text:
\section{Introduction}
\label{sec:intro}
Singular spectrum analysis
\cite{Broomhead.King1986, Vautard.etal1992, Elsner.Tsonis1996, Golyandina.etal2001, Ghil.etal2002, Golyandina.Zhigljavsky2012}
is a widely used method of time series analysis and image processing, which is based on singular value decomposition of Hankel and block-Hankel matrices. From the viewpoint of low-rank approximation, singular spectrum analysis is related to the problem of structured low rank approximation
\cite{Markovsky2012}.

SSA is closely related to subspace-based methods, where the signal estimation
is performed  with the help of the signal subspace. There is a lot of subspace-based methods of frequency estimation,
for example, the ESPRIT method \cite{Roy.Kailath1989} and its analogues \cite{Barkhuijsen.etal1987, VanHuffel.etal1994}.
The SVD of Hankel-like matrices is also widely used in system identification and model reduction (see \cite{Markovsky2012} for an overview).

Application of SSA is by far not limited to low-rank approximation and includes the problems of smoothing, trend extraction,
decomposition into a sum of trend, periodic and noise components, and many others. In general, SSA does not need the model given in advance
and can serve as a model-building tool \cite{Golyandina.etal2001,Golyandina.Zhigljavsky2012}.

SSA was originally proposed for time series and then was extended to 2D digital images (2D-SSA) \cite{Golyandina.Usevich2010,Golyandina.etal2013}. Related is the subspace-based 2D-ESPRIT method for estimation of 2D frequencies (see \cite{Rouquette.Najim01Estimation}). 2D-SSA and related subspace based methods find applications in texture analysis \cite{monadjemi04-towards}, seismology \cite{trickett08conf-f}, spatial gene expression data \cite{Holloway.etal2011}, medical imaging \cite{Shin.etal13MRiM-Calibrationless}, etc., and are gaining increasing popularity.

The SSA and 2D-SSA methods deal with series and rectangular images and use intervals and rectangles as moving windows.
This can limit applications of the methods; for example, the methods hardly process  circular-shaped images,
images with gaps, and so on. Due to this reason, in \cite{Golyandina.etal2013} Shaped 2D-SSA
was proposed, which can deal with images of arbitrary shape and can use arbitrary window shapes.

The SSA-based methods are supported by an effective implementation in the freely available \pkg{R}-package \pkg{Rssa} \cite{Korobeynikov.etal2013}. The implementation uses the fact that various SSA versions can be considered as particular cases of Shaped SSA.  This allows a unified implementation for ordinary SSA, multivariate SSA (MSSA),
2D-SSA, SSA for analysis of objects with gaps. The  details can be found in \cite{Golyandina.etal2013, Golyandina.Korobeynikov2013}.

The contribution of this paper is twofold. Firstly, we consider an extension of Shaped SSA
to circular topologies, such as cylindrical and toroidal. This extension is motivated by problems in gene expression data analysis  \cite{Holloway.etal2011}, where the data are obtained by cylindric projections of 3D surfaces. We discuss theoretical properties and effective implementation of this extension.
Secondly, we propose the Shaped ESPRIT method of parameter estimation for shaped images. The paper~\cite{Golyandina.etal2013} does not contain the description of Shaped ESPRIT and we compensate this here.
Several theoretical aspects are discussed. An implementation of both contributions is included in the \pkg{Rssa} package \cite{Korobeynikov.etal2013}.

In Section~\ref{sec:shaped_algo}, we describe the common scheme of SSA methods, which differ mainly in the
embedding step, and introduce the embedding operator for Shaped SSA, which includes both
planar and circular cases.
In Section~\ref{sec:implementation}, we discuss efficient implementation aspects and show that the methods developed for
the planar case are naturally extended to the circular case.
Section~\ref{sec:shaped_rank} contains several propositions about the Shaped-SSA rank of infinite arrays
and the common term of infinite arrays of finite Shaped-SSA rank.
In Section~\ref{sec:shaped_esprit}, we consider a common scheme of ESPRIT-like methods, then
present the algorithm of ESPRIT for the shaped circular case and prove its correctness.
In Section~\ref{sec:examples}, we consider two examples, for demonstration of the shaped version of 2D-ESPRIT and for
demonstration of the circular version of 2D-SSA.
Section~\ref{sec:conclusions} closes the paper with conclusions and further remarks.

