# Efficient Multiscale Object-based Superpixel Framework

Felipe Belém1,2,111felipe.belem@{ic.unicamp.br,esiee.fr}, Benjamin Perret2,
Jean Cousty2
Silvio J. F. Guimarães 3, Alexandre Falcão1
1 University of Campinas, Campinas, Brazil
2 Université Gustave Eiffel, Noisy-le-Grand, France
3 Pontifical Catholic University of Minas Gerais, Belo Horizonte, Brazil

###### Abstract

Superpixel segmentation can be used as an intermediary step in many
applications, often to improve object delineation and reduce computer
workload. However, classical methods do not incorporate information about the
desired object. Deep-learning-based approaches consider object information,
but their delineation performance depends on data annotation. Additionally,
the computational time of object-based methods is usually much higher than
desired. In this work, we propose a novel superpixel framework, named
Superpixels through Iterative CLEarcutting (SICLE), which exploits object
information being able to generate a multiscale segmentation on-the-fly. SICLE
starts off from seed oversampling and repeats optimal connectivity-based
superpixel delineation and object-based seed removal until a desired number of
superpixels is reached. It generalizes recent superpixel methods, surpassing
them and other state-of-the-art approaches in efficiency and effectiveness
according to multiple delineation metrics.

##  1 Introduction

Some algorithms partition an image into several disjoint groups of connected
pixels, named superpixels. The pixels for the superpixels share a common
property (e.g., color or texture). Thus, a superpixel-based image
representation provides more contextual information than a pixel-based one
while reducing the computer workload. Due to these properties, superpixel
segmentation is often used as an intermediary step in medical applications
[12, 43, 19], semantic segmentation [39], plant diseased-leaf detection [42],
pedestrian segmentation [40], among other examples.

Most methods model the superpixel segmentation problem as a pixel-clustering
task. The most popular one is Simple Linear Iterative Clustering (SLIC) [1]
whose efficiency and simplicity inspired several and more accurate algorithms
[21, 18]. Other methods, like Superpixel Hierarchy (SH) [34], Entropy Rate
Superpixels (ERS) [20], initial approaches based on Iterative Spanning Forest
(ISF) framework [33], and Dynamic ISF (DISF) [8], model the problem
differently and offer a more effective solution with slightly higher
computational cost. However, all the aforementioned methods are unaware of the
object of interest in the image. Thus, instead of ensuring the boundaries of
interest (i.e., relevant borders), they pursue the preservation of all
detectable ones, including many irrelevant ones.

Recently, several deep-learning-based approaches have been proposed for
generating superpixels in which the breakthrough of end-to-end trainable
methods was Superpixel Sampling Networks (SSN) [17], further improved in [38].
These methods incorporate object information during training, but they require
data annotation, which may be scarce in some applications (e.g., medical and
biological). Although unsupervised methods have also been proposed [41, 31],
they all present moderate boundary delineation considering different datasets,
even those of the same domain (e.g., natural image datasets).

Some algorithms consider the object information represented as a monochromatic
image (i.e., an object saliency map) in which pixel intensity denotes an
object membership value. These object-based methods use such information for
favoring the delineation of the object of interest at the expense of others,
resulting in a more accurate superpixel segmentation. However, exhaustively
exploiting saliency information can lead to higher computational cost and
saliency quality dependency, as seen in Object-based Iterative Spanning Forest
(OISF) [6, 7] variants. Conversely, Object-based DISF (ODISF) [5] overcomes
the latter drawback while still being slightly slower than object-unaware
methods.

Due to SLIC’s simplicity and efficiency, the vast majority of superpixel
methods adopt the same three-step pipeline: (i) seed sampling; (ii) superpixel
generation; and (iii) seed recomputation; in which step (i) is applied once
and, subsequently, steps (ii) and (iii) are repeated for a given number of
iterations. The major advantage of seed-based superpixel segmentation is being
applicable for any object, exploiting its inherent existence in the image.
Moreover, by recomputing the seeds (for the next iteration), the intra-
superpixel dissimilarity and, thus, errors are minimized. Despite SLIC
presents fair effectiveness, recent and more accurate approaches benefit from
variations of the same pipeline [18, 21, 33, 36].

This three-step pipeline presents several drawbacks. First, the number of
seeds in step (i) approximates the desired number of superpixels. While it may
not choose seeds that promote accurate object delineation (i.e., relevant),
especially when the object is small, using object-based strategies usually
lead to higher computational time [9]. Second, several executions of the above
pipeline are required to produce a multiscale superpixel segmentation [34].
Third, reallocating a seed may not guarantee its accurate delineation in a
previous iteration. Finally, imposing a strict number of iterations may force
the execution of unnecessary iterations, notably when no convergence is met.

A novel strategy based on the previous pipeline was recently proposed,
exploited by DISF and ODISF. In step (i), a high initial number of seeds is
selected for increasing the chances of sampling relevant seeds through faster
approaches. A portion of the most irrelevant seeds is removed at each
iteration in step (iii), based on an exponential curve, until reaching the
desired number of superpixels. Thus, the most relevant ones are rest unaltered
for accurate delineation in the next iteration. As one may notice, this new
approach of oversampling and iterative seed removal, by design, permits
obtaining a multiscale segmentation in a single execution. Moreover, as the
desired number of superpixels increases, fewer iterations are performed for
segmentation. However, although a strict quantity of iterations is not
imposed, such number is indirectly defined by the initial seed quantity,
making it challenging to be intuitively set.

In this work, we propose a generalization of DISF and ODISF, overcoming their
inability to directly control the number of iterations for segmentation. By
reformulating the seed removal step, our proposal, named Superpixels through
Iterative CLEarcutting (SICLE), is capable of limiting the algorithm to
perform a maximum number of iterations, such that it can obtain DISF’s and
ODISF’s equivalent results if the seed removal decay is appropriately set.
Moreover, by not incorporating saliency information during delineation, SICLE
is robust to incorrect saliency estimations while producing highly accurate
superpixels. Finally, SICLE can generate an object-based multiscale
segmentation on-the-fly if the quantity of seeds at each iteration is
previously defined (see Figure 1). We also extensively evaluate different
configurations (including new proposals) for each pipeline’s step. Results
show that SICLE can accurately delineate boundaries of various objects from
distinct datasets, being faster than SLIC in most practical cases.

Figure 1: Multiscale superpixel segmentation comparison between two state-of-
the-art methods and our proposal. It was required three scales (i.e., $k=3$):
500, 100, and 25 superpixels, respectively. The red lines indicate the object
boundary, whereas the cyan ones, superpixel borders.

Our main contributions may be summarized as follows:

Seed sampling

An equidistant seed sampling strategy that distributes seeds proportionally
along each image axis;

Seed removal

Regarding seed removal, we proposed a (i) new criteria that accurately
maintains those promoting effective object delineation; and (ii) a procedure
which establishes a maximum non-strict number of iterations for improving
speed by avoiding unnecessary iterations for segmentation;

Multiscale approach

A novel framework (SICLE) for the design of object-based multiscale superpixel
segmentation methods;

On-the-fly

SICLE-based methods that are robust to object saliency errors and dismiss
unnecessary segmentation iterations while generating multiscale boundaries on-
the-fly;

Easy-to-use

Optimization and extensive analysis of each SICLE’s step, leading to an easy-
to-use framework with accurate and fast delineation in different image
domains;

This paper is organized as follows. In Section 2, we present an overview of
the state-of-the-art in superpixel segmentation. We describe our theoretical
background in Section 3 and, subsequently, detail our proposed framework in
Section 4. Finally, in Sections 5 and 6, we discuss the experimental results
and draw conclusions and possible future work, respectively.

##  2 Related Works

In this section, we present some of the most recent and notable works in
superpixel segmentation. One may refer to [30] for a deeper discussion on this
topic. In Section 2.1, we review those methods that do not incorporate any
prior object information. Subsequently, we present those which use deep-
learning strategies for generating superpixels in Section 2.2. Lastly, in
Section 2.3, we discuss the algorithms that consider object information
represented by an object saliency map. Table 1 compares different state-of-
the-art methods against our proposal concerning some properties.

Properties | SLIC[1] | LSC[18] | SH[34] | SEAL[32]1 | OISF[7] | SICLE
---|---|---|---|---|---|---
Boundary Recall | 0.864 | 0.893 | 0.949 | 0.826 | 0.952 | 0.978
Under-segmentation Error | 0.016 | 0.013 | 0.011 | 0.015 | 0.008 | 0.010
Speed (ms) | 507 | 850 | 941 | -2 | 2832 | 468
Multiscale Segmentation | No | No | Yes | No | No | Yes
Object Information | No | No | No | Yes | Yes | Yes
Error Propagation | No | No | Yes | Yes | Yes | No

1 Deep-learning-based algorithm.
---
2 For a single segmentation, the time for loading the network exceeds OISF’s.

Table 1: Average SICLE performance compared to state-of-the-art superpixel
algorithms for 750 superpixels. It was considered a grayscale dataset whose
images are 512$\times$512.

###  2.1 Classical Methods

We may broadly classify the majority of the classical methods by their
strategy: (i) clustering-based; (ii) graph-based; and (iii) path-based. In the
first group, the superpixels are generated through pixel clustering. Simple
Linear Iterative Clustering (SLIC) [1] is the most popular method of such
group, and it uses an adapted K-means on a 5-dimensional feature space for a
fast but moderate delineation. Similarly, Linear Spectral Clustering (LSC)
[18] applies a weighted K-means in a deliberately tailored 10-dimensional
space for producing superpixels with high boundary adherence while slightly
increasing the computational time. Some works adopt different clustering
techniques for superpixel generation, such as Gaussian Mixture Models [4] or
Density-based Spatial Clustering of Applications with Noise [27]. A common
drawback of methods within such group is not ensuring the desired number of
superpixels.

We may list Entropy Rate Superpixels (ERS) [20] and the Superpixel Hierarchy
(SH) [34] as the most popular and the most effective graph-based methods,
respectively. In ERS, the superpixel segmentation is modeled as an
optimization problem of the entropy of a random walk in the graph’s topology.
While ERS presents high computational time, even when using greedy strategies,
SH uses the Borůvka algorithm for generating superpixels with high boundary
adherence in linear time. Given it is a hierarchical method, errors in the
lower levels are propagated to the upper ones, compromising its delineation
performance.

