text:
\section{Introduction}

Empirical mode decomposition (EMD) \cite{Huang1998} is a versatile data-driven method that quantitatively decomposes real-world signals, whereby the original signal is modeled as a linear combination of amplitude-modulated and frequency-modulated (AM-FM) functions, called intrinsic mode functions (IMFs) \cite{Sharpley2006}. Together with the Hilbert transform \cite{Tolwinski2007}, the IMFs yield meaningful instantaneous frequency estimates that give sharp identifications of imbedded structures. The complete process is the so-called Hilbert-Huang Transform (HHT) \cite{Huang2005}.
With no prior assumption on the nature of the data, EMD and its noise-assisted variants, such as ensemble EMD (EEMD) \cite{Wu2009} and complete ensemble EMD with adaptive noise (CEEMDAN) \cite{Torres2011},
have been put to use in numerous applications in diverse fields, such as in biomedical engineering \cite{Gallix2012}, functional neuroimaging \cite{Navarro2012}, image enhancement \cite{Celebi2012} or fault diagnosis of mechanical systems \cite{Lei2011}. However, these extensions are limited in applicability to
one-dimensional time series, which was a major incentive for the numerous variations of the standard EMD algorithm.

Recent advances in physics and engineering
have brought new problems to light which deal with complexity, uncertainty, non-linearity and multi-channel dynamics \cite{UmairBinAltaf2007}. Over the past decade, EMD has been successfully extended to be capable of handling multi-variate, multi-dimensional signal decomposition. These kinds of extensions can be collectively called multi-EMD algorithms. For multi-dimensional signals, extracting local extrema to estimate the envelopes (which is the key issue in EMD algorithms) is not as straightforward as for one-dimensional signals, because the complex and quaternion fields are not ordered \cite{Mandic2009}. The first attempts at a multi-variate EMD \cite{Tanaka2007, Rilling2007, UrRehman2010} were restricted to bivariate and trivariate cases, and were later extended to n-variate signals by means of the Multi-Variate EMD (MVEMD) \cite{Rehman2010}. These extensions use various projection techniques to help capture the envelopes, where the data channels are projected onto numerous direction vectors. On the other hand, the primitive multi-dimensional capabilities of EMD were also restricted to two-dimensional data (images) \cite{Rilling2007}. Pseudo-bidimensional EMD (pseudo BEMD) \cite{Wu2009a}, the first approach, performed one-dimensional EMD on successive slices of 2D data, but generated poor IMF components due to the fact that it ignored the correlation among the rows and/or columns of a 2D image. More appropriate bidimensional EMD algorithms, such as bidimensional EMD (BEMD) \cite{Nunes2003} and image EMD (IEMD) \cite{Linderhed2009}, employed more complicated interpolation methods to estimate the envelopes, such as radial basis functions and thin plate splines rather than the cubic spline interpolation of the standard EMD.

However, in multi-EMD it is generally required to find local extrema to interpolate these points in each iteration of the process. These kinds of interpolation schemes have to face the following issues:

\textbf{Dimensional expansion:} In a standard one-dimensional EMD, the IMFs must satisfy two criteria: (1) the number of extrema and the number of zero-crossings must either be equal or differ at most by one; and (2) the mean value of the envelope defined by the local maxima and local minima must be zero. The situation is more difficult in the case of a multi-EMD, as parameters required for defining an IMF, such as zero-crossings, are not so well defined in more than one dimension.

\textbf{Computational load:}
Morphological operations \cite{Huang1996}, such as projection techniques in multi-variate EMD algorithms or interpolation methods in multi-dimensional EMD algorithms, render the envelope identification process complicated and time consuming, as it requires finding an interpolation surface on high dimensional data space during each iteration. The computational load of such algorithms is prohibitive for large-scale and complex scenarios.

A few EMD extensions, focusing on optimizing the current poor envelope determination across expansive domains, have been proposed to overcome the large computational load requirements. Fast and adaptive BEMD (FABEMD) \cite{Bhuiyan2010} employed order statistic filters (OSFs) to get the upper and lower envelopes in the decomposition process instead of using surface interpolation. Green's function in tension BEMD (GiT-BEMD) \cite{Al-Baddai2016} introduced the Green's function as a surface interpolation technique, which in addition needed very few iterations to estimate each IMF. Most of these modifications have been specifically designed for bidimensional signals. However, general fast-speed n-variate or multi-EMD algorithms are still in their infancy, and further efforts are needed to speed up the signal decomposition.

In this study, we propose a simple and novel signal serialization technique,
through which we can concatenate multi-variate or multi-dimensional signals into a one-dimensional signal. This paves the way for performing multi-signal decomposition through one-dimensional EMD algorithms. Most favourably, this concatenation method can obviously reduce the computational time without altering the existing EMD algorithms. Furthermore, it can provide a new perspective to optimize current EMD algorithms, that is, transforming the structure of input signals instead of developing new envelope identification techniques or new multi-EMD algorithms. Thus, the proposed fast serial-EMD method can be a good alternative, providing efficient signal analysis and mode decomposition.

The rest of the paper is organized as follows: before introducing the concept of serial-EMD, the standard EMD and its variants will be briefly outlined in Section \ref{sec_02}. The description of the proposed serial-EMD algorithm will be discussed in Section \ref{sec_03}. Results and discussion on various EMD algorithms with simulated and real-world datasets will be given in Section \ref{sec_04}. Finally, a conclusion will be drawn in Section \ref{sec_05}.
\section{The classical EMD and its variants}
\label{sec_02}

The EMD and its variants can break down a signal into a finite sum of components. These components form a complete and nearly orthogonal basis for the original signal, which can be described in terms of IMFs.

\subsection{Standard one-dimensional EMD}

For a real-valued signal $x(k)$, the one-dimensional standard EMD \cite{Huang1998} finds a set of $N$ IMFs $\{c_i(k)\}^N_{i=1}$, and a residue signal $r(k)$, so that

\begin{equation}
    x(k) = \sum_{i=1}^N c_i(k) + r(k)
\end{equation}

IMFs $c_i(k)$ are defined so as to have symmetric upper and lower envelopes, with the number of zero crossings and the number of extrema differing at most by one. The standard one-dimensional EMD algorithm can be described as follows:

\begin{enumerate}[Step 1.]
    \item Find the locations of all the extrema of $x(k)$.
    \item Interpolate between extrema to obtain the lower (upper) signal envelope $e_{min}(k)$ ($e_{max}(k)$).
    \item Compute the local mean $m(k) = [e_{min}(k) + e_{max}(k)] / 2$.
    \item Compute the IMF candidate $s(k) = x(k) - m(k)$.
    \item If $s(k)$ satisfies some predefined stopping criterion, then define $c_i(k) = s(k)$ as an IMF, otherwise set $x(k) = s(k)$ and repeat the process from Step 1.
\end{enumerate}

The refinement process (steps 2 to 5), which is needed to extract every mode, requires a certain number of iterations and is named as the sifting process.

\subsection{Noise-assisted EMD}

One of the major drawbacks of the standard one-dimensional EMD is the frequent appearance of mode mixing \cite{Hu2012}, which is defined as either a single IMF consisting of signals of widely disparate scales, or a signal of a similar scale residing in different IMF components. The ensemble EMD (EEMD) \cite{Wu2009} was proposed to counteract the problem; it performs the decomposition over an ensemble of noisy copies of the original signal and obtains the final results by averaging. The EEMD algorithm can be described as follows:

\begin{enumerate}[Step 1.]
    \item Generate $x^{(m)} = x + \beta w^{(m)}$, where $w^{(m)}$ $(m=1,...,M)$ are different realizations of white noise.
    \item Decompose each $x^{(m)}$ $(m=1,...,M)$ completely by EMD, obtaining its modes ${c}_i^{(m)}$, where $i=1,...N$ indicates the modes.
    \item Assign $\overline{c_i}$ as the $i$th mode of $x$, obtained by averaging the corresponding modes.
\end{enumerate}

It should be noticed that in EEMD, every $x^{(m)}$ is decomposed independently from different realizations and for every one of them a residue $r_i(k) = x(k) - c_i(k)$ is obtained at each stage. The extraction of every $x^{(m)}$ $(m=1,...,M)$ requires a different number of sifting iterations. This situation can cause different realizations of signal plus noise, might produce different number of modes and residual noise might be involved in the modes.
Taking these drawbacks into account, the complete ensemble EMD with adaptive noise (CEEMDAN) \cite{Torres2011} was proposed. The general idea is the following:

\begin{enumerate}[Step 1.]
    \item Compute the first residue $r_1 = x - \widetilde{c_1}$, where $\widetilde{c_1}$ is the first CEEMDAN mode, namely, the first mode obtained by EEMD.