\section{Shaped SSA algorithm}
\label{sec:shaped_algo}

\subsection{Structure of SSA-like algorithms}
We will follow the common structure of SSA-like algorithms described in \cite{Golyandina.etal2013}.
This common structure consists of the embedding, decomposition, grouping and reconstruction steps.
The input of an SSA-like algorithm is an object $\mathbb{X}$ to be decomposed (time series, multivariate time series, image, etc.). The result of SSA-like algorithms is a decomposition of $\mathbb{X}$  on
identifiable components in the form $\mathbb{X} = \mathbb{X}_1 + \dots + \mathbb{X}_s$.

\smallskip
\textbf{1. Embedding step}:  construction of the trajectory matrix $\mathbf{X}=\mathcal{T} (\mathbb{X})\in \mathsf{H}$,
where $\mathsf{H}$ is a space of structured  Hankel-like matrices. The structure of the matrix $\mathbf{X}$ (and the space $\mathsf{H}$) depends on the object to be decomposed  and on the \emph{window}, which is the main algorithm parameter.
In a sense, the window size reflects the resolution of the method, i.e., larger windows lead to more detailed decompositions.

\smallskip
\textbf{2. Decomposition step}: Singular Value Decomposition (SVD) of the trajectory matrix $\mathbf{X}=\sum_{i=1}^d \sqrt{\lambda_i} U_i V_i^\top = \sum_{i=1}^d \mathbf{X}_i$. Here $(\sqrt{\lambda_i},U_i,V_i)$ are so called eigentriples (abbreviated as ET) and consist of singular values, left and right singular vectors of $\mathbf{X}$.
Also, $\lambda_i$ and $U_i$ are eigenvalues and eigenvectors of $\mathbf{X} \mathbf{X}^\top$, $\{U_i\}_{i=1}^d$ forms an orthonormal basis of the column space
of $\mathbf{X}$, while $\{V_i\}_{i=1}^d$ forms an orthonormal basis of the row space. Vectors $V_i$ are also called factor vectors.

\smallskip
\textbf{3. Grouping step}:
partition $\{1,\ldots,d\} = \bigsqcup_{j=1}^s I_j$ and grouping of summands in the SVD decomposition to obtain a grouped matrix decomposition $\mathbf{X}=\sum_{j=1}^s \mathbf{X}_{I_j}$, where $\mathbf{X}_I= \sum_{k\in I} \mathbf{X}_k$.
The grouping with  $I_j=\{j\}$ is called elementary. The aim of this step is to group the SVD components to obtain an interpretable decomposition of the initial
object. This can be performed by means of analysis of eigentriples.

\smallskip
\textbf{4. Reconstruction step}: decomposition of the initial object $\mathbb{X} = \mathbb{X}_1 + \ldots + \mathbb{X}_s$,
where $\mathbb{X}_j = \mathcal{T}^{-1} \mathcal{H} (\mathbf{X}_{I_j})$, $\mathcal{H}$ is the operator of projection
on the space $\mathsf{H}$ (e.g., hankelization in the 1D case); $\mathcal{H}(\mathbf{X}_I)=\sum_{i\in I}  \mathcal{H}(\mathbf{X}_i)$ holds.

In this paper, we consider the Shaped SSA algorithm for shaped arrays. In different basic variants of SSA, all the steps except Embedding are the same; therefore, we should describe the embedding step only (although, a variety of modifications of SVD step can be also considered, see e.g. \cite{Shlemov.Golyandina2014}).
The embedding step for Shaped SSA is described in detail in Section~\ref{sec:embedding_shaped}. For simplicity, we consider only the 2-dimensional Shaped SSA. Multidimensional generalizations can be obtained analogously.

\subsection{Comments on the SSA-like algorithms}
For one-dimensional SSA, the theory related to choice of window sizes and grouping can be found
in \cite{Golyandina.etal2001,Golyandina.Zhigljavsky2012}. For multidimensional extensions this theory can be generalized;
however, it is much more complex and not well elaborated.
The main notion is the separability of components of the initial object. The separability theory answers the question
if the SVD expansion at Decomposition step allows the grouping that provides a desirable decomposition of the initial object
(at least, approximately); e.g. decomposition on such identifiable components as trend and oscillations, signal and noise, and so on.
Fortunately, the conditions of approximate separability are not too restrictive and many of series components can be
separated by SSA.