Path-based approaches define superpixels by path searching from each seed to
its most similar pixels, ensuring superpixel connectivity. The methods based
on the Image Foresting Transform (IFT) [14] algorithm often present top object
delineation at the expense of not easily controlling the superpixel compacity
[30]. Iterative Spanning Forest (ISF) [33] is a three-step seed-based
framework that generates superpixels through iterations of IFT executions on
improved seed sets. Based on the latter, the authors in [15] proposed the
Recursive ISF (RISF) algorithm to obtain a sparse-hierarchical superpixel
segmentation with no efficiency deterioration. Recently, the authors in [8]
proposed Dynamic ISF (DISF), which starts from seed oversampling and
iteratively removes seeds based on a relevance criterion applied to the IFT-
based superpixel delineations, achieving highly accurate segmentation.
Although this group also contains non-IFT-based methods [2], they usually
present fair delineation performance.

Some methods aim to increase the number of superpixels in regions
characterized by higher content density (i.e., content-sensitive). For
instance, Intrinsic Manifold SLIC (IMSLIC) [21] maps every pixel to a
2-dimensional manifold and measures the superpixel density through its area.
Other examples, like Content-Adaptive Superpixels (CAS) [36] and Texture-Aware
and Structure-Preserving (TASP) [35], generate content-sensitive superpixels
by improving the separability in the feature space. However, differently from
IMSLIC, CAS and TASP require a post-processing step for ensuring superpixel
connectivity.

###  2.2 Deep-learning Methods

Although classical methods often present high delineation accuracy, they are
unaware of the user’s desire to segment a particular object. On the other
hand, deep-learning methods are driven by the information inferred during
training (e.g., important borders). For a period of time, some works [3] aimed
at generating superpixels using deep features, even though the benefits of
this strategy are not yet solid [32]. Thus, in [32], the authors proposed a
novel loss function named SEgmentation Aware Loss (SEAL) for learning the
pixel affinities for the underlying ERS algorithm. Such method, named SEAL-
ERS, was further improved in [25] in terms of speed. Still, as one may argue,
SEAL-ERS is dependable on the quality of estimating such affinities since
incorrect inferences can prejudice the entropy computation performed by its
backbone method.

Other common approach is to design end-to-end trainable networks for
superpixel segmentation, either supervisedly or unsupervisedly. Although both
are dependable on the quantity of data available for training, the latter does
not require any data annotation. We may cite the popular Superpixel Sampling
Networks (SSN) [17] and Fully Convolutional Network (FCN) [38] as examples of
supervised methods. Regarding unsupervised ones, we list Regular Information
Maximization (RIM) [31] and Edge-Aware Superpixel Segmentation (EASS) [41] as
two of the most recent representatives of this group.

###  2.3 Object-based Methods

We term an algorithm as object-based when it incorporates prior object
information, often represented by object saliency maps. Saliency-based
Superpixels (SS) [37] can be classified as such, even though such term was
first coined in [6]. Recent object-based solutions can be seen as
generalizations of previous methods for handling with saliency maps, such as
Object-based ISF (OISF) [6, 7]. In OISF, the user controls the superpixel
displacement and morphology concerning the map’s estimation. While it is slow
and it is highly dependable on the map quality (i.e., propagates the maps’
errors), Object-based DISF (ODISF) [5] offers a highly accurate and faster
solution with minimum influence of saliency errors.

##  3 Theoretical Background

In this section, we present important definitions regarding our proposal.
First, we recall basic graph notions in Section 3.1 and, subsequently in
Section 3.2, we detail the Image Foresting Transform (IFT) [14], the core
algorithm for delineation.

###  3.1 Image Graph

An image $I$ can be defined as a pair
$\left\langle\mathcal{P},\mathbf{F}\right\rangle$ in which
$\mathcal{P}\subset\mathbb{Z}^{2}$ is the set of picture elements (i.e.,
pixels) and $\mathbf{F}$ maps every $p\in\mathcal{P}$ to its feature vector
$\mathbf{F}(p)\in\mathbb{R}^{m}$, given $m\in\mathbb{N}^{*}$. If $m>1$, $I$ is
a colored image (e.g., CIELAB), otherwise, $I$ is grayscale. Lastly, an object
saliency map $O=\left\langle\mathcal{P},\mathbf{O}\right\rangle$ is a
grayscale image whose feature indicates a proportional likelihood
$\mathbf{O}(p)\in\left[0,1\right]$ of a pixel $p$ belonging to an object of
interest.

An image digraph $G$ derived from $I$ is a pair
$\left\langle\mathcal{V},\mathcal{A}\right\rangle$ such that
$\mathcal{V}\subseteq\mathcal{P}$ is the vertex set and
$\mathcal{A}\subset\mathcal{V}^{2}$ is the set of arcs. Two vertices $x,y$ are
said to be adjacents if $\exists\left\langle x,y\right\rangle$ or
$\left\langle y,x\right\rangle\in\mathcal{A}$. A common approach for
establishing the arcs in $\mathcal{A}$ is based on the Euclidean spatial
distance between the vertices. Therefore, for any $x,y\in\mathcal{V}$ and a
radius $r\in\mathbb{R}_{+}$, we may obtain the arc set
$\mathcal{A}^{r}=\left\\{\left\langle
x,y\right\rangle:\left\|x-y\right\|_{2}\leq r\right\\}$. As one may see,
$\mathcal{A}^{\sqrt{2}}$ contains the arcs defined in an 8-neighborhood.
Throughout this paper, $G$ has neither loops nor parallel edges (i.e., $G$ is
a simple digraph).

We may define a (directed) path $\rho$ as a sequence of distinct vertices
$\left\langle v_{i}\right\rangle_{i=1}^{k}$ in which $\left\langle
v_{i},v_{i+1}\right\rangle\in\mathcal{A}$ for $i<k$. $\rho$ is trivial when
$k=1$, and non-trivial, otherwise. For $1<i<k$, $v_{i}$ is said to be the
predecessor of $v_{i+1}$ and the sucessor of $v_{i-1}$. We may refer to
$v_{1}$ and $v_{k}$ as origin (or root) and terminal vertices of $\rho$,
respectively. For simplicity, we can exhibit both either by
$\rho_{v_{1}\rightsquigarrow v_{k}}$ or by $\rho_{v_{k}}$. Finally,
$\rho_{y}=\rho_{x}\odot\left\langle x,y\right\rangle$ denotes the path
$\rho_{y}$ resultant from concatenating $\rho_{x}$ with an arc $\left\langle
x,y\right\rangle$.

###  3.2 Image Foresting Trasform

The Image Foresting Transform (IFT) [14] is an elegant framework for
developing connectivity-based image operators, and can be computed using a
generalization of the Dijkstra’s algorithm. Due to its object delineation
performance, it has been used in several applications [23, 29, 11], especially
for superpixel segmentation [33, 8, 15]. In this work, we consider its seed-
restricted variant. That is, for a given seed set
$\mathcal{S}\subset\mathcal{V}$, the IFT find paths with optimum cost from any
seed to every $p\in\mathcal{V}\setminus\mathcal{S}$.

A path-cost function $\mathbf{f}_{*}$ assigns a cost
$\mathbf{f}_{*}(\rho)\in\mathbb{R}_{+}$ to any $\rho\in\mathscr{P}$,
considering $\mathscr{P}$ to be the set of all possible paths in $G$.
Similarly to $\mathbf{f}_{*}$, an arc-cost function
$\mathbf{w}_{*}(x,y)\in\mathbb{R}$ defines a cost to any arc $\left\langle
x,y\right\rangle\in\mathcal{A}$. A popular option of path-cost function is the
$\mathbf{f}_{\max}$ function (Equation 1) due to its accurate object
delineation performance [8, 10]:

$\displaystyle\mathbf{f}_{max}(\left\langle x\right\rangle)$
$\displaystyle=\begin{cases}\begin{aligned} 0,&\quad\text{if
}x\in\mathcal{S}\\\ +\infty,&\quad\text{otherwise}\\\
\end{aligned}\end{cases}$ (1)
$\displaystyle\mathbf{f}_{max}(\rho_{x}\odot\left\langle x,y\right\rangle)$
$\displaystyle=\max\left\\{\mathbf{f}_{max}(\rho_{x}),\mathbf{w}_{*}(x,y)\right\\}$

A path $\rho_{x}$ is said to be optimum if
$\mathbf{f}_{*}(\rho_{x})\leq\mathbf{f}_{*}(\tau_{x})$ for any other
$\tau_{x}\in\mathscr{P}$ irrespective of its origin.

By electing a unique optimum-path $\rho_{x}$ for all $x\in\mathcal{V}$, we
define the predecessor map $\mathbf{P}$ as an acyclic map which assigns $x$
either to its predecessor in $\rho_{x}$ or to a to a distinctive marker
$\blacktriangle\not\in\mathcal{V}$ when $x$ is the root of $\rho_{x}$ (and,
consequently, of $\mathbf{P}$). Note that we can recursively map $x$ to its
root $\mathbf{R}(x)\in\mathcal{S}$ through $\mathbf{P}$. Starting at the
seeds, the IFT framework builds $\mathbf{P}$ through path concatenation while
it minimizes a cost map
$\mathbf{C}(x)=\min_{\rho_{x}\in\mathscr{P}}\left\\{\mathbf{f}_{*}(\rho_{x})\right\\}$.
Even if $\mathbf{f}_{*}$ does not satisfy certain properties, the resulting
paths still present desirable properties for superpixel segmentation [22]. For
any $s\in\mathcal{S}$, $\mathbf{P}$ defines an optimum-path tree (or,
equivalently, a superpixel) $\mathbf{T}(s)=\left\\{x:\mathbf{R}(x)=s\right\\}$
rooted in $s$ whose paths are more closely connected to its seed than to any
other. Finally, we can calculate its mean feature vector
$\overline{\mathbf{F}}(\mathbf{T}(s))$ and mean saliency value
$\overline{\mathbf{O}}(\mathbf{T}(s))$ by
$\sum_{x\in\mathbf{T}(s)}\mathbf{F}(x)/|\mathbf{T}(s)|$ and
$\sum_{x\in\mathbf{T}(s)}\mathbf{O}(x)/|\mathbf{T}(s)|$, respectively.