    \item Obtain each first mode of $r_1 + \beta_1 E_1 (w^{(m)})$ by EMD, $m=1,...,M$, and compute the second CEEMDAN mode $\widetilde{c_2}$ by averaging these first modes.

    \item For $i=2,...,N$, calculate the $i$th residue $r_i = r_{(i-1)} - \widetilde{c_i}$.
    \item Obtain each first mode of $r_i + \beta_i E_i (w^{(m)})$ by EMD, $m=1,...,M$, and compute the $i$th CEEMDAN mode $\widetilde{c_2}$ by averaging these first modes

\end{enumerate}

\subsection{Multivariate EMD}

The general approach for $n$-variate signals, multivariate EMD (MVEMD) \cite{Rehman2010}, has the ability to extract common modes across the signal components, making it suitable for the fusion of information from multiple source. The MVEMD algorithm can be summarized as follows:

\begin{enumerate}[Step 1.]
    \item Choose a suitable point set for sampling on an $(n-1)$ sphere.
    \item Calculate a projection, denoted by $\{p^{\theta_j}(k)\}_{k=1}^K$, of the input signal $\{\mathbf{v}(k)\}_{k=1}^K$ along the direction vector $\mathbf{x}^{\theta_j}$, for all $j$ (the whole set of direction vectors), giving $\{p^{\theta_j}(k)\}_{j=1}^J$ as the set of projections.
    \item Find the time instants $\{t_i^{\theta_j}\}$ corresponding to the maxima of the set of projected signals $\{p^{\theta_j}(k)\}_{j=1}^J$.
    \item Interpolate $[t_i^{\theta_j}, \mathbf{v}(t_i^{\theta_j})]$ to obtain multivariate envelope curves $\{\mathbf{e}^{\theta_j}(k)\}_{j=1}^J$.
    \item For a set of $J$ direction vectors, calculate the mean $\mathbf{m}(k)$ of the envelope curves by averaging $\mathbf{e}^{\theta_j}(k)$.
    \item Extract the detail $d(k)$ using $d(k)=x(k)-m(k)$. If the detail $d(k)$ fulfills the stoppage criterion for a multivariate IMF, apply the above procedure to $x(k)-d(k)$, otherwise apply it to $d(k)$.
\end{enumerate}

\subsection{Multidimensional EMD}

The bidimensional EMD (BEMD) \cite{Nunes2003} is one of the most famous two-dimensional EMD algorithms on 2D image processing, and its variants \cite{Damerval2005, Nunes2009, Liu2005} with various interpolation methods have been thoroughly studied. A typical sifting process for a general BEMD can be described as follows:

\begin{enumerate}[Step 1.]
    \item Set $r(x, y) = X(x, y)$. Identify all local maxima and local minima of $r$.
    \item Interpolate the local minima (maxima) to obtain the lower (upper) envelope surface $e_{min}(x, y)$ ($e_{max}(x, y)$).
    \item Calculate the mean of the envelope surface $m(x, y)$ and subtract it from $r(x, y)$ to update $r(x, y)$.
    \item Repeat Step 1 to 3 until the stopping criterion is met.
\end{enumerate}

The state-of-the-art BEMD variant, Green's function in tension BEMD (GiT-BEMD) \cite{Al-Baddai2016}, which replaces the direct spline interpolation step by an interpolation based on Green's function \cite{Wessel1998}, turned out to be very efficient in practical applications such as image analysis \cite{Al-Baddai2020, Yang2016}. Interpolation using Green's function implies that the points of the interpolating envelope surface can be expressed as

\begin{equation}\label{eq:green_function}
    s(\mathbf{x}_u) = \sum_{n=1}^N w_n \phi(\mathbf{x}_u, \mathbf{x}_n)
\end{equation}

where $\mathbf{x}_u$ denotes any point where the surface is unknown, $\mathbf{x}_n$ presents the $n$th data constraint, $\phi(\mathbf{x}_u, \mathbf{x}_n)$ is the Green's function and $w_n$ is the respective weight in the envelope representation. The GiT-BEMD can be summarized in three steps:

\begin{enumerate}[Step 1.]
    \item Calculate the local extrema $\mathbf{s}(\mathbf{x}_n)$ by using an 8-connected neighbourhood strategy, and estimate the weights $\mathbf{w}$ using eq. \ref{eq:green_function} with locations $\mathbf{x}_n = {[x_n, y_n]}^T$.
    \item

    Using the interpolation eq. \ref{eq:green_function} for each of the known points, a linear system with $N$ equations is obtained. Then, solve for the weights $\mathbf{w}=\mathbf{G^{-1}c}$.
    \item Using the weights $\mathbf{w}$, the value $s(\mathbf{x}_u) \equiv c_u$ of the envelope surface can be estimated at any point $\mathbf{x}_u$ by solving eq. \ref{eq:green_function}, where the vector $\mathbf{\phi} = {[\phi(\mathbf{x}_u, \mathbf{x}_1), \phi(\mathbf{x}_u, \mathbf{x}_2) ... \phi(\mathbf{x}_u, \mathbf{x}_N)]}^T$ contains the Green's function values of all distances between the $N$ data constraints and the considered location.
\end{enumerate}

Various complicated envelope estimation techniques during each sifting iteration can be used to characterize these EMD extensions. The poor performance and huge difficulty when interpolating points across expansive domains are the primary reason for the heavy computational burden of multi-EMD algorithms. In this paper, the mentioned EMD algorithms, EMD, EEMD, CEEMDAN, MVEMD and GiT-BEMD will be used for performance comparison with the proposed serial-EMD.
\section{The proposed serial-EMD algorithm}
\label{sec_03}

With the aim of reducing the computational load during the sifting process for multivariate or multi-dimensional EMD algorithms, a novel serial-EMD approach is proposed here based on a serialization technique.
To start with, note that a multi-dimensional signal can be treated as being composed by multiple vectors (one-dimensional signals). Then, using a transition, which is calculated by using some part of the data from each dimension, the multi-dimensional signal can be serialized. After that, the serialized one-dimensional signal is decomposed by a standard one-dimensional EMD or its variants and IMFs for serialized signal are calculated. Finally, after removing the redundant transitions, the IMFs of the original multi-signals are extracted from the IMFs from the serialized signal by reshaping and slicing. The details of the signal concatenation and IMF deconcatenation techniques are discussed in the following subsection.

\subsection{Serial concatenation of two signals}

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{02}
    \caption{(a) The concatenation of signals $f(t)$ and $g(t)$ by inserting a transition signal $h(t)$ computed through eq. (\ref{eq:transition}). The concatenated signal maintains the continuity and characteristics of the original signals. The only parameter we need to choose is the length of transition $D$. (b) Details of the concatenation algorithm when applied to multiple signals arranged as columns of matrix $\mathbf{X}$. The final result is stored in matrix $\mathbf{T}$.}
    \label{fig:C2}
\end{figure}

Given two continuous functions $f(t)$ and $g(t)$ with $t\in [0,T]$, let us first consider the problem of how to concatenate those signals avoiding a discontinuity in the joint, i.e. assuming that in general $f(T)\neq g(0)$.

We propose to define a transition function $h(t)$ of length $D<T$, with $t\in [0,D]$, which is to be inserted in between $f(t)$ and $g(t)$ such that the resulting function $s(t)$, with $t\in [0,2T+D]$, is continuous. Specifically, $s(t)$ is defined as the serial concatenation of $f(t)$, $h(t)$ and $g(t)$ as follows:

\begin{equation}
    s(t) =  \left\{
    \begin{array}{ll}
        f(t)     & t\in[0,T]      \\
        h(t-T)   & t\in[T,T+D]    \\
        g(t-T-D) & t\in[T+D,2T+D]
    \end{array}
    \right.
\end{equation}

A simple way of obtaining such a transition function $h(t)$ is the following:

\begin{equation}\label{eq:transition}
    h(t) = \left(1 - \frac{t}{D}\right)f(T-t) + \frac{t}{D}g(D-t),
\end{equation}

\noindent with $t\in[0,D]$. It is easy to show that by using this transition function $h(t)$ the continuity of the resulting function $s(t)$ is guaranteed because $h(0)=f(T)$ and $h(D)=g(0)$. It is noted that $f(T-t)$ and $g(D-t)$, with $t\in[0,D]$, correspond to the flipped versions of the tail of $f(t)$ and the beginning part of $g(t)$ with lengths $D$. Figure \ref{fig:C2}(a) shows the effect of the proposed continuous concatenation of two signals.

