\section{Introduction}

Tables are widely present in a great variety of documents such as administrative documents, invoices, scientific papers, reports, or archival documents among others; and, therefore, techniques for table analysis are instrumental to automatically extract relevant information stored in a tabular form from several sources~\cite{Couasnon14}. The first step in table analysis is \emph{table detection} --- that is, determining the position of the tables in a document --- and such a step is the basis to later determine the internal table structure and, eventually, extract semantics from the table contents~\cite{Couasnon14}.

Table detection methods in digital born documents, such as readable PDFs or HTML documents, employ the available meta-data included in those documents to guide the analysis by means of heuristics~\cite{Oro09}. However, table detection in image-based documents, like scanned PDFs or document images, is a more challenging task due to high intra-class variability --- that is, there are several table layouts that, in addition, are highly dependent on the context of the documents --- low inter-class variability  --- that is, other objects that commonly appear in documents (such as figures, graphics or code listing among others) are similar to tables --- and the heterogeneity of document images~\cite{Embley06}. These three issues make difficult to design rules that are generalisable to a variety of documents; and, this has led to the adoption of machine learning techniques, and, more recently, deep learning methods.

Currently, deep learning techniques are the state of the art approach to deal with computer vision tasks~\cite{Pyimagesearch}; and, this is also the case for table detection in document images~\cite{tablebank,Schreiber17,Kerwat18}. The most accurate models for table detection have been constructed using \emph{fine-tuning}~\cite{Razavian14}, a \emph{transfer learning} technique that consists in re-using a model trained in a source task, where a lot of data is available, in a new target task, with usually scarce data. In the context of table detection, the fine-tuning approach has been applied due to the small size of table detection datasets that do not contain the necessary amount of images required to train deep learning models from scratch. In spite of its success, this approach has the limitation of applying transfer learning from natural images, a distant domain from document images. This makes necessary the application of techniques, like image transformations~\cite{Gilani17}, to make document images look like natural images.

In this work, we present the benefits of applying transfer learning for table detection from a close domain thanks to the {\LaTeX} part of the TableBank dataset~\cite{tablebank}, a dataset that consists of approximately 200K labelled images of academic documents containing tables --- a number big enough to train deep learning models from scratch. Namely, the contributions of this work are the following:

\begin{itemize}
    \item We analyse the accuracy of four of the most successful deep learning algorithms for object detection (namely, Mask-RCNN, RetinaNet, SSD and YOLO) in the context of table detection in academic documents using the TableBank dataset.
    \item Moreover, we present a comprehensive study where we compare the effects of fine-tuning table detection models from a distant domain (natural images) and a closer domain (images of academic documents from the TableBank dataset), and demonstrate the advantages of the latter approach. To this aim, we employ the 4 aforementioned object detection architectures and 7 heterogeneous table detection datasets containing a wide variety of document images.
    \item Finally, we show the benefits of using models trained for table detection on document images to detect other objects that commonly appear in document images such as figures and formulas.
\end{itemize}

As a by-product of this work, we have produced a suite of models that can be employed via a set of Jupyter notebooks (documents for publishing code, results and explanations in a form that is both readable and executable)~\cite{jupyter} that can be run online using Google Colaboratory~\cite{colab} --- a free Jupyter notebook environment that requires no setup and runs entirely in the cloud avoiding the installation of libraries in the local computer. In addition, the code for fine-tuning the models is also freely available. This allows the interested readers to adapt the models generated in this work to detect tables in their own datasets. All the code and models are available at the project webpage \href{https://github.com/holms-ur/fine-tuning}{https://github.com/holms-ur/fine-tuning}.

The rest of this paper is organised as follows. In the next section, we provide a brief overview of the methods employed in the literature to tackle the table detection task. Subsequently, in Section~\ref{sec:methods}, we introduce our approach to train models for table detection using fine-tuning, as well as the setting that we employ to evaluate such an approach. Afterwards, we present the obtained results along with a thorough analysis in Section~\ref{sec:results}, and the tools that we have developed in Section~\ref{sec:tools}. Finally, the paper ends with some conclusions and further work.

\section{Related work}\label{sec:related-work}

Since the early 1990s, several researchers have tackled the task of table detection in document images using mainly two approaches: rule-based techniques and data-driven methods. The former are focused on defining rules to determine the position of lines and text blocks to later detect tabular structures~\cite{Hirayama95,Jianying99,Zanibbi04}; whereas, the latter employ statistical machine learning techniques, like Hidden Markov models~\cite{Costa09}, a hierarchical representation based on the MXY tree~\cite{Cesari02} or feature engineering together with SVMs~\cite{Kasar13}. However, both approaches have drawbacks: rule-based methods require the design of handcrafted rules, that do not usually generalise to several kinds of documents; and, machine learning methods require manual feature engineering to decide the features of the documents that are feed to machine learning algorithms. These problems have been recently alleviated by using deep learning methods.

Nowadays, deep learning techniques are the state of the art approach to deal with table detection. The reason is twofold: deep learning techniques are robust for different document types; and, they do not need handcrafted features since they automatically learn a hierarchy of relevant features using convolutional neural networks (CNNs)~\cite{Goodfellow16}. Initially, hybrid methods combining rules and deep-learning models were suggested; for instance, in~\cite{Hao16} and~\cite{Borges17}, CNNs were employed to decide whether regions of an image suggested by a set of rules contained a table. On the contrary, the main approach followed currently consists in adapting general deep learning algorithms for object detection to the problem of table detection. Namely, the main algorithm applied in this context is Faster R-CNN~\cite{fasterrcnn}, that has been directly employed using different backbone architectures~\cite{tablebank,Schreiber17,Kerwat18}, and combined with deformable CNNs~\cite{Siddiqui18} or with image transformations~\cite{Gilani17}. Other detection algorithms such as YOLO~\cite{yolov3} or SSD~\cite{ssd} have also been employed for table detection~\cite{Kerwat18,Huang19}, but achieving worse results than the methods based on the Faster R-CNN algorithm. Nevertheless, training deep learning models for table detection is challenging due to the considerable amount of images that are necessary for this task --- up to recently, the biggest dataset of document images containing tables was the Marmot dataset with 2,000 labelled images~\cite{marmot}, far from the datasets employed by deep learning methods that consists of several thousands, or even millions, of images~\cite{ILSVRC15,pascalvoc}.

In order to deal with the problem of limited amount of data, one of the most successful methods applied in the literature is transfer learning~\cite{Razavian14}, a technique that re-uses a model trained in a source task in a new target task. This is the approach followed in~\cite{Schreiber17,Siddiqui18,Gilani17}, where they use models trained on natural images to fine-tune their models for table detection. However, transfer learning methods are more effective when there is a close relation between the source and target domains, and, unfortunately, there is only a vague relation between natural images and document images. This issue has been faced, for instance, by applying image transformations to make document images as close as possible to natural images~\cite{Gilani17}.

Another option to tackle the problem of limited data consists in acquiring and labelling more images, a task that has been undertaken for table detection in the TableBank project~\cite{tablebank} --- a dataset that consists of 417K labelled images of documents containing tables. The TableBank dataset opens the door to apply transfer learning to not only construct models for table detection in different kinds of documents, but also to detect other objects, such as figures or formulas, that commonly appear in document images. This is the goal of the present work.

\section{Materials and methods}\label{sec:methods}
In this section, we explain the fine-tuning method, as well as the object detection algorithms, datasets and evaluation metrics used in this work.

\subsection{Fine-tuning}
Transfer learning allows us to train models using the knowledge learned by other models instead of starting from scratch. The idea on which transfer learning techniques are based is that CNNs are designed to learn a hierarchy of features. Specifically, the lower layers of CNNs focus on generic features, while the final ones focus on specific features for the task they are working with. As explained in~\cite{Razavian14}, transfer learning can be employed in different ways, and the one employed in this work is known as fine-tuning. In this technique, the weights of a network learned in a source task are employed as a basis to train a model in the destination task. In this way, the information learned in the source task is used in the destination task. This approach is especially beneficial when the source and target tasks are close to each other.

In our work, we study the effects of fine-tuning table detection algorithms from a distant domain (natural images from the Pascal VOC dataset~\cite{pascalvoc}) and a close domain (images of academic documents from the TableBank dataset). To this aim, we consider the following object detection algorithms.

\subsection{Object detection algorithms}
Object detection algorithms based on deep learning can be divided into two categories~\cite{Girs, yolov3}: the \emph{two-phase algorithms}, whose first step is the generation of proposals of ``interesting'' regions that are  classified using CNNs in a second step. And the \emph{one-phase algorithms} that perform detection without explicitly generating region proposals. For this work, we have employed algorithms of both types. In particular, we have used the two-phase algorithm Mask R-CNN, and the one-phase algorithms RetinaNet, SSD and YOLO.

\paragraph{Mask R-CNN~\cite{fasterrcnn}} is currently one of the most accurate algorithm based on a two-phase approach, and the latest version of the R-CNN family. We have used a library implemented in Keras~\cite{matterport} for training models with this algorithm.

\paragraph{RetinaNet~\cite{retinaNet}} is a one-phase algorithm characterised by using the focal loss for training on a scarce set of difficult examples, and that prevents the large number of easy negatives from overwhelming the detector during training.We have used another library implemented in Keras~\cite{kerasRetina} for traning models with this algorithm.

\paragraph{SSD~\cite{ssd}} is a simple detection algorithm that completely eliminates proposal generation and encapsulates all computation in a single network. In this case, we have used the MXNET library~\cite{MxNET} for training the models.

\paragraph{YOLO~\cite{yolov3}} frames object detection as a regression problem where a single neural network predicts bounding boxes and class probabilities directly from full images in one evaluation. Although there are several versions of YOLO, the main ideas are the same for all of them. We have used the Darknet library~\cite{yolodarknet} for training models with this algorithm.

The aforementioned algorithms have been trained for detecting tables in a wide variety of document images by using the datasets presented in the following section.