##  4 SICLE Framework

In this section, we describe our proposed framework named Superpixels through
Iterative CLEarcutting (SICLE). Sections 4.1, 4.2, and 4.3 detail, in this
order, the steps of the pipeline shown in Figure 2.

Figure 2: SICLE framework starting from $N_{0}=1000$ and requiring $N_{f}=5$
superpixels.

###  4.1 Seed Oversampling

The first step of SICLE is to select $N_{0}$ initial seeds for generating
$N_{f}$ superpixels, given $N_{0},N_{f}\in\mathbb{N}^{*}$. Most seed-based
algorithms aim to build a set $\mathcal{S}\subset\mathcal{V}$ containing the
seeds that promote accurate object delineation (i.e., relevant) using the
orthodox approach of sampling $N_{0}\approx N_{f}$ seeds, making precision
crucial. However, while object-based methods use a precise but slow sampling
algorithm [6, 9], non-object-based ones paradoxically opt for a fast but
imprecise procedure [1, 21, 18, 33]. Thus, although precision can lead to more
effective delineation, it is inversely correlated to the algorithm’s
efficiency.

On the other hand, we argue that oversampling (i.e., $N_{0}\gg N_{f}$) offers
a better solution since it pursues recall over precision. Given such high
quantity, it is expected that a portion of the seeds in $\mathcal{S}$ are
relevant, and, since seeds with similar features and within the same region
tend to be similarly relevant, it is unnecessary to select all relevant ones.
Given that, we infer that the drawbacks of using fast but imprecise sampling
strategies are overcome by oversampling. Thus, the objective is to guarantee
the presence of the relevant seeds while removing the irrelevant ones
throughout the iterations for ensuring $N_{f}$ superpixels in the last
iteration (see Section 4.3).

A classic sampling strategy, hereafter named GRID [1], is to select equally-
distanced seeds in a grid-like pattern. However, in most works, the image’s
axis are assumed to be equal, leading to an uneven distribution of the seeds.
In this work, we reformulate such approach for taking into account any image
dimensionality. Let $\varrho_{*}\in\;]0,1[$ be the proportion of a given axis
over all image’s axis. Then, $\mathsf{N}=(\varrho_{X}\cdot
c)\cdot(\varrho_{Y}\cdot c)$ seeds are sampled considering a constant
$c=\sqrt{N_{0}/\left(\varrho_{X}\cdot\varrho_{Y}\right)}$. It is possible to
determine the stride between seeds in each axis by dividing its length by
$c\cdot\varrho_{*}$. As one may notice, $\mathsf{N}\approx N_{0}$. Unlike most
methods, in this work, we do not shift seeds to the lowest gradient position
(in the neighborhood), given their questionable importance in oversampling.

However, one may question the benefits of GRID in oversampling. That is, as
$N_{0}$ increases, the stride between seeds decreases sharply (i.e., they tend
to be closer) and, as consequence, dispersing seeds equidistantly may not lead
to improvements in delineation. Also, such strategy can become challenging if
the user provides a non-rectangular mask representing a region-of-interest to
be segmented: it is necessary to ensure the equidistance rule and best
approximate $\mathsf{N}$ to the initial number of seeds $N_{0}$ desired by the
user. Conversely, we argue in favor of random seed oversampling (RND), given
its simplicity to be implemented, especially for the latter case. Furthermore,
backed up by the premise of seed relevance redundancy, one may expect that
such approach can generate seed sets with accurate object delineation results.

###  4.2 Superpixel Generation

In SICLE, superpixels are generated by the seed-restricted version of the IFT
framework. Recently, an object-based path-cost function [6] that controls the
superpixel morphology with respect to $\mathbf{F}$ or $\mathbf{O}$ has been
proposed. However, aside from requiring parameter optimization, we claim no
need to use either such function or any prior object information in
delineation. Clearly, the ground-truth (i.e., the ideal saliency map) is
defined by the borders in $\mathbf{F}$ during annotation by the user, and
considering such map can only reinforce the preexistent borders. Also, given
it is yet unfeasible to generate the ideal map, subjecting delineation to a
piece of information produced by inference can deteriorate segmentation,
especially by reinforcing the borders derived from incorrect estimations.
Figure 3 shows the comparison between SICLE and OISF, which uses such object-
based path-cost function. Therefore, in this work, we consider the
$\mathbf{f}_{\max}$ function due to its reportedly accurate performance in
several works [8, 5, 10].

Figure 3: Comparison between SICLE and OISF regarding the dependability of the
object saliency map quality. The desired number of superpixels was $N_{f}=50$.
The red lines indicate the object boundary, whereas the cyan ones, superpixel
borders.

In [10], the authors propose a novel arc-cost function whose segmentation
results surpassed many state-of-the-art methods in terms of interactive
segmentation. The Dynamic IFT (DYN) estimates the arc-costs with respect to
the tree’s mean features when it reaches the respective vertex to be conquered
(i.e., while it is growing). More formally, the $\mathbf{w}_{1}$ can be
defined as
$\mathbf{w}_{1}(x,y)=\left\|\overline{\mathbf{F}}(\mathbf{T}(s))-\mathbf{F}(y)\right\|_{2}$
in which $s=\mathbf{R}(x)$.

However, $\mathbf{w}_{1}$ is highly unstable. As one may see, the tree’s mean
feature vector may differ for different tie-breaking policies in the IFT,
which can impact the tree’s composition and characteristics, and produce,
thus, distinct delineations. Conversely, we argue in favor of considering the
seed’s features over its tree’s mean vector. First, they promote stability
since their features are immutable throughout the execution of the IFT.
Furthermore, given that superpixels, by definition, minimize dissimilarity,
one may argue that the tree’s features may resemble its seed’s. Therefore,
both arc-cost functions may present similar delineations, especially for
regions with high color dissimilarity (e.g., object borders). In this work, we
evaluate the use of a root-based function
$\mathbf{w}_{2}(x,y)=\left\|\mathbf{F}(\mathbf{R}(x))-\mathbf{F}(y)\right\|_{2}$,
hereafter named ROOT.

###  4.3 Seed Removal Criterion

As presented in Section 4.1, $\mathcal{S}$ is produced through oversampling,
and it is necessary to remove $N_{0}-N_{f}$ irrelevant seeds for generating
$N_{f}$ superpixels in the last IFT execution. Since determining whether a
seed is relevant or not based on its particular features, in SICLE, we take
advantage of the information generated from the previous IFT execution. That
is, at each iteration, we let the seeds within $\mathcal{S}$ manifest their
possible relevance (through competition) during the IFT and include their
superpixel features in their relevance computation.

One may note that the relevance of a seed is relatively defined (e.g.,
proximity to the object). Thus, since the ideal final set of seeds is not
known beforehand, we determine the relevance of a seed by comparing it with
its pairs. Then, “perturb” the most relevant by removing a portion of the
least relevant, and test their relevance in the subsequent IFT execution. In
this approach, the most relevant seeds are consistently labeled as such
throughout the iterations.

At the first iteration, it is expected that, aside from numerous irrelevant
seeds, many regions contain several equally-relevant seeds. By ordering the
seeds by their relevance, the portion of the least ones should present
instances of the previous cases and, thus, removing a large quantity should
impact the object minimally. Throughout the iterations, the number of seeds
decreases, and the least relevant ones are more defined by relation (i.e.,
less relevant than its pairs) than by estimation (i.e., almost zero
relevance). Moreover, fewer seeds are directly associated with reducing the
seed competition, leading to more volatile superpixels and major delineation
errors (especially “leakings”). In such environment, removing smaller
quantities should prevent superpixel growth instability and minimize errors.
Therefore, we chose an exponential function $\mathbf{M}(i)\in\mathbb{N}^{*}$
for computing the quantity of relevant seeds to be maintained for the
subsequent iteration $i+1\in\mathbb{N}^{*}$.

The major drawback of this strategy is not bounding the number of iterations
for reaching the desired number $N_{f}$ of superpixels, possibly compromising
its efficiency. One may argue in favor of imposing a strict number of $\Omega$
iterations for segmentation for any quantity of superpixels desired. However,
we argue that the necessary number of iterations depends on $N_{f}$. For
instance, for high competition environments, intra-superpixel dissimilarity is
significantly minimized and, thus, applying more iterations for subtle
variations in such configuration should minimally impact the object
delineation. Conversely, as $N_{f}$ decreases, removing fewer portions of
seeds (i.e., more iterations) assists in stabilizing the superpixel growth.
Therefore, we limited the number of iterations of SICLE to be at most $\Omega$
by setting $\mathbf{M}(i)=\max\\{(N_{0})^{1-\omega i},N_{f}\\}$ in which
$\omega=1/(\Omega-1)$. For obtaining DISF and ODISF segmentations, one may set
$\omega=\log_{N_{0}}e$.

As mentioned, we exploit the optimum-path forest generated from the previous
iteration for computing the relevance of each seed $s\in\mathcal{S}$. That is,
its relevance $\mathbf{V}_{*}(s)\in\mathbb{R}_{+}$ is estimated based on the
characteristics of its superpixel $\mathbf{T}(s)$. Given that, in IFT, a
seed’s capability of conquering similar vertices is reflected on its
superpixel’s size, we propose a size-based criterion
$\mathbf{V}_{1}(s)=|\mathbf{T}(s)|/|\mathcal{V}|$ for computing the relevance
of $s$.