Thus, to perform SSA analysis for component extraction, one should choose window sizes based on the SSA theory
and then choose the set $I$ of eigentriple numbers (or the set of eigenvectors $\{U_i, i \in I\}$).
This set of eigenvectors can provide information about the object parametric model and also
allows the reconstruction of the extracted component, e.g., of a trend, a periodic component or a signal.

The model that underlies the most of the theory of SSA-like methods is associated with objects with rank-deficient trajectory matrices.
This model is related to the objects satisfying linear recurrence relations (e.g., $x_n=\sum_{i=1}^r a_i x_{n-i}$
for one-dimensional time series).

SSA-like methods are called non-parametric or free of model, since they do not use the model in the algorithm, what is very convenient
for the exploratory analysis, and also they allow to analyze the objects that satisfy the model only locally or
do not satisfy it at all; for example, SSA-like methods can be used for smoothing and filtering \cite{Golyandina.Zhigljavsky2012}.

\subsection{Shapes and actions with them}
\label{ssec:shapesandactions}

Formally, we call \emph{shape} $\mathfrak{B}$ a finite non-empty subset of $\{1,\ldots,T_x\}\times \{1,\ldots,T_y\}$,
where  $T_x, T_y \in \mathsf{N} \cup \{\infty\}$, $\mathsf{N}$ is the set of natural numbers.
The values $T_x$ and $T_y$ characterize the topology of the set containing this shape.
Finiteness of $T_x < \infty$ ($T_y < \infty$) means that the topology is circular and
the shapes are cyclic by the $x$-coordinate (the $y$-coordinate) with period $T_x$ ($T_y$).

A\emph{ $\mathfrak{B}$-shaped array }is a partially indexed array $\mathbb{X} = \mathbb{X}_\mathfrak{B} = (x_{(i, j)})_{(i, j) \in \mathfrak{B}}$.
Let us denote the space of $\mathfrak{B}$-shaped arrays as $\mathsf{R}^\mathfrak{B}$.
Obviously, there is an isomorphism $\mathsf{R}^\mathfrak{B} \sim \mathsf{R}^B$,
where $B = |\mathfrak{B}|$ is the cardinality of the set $\mathfrak{B}$.
This isomorphism is not unique, since the elements in the shape can be ordered in different ways.
However, the result of the algorithm does not depend on the particular order.

For convenience, we consider the lexicographical order, which fixes the isomorphism
\begin{equation}
\label{eq:iso}
\EuScript{J}_\mathfrak{B}:\,\mathsf{R}^\mathfrak{B} \mapsto \mathsf{R}^B.
\end{equation}
We call $\EuScript{J}_{\mathfrak{B}}$ \emph{vectorization} and its inverse $\EuScript{J}_{\mathfrak{B}}^{-1}$ \emph{shaping}.

Next, we introduce the operation of addition in the considered topology
for two pairs of indices $\ell = (\ell_x, \ell_y)$, $\kappa = (\kappa_x, \kappa_y)$.
For convenience, we omit the parameters $(T_x, T_y)$ and write $\oplus$ instead of $\oplus^{(T_x, T_y)}$:
\begin{equation*}
  \ell \oplus \kappa =
  \big((\ell_x + \kappa_x - 2) \mathop{\mathrm{mod}} T_x + 1, (\ell_y + \kappa_y - 2) \mathop{\mathrm{mod}} T_y + 1\big).
\end{equation*}
Here $\mathop{\mathrm{mod}}$ denotes the remainder of the integer division. Formally, $a \mathop{\mathrm{mod}} \infty = a$ for any $a$.
Note that $-2$ and $+1$ in the definition of $\oplus$ is the consequence of the index enumeration starting from 1 (such that $\{1\}\oplus\{1\} = \{1\}$).