\subsection{Benchmarking datasets}

For this project, we have used several datasets, see Table~\ref{datasets}. Namely, we have employed three kinds of datasets: the base datasets (which are used to train the base models), the fine-tune datasets for table detection, and the fine-tune dataset for detecting other objects in document images. The reason to consider several table detection datasets is that there are several table layouts that are highly dependent on the document type, and we want to prove that our approach can be generalised to heterogeneous document images.

\begin{table}
\centering
\begin{tabular}{lllp{4cm}}
\toprule
Datasets  & \#Train Images & \#Test Images & Type of images\\
\midrule
Pascal VOC&16,551&4,952& Natural images\\
TableBank&199,183&1,000& Academic documents\\
\midrule
ICDAR13 & 178 & 60  & Documents obtained from Google search\\
ICDAR17 &   1,200 & 400& Scientific papers   \\
ICDAR19 &   599 & 198 & Modern images \\
Invoices &  515 & 172 & Invoices\\
MarmotEn &  744 & 249  & Scientific papers \\
MarmotChi &   754 & 252 & E-books\\
UNLV &  302 & 101  & Technical reports, business letters, newspapers and magazines\\
\midrule
ICDAR17FIG &   1,200 & 400& Scientific papers  \\
ICDAR17FOR &   1,200 & 400 & Scientific papers \\
\bottomrule
\end{tabular}
\caption{Sizes of the train and test sets of the datasets}
\label{datasets}
\end{table}

\subsubsection{Base Datasets}
In this work, we have employed two datasets of considerable size for creating the base models that are later employed for fine-tuning.

\paragraph{The Pascal VOC dataset~\cite{pascalvoc}} is a popular project designed to create and evaluate algorithms for image classification, object detection and segmentation. This dataset consists of natural images which have been used for training different models in the literature. Thanks to the trend of releasing models to the public, we have employed models already trained with this dataset to apply fine-tuning from natural images to the context of table detection.

\paragraph{TableBank~\cite{tablebank}} is a table detection dataset built with Word and {\LaTeX} documents that contains 417K labeled images. For this project, we only employ the {\LaTeX} images (199,183 images) since the Word images contain some errors in the annotations. On the contrary to the Pascal VOC dataset, where there were available models trained for such a dataset, we have trained models for the TableBank dataset from scratch.

\subsubsection{Fine-tuning Datasets}

We have used several open datasets for fine-tuning; however, most table detection datasets only release the training set. Hence, in this project, we have divided the training sets into two sets (75\% for training  and 25\% for testing) for evaluating our approach. The dataset split are available in the project webpage, and the employed datasets are listed as follows.

\paragraph{ICDAR13~\cite{icdar13}} is one of the most famous datasets for table detection and structure recognition. This dataset is formed by documents extracted from Web pages and email messages. This dataset was prepared for a competition focused on the task of detecting tables, figures and mathematical equations from images. The dataset is comprised of PDF files which we converted to images to be used within our framework. The dataset contains 238 images in total, 178 were used for training and 60 for testing.

\paragraph{ICDAR17~\cite{icdar17}} is a data set prepared for a competition as ICDAR13. The dataset consists of 1.600 images in total, where we can find tables, formulas and figures. The training set consists of 1,200 images, while the rest of the 400 images are used for testing. This dataset has been employed three times in our work: for the detection of tables (from now on, we will call this dataset ICDAR17), for the detection of figures (from now on, we will call this dataset ICDAR17FIG) and for the detection of formulas (from now on, we will call this dataset ICDAR17FOR).

\paragraph{ICDAR19~\cite{icdar19}} is, as in the previous cases, a dataset proposed for a competition. The dataset contains two types of images: modern documents and archival ones with various formats. In this work we have only taken the modern images (797 images in total, 599 for training and 198 for testing).

\paragraph{Invoices} is a proprietary dataset of PDF files containing invoices from several sources. The PDF files had to be converted into images. This set has 515 images in the training set and 172 in the testing set.

\paragraph{Marmot~\cite{marmot}} is a dataset that shows a great variety in language type, page layout, and table styles. Over 1,500 conference and journal papers were crawled for this dataset, covering various fields, spanning from the year 1970. to latest 2011 publications. In total, 2,000 pages in PDF format were collected. The dataset is composed of Chinese (from now on MarmotChi) and English pages (from now on MarmotEn): the MarmotChi dataset was built from over 120 e-Books with diverse subject areas provided by Founder Apabi library, and no more than 15 pages were selected from each book, this dataset contains 993 images in total, 744 were used for training and 249 for testing. And the MarmotEn dataset was crawled from the Citeseer website, this dataset contains 1,006 images in total, 754 were used for training and 252 for testing.

\paragraph{UNLV~\cite{unlv}} is comprised of a variety of documents which includes technical reports, business letters, newspapers and magazines. The dataset contains a total of 2,889 scanned documents where only 403 documents contain a tabular region. We only used the images containing a tabular region in our experiments: 302 for training and 101 for testing.

Using the aforementioned algorithms and datasets, we have trained several models that have been evaluated using the following metrics.

\subsection{Performance measure}
In order to evaluate the constructed models for the different datasets, we employed the same metric used in the ICDAR19 competition for table detection~\cite{icdar19}. Considering that the ground truth bounding box is represent by GTP, and that the bounding box detected by an algorithm is represented by DTP; then, the formula for finding the overlapped region between them is given by:
$$
    IoU(GTP, DTP)  = \frac{area(GTP \bigcap DTP)}{area(GTP\bigcup DTP)}
$$
IoU(GTP, DTP) represents the overlapped region between ground truth and detected bounding boxes and its value lies between zero and one.

Now, given a threshold $T\in [0,1]$, we define the notions of True Positive at T, TP@T, False Positive at T, FP@T, and False Negative at T, FN@T. The TP@T is the number of ground truth tables that have a major overlap $(IoU\geq T)$ with one of the detected tables. The FP@T indicates the number of detected tables that do not overlap $(IoU < T)$ with any of the ground tables. And, FN@T indicates the number of ground truth tables that do not overlap $(IoU < T)$ with any of the detected tables. From these notions, we can define the Precision at T, P@T, Recall at T, R@T,  and F1-score at T, F1@T, as follows:
$$
   P@T = \frac{TP@T}{FP@T + TP@T}
$$
$$
    R@T = \frac{TP@T}{FN@T + TP@T}
$$
$$
    F1@T = \frac{2*TP@T}{FP@T +FN@T+ 2*TP@T}
$$
Finally, the final score is decided by the weighted average WAvgF1 value:
$$
    WAvgF1 = \frac{0.6\times F1@0.6+0.7\times F1@0.7+0.8\times F1@0.8+0.9\times F1@0.9}{0.6+0.7+0.8+0.9}
$$
In the above formula, and since results with higher IoUs are more important than those with lower IoUs, we use IoU threshold as the weight of each F1 value to get a definitive performance score for convenient comparison. Using these metrics we have obtained the results presented in the following section.

\section{Results}\label{sec:results}
In this section, we conduct a thorough study of our approach, see Tables 2--5. Each table corresponds with the results obtained from each object detection algorithm: Table~\ref{fast} contains the results that have been obtained using Mask R-CNN; Table~\ref{retina}, the results of RetinaNet; Table~\ref{ssd}, the results of SSD; and, finally, Table~\ref{yolo} contains the results of YOLO. The tables are divided into three parts: the first row contains the results obtained for the TableBank dataset, the next 9 rows correspond with the result obtained with the models trained by fine-tuning from natural image models, and the last 9 rows correspond with the models fine-tuned from the TableBank models. All the models built in this work were trained using the default parameters in each deep learning framework, and using K80 NVIDIA GPUs.

\begin{table}
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{llllllllllllllll}
\toprule
 & \multicolumn{3}{c}{@0.6}&\multicolumn{3}{c}{@0.7}&\multicolumn{3}{c}{@0.8}&\multicolumn{3}{c}{@0.9}&\\
&P@0.6&R@0.6&F1@0.6&P@0.7&R@0.7&F1@0.7&P@0.8&R@0.8&F1@0.8&P@0.9&R@0.9&F1@0.9&  WAvgF1 & Improvement\\
\midrule
TableBank&0.94&0.98&0.96&0.94&0.97&0.95&0.93&0.96&0.94&0.84&0.87&0.86&0.92\\
\midrule
ICDAR13&0.14&0.77&0.23&0.08&0.45&0.14&0.03&0.16&0.05&0&0&0&0.09\\
ICDAR17&0.32&0.85&0.46&0.28&0.75&0.41&0.17&0.47&0.25&0.04&0.1&0.06&0.27\\
ICDAR17FIG&0.29&0.61&0.39&0.22&0.46&0.3&0.13&0.27&0.17&0.01&0.03&0.02&0.19\\
ICDAR17FOR&0.18&0.45&0.26&0.09&0.24&0.13&0.03&0.07&0.04&0&0.01&0&0.09 \\
ICDAR19&0.6&0.64&0.62&0.48&0.51&0.5&0.24&0.25&0.25&0.02&0.02&0.02&0.31 \\
Invoices&0.38&0.56&0.45&0.28&0.42&0.34&0.16&0.23&0.19&0.02&0.03&0.02&0.22\\
MarmotEn&0.37&0.75&0.49&0.28&0.58&0.38&0.08&0.17&0.11&0&0.01&0&0.21\\
MarmotChi&0.52&0.83&0.64&0.48&0.76&0.59&0.32&0.51&0.39&0.07&0.11&0.08&0.39\\
UNLV&0.29&0.58&0.39&0.17&0.34&0.23&0.06&0.11&0.08&0.01&0.02&0.01&0.15\\

\midrule

ICDAR13&0.7&0.97&0.81&0.7&0.97&0.81&0.7&0.97&0.81&0.47&0.65&0.54&0.72 & 0.63\\
ICDAR17&0.72&0.95&0.82&0.7&0.93&0.8&0.68&0.9&0.78&0.49&0.64&0.56&0.72&0.45\\
ICDAR17FIG&0.36&0.69&0.47&0.33&0.63&0.43&0.23&0.43&0.3&0.05&0.09&0.07&0.29& 0.09\\
ICDAR17FOR&0.1&0.49&0.17&0.06&0.28&0.1&0.02&0.12&0.04&0&0.01&0&0.06 & -0.02\\
ICDAR19&0.76&0.85&0.81&0.74&0.83&0.79&0.67&0.75&0.71&0.38&0.42&0.4&0.65 & 0.34\\
Invoices&0.54&0.71&0.61&0.5&0.66&0.57&0.39&0.52&0.45&0.19&0.26&0.22&0.44 & 0.21\\
MarmotEn&0.72&0.93&0.81&0.7&0.9&0.79&0.67&0.87&0.76&0.46&0.6&0.52&0.70 & 0.48\\
MarmotChi&0.82&0.98&0.89&0.82&0.98&0.89&0.81&0.96&0.88&0.62&0.73&0.67&0.82 & 0.42\\
UNLV&0.66&0.83&0.74&0.63&0.8&0.7&0.55&0.69&0.61&0.24&0.3&0.27&0.55 & 0.39\\
\bottomrule
\end{tabular}}
\caption{Results using the Mask R-CNN algorithm}
\label{fast}
\end{table}