However, since the background superpixels are often larger than the object’s,
$\mathbf{V}_{1}$ may undesirably favor the former. On the other hand, the
typical homogeneity of the background could be differed by the heterogeneity
of the object through superpixel contrast. For that, we first describe the
color dissimilarity $\mathbf{G}_{\mathbf{F}}(s,t)\in\mathbb{R}_{+}$ between
two trees by
$\left\|\overline{\mathbf{F}}(\mathbf{T}(s))-\overline{\mathbf{F}}(\mathbf{T}(t))\right\|_{2}$,
given $s,t\in\mathcal{S}$. Then, we define the tree’s neighbors (or adjacents)
by $\mathbf{A}(s)=\left\\{t:\exists\left\langle
x,y\right\rangle\in\mathcal{A}\right\\}$ for
$x\in\mathbf{T}(s),y\in\mathbf{T}(t)$, and $s\neq t$. Finally, we can
formulate two novel color-based criteria: $\mathbf{V}_{2}(s)=\min_{\forall
t\in\mathbf{A}(s)}\left\\{\mathbf{G}_{\mathbf{F}}(s,t)\right\\}$ and
$\mathbf{V}_{3}(s)=\max_{\forall
t\in\mathbf{A}(s)}\left\\{\mathbf{G}_{\mathbf{F}}(s,t)\right\\}$; which aims
for the minimum and maximum contrast , respectively, amongst the seed’s
neighbors.

Similarly to $\mathbf{V}_{1}$, considering only $\mathbf{V}_{2}$ or
$\mathbf{V}_{3}$ may not be sufficient for selecting relevant seeds. High
contrast regions can indicate noises or a well-defined object border.
Conversely, low contrast ones are often located within objects or nearby
poorly-defined borders. Thus, given that both size and contrast are important
for describing the relevance of the seed, we propose two new criteria by
combining the size-based and contrast-based estimations. The
$\mathbf{V}_{4}(s)=\mathbf{V}_{1}(s)\cdot\mathbf{V}_{2}(s)$ favors large
superpixels and prioritizes the minimum contrast amongst the superpixel
neighbors, while $\mathbf{V}_{5}(s)=\mathbf{V}_{1}(s)\cdot\mathbf{V}_{3}(s)$
also favors large superpixels, but aims for the maximum contrast.

Still, the relevance of a seed $s$ is strongly tied to which object is
desired, making it irrelevant when it does not delineate the expected one. If
an object is desired, then more superpixels should be within or near it, in
order to promote its accurate delineation. In such case, the probable object
location can assist in removing seeds that are not within (or nearby) the
object of interest. First, analogously to $\mathbf{G}_{\mathbf{F}}(s,t)$, we
define
$\mathbf{G}_{\mathbf{O}}(s,t)=\left\|\overline{\mathbf{O}}(\mathbf{T}(s))-\overline{\mathbf{O}}(\mathbf{T}(t))\right\|_{2}$,
given $s,t\in\mathcal{S}$. Then, we propose an object-based criterion
$\mathbf{V}_{\text{obj}}(s)=\mathbf{V}_{*}(s)\cdot\max\left\\{\overline{\mathbf{O}}(\mathbf{T}(s)),\max_{t\in\mathbf{A}(s)}\left\\{\mathbf{G}_{\mathbf{O}}(s,t))\right\\}\right\\}$.
By setting $\mathbf{O}(p)=1\;\forall p\in\mathcal{V}$ whenever a prior object
information is abscent, one may see that
$\mathbf{V}_{\text{obj}}(s)=\mathbf{V}_{*}(s)$.

##  5 Experimental Results

In this section, we detail the experimental framework for evaluating our
proposal and discuss the obtained results. In Section 5.1, the experimental
setup is presented and, in Section 5.2, we perform an ablation study over all
SICLE’s steps. Finally, we compare our proposal against state-of-the-art
methods both quantitatively and qualitatively (i.e., Sections 5.3 and 5.4).

###  5.1 Experimental Setup

In order to evaluate the performance of each method for different domains, we
selected four datasets whose object of interest is clearly defined (i.e., not
ambiguous). The Extended Complex Saliency Scene Dataset (ECSSD) [28] is a
well-known salient object detection dataset composed of 1000 natural images
with distinct and complex objects. Another natural image dataset, Insects
[22], has 130 images of insects whose thin legs impose a significant challenge
in delineation. For evaluating objects with smooth borders, we opted for two
different medical image datasets. The Liver [33] dataset contains 40 CT slices
of the human liver, whose grayscale borders are difficult to define. Finally,
the Parasites [6] dataset has 72 colored images of helminth eggs in which an
impurity may be attached to it. We randomly selected 70% of the data for
testing and 30% for training. For generating the object saliency maps, we
opted for an effective deep learning approach, named U2-Net [26], suitable for
small datasets like Liver and Parasites. We trained such estimator using the
training set and considering its default parameters.

We selected several state-of-the-art superpixel methods based on their
effectiveness and efficiency: (i) SLIC [1]
222https://www.epfl.ch/labs/ivrl/research/slic-superpixels/; (ii) LSC [18]
333https://jschenthu.weebly.com/projects.html; (iii) SH [34]
444https://github.com/semiquark1/boruvka-superpixel; (iv) ERS [20]
555https://github.com/mingyuliutw/EntropyRateSuperpixel; (v) OISF-OSMOX [7]
666https://github.com/LIDS-UNICAMP/OISF; and (vi) SEAL-ERS [32]
777https://github.com/wctu/SEAL. For all methods, the recommended parameter
configuration was set. Our code will be publicly available
888https://github.com/LIDS-UNICAMP/SICLE after publishing. Given that it is
burdensome to evaluate all possible SICLE configurations, we elected the
following one as starting point for our analysis based on previous works. We
sampled using GRID with $N_{0}=8000$ seeds and considered $\mathbf{w}_{1}$ as
in [8], and used the $\mathbf{V}_{\text{obj}}$, considering $\mathbf{V}_{4}$,
as presented in [5]. Finally, similarly to most seed-based methods [1, 18,
33], we set $\Omega=10$.

Due to the high correlation between several superpixel segmentation metrics
[30], we chose the two most popular. The Boundary Recall (BR) [30] calculates
the percentage of the object boundaries correctly overlapped by a superpixel
border given a certain tolerance radius. The Under-Segmentation Error (UE)
[24] measures the error resultant from superpixels overlapping multiple
ground-truths. We aim for higher values of BR and lower values of UE. For all
experiments, we set $N_{f}\in\left[25,750\right]$.

###  5.2 Ablation Study

We analyze the impacts on the oversampling task by altering the strategy
(i.e., GRID or RND) and $N_{0}$, as Figure 4 shows. It is possible to see that
both strategies achieved a similar performance, irrespective of $N_{0}$,
leading to conclude that pattern has no substantial influence in SICLE. On the
other hand, we may also affirm that there are multiple ways to obtain a
relevant seed set, given that different strategies achieved equivalent
performance. That is, such result support the idea of relevance redundancy
amongst seeds.

In terms of $N_{0}$, it is possible to see that $1000$ seeds are sufficient
for accurate delineation in the ECSSD dataset but insufficient for the
remaining ones. We argue that such event is due to the images’ size: the
bigger the images, the more seeds are required in oversampling. However,
$8000$ is unnecessary for all datasets considered, especially given that
$5000$ achieves on par performance with fewer seeds. Thus, based on efficiency
and simplicity, we opt for randomly oversampling $3000$ seeds.

Figure 4: Results of SICLE variants with varying oversampling strategy and
$N_{0}$.

Figure 5 shows the results for varying arc-cost functions (i.e., DYN and ROOT)
and limiting $\Omega\in\left\\{3,5,10\right\\}$. The equal performance between
estimations can be explained by the possible similarity between the seed’s and
tree’s features. Interestingly, $\Omega=3$ is sufficient for homogeneous
objects (i.e., liver and parasite egg) but insufficient for more complex ones.
We argue that the number of iterations for seed relevance manifestation is
proportional to the object’s complexity. However, when comparing $\Omega=5$
and $\Omega=10$ in all datasets, there are no significant improvements,
indicating a possible upper bound (i.e., performing unnecessary iterations).
Therefore, we limited SICLE to perform, at most, five iterations using the
ROOT function, which allows future improvements [16, 13].

Figure 5: Results of SICLE variants with varying arc-cost functions and
$\Omega$.

We have also evaluated our proposed seed removal criteria
$\left\\{\mathbf{V}_{i}\right\\}^{5}_{i=1}$ subjected to
$\mathbf{V}_{\text{obj}}$ alongside a random selection criterion
$\mathbf{V}_{\text{rnd}}$ as baseline (Figure 6). Unsurprisingly, the random
criterion is the worst approach for all datasets. However, it is interesting
that, while contrast-only criteria performs fairly for datasets in which
contrast is crucial (i.e., Liver and Parasites), size-only criterion achieves
top performance in all datasets. We argue that $\mathbf{V}_{\text{obj}}$
manages to overcome their possible biases by constraining the seed selection
nearby (or within) the desired object. When combining both contrast and size
estimations, we can see a slight improvement for Liver and Parasites, and no
performance degradation for the remaining ones. Furthermore, if no map is
provided, such combination can overcome their particular biases [8]. Thus, we
opt for maintaining the $\mathbf{V}_{4}$ subjected to
$\mathbf{V}_{\text{obj}}$.

Figure 6: Results of SICLE variants with varying seed removal criteria.

Finally, as Figure 7 shows, we assessed SICLE considering maps of different
qualities for analyzing the influence of the map’s borders in delineation.
Aside from those generated by the U2-Net, used throughout this paper, we
considered the ground-truth (GT) and the object’s minimum bounding box (BB)
for emulating the generation of maps with ideal and rough object borders,
respectively. We emphasize that the GT maps are only used in this particular
analysis. It is important to note that, differently from other object-based
strategies [6, 7], the delineation performance of SICLE is not impacted by the
quality of the map’s border. As consequence, and setting aside that generating
ideal maps is yet an unfeasible task, we claim that such property is positive
since it disregards the need to improve the object saliency map whenever it is
not the problem’s main goal.

Figure 7: Results of SICLE variants with different object saliency maps.

###  5.3 Quantitative Analysis

We compared our best SICLE variant against several state-of-the-art methods,
as illustrated in Figure 8. It is possible to see that SICLE is consistently
accurate, for both metrics, in all datasets. That is, considering different
objects with distinct features, our approach’s strategy of oversampling and
object-based seed removal is proven more effective than the typical approach
of seed recomputation seen in SLIC, LSC and OISF. When analyzing the UE, the
results support the findings of [5, 9] claiming that object-based strategies
offer a better superpixel leaking prevention.

Figure 8: Results of SICLE against several state-of-the-art methods.