\subsection{The fast multiple serial concatenation algorithm}

Let's consider $N$ signals of length $M$ organized as columns $\in \mathbf{R}^M$ $\mathbf{x}_i$, ($i=1, \dots, N$) of matrix $\mathbf{X}$ of size $ M \times N $ as follows:

\begin{equation}
    \mathbf{X}_{(M \times N)}=\begin{bmatrix} \mathbf{x}_1 & \mathbf{x}_2 & \cdots & \mathbf{x}_N
    \end{bmatrix}
\end{equation}

Let $D$ be the number of points we add between the signals to define the transitions in their concatenation. Then, consider the vector $\mathbf{a}$ with elements $a_i=i/(D+1)$, $i=1, \dots, D$, of evenly spaced values between 0 and 1, but with both values not included. In order to compute the set of filling points, we define the sub-matrices of $\mathbf{X}$, $\mathbf{X_A}$ and $\mathbf{X_B}$, both of size $D \times N-1$ in terms of the indices of $\mathbf{X}$ as follows (see Figure \ref{fig:C2}(b)-left):

\begin{equation}
    \mathbf{X_A}_{(D \times N-1)}=\mathbf{X}_{1:D \mathbf{,} 2:N}
\end{equation}

\begin{equation}
    \mathbf{X_B}_{(D \times N-1)}=\mathbf{X}_{M-D+1:M \mathbf{,} 1:N-1}
\end{equation}

Then we compute the signal extensions in a matrix $\mathbf{E}$ as follows:

\begin{equation}
    \mathbf{E}_{(D \times N-1)} =
    \mathbf{X_A}^f \odot \mathbf{a} \mathbf{u}^T + \mathbf{X_B}^f \odot \mathbf{a}^f \mathbf{u}^T
\end{equation}

\noindent where $\mathbf{u}=(1,...,1)$ of dimension $(N-1)$. The symbol $\odot$
stands for the the Hadamard product. The super-index $T$ means transposition and the super index $f$ means the flip up-to-down permutation, which flipped the order of the elements upside down along the first dimension. Note that the latter operation is obtained from pre-multiplying the matrix/vector affected by the standard involutory permutation, also known as backward identity or exchange matrix. The $f$ operation, however, can be done by appropriately addressing the elements, thus avoiding multiplications.

 By using the zero-vector $\mathbf{z}$ of dimension $D$, we build the matrix $\mathbf{T}$ which contains the transitions introduced. Matrix $\mathbf{T}$ is defined in blocks as (see Figure \ref{fig:C2}(b)-right):

\begin{equation}
    \mathbf{T}_{(M+D \times N)} = \begin{bmatrix}[l]
        \mathbf{X}_{(M \times N)}                      \\
        \mathbf{E}_{(D \times N-1)} & \mathbf{z}_{(D)}
    \end{bmatrix}
\end{equation}

The vectorization of the matrix $\mathbf{T}$ provides the concatenated signals
through the added portions of the transitions.

\begin{equation}
    \mathbf{t}_{((M+D) \cdot N)} = \operatorname{vec} ({\mathbf{T}})
\end{equation}

Finally, we eliminate the $\mathbf{z}$ vector
introduced in $\mathbf{T}$ to obtain the useful samples in $\mathbf{x}$.

\begin{equation}
    \mathbf{x}_{(MN+DN-D)} = \mathbf{t}_{1:(M+D) \cdot N-D}
\end{equation}

Figure \ref{fig:C2} shows graphically an example of the concatenation of two consecutive signals, $\mathbf{x}_i$ and $\mathbf{x}_{i+1}$, through the $D$ filling samples $\mathbf{x_E}_{i}$ in terms of the elements involved in their formation. Note that using the vector $\mathbf{a}$, previously defined, and the $i$ column vectors $\mathbf{x_A}_i$, $\mathbf{x_B}_i$ of matrices $\mathbf{X_A}$, $\mathbf{X_B}$ respectively, the padding samples of column vector $\mathbf{x_E}_{i}$ of $\mathbf{X_E}$ can be written as: $\mathbf{x_E}_i = \mathbf{x_A}_i^f \odot \mathbf{a} + \mathbf{x_B}_i^f \odot \mathbf{a}^f$.

\subsection{The fast IMFs deconcatenation algorithm}

Let $F(\cdot)$ be any one-dimensional EMD algorithm or its variants, for the concatenated signals $\mathbf{x}$ without $D$ zeros attached. Its resulting IMFs $\mathbf{R}$ can be computed as follows:

\begin{equation}
    \mathbf{R}_{(MN+DN-D \times K)} = F(\mathbf{x}_{(MN+DN-D)})
\end{equation}

\noindent where $K$ is the number of IMFs. According to the definition of the EMD algorithm, the length of each IMF should be the same as the length of the signal to be decomposed. Thus, the transition part $\mathbf{E}$ in the concatenated signal $\mathbf{x}$ is also decomposed and included in the resulting IMFs $\mathbf{R}$.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{03}
    \caption{Detail of the deconcatenation algorithm applied to the resulting matrix $\mathbf{R}$ computed through EMD and its variants. The resulting matrix $\mathbf{R}$ can be filled with $D$ zeros for each column and can be reshaped to exclude the transitions. Notice that the matrix $\mathbf{Z}$ in the matrix $\mathbf{T}$ becomes the last column in the transitions after the reshaping operation. The pure IMFs of signals are stored in the matrix $\mathbf{IMF}$.}
    \label{fig:C3}
\end{figure}

In order to get the pure IMFs of every single signal, we build the matrix $\mathbf{T}$ to exclude the transitions of IMFs from the resulting IMFs $\mathbf{R}$, by using the matrix $\mathbf{Z}$ of size $D \times K$ of all-zero elements. Matrix $\mathbf{T}$ is defined in blocs as (see Figure \ref{fig:C3}-left):

\begin{equation}
    \mathbf{T}_{(MN+DN \times K)} = \begin{bmatrix}[l]
        \mathbf{R}_{(MN+DN-D \times K)} \\
        \mathbf{Z}_{(D \times K)}
    \end{bmatrix}
\end{equation}

Finally, matrix $\mathbf{T}$ can be reshaped into a new $(M+D) \times N \times K$ shape (see Figure \ref{fig:C3}-right). The IMFs of transitions in the concatenated signals are stored in the last $D$ columns after the reshaping operation. Thus, we can eliminate the $D \times N \times K$ transitions part introduced in $\mathbf{T}$ to get the pure IMFs.

\begin{equation}
    \mathbf{IMF}_{(M \times N \times K)} = reshape(\mathbf{T}, [M, N, K])
\end{equation}

The graphical description of the deconcatenation of IMFs can be found in Figure \ref{fig:C3}. Note that for the $i$th signal, its $j$th IMF can be written as: $\mathbf{imf}_{ij}=\mathbf{IMF}_{1:M \mathbf{,} i \mathbf{,} j}$.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=1.0\textwidth]{01.png}
    \caption{The description of the serial-EMD algorithm for 2D signals. The original multi-dimensional signals are concatenated by adding transitions. The IMFs for original multi-dimensional signals are extracted by removing the transitions.}
    \label{fig:C1}
\end{figure}

In summary, Figure \ref{fig:C1} explicitly outlines the steps involved in the serial-EMD algorithm. The original multi-signals are concatenated by using part of the information at the head and tail of each dimension or variate, and the IMFs of concatenated signal are deconcatenated by removing the transitions.
\section{Experimental results}
\label{sec_04}

A simulation dataset, as well as a real-world face image database, have been used to test and validate the proposed approach in order to show some of its potential applications. Several approaches are discussed: MVEMD, GiT-BEMD, standard one-dimensional EMD, EEMD, CEEMDAN and their corresponding proposed serial-versions. For the standard one-dimensional EMD and its variants, the input data was treated as a collection of one-dimensional slices, which enabled them to decompose the multi-signals. For the serial-EMD and its variants, on the other hand, such an operation is not necessary, as the input data has been naturally serialized to one dimension. The quality of IMFs and the time of decomposition are the two main performance indicators that have been used to evaluate these algorithms.