We start by analysing the results for the TableBank dataset, see the first row of the tables. Each model has its strenghts and weaknesses, and depending on the context we can prefer different models. The overall best model, that is the model with higher WAvgF1-score, is the Mask R-CNN model; the other three models are similar among them. If we focus on detecting as most tables as possible (R@0.6) and not detecting other artifacts as tables (P@0.6), the best model is YOLO. Finally, if we are interested in accurately detecting the regions of the tables (F1@0.9), the best model is RetinaNet. The strength of the SSD model is that it is faster than the others.

\begin{table}
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{llllllllllllllll}
\toprule
 & \multicolumn{3}{c}{@0.6}&\multicolumn{3}{c}{@0.7}&\multicolumn{3}{c}{@0.8}&\multicolumn{3}{c}{@0.9}&\\
&P@0.6&R@0.6&F1@0.6&P@0.7&R@0.7&F1@0.7&P@0.8&R@0.8&F1@0.8&P@0.9&R@0.9&F1@0.9&  WAvgF1 & Improvement\\
\midrule
TableBank&0.98&0.86&0.92&0.98&0.86&0.92&0.97&0.85&0.91&0.94&0.82&0.87&0.90\\
\midrule
ICDAR13 & 0.56&0.58&0.57&0.56&0.58&0.57&0.56&0.58&0.57&0.34&0.35&0.35&0.50\\
ICDAR17&0.65&0.86&0.74&0.64&0.85&0.73&0.58&0.77&0.67&0.48&0.63&0.55&0.66\\
ICDAR17FIG&0.57&0.61&0.59&0.7&0.76&0.73&0.73&0.79&0.76&0.74&0.8&0.77&0.72\\
ICDAR17FOR&0.63&0.06&0.12&0.63&0.06&0.12&0.6&0.06&0.11&0.4&0.04&0.07&0.10\\
ICDAR19&0.86&0.66&0.74&0.82&0.63&0.72&0.76&0.58&0.66&0.58&0.45&0.51&0.64\\
Invoices&0.90&0.59&0.71&0.88&0.58&0.70&0.86&0.56&0.68&0.70&0.46&0.56&0.65\\
MarmotEn&0.75&0.86&0.8&0.74&0.86&0.8&0.7&0.81&0.75&0.47&0.54&0.5&0.69\\
MarmotChi&0.78&0.85&0.81&0.75&0.81&0.78&0.73&0.79&0.75&0.5&0.54&0.52&0.70\\
UNLV &0.81&0.83&0.82&0.79&0.81&0.80&0.76&0.77&0.76&0.61&0.63&0.62&0.73\\
\midrule
ICDAR13&0.83&0.77&0.8&0.79&0.74&0.77&0.76&0.71&0.73&0.72&0.68&0.7&0.74&0.24\\
ICDAR17&0.92&0.87&0.89&0.92&0.87&0.89&0.89&0.84&0.86&0.79&0.75&0.77&0.84&0.18\\
ICDAR17FIG&0.76&0.79&0.77&0.74&0.77&0.76&0.72&0.75&0.74&0.64&0.66&0.65&0.72&0.001\\
ICDAR17FOR&0.26&0.35&0.3&0.24&0.32&0.27&0.19&0.26&0.22&0.08&0.1&0.09&0.20&0.11\\
ICDAR19&0.91&0.74&0.82&0.87&0.81&0.79&0.81&0.67&0.73&0.68&0.56&0.61&0.72&0.08\\
Invoices&0.92&0.59&0.72&0.92&0.59&0.71&0.87&0.55&0.68&0.74&0.47&0.58&0.66&0.01\\
MarmotEn&0.93&0.86&0.9&0.92&0.86&0.89&0.91&0.84&0.87&0.78&0.73&0.75&0.84&0.14\\
MarmotChi&0.87&0.87&0.87&0.85&0.85&0.85&0.83&0.83&0.83&0.69&0.7&0.69&0.80&0.10\\
UNLV&0.81&0.83&0.82&0.79&0.8&0.8&0.75&0.77&0.76&0.63&0.64&0.64&0.74&0.01\\
\bottomrule
\end{tabular}}
\caption{Results using the RetinaNet algorithm}
\label{retina}
\end{table}

Let us focus now on the table detection datasets. In the case of models fine-tuned using natural images, the algorithms that stand out are RetinaNet and YOLO, see Tables~\ref{fast} to 5 and Figure~\ref{fig:das}. Similary, the models that achieve higher accuracies when fine-tuning from the TableBank dataset are YOLO and RetinaNet, see Figure~\ref{fig:dasTrans}. As can be seen in Tables 2 to 5, fine-tuning from a close domain produce more accurate models that fine-tuning from an unrelated domain. However, the effects on each algorithm and dataset greatly differ. The algorithm that is more considerably boosted for table detection is Mask R-CNN, that improves up to a 60\% in some cases and 42\% in mean. In the case of RetinaNet, in mean it improves by 11\%, YOLO a 9\%, and SSD is the one with the least improvement, only a 5\%.

Finally, if we consider the results for the datasets containing figures and formulas, the improvement is not as remarkable as in the detection of tables. In this case, the algorithm that takes a bigger advantage of this technique is YOLO, since in both cases it improves up to a 10\%. In the case of RetinaNet, it improves the detection of formulas by 10\%, while that of figures barely improves. And in the case of SSD and Mask R-CNN, they are the ones with the least improvement and even have some penalty.

\begin{table}
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{llllllllllllllll}
\toprule
 & \multicolumn{3}{c}{@0.6}&\multicolumn{3}{c}{@0.7}&\multicolumn{3}{c}{@0.8}&\multicolumn{3}{c}{@0.9}&\\
&P@0.6&R@0.6&F1@0.6&P@0.7&R@0.7&F1@0.7&P@0.8&R@0.8&F1@0.8&P@0.9&R@0.9&F1@0.9& WAvgF1 & Improvement\\
\midrule
TableBank&0.96&0.97&0.96&0.94&0.95&0.95&0.92&0.92&0.92&0.82&0.82&0.82&0.90\\
\midrule
ICDAR13 &0.54&0.68&0.6&0.44&0.55&0.49&0.38&0.48&0.43&0.15&0.19&0.17&0.40\\
ICDAR17&0.49&0.71&0.58&0.41&0.59&0.48&0.34&0.49&0.4&0.22&0.32&0.4&0.45\\
ICDAR17FIG&0.7&0.8&0.75&0.68&0.77&0.72&0.61&0.69&0.65&0.34&0.38&0.36&0.59\\
ICDAR17FOR&0.44&0.64&0.52&0.34&0.49&0.4&0.19&0.27&0.22&0.03&0.05&0.22&0.32\\
ICDAR19&0.31&0.35&0.33&0.23&0.26&0.25&0.18&0.2&0.19&0.1&0.11&0.1&0.20\\
Invoices&0.87&0.84&0.85&0.8&0.78&0.79&0.63&0.61&0.62&0.27&0.26&0.26&0.59\\
MarmotEn &0.67&0.76&0.71&0.63&0.71&0.67&0.6&0.67&0.63&0.38&0.43&0.4&0.58\\
MarmotChi &0.57&0.7&0.63&0.48&0.60&0.53&0.36&0.45&0.4&0.25&0.31&0.28&0.44\\
UNLV &0.66&0.64&0.65&0.6&0.58&0.59&0.45&0.43&0.44&0.12&0.11&0.12&0.42\\

\midrule

ICDAR13 &0.62&0.68&0.65&0.62&0.68&0.65&0.5&0.55&0.52&0.32&0.35&0.34&0.52&0.12\\
ICDAR17&0.55&0.71&0.62&0.46&0.60&0.52&0.42&0.54&0.47&0.30&0.39&0.34&0.47&0.01\\
ICDAR17FIG&0.27&0.77&0.41&0.26&0.72&0.38&0.21&0.58&0.31&0.10&0.28&0.15&0.29&-0.3\\
ICDAR17FOR&0.5&0.74&0.6&0.41&0.6&0.49&0.29&0.42&0.34&0.07&0.11&0.09&0.35&0.03\\
ICDAR19&0.35&0.35&0.35&0.28&0.27&0.28&0.23&0.22&0.23&0.13&0.12&0.13&0.23&0.03\\
Invoices&0.91&0.86&0.89&0.87&0.81&0.84&0.71&0.67&0.69&0.37&0.35&0.36&0.66&0.07\\
MarmotEn&0.71&0.75&0.73&0.69&0.73&0.71&0.66&0.70&0.68&0.49&0.52&0.51&0.64&0.06\\
MarmotChi&0.61&0.67&0.64&0.50&0.55&0.52&0.42&0.46&0.44&0.28&0.31&0.29&0.45&0.01\\
UNLV &0.72&0.66&0.69&0.66&0.61&0.63&0.5&0.45&0.47&0.28&0.26&0.27&0.49&0.07\\
\bottomrule
\end{tabular}}
\caption{Results using the SSD algorithm}
\label{ssd}
\end{table}