For two shapes $\mathfrak{A}$ and $\mathfrak{B}$, we modify the definition of the Minkowski sum in the following way:
\begin{equation*}
  \mathfrak{A} \oplus \mathfrak{B} = \{\alpha \oplus \beta \mid \alpha \in \mathfrak{A}, \beta \in \mathfrak{B} \}.
\end{equation*}

\subsection{Embedding step}\label{sec:embedding_shaped}
Now we are ready to define the embedding operator for the Shaped SSA algorithm.
\subsubsection{Input data and parameters of the embedding}
The input data are the topology characteristics $(T_x, T_y)$,
the shape $\mathfrak{N}$, where $\mathfrak{N} \subset \{1, \dots, T_x \} \times \{1, \dots, T_y\}$,
the $\mathfrak{N}$-shaped array $\mathbb{X} \in \mathsf{R}^\mathfrak{N}$.
The parameter of the algorithm is a \emph{window shape} $\mathfrak{L} \subset \mathfrak{N}$.
It is convenient to consider
the window shape in the form $\mathfrak{L} = \{\ell_1, \dots, \ell_L\}$,
where $L=|\mathfrak{L}|$ and $\ell_i \in \mathsf{N}^2$ are ordered in lexicographical order.

For each $\kappa \in \mathfrak{N}$, we define a shifted $\mathfrak{L}$-shaped subarray as
$\mathbb{X}_{\mathfrak{L} \oplus \{\kappa\}} = (x_\alpha)_{\alpha \in \mathfrak{L} \oplus \{\kappa\}}$.
The index $\kappa$ is a position of the origin for the window. Consider the set of all possible
origin positions for $\mathfrak{L}$-shaped windows:
\begin{equation*}
  \mathfrak{K} = \{\kappa \in \mathfrak{N} \mid \mathfrak{L} \oplus \{\kappa\} \subset \mathfrak{N}\}.
\end{equation*}
We assume that $\mathfrak{K}$ is written as $\mathfrak{K} = \{\kappa_1, \dots, \kappa_K\}$, where $K=|\mathfrak{K}|\neq 0$ and $\kappa_j$ are ordered in lexicographical order.

If the shapes $\mathfrak{N}$ and $\mathfrak{L}$ are rectangles, then $\mathfrak{K}$ is also rectangular and we call the version of Shaped SSA \emph{rectangular}.

\subsubsection{Embedding operator}
The trajectory matrix $\mathbf{X}$ is constructed by the embedding operator $\mathcal{T}_\mathrm{ShSSA}: \mathsf{R}^\mathfrak{N} \to \mathsf{R}^{L \times K}$
\begin{equation*}
\mathcal{T}(\mathbb{X}) = \mathcal{T}_\mathrm{ShSSA}(\mathbb{X}) := \mathbf{X} = [X_1, \dots, X_K],
\end{equation*}
where the columns
\begin{equation*}
  X_j = (x_{\ell_i \oplus \kappa_j})_{i=1}^L
\end{equation*}
are vectorizations of the shaped subarrays $\mathbb{X}_{\mathfrak{L} \oplus \{\kappa_j\}}$.

The embedding operator $\mathcal{T}_\mathrm{ShSSA}$ is obviously linear.
Denote its range as $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}} \subset \mathsf{R}^{L \times K}$.
The subspace  $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}}$ consists of structured matrices, which we will call \emph{quasi-Hankel}. (In fact, they are generalizations of quasi-Hankel matrices from \cite{Mourrain.Pan2000}.)
If the operator $\mathcal{T}_\mathrm{ShSSA}$ is injective, then it sets the isomorphism of the spaces
$\mathsf{R}^\mathfrak{N}$ and $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}}$.

\begin{remark}
The embedding operator $\mathcal{T}_\mathrm{ShSSA}$ is injective if and only if $\mathfrak{L} \oplus \mathfrak{K} = \mathfrak{N}$, i.e.,
if each point of the initial shaped array $\mathbb{X}$ can be covered by a window of shape $\mathfrak{L}$.
\end{remark}

If there are uncovered points, we can remove them and consider only the decomposition of the restricted array
$\mathbb{X}' = (\mathbb{X})_{\mathfrak{N}'}$, where $\mathfrak{N}' = \mathfrak{L} \oplus \mathfrak{K}$.
Hereafter we will suppose that $\mathfrak{N} = \mathfrak{L} \oplus \mathfrak{K}$, i.e., all the points are covered and the operator $\mathcal{T}_\mathrm{ShSSA}$ is injective.