In the following subsections, we will detail the results under several situations (artificial multi-variate signals, artificial multi-dimensional images and face images). Both standard-version and serial-version EMD, EEMD and CEEMDAN were applied to these three situations, while MVEMD was only applied on artificial multi-variate signals and GiT-BEMD was only applied on artificial images and face datasets. For noise-assisted EMD algorithms, such as EEMD, CEEMDAN and GiT-BEMD, the noise standard deviation (NSTD), number of realizations (NR), maximum number of sifting iterations (MaxIter) and other hyper-parameters
are set as the values recommended in their respective original papers\cite{Wu2009, Torres2011, Al-Baddai2016}. All the experiments have been carried out on the same laptop (Intel(R) i7-6700HQ @ 2.6GHz).

\subsection{Artificial multi-variate signals}

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{04.pdf}
    \caption{Synthetic multi-variate signals (U, V, W, X, Y, Z) with multiple frequency modes (32Hz, 16Hz, 8Hz, 2Hz). }
    \label{fig:C4}
\end{figure}

Similarly to MVEMD \cite{Rehman2010}, synthetic multi-variate time series with multiple frequency modes were considered. Each variate, shown in the top row of Figure \ref{fig:C4} (denoted by U, V, W, X, Y and Z), was constructed from a set of four sinusoids ($f_1=32Hz, f_2=16Hz, f_3=8Hz, f_4=2Hz$). One sinusoid was made common to all the components, whereas the remaining three sinusoidal components were combined so that the resulting signal had a common frequency mode ($f_3=8Hz$). The \emph{pick-up mask} eq. \ref{eq:pickup} is introduced to indicate which sinusoidal components will be included in each variate.

\begin{equation}\label{eq:pickup}
    mask = \begin{bmatrix}[l]
        1 & 1 & 0 & 1 & 0 & 0 \\
        1 & 1 & 1 & 0 & 1 & 0 \\
        1 & 1 & 1 & 1 & 1 & 1 \\
        1 & 0 & 1 & 1 & 0 & 1 \\
    \end{bmatrix}
\end{equation}

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{05.pdf}
    \caption{Decomposition results of one of the synthetic multi-variate signals (U) exhibiting multiple frequency modes (with $f_1=32Hz, f_2=16Hz, f_3=8Hz, f_4=2Hz$) via the proposed approach applied to other EMD algorithms. The IMFs extracted using serial-version algorithms (green) are similar to the IMFs extracted using standard-version algorithms (blue). Note that the subsequent IMFs (IMF5, IMF6...) are omitted since the amplitude and frequency of these IMF components are small enough to be ignored.}
    \label{fig:C5}
\end{figure}

The results for the IMFs for all algorithms are presented in Fig. \ref{fig:C5}. It can be observed that in terms of the amplitude and frequency of the IMFs, the resulting IMFs generated by serial-version algorithms are similar to the IMFs decomposed by standard-version algorithms, which indicates that the serialization of data has little effect on the standard EMD itself. Compared with standard-MVEMD, the amplitude and frequency of the IMFs extracted by standard-version algorithms and by serial-version algorithms are more inconsistent, which is known as the mode mixing problem. However, this is due to the defects of the EMD algorithm itself, not because of the introduction of serialization methods.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{06.pdf}
    \caption{Comparison among various EMD algorithms for the artificial multi-variate signals (interval parameter, $D=50$) discussed in this paper in terms of the total time required. For each algorithm, the decomposition was repeatedly carried out 10 times. The black circles represent the outliers.}
    \label{fig:C6}
\end{figure}

Next,
the computational load of these algorithms is investigated for synthetic multi-variate signals, as presented in Fig. \ref{fig:C6}.
In this figure,
the bar ranges from Q1 (the first quartile) to Q3 (the third quartile) of the execution time distribution, the width represents the IQR (inter-quartile range), the median value is indicated by a red line across the bar and the outliers are shown in black circles. Compared with standard-MVEMD, the serial-version methods (EMD, EEMD, CEEMDAN) reduce the execution time by 99\%, 74\% and 26\%, respectively. Furthermore, the execution time of serial-CEEMDAN ($t=1154.97ms, std=46.32$), the most costly amongst the three serial-version algorithms, is less than that of standard-MVEMD ($t=1562.09ms, std=58.46$). For MVEMD, with the increase in the number of signals, the computational complexity shows
a geometrically increasing trend, which makes it difficult to apply it on n-variate signals \cite{Tanaka2007}. For serial-EMD algorithms, the computational complexity can be reduced significantly, because it simplify the calculation of the mean envelope with the serialized one-dimensional signal.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{07.pdf}
    \caption{The components of IMFs obtained using serial-EMD with different interval parameters from $D=1$ (the lowest value for the interval parameter) to $D=500$ (half of the length of signals) are displayed. Only the first two IMFs are shown in this figure.}
    \label{fig:C7}
\end{figure}

Then, the effect of the interval values on the IMFs decomposed by different algorithms are compared. Fig. \ref{fig:C7} presents the results for several interval parameters $D$ from 1 (the lowest value for the interval) to 500 (half of the length of the signal). In the beginning (from $D=1$ to $D=50$), the larger the interval value is, the more similar the two ends of the IMF are to the middle part in terms of amplitude and frequency (for example, IMF2 in Fig. \ref{fig:C7}), which indicates that the transition part can be used to avoid the discontinuity in concatenated signals. However, for a larger interval (from $D=100$ to $D=500$), it is hard to distinguish between these IMFs,
which means that there is no need to set a large interval between the signals.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{08.pdf}
    \caption{Comparison among various interval parameters from $D=1$ (the lowest value for the interval parameter) to $D=500$ (half of the length of signals) for serial-version algorithms in terms of the total time required. For each situation, the decomposition was repeatedly executed 10 times. The black circles represent the outliers.}
    \label{fig:C8}
\end{figure}

In addition, the effects of different interval values on the computational load of serial-EMD algorithms are also explored, as displayed in Fig. \ref {fig:C8}. From the beginning (from $D=1$ to $D=50$), the computational load decreases gradually and reaches its lowest value at $D=50$ (EMD: $t=10.96ms, std=0.92$, EEMD: $t=397.92ms, std=16.38$, CEEMDAN: $t=1154.97ms, std=46.32$). Increasing the interval value improves the continuity in concatenated signals, which speeds up the sifting process. However, for a larger interval (from $D=100$ to $D=500$), the computational load increases gradually, which indicates that the effect of the transition part reaches its bottleneck since the time cost is more related to the length of the signal.

In order to trade off between these two aspects, the hyper-parameter interval value $D$ should be set to 20\% of the length of the original signals. This value is also used in subsequent experiments.

\subsection{Artificial images}

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{09.pdf}
    \caption{Left: Component results from a superposition of a horizontal ATI and a vertical ATI. Right: The original artificial texture image (ATI) produced by summing up the three ATC images.}
    \label{fig:C9}
\end{figure}

Similarly to MVEMD \cite{Al-Baddai2016}, an artificial texture image (ATI) of size $101 \times 101$ pixels is considered, which is composed of a superposition of artificial texture components (ATCs) of the same size. The ATCs represent harmonically varying spatial oscillations (sinusoids) with horizontal and vertical spatial frequencies ($f1=20$, $f2=4$, $f3=1$) and unit amplitudes, which are shown in Fig. \ref{fig:C9} on the left. The first ATC contains the highest spatial frequency, whereas the last ATC contains the lowest spatial frequency. These artificial textures provide a good performance indicator of the algorithm, even though some imperfections cannot be avoided. The ATI, the summation of these three ATCs, is shown in Fig. \ref{fig:C9} right.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{10.pdf}
    \caption{Decomposition of the ATI using standard-version algorithms and serial-version algorithms (interval parameter, $D=20$). Note that the subsequent IMFs (IMF4, IMF5...) are omitted since the amplitude and frequency of these IMF components are small enough to be ignored.}
    \label{fig:10}
\end{figure}

Firstly, the IMFs decomposed using each algorithm are investigated. Based on the former result of the artificial multi-variate signals, the optimal value of the interval parameter should be set at 20\% of the length of the original signal. Thus, considering that the size of the ATC is $101 \times 101$, the interval parameter should be set as 20. The resulting IMFs are displayed in Fig. \ref{fig:10}. Similarly to the previous experiment, the resulting IMFs generated by serial-version algorithms are similar to the IMFs decomposed by standard-version algorithms. For standard-version and serial-version algorithms, one single ATC can be decomposed into two or more IMFs (for example, some components of the second ATC appear in IMF2, IMF3 and IMF4 for the serial-EEMD algorithm),
which are caused by the artifacts introduced during the sifting process. Note that even though the IMFs decomposed using the EMD, EEMD and CEEMDAN algorithms contain some striated texture
compared with those decomposed using GiT-BEMD, the serial-version algorithms keep the same characteristics of the original ones.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{11.pdf}
    \caption{Comparison among various EMD algorithms for the ATI discussed in this paper in terms of the total time required. For each situation, the decomposition was repeatedly executed 10 times. The black circles represent the outliers.}
    \label{fig:C11}
\end{figure}