\begin{table}
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{lllllllllllllll}
\toprule
 & \multicolumn{3}{c}{@0.6}&\multicolumn{3}{c}{@0.7}&\multicolumn{3}{c}{@0.8}&\multicolumn{3}{c}{@0.9}&\\
&P@0.6&R@0.6&F1@0.6&P@0.7&R@0.7&F1@0.7&P@0.8&R@0.8&F1@0.8&P@0.9&R@0.9&F1@0.9& WavgF1 & Improvement\\
\midrule
TableBank&0.98&0.99&0.98&0.98&0.99&0.98&0.96&0.97&0.96&0.74&0.75&0.75&0.90\\
\midrule
ICDAR13 &0.92&0.58&0.6&0.61&0.61&0.61&0.57&0.55&0.56&0.31&0.32&0.32&0.50\\
ICDAR17&0.9&0.94&0.92&0.88&0.93&0.9&0.78&0.82&0.8&0.39&0.41&0.4&0.72\\
ICDAR17FIG &0.88&0.84&0.86&0.85&0.82&0.84&0.75&0.72&0.74&0.23&0.22&0.23&0.63\\
ICDAR17FOR &0.9&0.85&0.87&0.82&0.77&0.79&0.54&0.5&0.52&0.1&0.09&0.1&0.52\\
ICDAR19&0.95&0.91&0.93&0.94&0.9&0.92&0.89&0.85&0.87&0.61&0.58&0.6&0.81\\
Invoices &0.89&0.87&0.88&0.84&0.82&0.83&0.7&0.68&0.69&0.26&0.26&0.26&0.63\\
MarmotEn &0.9&0.96&0.93&0.87&0.93&0.9&0.76&0.81&0.78&0.32&0.34&0.33&0.70\\
MarmotChi &0.95&0.96&0.96&0.93&0.94&0.94&0.88&0.89&0.89&0.61&0.62&0.61&0.83\\
UNLV &0.91&0.95&0.93&0.88&0.91&0.89&0.73&0.76&0.74&0.39&0.4&0.39&0.70\\
\midrule

ICDAR13&1&0.65&0.78&0.95&0.61&0.75&0.9&0.58&0.71&0.6&0.39&0.47&0.66&0.15\\
ICDAR17&0.94&0.94&0.94&0.93&0.94&0.93&0.89&0.89&0.89&0.61&0.62&0.61&0.82&0.09\\
ICDAR17FIG &0.91&0.83&0.87&0.89&0.82&0.86&0.83&0.76&0.8&0.44&0.4&0.42&0.71&0.07\\
ICDAR17FOR&0.94&0.85&0.89&0.88&0.79&0.83&0.65&0.59&0.62&0.19&0.17&0.18&0.59&0.06\\
ICDAR19&0.95&0.95&0.95&0.94&0.94&0.94&0.9&0.9&0.9&0.68&0.68&0.68&0.85&0.04\\
Invoices &0.9&0.89&0.89&0.87&0.85&0.86&0.76&0.74&0.75&0.39&0.39&0.39&0.69&0.06\\
MarmotEn &0.95&0.97&0.96&0.95&0.97&0.96&0.92&0.93&0.93&0.68&0.69&0.69&0.87&0.16\\
MarmotChi &0.97&0.93&0.95&0.96&0.93&0.94&0.92&0.89&0.91&0.69&0.67&0.68&0.85&0.02\\
UNLV&0.93&0.95&0.94&0.92&0.94&0.93&0.83&0.85&0.84&0.48&0.49&0.49&0.77&0.06\\
\bottomrule
\end{tabular}}
\caption{Results using the YOLO algorithm}
\label{yolo}
\end{table}

\begin{figure}
    \centering
    \includegraphics[scale=0.1]{das.png}
    \caption{Dispersion diagram using fine-tuning from natural images}
    \label{fig:das}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.1]{dasTrans.png}
    \caption{Dispersion diagram using fine-tuning dataset from the TableBank}
    \label{fig:dasTrans}
\end{figure}

As we have shown in this section, fine-tuning from the TableBank dataset can boosten table detection models. However, there is not a model that outperforms the rest, see Figures \ref{fig:das} and \ref{fig:dasTrans}. Therefore, we have released a set of tools to employ the trained models, and also employ them for constructing models using fine-tuning on custom datasets.

\section{Tools for table detection}\label{sec:tools}
Using one of the generated detection model with new images is usually as simple as invoking a command with the path of the image (and, probably, some additional parameters). However, this requires the installation of several libraries and the usage of a command line interface; and, this might be challenging for non-expert users. Therefore, it is important to create simple and intuitive interfaces that might be employed by different kinds of users; otherwise, they will not be able to take advantage of the object detection models.

To disseminate our detection models, we have created a set of Jupyter notebooks, that allows users to detect tables in their images. Jupyter notebooks~\cite{jupyter} are documents for publishing code, results and explanations in a form that is both readable and executable; and, they have been widely adopted across multiple disciplines, both for their usefulness in keeping a record of data analyses, and also for allowing reproducibility. The drawback of Jupyter notebooks is that they require the installation of several libraries. Such a problem has been overcome in our case by providing our notebooks in Google Colaboratory~\cite{colab}, a free Jupyter notebook environment that requires no setup and runs entirely in the cloud avoiding the installation of libraries in the local computer. The notebooks are available at \href{https://github.com/holms-ur/fine-tuning}{https://github.com/holms-ur/fine-tuning}.

In addition, in the same project webpage, due to the heterogeneity of document images containing tables, we have provided all the weights, configuration files, and necessary instructions to fine-tune any of the detection models created in this work to custom datasets containing tables.

\section{Conclusion and Further work}
In this work, we have shown the benefits of using fine-tuning from a close domain in the context of table detection. In addition to the accuracy improvement, this approach avoids overfitting and solves the problem of having a small dataset. Moreover, we can highlight that apart from the Mask R-CNN algorithm, other algorithms such as YOLO and RetinaNet can achieve a good performance in the table detection task.

Since table detection is the first step towards table analysis, we plan to use this work as a basis for determining the internal structure of tables, and, eventually, extracting the semantics from table contents. Moreover, we are also interested in extending these methods to detect forms in document images.

\bibliographystyle{splncs04}


\section{Introduction}\label{SecI}
In recent decades, many heavy-light mesons are observed in experiments~\cite{PDG2018}, which has attracted a lot of interest.
The study of heavy-light mesons provides a way to understand dynamical chiral symmetry broken (DCSB)~\cite{Eichten1974PRD,Pennington2005JPCS,Bashir2012PRC}--one of the most fascinating features of the quantum-chromo dynamics (QCD).
Nevertheless it is still very challenging.
The heavy-light mesons are two-body bound states of heavy and light (anti-)quarks, e.g., $c\bar{d}$, $c\bar{u}$, $c\bar{s}$, etc, which are highly flavor unsymmetric. Thus, the successful descriptions require a systematic and self-consistent understanding of strong interaction in both ultraviolet and infrared regions. Herein, non-perturbative approach of QCD is the key for solving the problem.

Several non-perturbative approaches of QCD have been developed, such as the lattice QCD (lQCD)~\cite{Wilson1974PRD,Ginsparg1982PRD,Batrouni1985PRD,Ryan2018LNP}, the Dyson-Schwinger equations (DSEs)~\cite{Williams1994PPNP,Roberts2000PPNP,Alkofer2001PR,Maris2003IJMP,Roberts2008PPNP,Bashir2012CTP,Cloet2014PPNP}, the functional renormalization group (FRG)~\cite{Polonyi2003CEJP,Pawlowski2007AP,Gies2012LNP}, and so on.
In this work, the QCD's DSE approach is implemented. Within the DSE framework, mesons are described by the two-body Bethe-Salpeter equation (BSE)~\cite{Salpeter1951PR,Nakanishi1969PTPS,Munczek1995PRD,Bender1996PLB,Alkofer2001PR,Qin2016FBS}.
As input, the quark propagator must be solved by the one-body gap equation.
The key point is that the one-body and the two-body equations must be constructed in a self-consistent way to preserve QCD's symmetries. For instance, the chiral symmetry is crucial to realize pion's twofold role as quark-antiquark bound state and Nambu-Goldstone boson~\cite{Munczek1995PRD,Bender1996PLB,Maris1998PLB}.

The simplest symmetry-preserving scheme is the so called rainbow-ladder (RL) truncation~\cite{Williams1994PPNP,Maris1997PRC,Maris1999PRC,Bhagwat2004PRC,Eichmann2008PRC,Qin2012PRC85.035202}, which approximates the dressed quark-gluon vertex as the bare one and the quark-antiquark scattering kernel as the one-gluon exchange form.
Since the bare vertex is flavor-blind and the one-gluon exchange lacks of spin-orbit repulsion, the RL truncation can work for ground state pseudoscalar and vector mesons with symmetric flavors,
e.g., the light-light mesons~\cite{Maris1997PRC,Maris1998PLB,Maris1999PRC,Qin2012PRC85.035202}
and the heavy-heavy mesons~\cite{Casalbuoni1997PR,Bali2001PR,Fischer2015EPJA,Ding2016EPJWC,Chen2017PRD}.

For heavy-light mesons, a sophisticated improvement of the RL truncation must be developed. Recently, following this direction, some progress has been made~\cite{Ivanov1999PRD,Rojas2014PRD,Gomez2015FBS,Gomez2015PRD,Gomez2016PRD,Hilger2017EPJA,Bedolla2018EPJC,Chen2019CPC,Binosi2019PLB,Yin2019PRD100.034008}. In this work, inspired by the axial-vector Ward Takahashi identity (AV-WTI)~\cite{Maris1998PLB,Maris1997PRC}, we propose a flavor-sensitive Bethe-Salpeter kernel beyond the simplest ladder approximation.
This kernel degenerates to the ladder approximation for flavor-symmetric systems, and thus can be considered as a generalization of the latter.
Without introducing any new parameter, the mass spectra and leptonic decay constants of heavy-light mesons in pseudoscalar and vector channels can be well reproduced, self-consistently and systematically.

The article is organized as follows:
In Sec.~\ref{SecII}, we describe the DSE approach and the setups.
In Sec.~\ref{SecIII}, we fit the parameters with the light-light and heavy-heavy systems first, and then present results of heavy-light mesons. The comparisons with the experimental data and results of lattice QCD are also included.
Section~\ref{SecIV} provides a summary and perspective.