Table 2 shows the computational time required for each method on a 64bit Intel
(R) Core(TM) i5-5200U PC with CPU speed of 2.20Ghz. In terms of time
complexity, SICLE is $\mathrm{O}\left(|\mathcal{V}|\log|\mathcal{V}|\right)$
bounded by the IFT executions. Although SLIC is
$\mathrm{O}\left(|\mathcal{V}|\right)$, we can see that, in practice, SICLE
tends to be faster, since it is limited by, at most, 5 iterations. One could
argue in favor of SH, which is $\mathrm{O}\left(|\mathcal{V}|\right)$, given
its property of computing a multiscale segmentation in a single execution.
However, SICLE has such property if one sets the desired quantity of
superpixels (i.e., seeds) at each iteration. Lastly, SICLE is a more efficient
object-based solution than OISF.

$N_{f}$ | SLIC | SH | OISF-OSMOX | SICLE
---|---|---|---|---
25 | 0.472$\pm$0.026 | 0.916$\pm$0.035 | 3.091$\pm$0.628 | 0.541$\pm$0.067
100 | 0.473$\pm$0.025 | 0.915$\pm$0.032 | 1.757$\pm$0.343 | 0.445$\pm$0.060
750 | 0.477$\pm$0.025 | 0.916$\pm$0.032 | 1.182$\pm$0.229 | 0.361$\pm$0.053
Table 2: Average execution time for each method in the ECSSD dataset. The best
performance for each quantity is depicted in bold.

###  5.4 Qualitative Analysis

Figure 9: Qualitative comparison between SICLE and several state-of-the-art
methods. The desired number of superpixels was $N_{f}=50$. The red lines
indicate the object boundary, whereas the cyan ones, superpixel borders.

We compare some segmentation results between SICLE and other state-of-the-art
methods in Figure 9. The ERS method achieves effective object delineation only
for the insect image, whereas for the other images, it performs fairly.
Conversely, we can see that SH present better and, more importantly, more
consistent results for all images. However, the lack of object information
compromises the object delineation of the desired object in several parts. For
example, both object-based methods increased the superpixel resolution within
(or nearby) the object and improved its delineation. Still, for OISF, the
incorrect estimations in the map assisted in deteriorating the object
delineation in several parts. Finally, it is possible to see that SICLE
achieves effective delineation for all objects irrespective of saliency
errors.

##  6 Conclusion and Future Work

In this work, we propose a novel object-based superpixel segmentation
framework named Superpixels through Iterative CLEarcutting (SICLE), which is a
generalization of two state-of-the-art methods [8, 5]. It starts off from
oversampling and, through several iterations, generates superpixels from the
seed set and remove a portion of irrelevant seeds for preserving the accurate
object delineation from the previous iteration. We show that SICLE is
consistently effective by evaluating its performance against several state-of-
the-art methods in different datasets. Moreover, results show that, in most
practical cases, SICLE can be faster than SLIC. For future endeavors, we
intent to extend SICLE for 3D and video segmentation, while exploring its
capability of being an interactive segmentation tool.

## Acknowledgment

The authors thank the Conselho Nacional de Desenvolvimento Científico e
Tecnológico – CNPq – (Universal 407242/2021-0, PQ 303808/2018-7,
310075/2019-0), the Fundação de Amparo a Pesquisa do Estado de Minas Gerais –
FAPEMIG – (PPM-00006-18), the Fundação de Amparo a Pesquisa do Estado de São
Paulo – FAPESP – (2014/12236-1) and the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior – CAPES – Finance code 001 (COFECUB
88887.191730/2018-00) for the financial support.

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# Effect of quintessence dark energy on the shadow of Hayward black holes with
spherical accretion

Malihe Heydari-Fard
_Department of Physics, The University of Qom, 3716146611, Qom, Iran_
Electronic address<EMAIL_ADDRESS>
###### Abstract

It is believed that the astrophysical black holes are surrounded by a luminous
accretion flow that is a necessary ingredient for imaging a black hole. In
this paper, we study the influence of quintessence dark energy on the shadow
images of a Hayward black hole surrounded by the static and infalling
spherical accretion flow. We find the effect of the magnetic charge and the
state parameter of quintessence matter on the horizons, photon sphere and
impact parameter of the quintessence Hayward black hole. The observed specific
intensity of the shadow and also the shadow and photon ring luminosities of
the quintessence Hayward black hole in two different spherically accretion
flow are investigated. Comparing the results of spherically accreting Hayward
black hole with and without quintessence matter shows that in the presence of
quintessence dark energy the shadow and photon rings radius and also their
luminosity increase, while the observed specific intensity of the shadow
decreases.
Keywords: Physics of black holes, Black hole shadow, Photon sphere, Spherical
accretion, Modified theories of gravity

##  1 Introduction

The Event Horizon Telescope (EHT) collaboration has released Very Long
Baseline Interferometry (VLBI) observations at the core of the M87* galaxy
[1]–[6] and Milky Way galaxy [7] with angular resolution comparable to that
expected of a supermassive black hole. The EHT images show a dark central area
called the black hole shadow which surrounded by a bright ring, the photon
ring [8]. These results consistent with predictions of the theory of general
relativity (GR) [9] and thus the EHT observations provide another strong
evidence to Einstein’s theory. Synge was the first to investigate the
deflection of light rays around a gravitationally intense star [10]. Bardeen
then argued that the shadow radius of static Schwarzschild black hole is
$r_{s}=3M$, and also found that the angular momentum of rotating black holes
deformed the shape of the shadow so that it’s not a perfect circle as the
static case [11]. Moreover, in recent years many studies about the black hole
shadow in the context of different theories of gravity have been done
[12]–[28].

On the other hand, it is generally believed that the astrophysical black holes
are not in an empty space but instead surrounded by an extremely luminous
accretion flow which definitely affects the obtained image of the black hole.
There are many papers and interesting discussions, show how the presence of
accretion material modifies the black hole shadow. In 1978, Luminet extends
the Synge’s work to a more realistic case in which the Schwarzschild black
hole is surrounded by an accretion flow [29]. He studied the light deflection
of a black hole surrounded by a geometrically thin, optically thick accretion
disk and showed that the optical appearance of the black hole and the photon
ring depend on both the location and profile of the accretion flow. In an
interesting viewpoint, Gralla et al. by studying the rings around black hole
shadow of M87* showed that depending on the intersection points of light rays
with the plane of the disk there exist different rings called direct emission,
lensing ring and photon ring. In this model, the accretion flow around the
black hole is optically and geometrically thin [30]. Moreover, the shadow of
the Schwarzschild black hole surrounded by a geometrically thick, optically
thin accretion flow has been studied by Cunha et al. in [31]. A simple
spherical model of optically thin accretion in the space-time of the
Schwarzschild black hole, and the study of properties of its shadow have been
done in [32]. Narayan et al. argue that the behavior of the black hole image
in spherical model is different from geometrically thin disk model regarding
that the inner edge of the disk can leave an remarkable imprint on the image,
especially when the edge of disk is out of the photon orbit. The shadow of
quintessence Schwarzschild black hole surrounded by accretion flow for two
types of the accretion flow, optically and geometrically thin disk accretion
flow, and also spherically accretion flow, is studied in [33]. Authors have
studied the effects of quintessence state parameter and accretion flow on the
optical appearance of black hole [33]. Also, authors have studied the similar
behavior for 4-dimensional Gauss-Bonnet black holes and explored the influence
of both the Gauss-Bonnet parameter and spherical accretion flow on the
properties of the black hole shadow [34]. The optical appearance of black
holes in the context of Rastall gravity with different spherical accretions
has been also studied in Ref. [35]. For further studies about the effect of
the location and profile of accretion flow which surrounded the black hole
space-times in alternative gravity theories see, –[57].

The astronomical observations show that our universe is in an accelerated
expansion phase caused by unknown component with negative pressure and
positive energy density, is called dark energy [58]–[59]. One candidate
interpret the negative pressure is the quintessence dark energy. The state
equation of quintessence dark energy is $p=\omega\rho$ with $\omega$ being the
quintessence state parameter in the range $-1<\omega<-1/3$ [60]–[61]. The
black hole solutions in the presence of quintessence dark energy have been
extensively studied. The first static and spherically symmetric black hole
solution containing the quintessence matter is obtained by Kiselev [62]–[63].

In addition to irregular black holes which have intrinsic singularity in the
origin of space-time, black hole solutions without space-time singularity were
first introduced by Bardeen [64]–[65] and later another type of regular black
holes was presented by Hayward [66]. Rotating Hayward and charged Hayward
black holes are also constructed in [67] and [68]. Similar to irregular black
holes studied in the presence of quintessence matter, Nam obtained a static
Hayward black hole solution with quintessence matter [69]. The geodesic motion
in the space-time of such a black hole is studied in [70]–[71] and extended to
the rotating case in [72]. Also, for studying the shadow of rotating Hayward
black holes in the absence and presence of quintessence matter see, [73]–[78].
Guo et al. have studied the shadow of Hayward black hole with an accretion
flow . Now, in the present paper, we study the shadow of Hayward black holes
surrounded by quintessence matter obtained by Nam [69], in the presence of
spherical accretion flow; Namely we generalize Guo’s work to the case of
accreting Hayward black hole surrounded by quintessence dark energy and
investigate the effects of both the magnetic charge and quintessence state
parameter on the physical parameters of black hole. We discuss the specific
intensity of the shadow and the luminosities of the quintessence Hayward black
hole shadow and photon sphere, and also compare our results with the results
corresponding to the Schwarzschild, quintessence Schwarzschild, and Hayward
black holes.

The paper is structured as follows. In section 2, we present a brief review of
Hayward black holes surrounded by quintessence matter and some of their
properties. In section 3 by employing the ray-tracing method we discuss the
photon trajectories in the space-time of quintessence Hayward black holes and
investigate the effect of the model parameters on them. In section 4 we
present the shadow images of quintessence Hayward black hole with static and
spherical accretion flow. The paper ends with drawing conclusions.