Subsequently, the computational loads of theses algorithms for ATI decomposition are explored, as presented in Fig. \ref{fig:C11}. When compared with standard GiT-BEMD, the serial-version methods (EMD, EEMD, CEEMDAN) reduce the execution time by 99\%, 87\% and 60\%, respectively. Moreover, the execution time of serial-EEMD ($t=6.41s, std=0.04$) is 1.5 times faster than that of standard-EEMD ($t=10.15s, std=0.69$). In addition, the execution time of serial-CEEMDAN ($t=20.58s, std=0.09$), the most costly amongst the three serial-version algorithms, is less than that of standard GiT-BEMD ($t=51.59s, std=0.19$). All these results clearly show that the serial-version algorithms are better than the standard-version algorithms in terms of computational load.

\subsection{Face database}

To demonstrate the validity of the serialization method, real-world face images as the ones used with GiT-BEMD in \cite{Al-Baddai2020} were considered. The face database (AT \& T database) contains ten different images for each of the forty subjects in the database, which represents a total of 400 different images. All images were taken against a dark background, with the subjects in a frontal facial pose and with a varying orientation of the faces. All images are in a gray scale of 256 values, with an original size of $92 \times 112$ pixels.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{12.pdf}
    \caption{Top:
    Original images of different people from an AT \& T face database. Bottom: Corresponding noisy images with \emph{Speckle} noises at $SNR=-6$ dB.}
    \label{fig:C12}
\end{figure}

The authors from \cite{Al-Baddai2020} reported that the worst result of the face recognition system was in the case when face images were corrupted by \emph{Speckle} noise with $SNR=-6$ dB. The classification system used IMFs extracted by GiT-BEMD.
The \emph{Speckle} noise adds multiplicative noise to the image elements $I(x,y)$ in the form $I(x,y)=I(x,y)+I(x,y)\hat{A} \cdot N(x,y)$, where $N(x,y)$ is a uniformly distributed zero-mean random noise with standard deviation $\sigma$, as provided in eq. \ref{eq:std_theta}. The presence of \emph{Speckle} noise in the images clearly has an effect on the edges and fine details which limit the resolution contrast. The original images and their corresponding noisy images are shown in Fig. \ref{fig:C12}. Note that only images with $SNR=-6$ dB \emph{Speckle} noise are used in the subsequent experiments.

\begin{equation}\label{eq:std_theta}
    f_U(n)=\left\{
    \begin{array}{rcl}
        1 / \sqrt{2\sigma} &  & for \quad n \in [-\sigma/\sqrt{2}, \sigma/\sqrt{2}] \\
        0                  &  & otherwise                                           \\
    \end{array}
    \right.
\end{equation}

In Fig. \ref{fig:C13},
the resulting IMFs of the same noisy image of a face are presented when decomposed using both GiT-BEMD and serial-version algorithms. Considering that the size of the image is $92 \times 112$, the interval parameter should be set to 20. Similarly to the results for GiT-BEMD, the noise can also be separated from the original image using all the serial-version algorithms, and it is mostly captured by the first two IMFs. After removing the first few IMFs and summing up the remaining ones, the relatively clear images are presented in the summation column.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{13.pdf}
    \caption{Examples of noisy images ($SNR=-6$ dB, \emph{Speckle} noise) and their corresponding IMFs through GiT-BEMD and serial-version algorithms. Note that the subsequent IMFs (IMF5, IMF6, ..., IMF12) are not shown in this figure. Summation column: summing up all the IMFs excluding the first two.

    }
    \label{fig:C13}
\end{figure}

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{14.pdf}
    \caption{Left: Comparison between various algorithms for the processing of one single image in terms of the average time required. We calculated the average computational load per image. The black circles represent the outliers. Right: Comparison between various algorithms for the processing of the whole AT \& T database in terms of the total time required.}
    \label{fig:C14}
\end{figure}

The computational load of these algorithms for face images with \emph{Speckle} noise is also investigated, as displayed in Fig. \ref{fig:C14}. For all the serial-version algorithms, the average computational load per image (EMD: $t=0.024s, std=0.004$, EEMD: $t=0.578s, std=0.033$, CEEMDAN: $t=1.599ms, std=0.077$) is much less than that of GiT-BEMD ($t=11.86s, std=2.15$), as shown in Fig. \ref{fig:C14} (left). On top of that, the total execution times for the processing of the whole AT \& T database with noise are also presented in Fig. \ref{fig:C14} (right), which shows that the serial-version algorithms are greatly superior to GiT-BEMD when processing large amounts of data in terms of the execution time.

More specifically,
to decompose the face AT \& T database, which contains 400 images, standard GiT-BEMD takes about 90 minutes, while serial-EMD, serial-EEMD, serial-CEEMDAN take about 9 seconds, 4 minutes and 10 minutes instead, which reduces the execution time by 99\%, 95\% 88\%, respectively.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{15.png}
    \caption{Schema of the face recognition experiment. All the IMFs except the first few are summed-up together to feed the classifier. When compared with the original experimental paradigm \cite{Al-Baddai2020}, our paradigm employed the summed-up way to process these IMFs instead of concatenating them.}
    \label{fig:C15}
\end{figure}

To explore
the ability to capture noise of the serial-version algorithms, the similar experimental paradigm as in \cite{Al-Baddai2020} is used, which is slightly different to the original paradigm to reduce the dimensions of the feature space.
For image data with noise, its first few IMFs contain the high-frequency components, and its last few IMFs contain the low-frequency components. Since the \emph{Speckle} noise may be captured in the first few IMFs, a reasonable way to improve the face recognition performance is by eliminating them or applying further filters to these IMFs. The schema of this experiment is shown in Fig. \ref{fig:C15}. Images which have been corrupted by \emph{Speckle} noise at $SNR = -6$dB are first decomposed by serial-version algorithms. Since the first few extracted modes capture almost all of the noise contained in the image, these modes need to be removed. A new data vector is created by using KNN as the classification approach. All the experimental conditions, such as classification strategy, cross-validation strategy and its hyper-parameters, are consistent with those described in paper \cite{Al-Baddai2020}, except for the algorithm used for image decomposition and the number of summed-up IMFs.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{16.pdf}
    \caption{Highest face recognition accuracies for the standard GiT-BEMD and the proposed serial-version approaches by 10-fold cross-validation. Notice that the picked-up IMFs used for classification are different in each method. To achieve the highest accuracy, we summed up the 2nd to 5th IMFs, the 2nd to 7th IMFs, the 3rd to 6th IMFs and the 5th to 8th IMFs, respectively, which correspond to the red rectangle in Fig. \ref{fig:C17}.}
    \label{fig:C16}
\end{figure}

Face recognition performance results for the standard GiT-BEMD and for the proposed serial-version algorithms are presented in Fig. \ref{fig:C16}. In this case, several IMFs are summed up and flattened to reform a new one-dimensional vector to feed the KNN classifier. With 10-fold cross-validation,
it can be seen that when compared with GiT-BEMD (accuracy=91.25\%, std=4.63), serial-EMD (accuracy=92.50\%, std=3.23), serial-EEMD (accuracy=93.24\%, std=3.91) and serial-CEEMDAN (accuracy=93.49\%, std=3.01) can all achieve a higher accuracy.

\begin{table}[ht]
    \caption{The distribution of the number of IMFs decomposed by serial-version algorithms}
    \label{tbl:imf_distribution}
    \begin{center}
        \begin{tabular}{p{100pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}}
            \hline
            number of IMFs & 9 & 10 & 11 & 12 \\
            \hline
            number of trials & 66 & 960 & 172 & 2 \\
            \hline
        \end{tabular}
    \end{center}
\end{table}

For one single image, the number of IMFs decomposed by the three serial-version algorithms (EMD, EEMD, CEEMDAN) may be inconsistent. For different images, the number of IMFs decomposed by one algorithm may also change. Therefore, table \ref{tbl:imf_distribution}
shows the distribution of the number of IMFs obtained when decomposing the 400 images (1200 total trials) using the three serial-version algorithms. Since the image can be decomposed into 10 IMFs in most of the trials, the maximum number of IMFs is set to 10 in subsequent experiments. For those images with less than 10 IMFs, we use zeros to fill the last IMF. If the number of IMFs exceeds 10, then the last few are discarded until only 10 remain.

\begin{figure}[ht]
    \centering{}
    \includegraphics[width=0.8\textwidth]{17.pdf}
    \caption{Accuracy heat map for all possible combinations of summed-up IMFs. The lowest-frequency IMF and the highest-frequency IMF refer to where to start picking up IMFs from and where to end the selection. The black square is a $2\times2$ regions which contains the highest accuracy. The topmost accuracy in each method is indicated with a red square in the sub-figure.