\section{Theoretical Framework}\label{SecII}
\subsection{The Gap Equation}\label{SecIIA}

The quark gap equation reads~\cite{Williams1994PPNP}
\begin{equation}\label{DSE}
S^{-1}(p)=Z_2(i\slashed{p}+Z_m m)+\Sigma(p),
\end{equation}
with the self energy
\begin{equation}
\Sigma(p)= g^2 Z_1\int_q^{\Lambda} D_{\mu\nu}(p-q)  \frac{\lambda^a}{2}\gamma_{\mu}S(q) \frac{\lambda^a}{2}\Gamma_{\nu}(q,p),
\end{equation}
where $Z_1$, $Z_2$, and $Z_m$ are the vertex, quark wave-function, and mass renormalization constants, respectively; $m$ is the renormalized current quark mass; ${\lambda^a}$ are the color matrices; $\int_q^{\Lambda}$ represents a Poinc$\acute{a}$re invariant regularization of the four-dimensional integral, with $\Lambda$ the ultraviolet regularization mass scale; $\Gamma_{\nu}$ and $D_{\mu\nu}$ are the dressed quark-gluon vertex and the dressed gluon propagator, respectively.

The solution of the gap equation, i.e., the dressed quark propagator, can be decomposed as
\begin{eqnarray}
S(p)&=&-i \slashed{p} \sigma_v(p^2) + \sigma_s(p^2), \label{quark_lorentz1}\\
S^{-1}(p)&=&i \slashed{p} A(p^2) +B(p^2),    \label{quark_lorentz2}
\end{eqnarray}
with the momentum subtraction renormalization condition
\begin{equation}
S^{-1}(p)|_{p^2=\zeta^2}=i\slashed{\zeta} + m^\zeta,
\end{equation}
where $\zeta$ is the renormalization point and $m^\zeta$ the renormalized current-quark mass. The dressed quark mass function reads then
\begin{equation}\label{MassFunc}
 M(p^2)=B(p^2,\zeta^2)/A(p^2,\zeta^2),
\end{equation}
which is independent of the renormalization point $\zeta$. Note that $m^\zeta$ is the mass function evaluated at the renormalization point $m^\zeta=M(\zeta^2)$.

To solve the gap equation, the dressed quark-gluon vertex and the dressed gluon propagator must be specified. The rainbow part of the RL approximation can be expressed as $(k=p-q)$
\begin{align}
Z_1 g^2 D_{\mu\nu}(k)\Gamma_{\nu}(q,p) = k^2 \mathcal{G}(k^2)D_{\mu\nu}^{\text{free}}(k)\gamma_{\nu},\label{InteractionModel}
\end{align}
where $D_{\mu\nu}^{\text{free}}(k)=(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^2})\frac{1}{k^2}$ is the Landau-gauge free gluon propagator. The interaction model $\mathcal{G}(k^2)$ is written as
\begin{align}
k^2 \mathcal{G}(k^2) = k^2\mathcal{G}_{IR}(k^2)+4\pi\tilde{\alpha}_{pQCD}(k^2)\,,
\end{align}
where $\tilde{\alpha}_{pQCD}(k^2)$ is a bounded and monotonically decreasing regular continuation of the perturbative QCD running coupling to all values of spacelike-$k^2$; and $\mathcal{G}_{IR}(k^2)$ is an $ansatz$ for the interaction at infrared region and dominates in the region $|k|<\Lambda_{QCD}$. The form of $\mathcal{G}_{IR}(k^2)$ determines whether the DCSB and/or confinement can be realized.

With the Qin-Chang (QC) model~\cite{Qin2011PRC}, the interaction is expressed as ($s=k^2$)
\begin{equation}\label{QC}
\mathcal{G}(s)=\frac{8\pi^2}{\omega^4}D e^{-s/\omega^2}+\frac{8\pi^2\gamma_m\mathcal{F}(s)}{\ln[\tau+(1+s/\Lambda_{QCD}^2)^2]},
\end{equation}
where $\mathcal{F}(s)=(1-e^{-s/4m_t^2})/s$ with $m_t=0.5$ GeV; $\gamma_m=12/(33-N_f)$ is the dimension anomaly with the flavor number $N_{f}$; $\tau=e^{2} -1$ is a constant.
Following Ref.~\cite{Qin2018PRD}, we take $N_{f} =5$ and $\Lambda_{QCD}=0.36\,$GeV.

The pointwise behavior of Eq.~\eqref{QC} can also be parameterized as~\cite{Aguilar2009PRD,Aguilar2012PRD}
\begin{equation}
\mathcal{G}(k^2)=\frac{4\pi\alpha(k^2)}{k^2+m_g^2(k^2)},\quad\quad m_g^2(k^2)=\frac{M_g^4}{k^2+M_g^2},
\end{equation}
where $m_g(k^2)$ is the gluon mass and $M_g$ is the mass scale. The interaction achieves its maximum value at the deep infrared $k^2=0$. The running behavior and the gluon mass scale are consistent with the modern DSE and lattice-QCD results \cite{Bowman2004PRD,Oliveira2011JPG,Binosi2015PLB,Binosi2017PRD,Rodriguez2018FBS}.

Many studies (e.g., Refs.~\cite{Qin2012PRC85.035202,Xin2014PRD,Gao2014PRD89.076009,Qin2016FBS,Chen2017PRD,Chen2018PRD,Qin2018PRD,Qin2019FBS}) have shown that the above mentioned interaction model is quite successful.
The parameters $D$ and $\omega$ control the strength and the width of the interaction, respectively.
In fact, observable properties of light-quark ground-state vector- and isospin-nonzero pseudoscalar mesons are insensitive to variations of $\omega \in[0.4,0.6]\,$GeV, as long as
\begin{equation}
\varsigma^3:=D\omega = {\rm constant}\,.
\end{equation}
Since the interaction model absorbs the dressing effects of the quark-gluon vertex and the gluon propagator, for light quark systems, it has much larger strength compared with that of the realistic gauge-sector interaction at infrared momenta~\cite{Qin2011PRC} and the one determined by solving the coulped equations of the quark propagator and the quark-gluon interaction vertex~\cite{Tang2019PRD}.
However, straightforward analysis shows that corrections to the RL truncation almost vanish in the heavy-quark limit~\cite{Bhagwat2004PRC}; hence, the aforementioned agreement entails that the RL truncation should provide a reasonable approximation for the systems involving only heavy quarks so long as one employs a smaller strength. This will be discussed in detail in Sec.~\ref{SecIII}.

\subsection{The Bethe-Salpeter Equation}\label{SecIIB}
In terms of Green functions, properties of mesons are encoded in the quark-antiquark scattering matrices. The corresponding bound-state equation for mesons is the homogenous Bethe-Salpeter equation, which reads
\begin{equation}\label{HBSE}
\Gamma(k;P)=\int_q^{\Lambda}S_f(q_+)\Gamma(q;P)S_g(q_-)K(k,q;P)\,,
\end{equation}
where $P^2=-M^2$ with the meson mass $M$; $q_+=q+\alpha P$ and $q_-=q-(1-\alpha) P$ with the momentum fraction $\alpha\in[0,1]$; $f$ and $g$ denote the quark and antiquark flavors; $\Gamma(k,P)$ is the Bethe-Salpeter amplitude (BSA); $K(k,q;P)$ is the quark-antiquark scattering kernel. The dressed quark propagators are fed with the solution of the gap equation.

According to the $J^P$ quantum numbers, the BSA of the pseudoscalar ($J^P=0^-$) and vector ($J^P=1^-$) mesons can be expanded with a set of basis~\cite{Krassnigg2009PRD}:
\begin{eqnarray}
\Gamma_{0^-}(k;P)&=&\sum_{i=1}^{4}\tau_{0^-}^i(k;P) \mathcal{F}_i(k^2,z_k;P^2)\label{BSA_PS},\\
\Gamma_{1^-}(k;P)&=&\sum_{i=1}^{8}\tau_{1^-}^i(k;P) \mathcal{F}_i(k^2,z_k;P^2)\label{BSA_VC}.
\end{eqnarray}
The pseudoscalar basis is written as \cite{Chen2017PRD}
\begin{align}\label{Basis_PS}
 \tau_{0^-}^1&=i\gamma_5,        & \tau_{0^-}^2&=\gamma_5 \slashed{P}, \notag\\
 \tau_{0^-}^3&=\gamma_5\slashed{k}(k\cdot P),      &\tau_{0^-}^4&=i\gamma_5\sigma^{P,k}.
\end{align}
The vector basis reads~\cite{Chen2017PRD}
\begin{align}\label{Basis_VC}
\tau_{1^-}^1&=i\gamma_{\mu}^T,     &\tau_{1^-}^2&=ik_{\mu}^T\slashed{k},  \notag   \\
 \tau_{1^-}^3&=ik_{\mu}^T\slashed{P}(k\cdot P), &\tau_{1^-}^4&=\gamma_5\epsilon_{\mu\nu\alpha\beta}^T\gamma_{\nu}k_{\alpha}P_{\beta} ,     \notag        \\
 \tau_{1^-}^5&=k_{\mu}^T,      &\tau_{1^-}^6 &=\sigma_{\mu\nu}^T k_{\nu}(k\cdot P),   \\
 \tau_{1^-}^7&=\sigma_{\mu\nu}^T P_{\nu}      &\tau_{1^-}^8 &=k_{\mu}^T\sigma_{\alpha\beta}^T k_{\alpha}P_{\beta}\,, \notag
\end{align}
where $l_{\mu}^T = P_{\mu\nu}l_{\nu}$ with $P_{\mu\nu}=\delta_{\mu\nu}-\frac{P_{\mu}P_{\nu}}{P^2}$ as the transverse projector on the meson momentum $P$.