##  2 Hayward black holes with quintessence matter

The geometry of a Hayward black hole surrounded by quintessence dark energy
can be expressed as [69]

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (1)

where

$f(r)=1-\frac{2Mr^{2}}{r^{3}+g^{3}}-\frac{a}{r^{3\omega+1}},$ (2)

that $M$ is the mass of black hole, $g$ is the magnetic charge of black hole
and $a$ and $\omega$ are the normalization factor and the state parameter of
the quintessence matter, respectively. In the case of $a=0$, the above metric
reduces to the Hayward black hole obtained by Hayward [66], and for $g=0$
represents the Schwarzschild black hole surrounded by quintessence matter that
was initially obtained by Kiselve in [62]. Also, for $a=g=0$ behaves like the
Schwarzschild black hole that has only one event horizon. We also note by
choosing different values of $\omega$, the different black hole solutions can
be obtained. For instance, when $\omega=-\frac{1}{3}$ the solution (1)
corresponds to the Hayward black hole [66] and for $\omega=-1$ and
$a=\frac{\Lambda}{3}$ reduces to the Hayward de Sitter solution. Moreover, in
the case of $\omega=\frac{1}{3}$ and $a=-Q^{2}$ corresponds to the charged
Hayward black hole [68].

The behavior of $f(r)$ function and thus the black hole horizons depend on the
magnetic charge $g$, $a$ and $\omega$ parameters. For quintessence dark energy
the state parameter takes the values in the interval $-1<\omega<-\frac{1}{3}$
and the energy density of quintessence matter is given by

$\rho=-\frac{3\omega a}{2r^{3\omega+1}}.$ (3)

As is clear, for a positive energy density the parameter $a$ should be
positive. In the present work, we take $\omega=-0.5$ and $\omega=-0.7$ and
thus we investigate the behavior of horizons for two cases. The interested
reader is referred to [69] for more discussions about properties of the black
hole horizons. In what follows, we find the conditions that the Hayward metric
surrounded by quintessence matter could have three horizons. From equation
$f(r)=0$ one can obtain the relationship between the black hole mass and the
event horizon radius as follows [69]

$M=\frac{r^{3}+g^{3}}{2r^{2}}(1-\frac{a}{r^{3\omega+1}}).$ (4)

Choosing suitable values of the constants $g$, $a$ and $\omega$ the function
$M$ is plotted in Figure 1. It can be seen that the function $M$ has a local
maximum and minimum both located above the horizontal axis. For values of $M$
in the interval $M_{\rm min}<M<M_{\rm max}$ the black hole has three horizons.
The radii of inner and event horizons are represented by $r_{-}$ and $r_{h}$
respectively, while $r_{c}$ is the cosmological horizon. There exist a
critical value of normalization factor, $a$, for which $r_{-}=r_{h}=r_{c}$.
This critical value $a_{\rm critical}$ is given by [69]

$a_{\rm
critical}=-\frac{4+3\omega+\sqrt{16+9\omega^{2}}}{2-3\omega+\sqrt{16+9\omega^{2}}}\left(\frac{\omega-4-\sqrt{16+9\omega^{2}}}{2\omega}\right)^{\omega+1/3}\frac{g^{3\omega+1}}{3\omega},$
(5)

when $a>a_{\rm critical}$, there exist no black hole solution for any values
of $M$, but in the case of $a<a_{\rm critical}$ the black hole solutions exist
for $M_{\rm min}<M<M_{\rm max}$.

In Table 1, we present the values of $M_{\rm min}$, $M_{\rm max}$ for
$a=0.05$, $\omega=-0.7$ and different values of $g$. Also from equation (5)
the value of $a_{\rm critical}$ is obtained for $\omega=-0.7$ and different
values of $g$. In this work, we take $a=0.05$, and $M=1$ which is in interval
$M_{\rm min}$ and $M_{\rm max}$. In this case the value of $a$ is smaller than
$a_{\rm critical}$ for all values of $g$ and thus we have the black hole
solutions with three horizons.

Figure 1: The black hole mass as a function of the radial coordinate $r$ for $g=0.9$, $a=0.05$ and $\omega=-0.7$. Table 1: The value of $a_{\rm critical}$ and also the values of $M_{\rm min}$, $M_{\rm max}$ for $a=0.05$, $\omega=-0.7$ and different values of $g$. $\omega$ | $g$ | $a_{\rm critical}$ | $M_{\rm min}$ | $M_{\rm max}$
---|---|---|---|---
\- 0.7 | 0.1 | 1.81558 | 0.47548 | 2.03223
| 0.3 | 0.54223 | 0.48783 | 2.03234
| 0.5 | 0.30914 | 0.45812 | 2.03277
| 0.7 | 0.21351 | 0.63224 | 2.03372
| 0.9 | 0.16194 | 0.80067 | 2.03540

The behavior of the function $f(r)$ for $M=1$, $a=0.05$ and $\omega=-0.7$ is
plotted in Figure 2. As the figure shows, $f(r)$ function has three real roots
corresponding to $r_{-}$, $r_{h}$ and $r_{c}$, respectively [69]. Also the
difference between event horizon and the cosmic horizon decreases with the
increases of absolute value of $\omega$ parameter, see Table 2.

Figure 2: The metric coefficient $f(r)$ as a function of the radial coordinate
$r$ for $M=1$, $a=0.05$ and $\omega=-0.7$ with different values of $g$

.

##  3 Light deflection of the quintessence Hayward black hole

To obtain the light deflection of quintessence Hayward black holes we study
the null geodesics in the space-time of such black holes. Thus, we consider
the Euler-Lagrange equation as follows

$\frac{d}{ds}\left(\frac{\partial{\cal
L}}{\partial{\dot{x}^{\mu}}}\right)-\frac{d{\cal L}}{dx^{\mu}}=0,$ (6)

where the Lagrangian density ${\cal L}$ is given by

${\cal
L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-f(r)\dot{t}^{2}+\frac{1}{f(r)}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}{\theta}\dot{\varphi}^{2}\right),$
(7)

$\dot{x}^{\mu}$ is the photon four-velocity, i.e.,
$\dot{x}^{\mu}=\frac{\partial x^{\mu}}{\partial s}$ where $s$ denotes the
affine parameter. Without loss of generality, we consider the motion in the
equatorial plane and impose two conditions $\theta=\frac{\pi}{2}$ and
$\dot{\theta}=0$ . Also, since the metric coefficients have no explicit
dependence on $t$ and $\varphi$ coordinates, there are two conserved
quantities corresponding to the energy $E$ and angular momentum $L$, of the
photon. Now, using equations (2), (6) and (7) one can find

$\dot{t}=-\frac{E}{1-\frac{2Mr^{2}}{r^{3}+g^{3}}-\frac{a}{r^{3\omega+1}}},$
(8) $\dot{\varphi}=\frac{L}{r^{2}},$ (9)
$\dot{r}^{2}+\left(1-\frac{2Mr^{2}}{r^{3}+g^{3}}-\frac{a}{r^{3\omega+1}}\right)\left(\frac{L^{2}}{r^{2}}+h\right)=E^{2},$
(10)

where $h=0$ and $h=1$ correspond to the null-like and time-like geodesics,
respectively. Considering the null geodesics with $h=0$ and using equation
(10), the effective potential can be rewrite as

$V_{\rm
eff}(r)=\frac{L^{2}}{r^{2}}f(r)=\frac{L^{2}}{r^{2}}\left(1-\frac{2Mr^{2}}{r^{3}+g^{3}}-\frac{a}{r^{3\omega+1}}\right).$
(11)

In Figure 3, the effective potential of quintessence Hayward black holes is
plotted for different values of magnetic charge, $g$. The state parameter set
to $\omega=-0.5$ and $\omega=-0.7$ in the left and right panels, respectively.
It can be seen that increasing $g$ leads to an increase in the peak of the
potential, while with increasing the absolute value of $\omega$ the maximum of
the potential decreases. The effect of quintessence parameter on the effective
potential is displayed in Figure 4, showing that in the presence of
quintessence dark energy the maximum of the effective potential decreases, and
for all values of $g$ the peak of the potential for quintessence Hayward black
holes is less than that for Hayward and Schwarzschild black holes.

Figure 3:  The effective potential as a function of the radial coordinate $r$
for different values of magnetic charge $g$ with $\omega=-0.5$ (left panel)
and $\omega=-0.7$ (right panel). Figure 4: The effective potential as a
function of the radial coordinate $r$ for Schwarzschild space-time $g=a=0$,
Hayward black hole for $g=0.9$, $a=0$, and Hayward black hole surrounded by
quintessence matter with $g=0.9$, $a=0.05$, $w=-0.7$ and $M=1$.

Now, we are going to obtain the photon sphere and impact parameter of photon
sphere for quintessence Hayward black holes by employing the ray-tracing
method. The photon orbit occurs at $r=r_{ph}$ and thus from equation (10) the
conditions for such orbits are given by

$V_{\rm eff}(r_{ph})=E_{ph}^{2},\hskip 14.22636ptV^{{}^{\prime}}_{\rm
eff}(r_{ph})=0,$ (12)

where the prime represents differentiation with respect to the radial
coordinate $r$. Use of equation (11) leads to the following relation

$r{f^{{}^{\prime}}}(r)-2f(r)=0.$ (13)

Substituting $f(r)$ from equation (2) we obtain the following equation for the
photon sphere

$r^{6}-3mr^{5}-4g^{3}r^{3}-2g^{6}+3a(1+\omega)(g^{3}+r^{3})^{2}r^{-(3\omega+1)}=0.$
(14)

In the general case of $a\neq 0$ and $g\neq 0$ the above equation does not
have an analytical solution and thus we numerically obtain the roots of that.
For $a=g=0$, we find $r_{ph}=3M$ which is the photon radius for the
Schwarzschild space-time. Also, in the case of $a=0,g\neq 0$ namely for the
Hayward black holes the above equation does not have an analytical solution
while, for the Schwarzschild black holes in the presence of quintessence
matter with $a\neq 0$ and $g=0$ there is only an exact solution for
$\omega=-\frac{2}{3}$ as follows

$r_{ph}=\frac{1-\sqrt{1-6Ma}}{a}.$ (15)

Moreover, the impact parameter of the photon sphere defined as
$b_{ph}=L_{ph}/E_{ph}$, is given by $b_{ph}=r_{ph}f(r_{ph})^{-1/2}$.