    }
    \label{fig:C17}
\end{figure}

To investigate the power of the new approach in depth, all the experiments were carried out to test all the possible different ranges of IMFs. To do this, we use the highest/lowest-frequency, which are the first and last IMFs of the summed-up range, respectively. For this part, the range of highest-frequency IMF is from 1 to 10, which is the same as that of the lowest-frequency IMF.
For example, when the highest-frequency IMF is set to 2 and the lowest-frequency IMF is set to 6, then all the IMFs from the 2nd IMF to the 6th IMF are selected to be summed up together and are fed to the KNN model.

Fig. \ref{fig:C17} shows the heat maps of the face recognition accuracies for different ranges of IMFs using the serial-version algorithms. In this figure, the darker the red, the higher the accuracy. The zoomed sub-figure of each heat map contains the best accuracy and shows the highest/lowest-frequency IMF corresponding to that highest accuracy.

Firstly, for serial-EMD, it can be observed that the highest accuracy is achieved with the combination of when the highest-frequency IMF is the 2nd IMF and the lowest-frequency is the 7th IMF (highest accuracy = 92.50\%). Likewise, for serial-EEMD (highest accuracy = 93.24\%) and serial-CEEMDAN (highest accuracy = 93.49\%), a similar result to serial-EMD can be observed, but with differently selected highest/lowest-frequency IMFs. When compared with the standard GiT-BEMD (highest accuracy = 91.25\%), all these serial-version methods can achieve a higher accuracy, which means that our approaches are better at separating noise from the synthetic images than GiT-BEMD.

Hence, it can be observed that the middle-range IMFs can achieve a higher accuracy than in other conditions, which shows that the IMFs in the middle range contain the most amount of information of the face images.

If we only use a single IMF to train the classifier, which corresponds to the diagonal in each heat map in Fig. 17, it can be seen that the first IMF achieves the highest accuracy out of any other IMF for all the algorithms. This indicates that the first IMF not only contains noise but also contains information of the original image. However, summing up multiple IMFs can achieve a higher accuracy, and note that in that case, the first IMF is not included in the ideal IMF range, which again may indicate that most of the information is kept in the middle range of the IMFs.

In summary, our proposed method can not only achieve a higher accuracy than GiT-BEMD, but also has a great reduction in computational load in comparison, which shows that our methods are efficient in separating the noise from these synthetic face images and could be used in real-time applications.
\section{Conclusion}
\label{sec_05}

Empirical mode decomposing (EMD) represents a fully unsupervised data-driven technique that allows us to analyze non-linear and non-stationary data sets such as images, EMG signals and fMRI data. EMD encompasses a sifting process for estimating the intrinsic modes into which the data can be decomposed. In most of the existing implementations of EMD,
the sifting process allows for the frequency estimation of the upper and lower envelope surfaces interpolating extreme data points. The computation of these envelope surfaces becomes a problem in existing EMD variants due to its high computational load - the computation time grows exponentially with the data dimension,
which also causes the existing multi-EMD to be unsuitable for higher dimensions.

With the requirement of reducing the computational load during the sifting process for multi-variate or multi-dimensional EMD algorithms, we propose a novel approach, serial-EMD, based on the signal serialization technique. Firstly, the original multi-signals are concatenated using transitions, which are calculated using part of the information at the head and tail of each dimension or variate in the original signal. Then, after vectorization, the serialized one-dimensional signal is decomposed using EMD or its variants and IMFs for serialized signal are calculated. Finally, after removing the redundant transitions, the IMFs of the original multi-signals are extracted from the IMFs of the serialized signal by reshaping and slicing.

We have shown experimental results for both real and artificial signals in terms of the quality of IMFs and the computational load, thus demonstrating that serial-EMD can be applied to any multi-signal and for real-time applications. However, the serial-EMD is a double-edged sword:
On the one hand, this algorithm only changes the shape of the data, so it is easy to combine with the existing EMD algorithms; on the other hand, since this method does not
modify the interpolation step, the resulting IMFs obtained by this algorithm are
more difficult to optimize compared with other existing algorithms.

Altogether, we believe that serial-EMD offers a highly competitive alternative to existing multi-EMD algorithms and represents a promising technique for fast signal analysis.

\section{Acknowledgments}

This work was supported by the National Key R\&D Program of China (No. 2017YFE0129700), the National Natural Science Foundation of China (Key Program) (No. 11932013), the National Natural Science Foundation of China (No. 61673224), the Tianjin Natural Science Foundation for Distinguished Young Scholars (No. 18JCJQJC46100), and the Tianjin Science and Technology Plan Project (No. 18ZXJMTG00260).

redpajama_set_name: RedPajamaArXiv


text:
Inside pictures of Sonam Kapoor & Anand Ahuja's wedding & reception venues
Last updated on - May 4, 2018
Sonam Kapoor and beau Anand Ahuja, who are all set to get married, are again making headlines as the pictures of their wedding and reception ... Read More
Sonam Kapoor and beau Anand Ahuja, who are all set to get married, are again making headlines as the pictures of their wedding and reception venues have gone viral on social media. Let's take a look at the photos...Read Less
/celebs/bollywood/sonam-kapoor-photos/Inside-pictures-of-Sonam-Kapoor-Anand-Ahujas-wedding-reception-venues/eventshow/64027344.cms
Bollywood actress Sonam Kapoor, who is all set to take wedding vows with beau Anand Ahuja, is making waves on social media with her viral pictures of wedding and reception venues. The Kapoors, who made an official announcement about the date, are busy gearing up for the gala wedding. Sonam's house has been decked up with lights and flowers. As the countdown has begun, the wedding cards are also out. The pictures of Sonam and Anand's wedding cards have flooded the internet. Sonam, who will soon be seen in Bollywood movie 'Veere Di Wedding', is all set for her big fat Indian wedding. As the official statement read, "The Kapoor and Ahuja families, take great joy and pride, in announcing the marriage of Sonam and Anand. The wedding will take place on the 8th of May in Mumbai. Since it is an intimate affair, we request you to respect the family's need for privacy. Thank you for all your blessings and love, as we celebrate this special moment in our lives." Sonam Kapoor will enter wedlock at aunt Kavita Singh's sprawling bungalow Rockdale on May 08, 2018. The wedding will be an afternoon affair and the duo will exchange garlands at the temple in the bungalow. Sonam's mehendi ceremony will be held on May 07, 2018, at the banquet hall of her new apartment, Sunteck, Signature Island. Reportedly, this duplex costs a whopping Rs 35 crores. The Kapoors have also planned for a grand wedding reception at The Leela Hotel, Mumbai. As per reports, with around 300 guests on the list, price of the food at the hotel will cost Rs 10 lakhs. Sonam and Anand's wedding is the most-awaited wedding of the year.
/celebs/bollywood/sonam-kapoor-photos/eventshow/64027336.cms
Inside pictures of Sonam Kapoor & Anand Ahuja's wedding & reception venues Pics | Inside pictures of Sonam Kapoor & Anand Ahuja's wedding & reception venues Photos | Inside pictures of Sonam Kapoor & Anand Ahuja's wedding & reception venues Portfolio Pics | Inside pictures of Sonam Kapoor & Anand Ahuja's wedding & reception venues Personal Photos - ETimes Photogallery
Sonam Kapoor's wedding venue. (Photo: Yogen Shah)
See more of : Sonam Kapoor
Sonam Kapoor's wedding venue.
Sonam Kapoor's house
Sonam Kapoor's house.
Anand Ahuja and Sonam Kapoor.
See more of : Sonam Kapoor, Anand Ahuja
Sonam Kapoor, Anand Ahuja