The $P$ and $C$ transformation for the BSA are defined as follows,
\begin{align}
\Gamma(k;P)&\stackrel{P}{\longrightarrow}\hat{P}\Gamma(\tilde{k};\tilde{P})\hat{P}^{-1} \label{PTrans},\\
\Gamma(k;P)&\stackrel{C}{\longrightarrow}\bar{\Gamma}(k;P)=\hat{C}\Gamma^t(-k;P)\hat{C}^{-1}\label{CTrans},
\end{align}
with $\tilde{k}=(k_4,-\vec{k})$, $\hat{P}=\gamma_4$, and $\hat{C}=\gamma_2\gamma_4$. The basis are chosen to have specific $P$- and $C$-parity. The coefficient functions $\mathcal{F}_i(k^2,z_k;P^2)$ can be expanded in terms of the Chebyshev polynomials
\begin{equation}\label{ChebyshevExpansion}
\mathcal{F}^{JP}(k^2,k\cdot P)=\sum_{j=0}a_j(k^2)U_j(k\cdot P/\sqrt{k^2P^2}).
\end{equation}

To normalize the BSA, we apply the Nakanish normalization condition~\cite{Nakanishi1965PRBI,Nakanishi1965PRBII}
\begin{equation}\label{Nakanishi Normalization}
\left(\frac{\partial \ln(\lambda)}{\partial P^2}\right)^{-1} = tr\int_k^{\Lambda}\bar{\Gamma}(k;-P)S(k_+)\Gamma(k;P)S(k_-),
\end{equation}
where $\bar{\Gamma}(k;-P)$ is the charge conjugation of $\Gamma(k;-P)$ as defined in Eq.~\eqref{CTrans}. With the normalized BSAs, the leptonic decay constants of pseudoscalar and vector mesons are defined as
\begin{eqnarray}
f_{0^-}P_{\mu} &=&\frac{Z_2}{\sqrt{2}} tr\int i\gamma_5\gamma_{\mu}S_f(k_+)\Gamma_{0^-}(k,P)S_g(k_-)\,,\label{fDecayPS}\\
f_{1^-} M  &=&\frac{Z_2}{3\sqrt{2}} tr\int \gamma_{\mu}S_f(k_+)\Gamma_{1^-}^{\mu}(k,P)S_g(k_-) \,. \label{fDecayVC}
\end{eqnarray}

\subsection{The Scattering Kernel}\label{SecIIC}

Now the scattering kernel is the only unknown piece of the whole puzzle. To construct it, we recall the axial-vector Ward-Takahashi identity (AV-WTI)~\cite{Maris1998PLB,Maris1997PRC} derived from the QCD chiral transformation, which reads
\begin{align}\label{AV-WTI}
P_{\mu}\Gamma_{5\mu}^{fg}(k;P)+i(m_{f}&+m_{g})\Gamma_{5}^{fg}(k;P)\notag\\
                                      & =S_{f}^{-1}(k_{+})i\gamma_{5}+i\gamma_{5}S_{g}^{-1}(k_{-})\,,
\end{align}
where $\Gamma_{5\mu}^{fg}(k;P)$ and $\Gamma_{5}^{fg}(k;P)$ are the axial-vector and pseudoscalar vertex, respectively, with two quark flavors denoted by $f$ and $g$. Note that the identity is model-independent and scheme-independent.

In the chiral limit, the current mass term vanishes. Inserting the Lorentz structure of the pion BSA, i.e.,
\begin{align}
\Gamma_{\pi}(k;P)=&\gamma_5[ i E_\pi(k;P)+\slashed{P} F_\pi(k;P)\notag\\
            &+\slashed{k}(k\cdot P)G_\pi(k;P)+\sigma_{\mu\nu}k_{\mu}P_{\nu}H_\pi(k;P)],
\end{align}
into the AV-WTI in Eq.~\eqref{AV-WTI}, we can obtain
\begin{equation}\label{AV-WTI-1}
f_{\pi}E_{\pi}(k;0)=B(k^2)\,,
\end{equation}
where $B(k^{2})$ is the scalar function defined in Eq.~\eqref{quark_lorentz2}.
It is apparent that the two-body problem is related to the one-body one and reveals that the pion can exist if and only if the chiral symmetry is dynamically broken.

The kernel with the one-gluon exchange form, which is called the ``ladder" approximation~\cite{Williams1994PPNP,Munczek1995PRD,Bender1996PLB}, can be expressed as ($l=k-q$)
\begin{equation}\label{Kernel}
K_{tu}^{rs}(q,k;P) = -\mathcal{G}(l^2) l^2 D_{\mu\nu}^{\text{free}}(l)\left(\gamma_{\mu}\frac{\lambda^a}{2}\right)_{tr}\left(\gamma_{\nu}\frac{\lambda^a}{2}\right)_{su}\,,
\end{equation}
where $t,u,r,s$ denote the Dirac and color indices. With the RL approximation, i.e., Eqs.~\eqref{InteractionModel} and \eqref{Kernel}, the solved quark propagator and vertices satisfy the identity in Eq.~\eqref{AV-WTI}. In other words, the RL approximation is a symmetry-preserving truncation scheme.

In order to construct a kernel beyond the ladder approximation, we first insert the gap equation and the inhomogeneous BSE into the AV-WTI, and then obtain
\begin{align}\label{Kernel_b2}
&\int_q K(q,k;P)[S_f(q_+)\gamma_5 + \gamma_5S_g(q_-)] \notag\\
                   & = -\int_q \gamma_{\mu} [D^f_{\mu\nu}(l)S_f(q_+)\gamma_5 + D^g_{\mu\nu}(l)\gamma_5S_g(q_-)]\gamma_{\nu}\,,
\end{align}
where the rainbow approximation has been used and all indices are suppressed for simplicity. Since the quark propagators with different flavors may have different effective interaction strengths, we use $D^{f}_{\mu\nu}$ and $D^{g}_{\mu\nu}$ for clarity.

Inspired by Eq.~\eqref{Kernel_b2}, we propose the BSE kernel as (the color matrices are suppressed)
\begin{align}\label{KernelModify}
K=-\frac{1}{2}\gamma_\mu\otimes\Sigma\cdot D_{\mu\nu}^{\Sigma}\cdot\gamma_\nu -\frac{1}{2}\gamma_\nu\cdot D_{\mu\nu}^{\Sigma}\cdot\Sigma\otimes\gamma_\mu,
\end{align}
with
\begin{align}\label{KernelModifyElem}
D_{\mu\nu}^{\Sigma}&=[D_{\mu\nu}^f(l) S_f(q_+) + D_{\mu\nu}^g(l) S_g(q_-)]\gamma_5, \notag\\
\Sigma &=\gamma_5[S_f(q_+) + S_g(q_-)]^{-1}.
\end{align}
The kernel can be illustrated by the Feynmann diagram as shown in Fig.~\ref{fig: KernelModifyFeyn}. If the two quarks have the same flavor, i.e., $D^{f} = D^{g}$, then the kernel degenerates into Eq.~\eqref{Kernel} and thus can be considered as a parameter-free generalization of the ladder approximation.
 \begin{figure}
 \centering
 \includegraphics[scale=0.5]{BeyondLadder.eps}
 \caption{The Feynmann diagram for the kernel in Eq.~\eqref{KernelModify}. The black blobs denotes the elements in Eq.~\eqref{KernelModifyElem}.}
 \label{fig: KernelModifyFeyn}
 \end{figure}

Straightforward analysis suggests that the kernel actually expresses an average of the effective interactions of the two flavors. For example, in the ultraviolet limit, i.e., $q \to \infty$, we obtain
\begin{equation}
K \rightarrow -\gamma\otimes\gamma\left(\frac{D^f+D^g}{2}\right),
\end{equation}
which is obviously a simple average of $D^{f}$ and $D^{g}$.
As another example, in the infrared limit, i.e., $q \to 0$, we have the kernel for $P \sim 0$ as
\begin{equation}
K \rightarrow -\gamma\otimes\gamma\left(\frac{D^f\sigma^f_{s}(0)+D^g\sigma^g_{s}(0)}
{\sigma^f_{s}(0)+ \sigma^g_{s}(0)}\right),
\end{equation}
which is also an average of $D^{f}$ and $D^{g}$ but weighted by the corresponding quark dressing functions.

\section{Results and Discussions}\label{SecIII}

\subsection{Light Quark Mesons}
Following Ref. \cite{Qin2018PRD}, for the mesons consisted of light quarks $u,d,s$, we take the interaction parameters as
\begin{equation}
\omega_q=0.5~\text{GeV}, \quad \varsigma_q=0.8~\text{GeV}.
\end{equation}
In the isospin symmetric limit: $m_{u} = m_{d}$, fitting with the underlined data in Table~\ref{table:mesonmassPS}, we take
\begin{equation}
m_{u/d}^{\zeta_{19}}=3.3~\text{MeV},\quad m_s^{\zeta_{19}}=74.6~\text{MeV},
\end{equation}
with which the masses and leptonic decay constants of light mesons are computed and listed in Tables~\ref{table:mesonmassPS} and ~\ref{table:mesonmassVC}. The renormalization-group-invariant masses are $\hat{m}_{u/d}=6.3$~MeV, $\hat{m}_s=146$~MeV. The one-loop-evolved masses at $2$~GeV are $m_{u/d}^{2\text{GeV}}=4.8~\text{MeV}$, $ m_s^{2\text{GeV}}=110~\text{MeV}$. The Euclidean constant quark masses are obtained
\begin{equation}
M_{u/d}^{E} = 0.41~\text{GeV}, \quad  M_s^{E} = 0.57~\text{GeV}\,,
\end{equation}
defined as $M_{q}^{E} =\{k|M_q(k^2)=k\}$, where $M_q(k^2)$ is the quark mass function.

\subsection{Heavy Quark Mesons}
For the heavy quark mesons, we follow the guide aforementioned and take the interaction parameters as
\begin{equation}
\omega_Q=0.8~\text{GeV}, \quad \varsigma_Q=0.6~\text{GeV}\,,
\end{equation}
which shape a much weaker effective interaction compared with that for the light quark mesons.

With the heavy quark current masses (fitted with the underlined data in Table~\ref{table:mesonmassPS})
\begin{equation}
m_{c}^{\zeta_{19}}=0.82~\text{GeV},\quad m_b^{\zeta_{19}}=3.59~\text{GeV},
\end{equation}
we obtain the masses and leptonic decay constants of of heavy quark mesons which are listed in Tables~\ref{table:mesonmassPS} and ~\ref{table:mesonmassVC}. The renormalization-group-invariant masses are $\hat{m}_{c}=1.61$~GeV, $\hat{m}_{b}=7.16$~GeV. The one-loop-evolved masses at $2$~GeV are $m_{c}^{2\text{GeV}}=1.22~\text{GeV}$, $m_b^{2\text{GeV}}=5.41~\text{GeV}$. The Euclidean constant quark masses are
\begin{equation}
M_{c}^{E} = 1.32~\text{GeV}, \quad M_b^{E} = 4.22~\text{GeV}.
\end{equation}

\subsection{Heavy-light Mesons}\label{SecIIIB}

With the parameters fixed in previous subsections, we can directly compute the properties of heavy-light mesons since the kernel beyond the ladder approximation, i.e., Eq. \eqref{KernelModify}, does not introduce any new parameters. In this work, we mainly focus on the mass spectrum and the leptonic decay constants.