\subsection{Particular cases}
As shown in \cite{Golyandina.etal2013}, many variants of SSA can be considered as special cases of Shaped SSA. Basic SSA corresponds to the shapes
\begin{equation*}
  \mathfrak{N} = \{1, \dots, N\} \times \{1\}, \qquad \mathfrak{L} = \{1, \dots, L\} \times \{1\}.
\end{equation*}
The shaped arrays $\mathbb{X} \in \mathsf{R}^\mathfrak{N}$ in this case are time series, and the set  $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}}$  is the set of ordinary Hankel matrices. SSA for time series with  missing values is obtained by removing the indices of missing elements from the shape $\mathfrak{N}$.

Multivariate SSA (MSSA) for multivariate time series of possibly different lengths
can be considered as Shaped SSA for a shaped 2D array consisting of stacked series
and a rectangular window shape $\mathfrak{L} = \{1, \dots, L\} \times \{1\}$.
Then
the space $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}}$ is the space of concatenated Hankel matrices.

The 2D-SSA algorithm \cite{Golyandina.Usevich2010} corresponds to Shaped SSA with
\begin{gather*}
  \mathfrak{N} = \{1, \dots, N_x\} \times \{1, \dots, N_y\},\\
  \mathfrak{L} = \{1, \dots, L_x\} \times \{1, \dots, L_y\}.
\end{gather*}
The shaped arrays $\mathbb{X} \in \mathsf{R}^\mathfrak{N}$ are rectangular images, the windows $\mathfrak{L}$ are rectangular, and the set $\mathsf{H}^{\mathfrak{L} \times \mathfrak{K}}$ is the set of \emph{Hankel-block-Hankel} matrices. For more details on different SSA variants as special cases of Shaped SSA, we refer the reader to \cite[Sec. 5.3]{Golyandina.etal2013}.

All these SSA versions can be extended by allowing circular topology. In order to distinguish between different variants, we use the following terminology. If both $T_x$ and $T_y$ are infinite,
then we call the SSA version \emph{planar} (see Figure~\ref{fig:shaped_demo}, left); otherwise the term \emph{``circular''} is used. The case, when only one of $T_x$ and $T_y$ equals infinity, can be named
\emph{cylindrical} (see Figure~\ref{fig:shaped_demo}, center), while the case, when both of them are finite, is \emph{toroidal} (see Figure~\ref{fig:shaped_demo}, right).

\begin{figure}
  \centering
  \vspace{0.3cm}
  \begin{subfigure}{.32\columnwidth}
    \centering
    \includegraphics[width=0.6\columnwidth]{shaped_pict/shaped.pdf}
  \end{subfigure}
  \begin{subfigure}{.32\columnwidth}
    \centering
    \includegraphics[width=0.6\columnwidth]{shaped_pict/cylinder.pdf}
  \end{subfigure}
  \begin{subfigure}{.32\columnwidth}
    \centering
    \includegraphics[width=0.6\columnwidth]{shaped_pict/torus.pdf}
  \end{subfigure}
  \caption{Left: shaped image and window; center: cylindrical topology; right: toroidal topology}
  \vspace{-0.5cm}
  \label{fig:shaped_demo}
\end{figure}

\section{Implementation issues}
\label{sec:implementation}
\subsection{Approaches to fast implementation}
The fast implementation of SSA-related methods, which was suggested in \cite{Korobeynikov2010} and is used in the \pkg{Rssa} package
\cite{Korobeynikov.etal2013},
relies on the following approaches:
\begin{enumerate}
  \item The truncated SVD calculated by Lanczos methods \cite[Ch. 9]{Golub.VanLoan1996} is used, since only a number of leading
  SVD components correspond to the signal and therefore are used at Grouping step of the SSA algorithm.
   \item Lanczos methods do not use an explicit representation of a decomposed matrix $\mathbf{A}$.
   They need only the results of multiplication of $\mathbf{A}$ and $\mathbf{A}^\top$ on some vectors.
   Due to special Hankel-type structure of $\mathbf{A}$ in the SSA-related algorithms,
   multiplication by a vector can be implemented very efficiently with the help of Fast Fourier Transform (FFT).