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Interpreting PIC Instruction Words I'm a second year Electrical & Computer Engineering student at the UWI.
I am currently learning the PIC 16F877 Microcontroller and its programming in mainly ASM (with little C) and am confused regarding symbols in its instruction set. Now I understand that instructions may follow varying formats (depending on number of operands, word-size of instruction), however I cannot seem to figure out what the (highlighted) symbols are referring to (yes I understand it's the MSB and LSB of the word, just that I don't know why they're labelled in the way they are, or what they mean)
Section 5 -Table 5-1: Mid-Range MCU Instruction Set, PIC Micro Mid-Range Reference Manual:

A: It's the opcode structure in binary, 14 bits per instruction.
I.e. ADDWF f, d for f=127 and d=1 will be encoded as 00 0111 1111 1111 (07FF opcode in hexadecimal), the "f" and "d" symbols are the arguments of the instruction (for f=48 and d=0 the ADDWF opcode would be 00 0111 0011 0000 = 0730 in hexadecimal).
The "x" in CLRW means probably "anything", the already defined bits are decisively identifying the CLRW instruction and the remaining "x" bits are ignored.
I don't know PIC assembly, so I may be wrong with the "x", but I would be super surprised. Is it this CPU with non-8/16/32 word memory structure, addressing memory by 14 bit words? Makes a bit more difficult to quickly asses memory amount, if you are used to 8 bit bytes and count data/memory according to those.

A: I'm not familiar with that particular document, but I've used the equivalent document for other families of PIC devices. If you look above the table you have highlighted, you should find a section/table describing the meaning of the various shorthand codes used to describe the instruction set (d, f, etc.).
===============
Edit - someone downvoted this answer, so I looked at the document in question. I was wrong: the answer to the question was not above, it was below.
On the page immediately following the one quoted by the OP, in section 5.2, we find figure 5.1 "General Format for Instructions" which explains the meaning of d, f, k, etc.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Python: Subprocess works different to terminal. What I have to change? I have to Python scripts: Tester1.py and Tester2.py.
Within Tester1 I want to start from time to time Tester2.py. I also want to pass Tester2.py some arguments. At the moment my code looks like this:
Tester1:
subprocess.call(['python3 Tester2.py testString'])

Tester2:
def start():
    message = sys.argv[1]
    print(message)

start()

Now my problem: If I run with my terminal Tester2 like 'python3 Tester2.py testString'my console prints out testString. But if I run Tester1 and Tester1 tries to start Tester2, I get an IndexError: "list index out of range".
How do I need to change my code to get everything working?
EDIT:
niemmi told me that I have to change my code to:
subprocess.call(['python3', 'Tester2.py', 'testString'])

but now I get a No such file or directory Error although both scripts are in the same directory. Someone knows why?

A: You need to provide the arguments either as separate elements on a list or as a string:
subprocess.call(['python3', 'Tester2.py', 'testString'])
# or
subprocess.call('python3 Tester2.py testString')

Python documentation has following description:

args is required for all calls and should be a string, or a sequence of program arguments. Providing a sequence of arguments is generally preferred, as it allows the module to take care of any required escaping and quoting of arguments (e.g. to permit spaces in file names). If passing a single string, either shell must be True (see below) or else the string must simply name the program to be executed without specifying any arguments.


text:
Wiebelsbach ist ein geographischer Name:

Orte:
 Wiebelsbach, Stadtteil und Dorf von Groß-Umstadt, Landkreis Darmstadt-Dieburg, Hessen
 Lützel-Wiebelsbach, Ortsteil der Gemeinde Lützelbach im Odenwaldkreis, Hessen

Gewässer:
 Wiebelsbach (Heilbach), rechter Zufluss des Heilbachs (zum Wörther Altrhein) bei Wörth am Rhein, Landkreis Germersheim, Rheinland-Pfalz
 Wiebelsbach (Lützelbach), rechter Zufluss des Lützelbachs (zum Breitenbach, zur Mümling) in Lützel-Wiebelsbach, Gemeinde Lützelbach, Odenwaldkreis, Hessen
 Wiebelsbach (Murg), rechter Zufluss der Nordschwarzwälder Murg (zum Oberrhein) in Ottenau, Stadt Gaggenau, Landkreis Rastatt, Baden-Württemberg
 Wiebelsbach (Ohlebach), auch Pferdsbach, linker Zufluss des dort noch Pferdsbach genannten Ohlebachs vor der Langheinzenmühle, Stadt Groß-Umstadt, Landkreis Darmstadt-Dieburg, Hessen