The numerical results of the inner horizon radius $r_{-}$, event horizon
$r_{h}$, cosmological horizon $r_{c}$, the radius of the photon sphere
$r_{ph}$ and the impact parameter of photon sphere $b_{ph}$ for different
values of $g$ and $\omega$ with $M=1$ and $a=0.05$ are presented in Table 2.
When we fixed the value of $\omega$ and increased the parameter $g$, $r_{h}$,
$r_{c}$, $r_{ph}$ and $b_{ph}$ decrease, however the rate of this decrease is
very small for cosmic horizon $r_{c}$. We also see that for a fixed $g$ with
increasing the absolute value of $\omega$ the values of $r_{h}$ and $b_{ph}$
increase but the cosmological horizon $r_{c}$ decreases. Moreover, the photon
radius $r_{ph}$ increases and then decreases around $\omega=-0.7$.

In Table 3, we compare the numerical values of $r_{h}$, $r_{c}$, $r_{ph}$ and
$b_{ph}$ for the Schwarzschild space-time, Hayward black hole, Schwarzschild
black hole surrounded by quintessence and quintessence Hayward black hole. The
first row corresponds to the results of Schwarzschild black hole and the
second row to those of Hayward regular black hole. The third and fourth rows
correspond to the Schwarzschild black hole surrounded by quintessence and
quintessence Hayward black hole, respectively.

For quintessence Hayward black holes under consideration, the dependence of
the photon radius on the parameters $a$ and $g$ with $\omega=-\frac{2}{3}$ is
plotted in Figure 5. As the figure shows, for a fixed value of $a$ with
increasing $g$ the photon radius decreases, while at fixed $g$ by increasing
$a$ the photon radius increases. Also, it can be seen that for $a=g=0$,
$r_{ph}=3M$ which is the radius of the photon sphere for the Schwarzschild
black hole.

Figure 5: The behavior of the photon radius as a function of the magnetic charge and normalization factor for $\omega=-\frac{2}{3}$. Table 2: The values of inner horizon, $r_{-}$, event horizon, $r_{h}$, cosmological horizon, $r_{c}$, photon radius, $r_{ph}$, impact parameter, $b_{ph}$, for different values of $g$ and $\omega$, with $M=1$, $a=0.05$. The results in the first row of $\omega$ corresponds to the quintessence schwarzschild black hole. $\omega$ | $g$ | $r_{-}/M$ | $r_{h}/M$ | $r_{c}/M$ | $r_{ph}/M$ | $b_{ph}/M$
---|---|---|---|---|---|---
\- 0.4 | 0 | $-$ | 2.12343 | 3199990 | 3.18038 | 5.72947
| 0.1 | 0.02222 | 2.12320 | 3199990 | 3.18018 | 5.72929
| 0.3 | 0.11768 | 2.11733 | 3199990 | 3.17496 | 5.72455
| 0.5 | 0.26227 | 2.09458 | 3199990 | 3.15491 | 5.70643
| 0.7 | 0.45862 | 2.03996 | 3199990 | 3.10808 | 5.66460
| 0.9 | 0.72981 | 1.92392 | 3199990 | 3.01594 | 5.58440
\- 0.5 | 0 | $-$ | 2.15857 | 395.9690 | 3.21631 | 5.98805
| 0.1 | 0.02240 | 2.15835 | 395.9690 | 3.21610 | 5.98786
| 0.3 | 0.11869 | 2.15250 | 395.9690 | 3.21087 | 5.98280
| 0.5 | 0.26438 | 2.12987 | 395.9690 | 3.19078 | 5.96347
| 0.7 | 0.46177 | 2.07564 | 395.9690 | 3.14389 | 5.91886
| 0.9 | 0.73313 | 1.96101 | 395.9690 | 3.05175 | 5.83346
\- 0.7 | 0 | $-$ | 2.28304 | 13.10270 | 3.27103 | 7.23434
| 0.1 | 0.22475 | 2.28281 | 13.10270 | 3.27082 | 7.23403
| 0.3 | 0.11944 | 2.27687 | 13.10280 | 3.26541 | 7.22597
| 0.5 | 0.26646 | 2.25391 | 13.10290 | 3.24462 | 7.19522
| 0.7 | 0.46563 | 2.19916 | 13.10310 | 3.19610 | 7.12450
| 0.9 | 0.73795 | 2.08509 | 13.10360 | 3.10074 | 6.99004
\- 0.8 | 0 | $-$ | 2.41487 | 6.55098 | 3.25476 | 9.22055
| 0.1 | 0.02248 | 2.41463 | 6.55099 | 3.25455 | 9.21989
| 0.3 | 0.11983 | 2.40832 | 6.55127 | 3.24896 | 9.20271
| 0.5 | 0.26737 | 2.38396 | 6.55231 | 3.22748 | 9.13747
| 0.7 | 0.46781 | 2.32619 | 6.55462 | 3.17731 | 8.98918
| 0.9 | 0.74126 | 2.20753 | 6.55869 | 3.07838 | 8.71353
Table 3: The values of event horizon, $r_{h}$, cosmological horizon, $r_{c}$, photon radius, $r_{ph}$, impact parameter, $b_{ph}$, for Schwarzschild space-time, Hayward black hole, Schwarzschild and Hayward black holes surrounded by quintessence matter. $Type$ | $\omega$ | $a$ | $g$ | $r_{h}/M$ | $r_{c}/M$ | $r_{ph}/M$ | $b_{ph}/M$
---|---|---|---|---|---|---|---
Schwarzschild | $-$ | 0 | 0 | 2 | $-$ | 3 | 5.19615
Hayward | $-$ | 0 | 0.5 | 1.96772 | $-$ | 2.97162 | 5.17169
Quintessence Schwarzschild | \- 0.5 | 0.05 | 0 | 2.15857 | 395.9690 | 3.21631 | 5.98805
Quintessence Hayward | \- 0.5 | 0.05 | 0.5 | 2.12987 | 395.9690 | 3.19078 | 5.96347

Figure 6: The polar plot of light rays around a schwarzschild black hole with
and without quintessence matter (top row) and a Hayward black hole with and
without quintessence matter (bottom row) for $g=0.9$ and $a=0.05$,
$\omega=-0.7$ with $M=1$ . The yellow, blue and orange curves correspond to
the trajectory of light rays with $E=E_{ph}$, $E<E_{ph}$ and $E>E_{ph}$,
respectively.

Using equations (9) and (10) and new variable $u=\frac{1}{r}$ we find

$\frac{du}{d\varphi}=\sqrt{\frac{1}{b^{2}}-u^{2}\left(1-\frac{2Mu}{1+g^{3}u^{3}}-au^{3\omega+1}\right)}.$
(16)

Assuming that light rays approach the black hole from the right side, shadow
and photon sphere are formed from the deflection of light corresponds to the
Figure 6. The light trajectories in the space-time of Schwarzschild,
quintessence Schwarzschild, Hayward and quintessence Hayward black holes are
plotted in panels (a) to (d). As can be seen, by increasing the magnetic
charge $g$ the shadow radius decreases, while in the presence of quintessence
matter an enhancement in the absolute value of $\omega$ increases the shadow
radius which is in agreement with Table 2. The top row shows the results of
the Schwarzschild and quintessence Schwarzschild black holes while the bottom
row shows the results of regular Hayward black holes in the absence and
presence of quintessence matter. By comparing figures (a) with (c) and also
(b) with (d), we find that by increasing the magnetic charge $g$ the light
deflection is smaller, namely the presence of magnetic charge cause to
decrease the strength of the gravitational field and thus diminishes the light
bending angle [79]. On the other hand, comparing the results of figures (a)
with (b) and also results of (c) with (d) show that the presence of
quintessence matter increases the deflection of light rays [80].

##  4 Effect of black hole spherical accretion flow on shadow and photon
sphere

Now, we are going to study the optical appearance of Hayward black hole
surrounded by the quintessence matter with the spherical accretion flow and
investigate the influence of magnetic charge, $g$, and quintessence state
parameter, $\omega$, on it. First, we consider a static spherical accretion
model of optically and geometrically thin accretion flows and then focus on
infalling spherical accretion flows.

###  4.1 The static spherical accretion

First we investigate the shadow and photon sphere in the background of a
quintessence Hayward black hole surrounded by static spherical accretion flow.
The observed specific intensity can be obtained by integrating the specific
emissivity along any ray path [81]–[82]

$I(\nu_{\rm obs})=\int_{\gamma}g^{3}j(\nu_{\rm em})dl_{p},$ (17)

with

$g\equiv\frac{\nu_{\rm obs}}{\nu_{\rm em}}.$ (18)

In the above equation, $g$ is the redshift factor, while $\nu_{\rm obs}$ and
$\nu_{\rm em}$ denote the observed photon frequency and radiated photon
frequency, respectively. In the quintessence Hayward black hole space-time
(1), the redshift factor is $g=f(r)^{1/2}$. Assuming that the emission is
monochromatic with rest frame frequency $\nu_{\rm t}$ and the emission radial
profile is $1/r^{2}$ [82], the specific emissivity takes the form

$j(\nu_{\rm em})\propto\frac{\delta(\nu_{\rm em}-\nu_{t})}{r^{2}}.$ (19)

Also, according to equation (1), the proper length measured in the rest frame
of the emitter is given by

$dl_{p}=\sqrt{\frac{1}{f(r)}dr^{2}+r^{2}d\varphi^{2}}=\sqrt{\frac{1}{f(r)}+r^{2}\left(\frac{d\varphi}{dr}\right)^{2}}dr.$
(20)

Thus, by substituting equation (16) into equation (20), we obtain the specific
intensity observed by a distant observer as

$I(\nu_{\rm
obs})=\int_{\gamma}\frac{f(r)}{r^{2}}\sqrt{1+\frac{b^{2}f(r)}{r^{2}-b^{2}f(r)}}dr.$
(21)

From equation (16), it is clear that the trajectory of light rays depends on
impact parameter $b$, so that in the case of $b=b_{ph}$, ($E=E_{ph}$), the
photons will revolve around the black hole several times and for $b>b_{ph}$,
($E<E_{ph}$), the light rays deflect by it. Moreover, the photons with
$b<b_{ph}$, ($E>E_{ph}$), eventually fall into the black hole singularity.
Therefore, in order to study the photon intensity which associate with the
light trajectories, we plotted the observed intensity as a function of impact
parameter $b$. In Figure 7, the plots of intensity as a function of $b$ for
different values of magnetic charge, $g$, with $\omega=-0.5$ (left panel) and
$\omega=-0.7$ (right panel) are presented. Clearly, the intensity increases
with impact parameter, sharply peak at $b=b_{ph}$ and then decreases with $b$.
The reason for maximum intensity at $b=b_{ph}$ is that the photon revolves
around the black hole several times as it approaches the photon sphere.

