319 (2001), no. 3, 483–491.
  * [6] Chern, S. S., On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117–126.
  * [7] Demailly, J.-P., Peternell, T., Schneider, M., Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345.
  * [8] Dimca, A., Sheaves in Topology, Universitext, Springer-Verlag, Berlin, 2004.
  * [9] Elduque, E., Geske, C., Maxim, L., On the signed Euler characteristic property for subvarieties of abelian varieties, J. Singul., 17 (2018) 368–387.
  * [10] Eyssidieux, P., La caractéristique d'Euler du complexe de Gauss-Manin, J. Reine Angew. Math. 490 (1997), 155–212.
  * [11] Fulton, W., Lazarsfeld, R., Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), no. 1, 35–60.
  * [12] Franecki, F., Kapranov, M., The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J. 104 (2000), no. 1, 171–180.
  * [13] Griffiths, P., Hermitian differential geometry, Chern classes, and positive vector bundles, 1969 Global Analysis (Papers in Honor of K. Kodaira), pp. 185–251, Univ. Tokyo Press, Tokyo.
  * [14] Gromov, M., Kähler hyperbolicity and $L_{2}$-Hodge theory, J. Differential Geom. 33 (1991), no. 1, 263–292.
  * [15] Jost, J., Zuo, K., _Vanishing theorems for $L^{2}$-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry_, Comm. Anal. Geom. 8 (2000), no. 1, 1–30.
  * [16] Kashiwara, M., Index theorem for constructible sheaves, Differential systems and singularities (Luminy, 1983), Astérisque No. 130 (1985), 193–209.
  * [17] Kashiwara, M., Schapira, P., Sheaves on manifolds, Springer-Verlag, Berlin, 1990.
  * [18] Kollár, J., Shafarevich maps and automorphic forms, Princeton Univ. Press, Princeton, NJ, 1995.
  * [19] Kratz, H., Compact complex manifolds with numerically effective cotangent bundles, Doc. Math. 2 (1997), 183–193.
  * [20] Lazarsfeld, R., _Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals_ , Ergebnisse der Mathematik und ihrer Grenzgebiete 49. Springer-Verlag, Berlin, 2004.
  * [21] Liu, Y., Maxim, L., Wang, B., Aspherical manifolds, Mellin transformation and a question of Bobadilla-Kollár, J. Reine Angew. Math. 781 (2021), 1–18.
  * [22] MacPherson, R., _Chern classes for singular algebraic varieties_ , Ann. of Math. 100 (1974), 423–432.
  * [23] Maxim, L., Schürmann, J., Constructible sheaf complexes in complex geometry and Applications, Handbook of Geometry and Topology of Singularities, vol. III (in press), Springer; arXiv:2105.13069.
  * [24] Mok, N., The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179–214.
  * [25] Mori, S., Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606.
  * [26] Schürmann, J., Topology of singular spaces and constructible sheaves, Birkhäuser Verlag, Basel, 2003.
  * [27] Siu, Y.-T., Yau, S.-T., Compact Kähler manifolds of positive bisectional curvature, Inv. Math. 59 (1980), 189–204.
  * [28] Wang, Y., Ramified cover of varieties with nef cotangent bundle, arXiv:2202.11165.
  * [29] Xie, S.-Y., On the ampleness of the cotangent bundles of complete intersections, Invent. Math. 212 (2018), no. 3, 941–996.
  * [30] Yau, S. T., Problem section, Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102 (1982), Princeton, N.J.: Princeton University Press, pp. 669–706.
  * [31] Zheng, F., Complex differential geometry, AMS/IP Studies in Advanced Mathematics, 18. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000.