Siehe auch:
 Wiebelbach

redpajama_set_name: RedPajamaWikipedia


Anyone familiar with Facebook, Twitter and other social media knows that these communication tools can be used to air grudges, disseminate questionable information and hurt feelings. But when do embarrassing claims and potentially damaging statements cross the line, legally, to the level of defamation, slander, libel and illegality?
An international civil case that resolved in March of this year helps illustrate what constitutes defamation conducted via social media.
In 2008, the head of an Australian school's music department left his position for health reasons. He was replaced by veteran instructor Christine Mickle. For reasons that remain unclear, the retired department leader's son made numerous tweets suggesting that Mickle bore responsibility for his father's need to leave the school. Mickle sued for defamation.
The type of communication used (Twitter) had an "evil," "grapevine effect" that allowed lies to spread easily.
The ease of which defamatory statements can spread via "the simple manipulation of mobile phones and computers" may have played a role in the judge's decision.
What Steps Can You Take in the Face of Defamatory Statements?
Everyone has a right to their opinion, but when businesses and careers are hurt by inaccurate and damaging misstatements and overt untruths, victims often have a right to financial compensation. The attorneys of Manos Alwine, PL have extensive commercial litigation experience in the Miami and all south Florida communities. If you need legal help regarding a possible defamation case involving Facebook, Twitter or any other form of print, verbal or electronic communication, contact us for a consultation by calling 305-341-3100.
Yet, the lines can seem blurred between free speech, liability, and the like when there is a possible defamation case at hand. Do potentially defaming statements truly equal defamation or are they perhaps merely offensive? Did damages occur and, if so, what are their value? And how can potential legal issues be resolved?
A recent case covered by the Miami Herald demonstrates that, despite a vigorous defense, potentially defaming statements based on questionable assertions can cause real damage and lead to potentially high-value legal outcomes.
Named only one producer of lean, finely textured beef- that producer was Beef Products Inc.
Following the ABC story, Beef Products Inc.'s sales plummeted, leading to the closing of three production plants, the laying off of 700 workers and reduced profits. In response, Beef Products Inc. sued ABC for $1.2 billion. ABC claimed, in response to the lawsuit, that it "never quoted critics saying the product is unsafe," that the term "pink slime" is not incorrect and that Beef Products Inc. "doesn't get to choose ABC's words." Despite ABC's defense, the judge handling the case refused to throw out the case, stating that ABC wasn't protected against liability by claiming in its news reports that lean, finely textured beef is beef and that it is safe and nutritious. Though the case has not yet concluded, Beef Products Inc. may potentially recover significant compensation for its losses.
As this case illustrates, laws against defamation have the ability to offer real protection. If you or your organization has suffered losses from a damaging statement made by another party, or you suspect that an individual or entity has made defaming statements about you, protect your rights and bottom line by obtaining sound legal advice. The attorneys of Manos Alwine PL in Miami have extensive civil litigation experience, including experience successfully handling defamation claims, and can be contacted at 305-341-3100.
Things just got surreal for the museum that houses the artwork of Salvador Dalí, "the father of surrealism." The Dalí Museum is being sued by consulting company Markalyst for breaching their contract agreement and abandoning an unpaid balance of more than $230,000.
Are oral contracts considered valid? Yes, but a plaintiff must allege offer, acceptance and consideration.
According to the lawsuit, the museum hired the consulting company to develop recommendations for new marketing objectives, strategies and processes, for which the museum originally planned to pay $300,000 over an extended period of time. Payments were to be made in installments due to the museum's "cash position" - a point which was not elaborated in the suit. Though Markalyst completed the work in July 2013, the museum told the company they would not be making any further payments on the project, despite the remaining balance of $233,200. The museum only paid a fraction, just $66,800, of the project's total cost, between January and August of that year.
If you are concerned you might be the victim of a contract breach, call the offices of Manos, Alwine, PL in Miami, Florida at 305-341-3100.
A recent case in South Florida illustrates that, in the case of blatant trademark infringement, legal action can bring both a fair outcome and a high-value result.
For years, booth holders at the Fort Lauderdale-based Swap Shop flea market allegedly served as a source for counterfeit luxury items, including Coach purses, handbags and other items. Earlier this year, Coach responded to the alleged trademark infringement by filing a lawsuit not against alleged sellers and makers of the knock-off goods but against Swap Shop owners Preston and Betty Henn. The Henns, Coach claimed, "willfully turned a blind eye" to vendors illegally selling counterfeit Coach products on their property.
After just two days in court, the Henns capitulated and agreed to a settlement that included a $5.5 million payment to Coach due no later than December 27, 2013. For its legal efforts Coach obtained intellectual property protection, sent a message to other would-be trademark infringers and won a high-value sum that will significantly boost its 2013 bottom line.
The attorneys of the law office of Manos &Alwine, P.L., applaud the outcome of this lawsuit. Coach expends vast sums annually designing, marketing and distributing its coveted products. It deserves compensation from any party complicit in trademark infringement.
If your business is involved in a trademark dispute or other commercial dispute in Miami, Fort Lauderdale or elsewhere in South Florida, contact one of our business dispute attorneys to discuss your concerns. Our team of commercial litigation attorneys has resolved hundreds of trademark and contract disputes involving licensed material, unfair competition, insurance, entertainment law, distribution agreements and more. We represent both plaintiffs and defendants and have achieved numerous six- and seven-figure outcomes both in and out of court. To contact us, call 305-341-3100.
Dunkin' Donuts is a publicly traded company with many locations throughout Florida. The donut shops are typically franchises owned and operated by independent franchisees.
One franchisee has filed a lawsuit against the coffee and donut chain recently claiming that she has suffered racial discrimination by the franchise. This franchise owner happens to be a female of Indian descent. Though many Indian women work for Dunkin' Donuts, she is apparently the only female Indian multi-unit manager in her area.
Her experience is not limited to making coffee and donuts, however. This woman was a former business analyst at Ernst & Young. She currently operates two Dunkin' Donuts restaurants and was hoping to expand.
However, she alleges that Dunkin' Donuts forced her to operate her two stores 24 hours a day, even though her locations are located in high crime areas, posing a serious threat to her personal safety. She also claimed that she did not make enough money in the evening and early morning hours to justify the expense of hiring employees to work around the clock.
Despite these difficulties, the franchisee also hoped to open a third franchise. However, management at Dunkin' Donuts denied her proposal citing the lack of the possibility of a drive-thru window. However, she believes she was discriminated against because Dunkin' Donuts then allowed a male franchisee to open a store in the same location without a drive-thru window.
Now it will be left for a court to decide whether Dunkin' Donuts is violating its contract with this franchisee, or whether it is violating discrimination laws by its actions against her.
Given the vibrant economic climate and recreational opportunities in Miami, there are many luxury hotels to serve visitors to the area. Of course some of them are better appointed and managed than others. But one hotel management company was taken completely by surprise earlier this year when they were cast out by the owner of the hotel for alleged poor performance. After kicking the management company out, the owners installed a new management company to run the hotel.
But the former management company claims that the ouster was a complete surprise and has initiated commercial litigation alleging breach of contract by the owner. They specifically claim that they were never given written notice of any perceived problems nor any opportunity to correct any issues. The former management company also claims that the actions of the owner damaged their reputation.
For its part, the new management company that was brought in to run the hotel does not seem interested in having any part in the dispute. Instead, in public statements they have focused on potential improvements and renovations to the property, including a revamping a restaurant that has up till now only been available for private functions.
The details of this conflict will have to be sorted out through the lawsuit or some process of arbitration. But in many cases, there is narrow distinction between motives when one party to contract accuses the other of failure to perform. It may be that they have a good faith belief that the there has been a failure and that the only path forward is to terminate the relationship. In other cases, the accusations of failure to perform are more perfunctory and while still potentially defensible, are actually a means of negating a contract that they feel no longer best serves their interests.
National Football League wide receiver Randy Moss has made his way around the league and the country. He has used his remarkable athletic ability to make great plays on the field, often out-leaping and outrunning defenders for big plays. But off the field, Moss has not proven to be quite as fleet of foot, running into several obstacles that have not been as easy to overcome as an opposing cornerback.
As you may know, Moss spends the offseason training and working out South Florida. The Miami Herald reported recently that Moss has now filed a lawsuit against the owners of a barber shop in which he invested.
According to a lawsuit Moss has filed in this matter, he invested $160,000 into a barbershop as part of an agreement that was supposed to see Moss' investment repaid in addition to receiving up to one quarter of the revenues from the barbershop.
But Moss contends that the owner of the barbers shop initially tried to repay him with checks that were worthless. Moss says the owner then tried to engage in a shell game in which he changes the name of the legal entity that owned the barbershop in an attempt to avoid paying Moss.
It is great when athletes and other entertainers that have experienced some success decide to diversify their income streams by seeking out opportunities outside of the professional sports and entertainment industries. It remains to be seen what the court decides in thisa matter, but successful athletes and entertainers must tread carefully as there are many individuals who seem them less as business partners and more as meal tickets.
South Florida is a popular area for those with means to come and enjoy the sun and the water. For actors and celebrities such as Eva Longoria and Jennifer Lopez, one of best places to relax in the Miami area was right offshore. The Utopia mega-yacht caters to singers and entertainers who want to spend their time in the Miami area in luxury.
But now, Market America is the defendant in a lawsuit filed by the Miami chef they employed to create fine meals for the celebrities and guests on the luxury boat.
The chef claims that the owners of the boat physically abused him while he was working on the yacht. In the lawsuit, the chef alleges that the owner would become intoxicated and throw kitchen utensils at the chef. Including at least one occasion on which the chef was knocked to the ground by the owner of the boat. The chef claims that he has sustained serious back injuries as a result of the abuse and it is now difficult for him to work.
Market America, the company that owns the boat, not surprisingly disagrees with the chef's version of events. According to a spokesperson for the company, they allege that the chef has a history of dishonesty and frivolous litigation. It should be interesting to see how this case plays out over the next few months.
The famed televangelists behind the Trinity Broadcasting empire are facing a $50 million lawsuit brought by their granddaughter, the company's former chief financial officer. She claims that she was fired after exposing illegal financial decisions. The suit alleges that the couple, Paul and Jan Crouch, misspent company funds on luxury mansions throughout the nation, private jets, and a mobile home for their dogs. A supporting suit has been filed by another unnamed family member.
The husband of the woman bringing the suit was accused of embezzling about $1 million from the Trinity Broadcasting Network by a debt collection agency. The agency later targeted the woman, along with two of her relatives. The woman's father, who is Paul and Jan's oldest son, resigned from the company last year. He had been the vice president and chief of staff. Experts postulate that the man's departure was attributable to the outbreak of legal action.
Despite conflicting reports, the Crouch's legal team said that the case is simple: The woman and her husband who brought the suit stole from TBN. Attorneys for the televangelists said that couple is attempting to divert attention from their crimes in order to escape punishment.
TBN is one of the most popular religious-based entertainment networks, and one of the most profitable, with the company claiming $800 million in assets in 2010. Critics have often wondered how the money is spent, but the televangelists file tax forms separately through a variety of other companies that they own and operate.
TBN has been fraught with scandal before. In 1998, a lead church official paid an accuser more than $400,000 to hide evidence about a homosexual encounter. Federal Communications Commission authorities also investigated the company as part of a five-year legal battle, during which TBN was accused of monopolizing broadcast stations.


• Date: Thursday, December 23, 2004, at 11:50 a.m.
> christmas lovers out there!!!!!
i ythink i can top you for teh love of this movie. i have been JAck for Halloween for several years & even won best costume at some parties & have watched teh movie soo many times i have seen a few goof ups, where the movements are not quite perfect.


Track employees and kids using location services in their iPhone or iPad. After installing this app on their mobile device, you can see their movements on the map in real-time.
IMPORTANT NOTICE: iTAGu Tracker App is explicitly tied to iTAGu GPS tracking service. The system will provide you with the device identification number (ID) which you should use when you launch the app for the first time.
The observer can view all the information about movements online in his personal account – using Web-based service or iTAGu Monitor mobile app. Corporate users can access business features for mobile workforce management: visits scheduling and checking, geo-based events and notifications, analytic reports, etc.
Disclaimer: The app can run in background. Continued usage of location services while the app is in the background can excessively drain battery.


Instead of bombs and rockets. Another provocation from Pyongyang.
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