     Fast SVD algorithms are implemented in the \pkg{R}-package \pkg{svd} \cite{Korobeynikov2013} in such a way that
     their input is the function of a vector which yields the result of fast multiplication of the vector by the trajectory matrix.
     Therefore, the use of \pkg{svd} in \pkg{Rssa} allows
     a fast and space-efficient implementation of the SSA-related algorithms.
   \item At Reconstruction step, hankelization or quasi-hankelization of a matrix of rank $1$ stored as  $\sqrt{\lambda}UV^\mathrm{T}$
  can be written by means of the convolution operator and therefore is also can be effectively implemented.
\end{enumerate}
The overall complexity of the computations is $O(r N \log (N))$, where $N = |\mathfrak{N}|$ is the number of elements in the shaped array and $r$ is the number of considered eigentriples, see details in \cite{Korobeynikov2010} and in \cite{Golyandina.etal2013}. This makes the computations feasible for large data sets and large window sizes.  For example, the case of images of size $299 \times 299$ and a window size $100\times 100$ can be handled easily, whereas the conventional algorithms (e.g., the full SVD \cite{Golub.VanLoan1996}) are impractical, because the decomposed matrix is of size $10^4 \times 4\cdot 10^4$. Using larger window sizes is advantageous, since, for example, separability of signal from noise (in the ``signal+noise'' scenario) can be significantly improved \cite{Golyandina2010}.

These approaches were implemented for 1D, 2D-SSA
and planar version of Shaped 2D-SSA in the version of the \pkg{Rssa} package described in \cite{Golyandina.etal2013}.

\subsection{Fast implementation for circular topologies}
In this section, we show how the same approaches can be used for Shaped SSA with circular topologies. As
 shown next, it is not difficult to extend the expressions in \cite[Sec. 6]{Golyandina.etal2013} to the general case. This allows us to incorporate circular topologies in the new version of the \pkg{Rssa} package \cite{Korobeynikov.etal2013}.

\subsubsection{Use of the Lanczos methods}
Since the circular version differs by the form of the embedding operator only,
the Lanczos methods for the truncated SVD can accelerate the circular version of SSA algorithms in the same way
as they do it for the planar case.

\subsubsection{Fast multiplication of a vector on the trajectory matrix}
The fast implementation in the planar case is based on the representation of the trajectory matrix with the help of
a Hankel or Hankel-block-Hankel circulant. This was suggested for 1D case in \cite{Korobeynikov2010}. It was shown in \cite{Golyandina.etal2013}
that the same approach is valid for the shaped 2D case.

The first step is the embedding of the initial array into the Hankel-block-Hankel circulant $\mathbf{C}_{\rm HbH}$.
It is performed in a straightforward way for the rectangular case. For the shaped case, the $\mathfrak{N}$-shaped array  is
padding by zeros to the rectangular array of sizes $N_x$ and $N_y$  such that
\begin{equation}
  \label{eq:NxNyCorrectness}
  \mathfrak{N} \subset \{1, \dots, N_x\} \times \{1, \dots, N_y\}.
\end{equation}
In \cite{Golyandina.etal2013} the $\mathfrak{L}$-trajectory matrix of a shaped 2D array $\mathbb{X}$ is presented in the form
\begin{equation*}
  \mathbf{X} = \mathcal{T}_{\rm ShSSA}(\mathbb{X}) = P^\mathrm{T}_\mathfrak{L} \mathbf{C}_{\rm HbH} P_\mathfrak{K},
\end{equation*}
where $P_\mathfrak{L}\colon \mathsf{R}^\mathfrak{L} \to \mathsf{R}^{N_x \times N_y}$ and $P_\mathfrak{K}\colon \mathsf{R}^\mathfrak{K} \to \mathsf{R}^{N_x \times N_y}$.
A shape  $\mathfrak{A} = \{\alpha_1, \dots, \alpha_A\}$ is defined as
\begin{equation}
\label{eq:circ_rep}
  P_{\mathfrak{A}} = [\mathop{\mathrm{vec}}(\bfE_{\alpha_1}) : \cdots : \mathop{\mathrm{vec}}(\bfE_{\alpha_A})],
\end{equation}
$\bfE_{\alpha_i}$ is the $\mathfrak{A}$-shape with $1$ on the $\alpha_i$th entry.
Then multiplication by $\mathbf{X}$ is reduced to multiplication by $\mathbf{C}_{\rm HbH}$, which is performed effectively by means of 2D FFT;
the projectors $P_\mathfrak{L}$ and $P_\mathfrak{K}$ are trivially calculated.