\subsubsection{Mass Spectra}

\begin{figure}
 \centering
 \includegraphics[scale=0.33]{fg-PS1-ver2.eps}
 \caption{Masses of pseudoscalar mesons calculated herein compared with those from the equal spacing rule herein (ESR herein), the lQCD and the experiments. The vertical blue dashed line is used to separate the light-light and heavy-heavy mesons (in the left region) with the heavy-light mesons (in the right region). The detailed numbers are list in Table~\ref{table:mesonmassPS}.}
 \label{fig:mesonmassPS}
 \end{figure}
 \begin{figure}
 \centering
 \includegraphics[scale=0.35]{fg-VC1-ver2.eps}
 \caption{Masses of vector mesons calculated herein compared with those from the ESR herein, the lQCD and the experiments. The vertical blue dashed line is used to separate the light-light and heavy-heavy mesons (in the left region) with the heavy-light mesons (in the right region). The detailed numbers are list in Table~\ref{table:mesonmassVC}.}
 \label{fig:mesonmassVC}
 \end{figure}

\begin{table}
 \centering
\caption{Masses of pseudoscalar mesons calculated herein compared with those from the lQCD and experimental data. The ESR results are also presented for comparison. The lQCD results are taken from: $M_{B_c}$ - Ref.~\cite{Mathur2018PRL}; $M_D$ and $M_{D_s^{\pm}}$ - Ref.~\cite{Cichy2016PRD}; $M_B$ and $M_{B_s^{\pm}}$ - Ref.~\cite{Dowdall2012PRD}. The experimental data are from Ref.~\cite{PDG2018}. The underlined data are the experimental value used to fit the interaction parameters.}
 \label{table:mesonmassPS}
 \setlength{\tabcolsep}{2.5mm}{\begin{tabular}{c|c|c|c|c}
 \hline
 Meson        &Herein                &ESR Herein    &lQCD           &Exp.           \\ \hline
 $\pi$        &$\underline{0.138}$   &$0.138$       &$/$            &$0.138(1)$     \\
 $K$          &$\underline{0.495}$   &$0.416$       &$/$            &$0.495(1)$     \\
 $\eta_c$     &$\underline{2.984}$   &$2.984$       &$/$            &$2.984(1)$     \\
 $\eta_b$     &$\underline{9.399}$   &$9.399$       &$/$            &$9.399(1)$     \\
 $B_c$        &$6.388$               &$6.192$       &$6.276(7)$     &$6.275(1)$     \\
 $D$          &$1.771$               &$1.561$       &$1.868(3)$     &$1.868(1)$     \\
 $D_s^{\pm}$  &$1.981$               &$1.839$       &$1.968(4)$     &$1.968(1)$     \\
 $B$          &$5.324$               &$4.769$       &$5.283(8)$     &$5.279(1)$     \\
 $B_s^{\pm}$  &$5.478$               &$5.047$       &$5.366(8)$     &$5.367(1)$     \\ \hline
 \end{tabular}}
 \end{table}

\begin{table}
 \centering
 \caption{Masses of vector mesons calculated herein compared with those from the ESR herein, the lQCD and the experiment. The lQCD results are taken from: $M_{\rho}$ - Ref.~\cite{Fu2016PRD}; $M_{K^*}$ - Ref.~\cite{Dudek2014PRL}; $M_{\phi}$ - Ref.~\cite{Donald2014PRD}; $M_{J/\Psi}$ - Ref.~\cite{Donald2012PRD}; $M_{B_c}$ - Ref.~\cite{Mathur2018PRL}; $M_{D^*}$, $M_{D_s^{\pm*}}$, $M_{B^*}$ and $M_{B_s^{\pm*}}$ - Ref.~\cite{Lubicz2017PRD}. The experimental data are from Ref.~\cite{PDG2018}.}
 \label{table:mesonmassVC}
 \setlength{\tabcolsep}{2.5mm}{\begin{tabular}{c|c|c|c|c}
 \hline
 Meson        &Herein     &ESR Herein      &lQCD           &Exp.         \\ \hline
 $\rho$      &$0.749$    &$0.749$         &$0.780(16)$    &$0.775(1)$   \\
 $K^*$       &$0.953$    &$0.919$         &$0.993(1)$     &$0.896(1)$   \\
 $\phi$      &$1.089$    &$1.089$         &$1.032(16)$    &$1.019(1)$   \\
 $J/\Psi$    &$3.122$    &$3.122$         &$3.098(3)$     &$3.097(1)$   \\
 $\Upsilon$  &$9.497$    &$9.497$         &$/$            &$9.460(1)$   \\
 $B_c^*$     &$6.542$    &$6.310$         &$6.331(7)$     &$/$          \\
 $D^*$       &$1.988$    &$1.936$         &$2.013(14)$    &$2.009(1)$   \\
 $D_s^{\pm*}$&$2.206$    &$2.106$         &$2.116(11)$    &$2.112(1)$   \\
 $B^*$       &$5.501$    &$5.123$         &$5.321(8)$     &$5.325(1)$   \\
 $B_s^{\pm*}$&$5.635$    &$5.293$         &$5.412(6)$     &$5.415(2)$   \\ \hline
 \end{tabular}}
 \end{table}

We first compute the masses of the pseudoscalar and vector mesons. The comparisons with experimental data and lattice-QCD results are plotted in Fig.~\ref{fig:mesonmassPS} and Fig.~\ref{fig:mesonmassVC}.
The concrete data are summarized in Table~\ref{table:mesonmassPS} and Table~\ref{table:mesonmassVC}.
Among them, $\pi$, $K$, $\eta_c$, $\eta_b$ are taken as the calibration to determine the parameters.
Except the heavy-light mesons, i.e., $D$, $D^{\pm}_s$, $B$ and $B^{\pm}_s$, the kernels used for others are actually the ladder approximation because the interactions of the two flavors are the same in Eq.~\eqref{KernelModify}.

Following Ref.~\cite{Qin2018PRD}, we also include the results using the equal spacing rule (ESR). The key of the ESR is to define a constituent-quark passive-mass via
\begin{equation}
	M_X^f = \frac{1}{2}m_{X_{f\bar{f}}}\,,
\end{equation}
where $m_{X_{f\bar{f}}}$ denotes the mass of the $f$-flavor-symmetric meson in the channel $X$. The computed values (in GeV) are:
\begin{equation}
\label{EqMfPS}
{\rm PS:}\quad
\begin{array}{l|cccc}
f & u=d & s & c & b \\\hline
M_X^f & 0.07 & 0.35 & 1.49 & 4.70
\end{array}\,,
\end{equation}
and
\begin{equation}
\label{EqMfVC}
{\rm VC:}\quad
\begin{array}{l|cccc}
f & u=d & s & c & b \\\hline
M_X^f & 0.37 & 0.54 & 1.56 & 4.75
\end{array}\,,
\end{equation}
where PS, VC stands for the pseudo-scalar, vector mesons, respectively.
Then, the masses of flavor-unsymmetric mesons can be obtained by a straightforward interpolation as
\begin{eqnarray}
	m_{X_{f\bar{g}}} = M_X^f + M_X^g\,.
\end{eqnarray}

It is found that the masses of all pseudoscalar and vector mesons can be well reproduced with the BSE beyond the ladder approximation. The relative errors between the results herein and the experimental data are at the level of 5\%. For the $B_c^*$ meson, as the lack of the experimental result, we compare the result herein with the lattice-QCD and obtain the relative error only 4\%. On the other hand, the masses obtained with the ESR are comparable with the experimental data and the lattice-QCD, and the corresponding errors are just slightly larger than those with the BSE. Thus, the ESR can be an approximation for meson mass spectra.

As mentioned above, the kernel, i.e., Eq. \eqref{KernelModify}, actually expresses the averaging interaction of the light and heavy quarks. To demonstrate how the average works, we can define the effective arithmetic average as
\begin{equation}
K = -\gamma\otimes\gamma\left[\eta D^q+ (1-\eta) D^Q\right],
\end{equation}
where $\eta \in [0,1]$ is the weight parameter. For $\eta=1$, the kernel is the same as the one for flavor-symmetric light quarks. In this case, the interaction strength of the kernel is quite large and the binding between the light and heavy quarks are too strong, accordingly. The resulting meson mass is smaller than the experimental data. On the other hand, for $\eta=0$, the kernel is the same as the one for flavor-symmetric heavy quarks. In this case, the interaction strength of the kernel is rather small and the binding between the light and heavy quarks are too weak, accordingly. The resulting meson mass is heavier than the experimental data.

 \begin{figure}
 \centering
 \includegraphics[scale=0.4]{fg-eta.eps}
 \caption{Masses of $D$ meson with the weight parameter of the arithmetic average of the kernel, where $\eta^*$ denotes the value corresponding to the experimental mass.}
 \label{fig:D_eta}
 \end{figure}

Now we can take $D$ meson as an example to analyze the mass with the running of the weight parameter. The result is shown in Fig.~\ref{fig:D_eta}. It is found that the weight parameter $\eta^*$ corresponding to the experimental mass indeed lies between the two limits, i.e., $0 < \eta^* < 1$. Thus, the kernel beyond the ladder approximation, i.e., Eq.~\eqref{KernelModify}, which produces the experimental mass, has the interaction strength equivalent to that weighted by $\eta^*$. However, Eq.~\eqref{KernelModify} is inspired by the AV-WTI and has no additional parameters. In other words, it expresses an implicit and automatic average. Moreover, the well-produced full mass spectra suggest that this kind of average is generally appropriate.