Meanwhile, at fixed $\omega$ the intensity increases with increasing $g$ so
that the quintessence Schwarzschild black hole with $g=0$ has the smallest
value. Also, for a given value of $g$ the larger absolute value of $\omega$
has the smaller intensity, namely the case of $\omega=-0.7$ has the smaller
intensity in comparison with $\omega=-0.5$. It shows that in the presence of
quintessence dark energy the observed intensity decreases.

Figure 7:  The observed intensity $I(\nu_{\rm obs})$ for static spherical
accretion flow around a quintessence Hayward black hole $\omega=-0.5$ (left
panel) and $\omega=-0.7$ (right panel) for different value of $g$, $a=0.05$
and $M=1$.

Two-dimensional plot of the shadow for different values of $g$ with
$\omega=-0.5$ and $\omega=-0.7$ are shown in Figure 8. It is clear that for a
fixed value of $\omega$ in each figure, the luminosity of quintessence Hayward
black holes decreases with increase of $g$, so that the quintessence
Schwarzschild black hole with $g=0$ has the most luminosity in comparison with
quintessence Hayward black holes. This is due to the fact that the
gravitational field of quintessence Schwarzschild black hole with $g=0$ is
stronger which increases the light deflection and leads to more photons enter
the photon ring, makes it more luminance compared to quintessence Hayward
black holes. Also, comparing plots in the top row of the figure with those in
the bottom row shows the effect of the state parameter of dark energy $\omega$
on the black hole shadow and photon ring; For a fixed value of $g$ the
luminosity of the photon sphere increases faintly with the absolute value of
$\omega$.

Figure 8: The two-dimensional plot of the black hole shadows and photon sphere
for static spherical accretion flow, $\omega=-0.5$ (top row) and $\omega=-0.7$
(bottom row) for different values of $g$, $a=0.05$ and $M=1$.

###  4.2 The infalling spherical accretion

Since in our real universe the most of accretions are moving, in this section
we consider the black hole shadow and photon sphere in the context of
infalling spherical accretion. Note that the observed intensity for a distant
observer in the case of infalling spherical accretion is still expressed as
equation (21), but the redshift factor is different from static model and is
related to the velocity of accretion flow

$g=\frac{k_{\alpha}u_{\rm obs}^{\alpha}}{k_{\beta}u_{\rm em}^{\beta}},$ (22)

where $k^{\mu}\equiv\dot{x_{\mu}}$, $u_{\rm obs}^{\mu}=(1,0,0,0)$ and $u_{\rm
em}^{\mu}$ are the photon four-velocity, distant observer four-velocity and
accretion four-velocity, respectively. From equation (8) we know that
$k_{t}=1/b$ is a constant and $k_{r}$ can be obtained from condition
$k_{\mu}k^{\mu}=0$. Therefore, we find

$\frac{k_{r}}{k_{t}}=\pm\sqrt{\frac{1}{f(r)}\left(\frac{1}{f(r)}-\frac{b^{2}}{r^{2}}\right)},$
(23)

where the upper and lower sign denote to the case that the photons approach or
away from the black hole, respectively. The four-velocity of the infalling
accretion is
$(u_{em}^{t},u_{em}^{r},u_{em}^{\theta},u_{em}^{\varphi})=(\frac{1}{f(r)},-\sqrt{1-f(r)},0,0)$.
Using these equations, the redshift factor in equation (22) can be obtained as

$g=\frac{1}{u_{em}^{t}+\left(\frac{k_{r}}{k_{t}}\right)u_{em}^{r}}.$ (24)

Moreover, the proper distance is defined as

$dl_{p}=k_{\alpha}u^{\alpha}_{em}ds=\frac{k_{t}}{g^{3}\mid k_{r}\mid}dr,$ (25)

where $s$ is the affine parameter along the photon path. By assuming that the
specific emissivity is monochromatic the specific intensity $I(\nu_{\rm obs})$
in the case of infalling spherical accretion thus can be written as

$I=\int\frac{g^{3}}{r^{2}\sqrt{\frac{1}{f(r)}\left(\frac{1}{f(r)}-\frac{b^{2}}{r^{2}}\right)}}dr.$
(26)

The observed intensity with respect to parameter $b$ with $\omega=-0.5$ (left
panel) and $\omega=-0.7$ (right panel) is plotted in Figure 9. It is easy to
see that the behavior of intensity is similar to the case of static spherical
accretion flow and peak is at $b_{ph}$, but the maximum of intensity in this
case is less than that of static case.

Two-dimensional plots of shadow cast for $\omega=-0.5$ and $\omega=-0.7$ are
shown in Figure 10. Clearly, in both cases at fixed $\omega$ with increasing
parameter $g$, the photon sphere luminosity becomes darker. Moreover, by
comparing plots in the top row of the figure with those in the bottom row we
find that at a given $g$, for instance $g=0.8$, the shadow region becomes
brighter with the absolute value of state parameter, namely the case of
$\omega=-0.7$ is brighter than $\omega=-0.5$. Note that the radius of shadow
and photon sphere only depend on the space-time geometry, while the properties
of the black hole shadow and photon ring not only depend on the space-time
geometry but also on the accretion flow property. In the other words, the
radius of shadow and photon sphere do not change in the case of static and
infalling spherical accretion, but the central shadow region for the infalling
accretion is darker than that for static accretion which is caused by the
Doppler effect in this case.

Finally, the observed intensity in the space-times of Schwarzschild black
hole, Hayward and quintessence Hayward black holes has been compared in the
left and right panels of Figure 11 for static and infalling cases,
respectively.

Figure 9:  The observed intensity $I(\nu_{\rm obs})$ for infalling spherical
accretion flow around a quintessence Hayward black hole $\omega=-0.5$ (left
panel) and $\omega=-0.7$ (right panel) for different value of $g$, $a=0.05$
and $M=1$.

Figure 10:  The two-dimensional plot of the black hole shadows and photon
sphere for infalling spherical accretion flow, $\omega=-0.5$ (top row) and
$\omega=-0.7$ (bottom row) for different value of $g$, $a=0.05$ and $M=1$.

Figure 11:  The observed intensity $I(\nu_{\rm obs})$ for static (left panel)
and infalling (right panel) spherical accretion flow around a Schwarzschild,
Hayward and quintessence Hayward black hole for $g=0.9$, $a=0.05$ and $M=1$.

##  5 Conclusions

The effects of accretion flow and magnetic charge on the features of observed
shadow and photon rings of Hayward black holes have been studied in . On the
other hand, Hayward black hole solution surrounded by quintessence matter has
constructed in [69]. In this paper, we considered Hayward black holes
surrounded by quintessence matter and similar to Ref. explored the effects of
accretion flow, magnetic charge and also quintessence state parameter on the
optical appearance of these black holes.

First, we discussed the event horizon properties of quintessence Hayward black
holes. It was shown that in the presence of quintessence matter there could
exist three horizons; inner horizon $r_{-}$, event horizon $r_{h}$, and
cosmological horizon $r_{c}$. We found that the difference between event
horizon and cosmological horizon depends on quintessence state parameter
$\omega$ and with increasing absolute value of $\omega$, this difference
decreases. Cosmological horizon has an important role in the observed shadow
of quintessence Hayward black holes, since we consider the distant observer
nearby the cosmological horizon and the accretion flow near the event horizon.
We also calculated the numerical values of the inner horizon $r_{-}$, event
horizon $r_{h}$, cosmological horizon $r_{c}$, the radius of the photon sphere
$r_{ph}$, and impact parameter of the photon sphere $b_{ph}$ for various
values of magnetic charge $g$ and quintessence state parameter $\omega$, with
the results summarized in Table 2. According to Table 2 for a fixed $\omega$
with increasing $g$ the values of $r_{ph}$ and $b_{ph}$ decrease, while at
fixed $g$ when we increase the absolute value of $\omega$, the values of
$r_{ph}$ and $b_{ph}$ increases too. Also, we plotted the trajectory of light
rays around Hayward black holes and also Hayward black holes surrounded by
quintessence matter for different values of energy and showed the effect of
parameter $\omega$ on the light deflection.

Furthermore, we assumed that quintessence Hayward black hole surrounded by
spherical accretion flow and considered two relativistic models: the static
and infalling spherical accretion flow. Then, in Figures 7 and 9 we plotted
the profile of observed intensity as a function of impact parameter for
$\omega=-0.5$ and $\omega=-0.7$ with different $g$, in the model with gas at
rest and, in radially infalling gas accretion scenario, respectively. The
figures show that the peak value of intensity is at $b_{ph}$ and increases
with increase of magnetic charge $g$ at fixed $\omega$, while it decreases
when at fixed $g$ we increase the absolute value of state parameter $\omega$.
The two dimensional plots of the black hole shadow and photon sphere for
$\omega=-0.5$ and $\omega=-0.7$ are shown in Figures 8 and 10. It is found
that for both spherical accretion flows, static and infalling, the shadow
luminosity decreases with increase of magnetic charge at fixed $\omega$, while
it increases when we increase the absolute value of state parameter $\omega$
for a fixed value of $g$.

Although the black hole shadow is a signature of space-time geometry, the
profile and position of accretion flow does not affect on the shape of shadow
and its size. Comparing results of spherically accreting Hayward black hole
and spherically accreting quintessence Hayward black hole shows that in the
presence of quintessence dark energy not only the radius of the shadow and
photon rings change but also the luminosity of this black hole shadow and the
specific intensity of the shadow change. Also, comparing two various cases of
spherically accretion flow show that for infalling accretion flow the central
region of shadow is darker and the luminosity of photon sphere is brighter
than the case of static spherical accretion flow.

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