\subsubsection{Leptonic Decay Constants}

We then compute the leptonic decay constants of the pseudoscalar and the vector mesons. The obtained results are presented in Table~\ref{table:fDecayPS}, Table~\ref{table:fDecayVC}, respectively.
The comparisons with experimental data and lattice-QCD results are plotted in Fig.~\ref{fig:fDecayPS} and Fig.~\ref{fig:fDecayVC}.
Following Ref.~\cite{Qin2018PRD}, we also include the leptonic decay constants using the ESR,
which is defined as
\begin{eqnarray}
	f_{qQ} \approx f^{\rm interp}_{qQ} := \frac{1}{2} \left( f_{qq} + f_{QQ} \right) \,.
\end{eqnarray}

The first observation is that the leptonic decay constants are not as stable as the masses and the discrepancies from different sources become sizable for some mesons.
For the pseudoscalar mesons of light, strange, and charm flavors (i.e., $\pi$, $K$, $\eta_c$, $D$, and $D_s^\pm$), the theoretical methods produce similar values which are comparable with the experimental data. However the bottom-flavored pseudoscalar mesons (i.e., $\eta_b$, $B_c$, $B$, and $B_s^\pm$) lack of experimental data and have very different theoretical results. The values produced with the BSE and the ESR are larger than lattice-QCD ones.
Especially, for $B$ and $B_{s}^{\pm}$, we have $f_{\rm ESR} > f_{\rm BSE} > f_{\rm lQCD}$.
The reason why the ESR and the BSE produce large decay constants is that the interaction strength for the bottom sector is too strong since it is the same as that for the charm sector. This can be understood from the truth that $f_{\eta_b}$ obtained by the ESR and the BSE is significantly larger than that by lattice-QCD. If tuning $f_{\eta_b}$ down by decreasing the interaction strength for the bottom sector, the values of the BSE move close to the lattice-QCD results. However, the ESR still gives very large $f_{B}$ and $f_{B_s^\pm}$. This means that the ESR always fails for extremely flavor-unsymmetric mesons.

For vector mesons, the situation is analogous and the differences between the BSE/ESR and lattice-QCD results are generally similar to those for pseudoscalar mesons.
However, for the flavor-unsymmetric vector mesons, i.e., $D^*$ and $B^*$, the differences are significant. Note that $f_\Upsilon$ obtained by the BSE/ESR is larger than that by lattice-QCD just as $f_{\eta_b}$.
This is also a signature for that the interaction strength of the BSE/ESR is stronger than that of lattice QCD. If decreasing the interaction strength for the bottom sector as aforementioned, $f_\Upsilon$ decreases, and so does $f_{B^*}$. However, the differences between the BSE/ESR and lattice-QCD results cannot be removed completely and remain noticeable. The reason is that the kernel in Eq.~\eqref{KernelModify} only considers the AV-WTI which does not constrain the vector channel directly. Therefore, incorporating the axial-vector and vector WTIs to develop a universal BSE kernel for all channels could be the solution.

 \begin{table}
 \centering
 \caption{Decay constants of pseudoscalar mesons calculated herein compared with those from the ESR herein, the lQCD and the experiments. The lQCD results are taken from: $f_{\pi}$ and $f_K$ - Ref.~\cite{Follana2008PRL}; $f_{\eta_c}$, $f_{\eta_b}$ and $f_{B_c}$ - Ref.~\cite{McNeile2012PRD}; $D$, $D_s^{\pm}$, $B$, $B_s^{\pm}$ - Ref.~\cite{Bazavov2018PRD}. The experimental data are from Ref.~\cite{PDG2018,Edwards2001PRL}.
 }
 \label{table:fDecayPS}
 \setlength{\tabcolsep}{2.5mm}{\begin{tabular}{c|c|c|c|c}
 \hline
 Meson       &Herein        &ESR Herein       &lQCD          &Exp. \\ \hline
 $\pi$       &$0.095$       &$0.095$          &$0.093(1)$    &$0.092(1)$ \\
 $K$         &$0.113$       &$0.115$          &$0.111(1)$    &$0.110(1)$ \\
 $\eta_c$    &$0.277$       &$0.277$          &$0.278(2)$    &$0.237(52)$\\
 $\eta_b$    &$0.559$       &$0.559$          &$0.472(5)$    &$/$  \\
 $B_c$       &$0.429$       &$0.418$          &$0.307(10)$   &$/$  \\
 $D$         &$0.169$       &$0.186$          &$0.150(1)$    &$0.144(4)$ \\
 $D_s^{\pm}$ &$0.212$       &$0.206$          &$0.177(1)$    &$0.182(3)$  \\
 $B$         &$0.212$       &$0.327$          &$0.134(1)$    &$0.133(18)$  \\
 $B_s^{\pm}$ &$0.248$       &$0.347$          &$0.163(1)$    &$/$  \\ \hline
 \end{tabular}}
 \end{table}

 \begin{table}
 \centering
 \caption{Decay constants of vector mesons calculated herein compared with those from the ESR herein,
 the lQCD and the experiments. The lQCD results are taken from: $f_{\phi}$ - Ref.~\cite{Donald2014PRD}; $f_{J/\Psi}$ - Ref.~\cite{Donald2012PRD}; $f_{\Upsilon}$ - Ref.~\cite{Colquhoun2015PRD91.074514}; $f_{B_c^*}$ - Ref.~\cite{Colquhoun2015PRD91.114509}; $f_{D^*}$, $f_{D_s^{\pm*}}$, $f_{B^*}$, $f_{B_s^{\pm*}}$ - Ref.~\cite{Lubicz2017PRD}. The experimental data are from Ref.~\cite{PDG2018,Maris1999PRC}.}
 \label{table:fDecayVC}
 \setlength{\tabcolsep}{2.5mm}{\begin{tabular}{c|c|c|c|c}
 \hline
 Meson        &Herein         &ESR Herein          &lQCD              &Exp.   \\ \hline
 $\rho$       &$0.149$        &$0.149$             &$/$               &$0.153(1)$ \\
 $K^*$        &$0.179$        &$0.170$             &$/$               &$0.159(1)$ \\
 $\phi$       &$0.190$        &$0.190$             &$0.170(13)$       &$0.168(1)$ \\
 $J/\Psi$     &$0.296$        &$0.296$             &$0.286(4)$        &$0.294(5)$ \\
 $\Upsilon$   &$0.526$        &$0.526$             &$0.459(22)$       &$0.505(4)$ \\
 $B_c^*$      &$0.483$        &$0.411$             &$0.298(9)$        &$/$    \\
 $D^*$        &$0.199$        &$0.222$             &$0.158(6)$        &$/$    \\
 $D_s^{\pm*}$ &$0.256$        &$0.243$             &$0.190(5)$        &$/$    \\
 $B^*$        &$0.246$        &$0.338$             &$0.131(5)$        &$/$    \\
 $B_s^{\pm*}$ &$0.283$        &$0.358$             &$0.158(4)$        &$/$    \\ \hline
 \end{tabular}}
 \end{table}

 \begin{figure}
 \centering
 \includegraphics[scale=0.33]{fg-PS2-ver2.eps}
 \caption{Decay constants of pseudoscalar mesons calculated herein compared with those from the ESR herein, the lQCD and the experiments. The vertical blue dashed line is used to separate the light-light and heavy-heavy mesons (in the left region) with the heavy-light mesons (in the right region). The detailed numbers are list in Table~\ref{table:fDecayPS}.}
 \label{fig:fDecayPS}
 \end{figure}

 \begin{figure}
 \centering
 \includegraphics[scale=0.35]{fg-VC2-ver2.eps}
 \caption{Decay constants of vector mesons calculated herein compared with those from the ESR herein, the lQCD and the experiments. The vertical blue dashed line is used to separate the light-light and heavy-heavy mesons (in the left region) with the heavy-light mesons (in the right region). The detailed numbers are listed in Table~\ref{table:fDecayVC}.}
 \label{fig:fDecayVC}
 \end{figure}

 \section{Summary}\label{SecIV}
In summary, the heavy-light mesons in pseudoscalar and vector channels are investigated within the Dyson-Schwinger equation framework. The rainbow-ladder truncation is modified under the consideration of the symmetries of QCD. It combines the effects of the light and heavy sectors. Moreover, the modified kernel involves no new parameter and can degenerate into the RL truncation for flavor-symmetric heavy-heavy or light-light mesons.

With the kernel beyond the ladder approximation, the calculated masses of the heavy-light mesons in pseudoscalar and vector channels agree with the experimental data at the level of $5\%$. The masses obtained with the ESR are comparable with the experimental data and the lattice-QCD results while the corresponding errors are slightly larger than those with the BSE. From our analysis, the modified kernel expresses an implicit and automatic average with no additional averaging parameter. These facts shows that the modified kernel is appropriate to describe the mass spectrum of the ground states mesons.

For the leptonic decay constants, our calculated results are not as stable as those for the masses. For the pseudoscalar mesons of light, strange and charm flavors, our theoretical methods produce similar values that are comparable with the experimental data. While for the bottom-flavored mesons, both the ESR and the BSE produce larger values than the lattice-QCD. The reason is that the interaction strength for the bottom sector is too strong since it is the same as that for the charm sector. For vector mesons, the differences between the BSE/ESR and lattice-QCD results are generally analogous but larger than those for pseudoscalar mesons. Besides the interaction strength, the missing constraint from the vector Ward-Takahashi identity is a factor to explain the larger differences.

From the discussion above, we can infer that the kernel beyond the ladder approximation is a promising tool to depict the quark-antiquark interactions. The mass spectra and the leptonic decay constants of the light-light, heavy-heavy and heavy-light mesons in the pseudoscalar and vector channels can be well described, systematically. The modified kernel can be further improved by incorporating the axial-vector and vector Ward-Takahashi identities together and used to study meson and baryon properties. The related work is under progress.\\

\begin{acknowledgments}
The authors are grateful for the fruitful discussion with C.D. Roberts. The work was supported by the National Natural Science Foundation of China under Contracts No. 11435001, No. 11775041, No.11847301, No. 11805024, and the National Key Basic Research Program of China under Contract No. 2015CB856900.
This work was also supported in part by the Fundamental Research Funds for the Central Universities under Grant No. 2019CDJDWL0005.
\end{acknowledgments}

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