Let us consider the peculiarities of the circular case.
First, in the planar case $N_x$ and $N_y$ have only low bounds given by \eqref{eq:NxNyCorrectness}.
In the circular case by $x$ we should take $N_x = T_x$ and in the circular case by $y$ it is necessary to set $N_y = T_y$.
Then the circular topology transfers to the cyclic structure of the circulant.
Note that the projector $P_{\mathfrak{L}}$ does not depend on either planar or circular case is considered,
while the matrix $P_{\mathfrak{K}}$ is generally larger for the circular case.

\subsubsection{Rank-one quasi-Hankelization}
Extension of fast Hankelization is also quite straightforward.
The formulas from \cite{Korobeynikov.etal2013} are valid with appropriate choice of $N_x$ and $N_y$ equal
to $T_x$ and/or $T_y$ in the circular case.

\section{Rank of shaped arrays}
\label{sec:shaped_rank}
Following the definition of finite-rank time series \cite{Golyandina.etal2001} and finite-rank 2D arrays
\cite{Golyandina.Usevich2010}, let us generalize the notion of rank to the shaped and circular shaped cases.
Note that generally the theory of infinite arrays of finite rank is closely related to multidimensional arrays satisfying
linear recurrence relations.

\begin{definition}
The \emph{$\mathfrak{L}$-trajectory space} $\mathsf{S}^\mathfrak{L}$ of a shaped array $\mathbb{S} \in \mathsf{R}^\mathfrak{N}$ is defined as
\begin{equation*}
  \mathsf{S}^\mathfrak{L}(\mathbb{S}) = \mathop{\mathrm{span}} \{ (\mathbb{S})_{\mathfrak{L} \oplus \{\kappa\}}  \}_{\kappa \mid \mathfrak{L} \oplus \{\kappa\} \subset \mathfrak{N}}.
\end{equation*}
\end{definition}

The introduced trajectory space corresponds to the column space of the trajectory matrix $\mathbf{S}=\mathcal{T}_\mathrm{ShSSA}(\mathbb{S})$.
Due to the isomorphism $\EuScript{J}_\mathfrak{L}$, $\{\mathbb{P}_1,\ldots,\mathbb{P}_r\}$ is a basis of $\mathsf{S}^\mathfrak{L}(\mathbb{S})$ if and only if
 $\{P_1,\ldots,P_r\}$, where $P_k=\EuScript{J}_\mathfrak{L}(\mathbb{P}_k)$, is a basis of the column space of $\mathbf{S}$.
 Let $\{Q_1,\ldots,Q_r\}$  be a basis of the row space of $\mathbf{S}$, that is, of the row trajectory space.
Then $\{\mathbb{Q}_1,\ldots,\mathbb{Q}_r\}$, where $\mathbb{Q}_k=\EuScript{J}_\mathfrak{K}^{-1}(Q_k)$, forms the basis of a space, which can be denoted as $\mathsf{S}^\mathfrak{K}(\mathbb{S})$.
Thus, we use the terms \emph{column}/\emph{row} \emph{trajectory spaces} and \emph{column}/\emph{row} \emph{shaped trajectory spaces,}
 depending on the context.
\begin{definition}
The \emph{$\mathfrak{L}$-rank} of a shaped array $\mathbb{S}$ is defined as dimension of its $\mathfrak{L}$-trajectory subspace:
\begin{equation*}
  \mathop{\mathrm{rank}}\nolimits_\mathfrak{L} \mathbb{S} = \dim \mathsf{S}^\mathfrak{L}(\mathbb{S}) = \mathop{\mathrm{rank}} \mathbf{S}.
\end{equation*}
\end{definition}

\subsection{Ranks of finite and infinite arrays}
In order to describe the general form of arrays of finite rank, we consider infinite arrays (and their trajectory spaces) in both sides for both dimensions (i.e., $\mathfrak{N}=\mathsf{Z} \times \mathsf{Z}$).
\begin{definition}
Infinite (in both sides) array $\mathbb{S}_\infty$ is called the \emph{array of finite shaped rank}
if $r = \max\limits_\mathfrak{L} \mathop{\mathrm{rank}}_\mathfrak{L} \mathbb{S}_\infty$, where maximum is taken over $\mathfrak{L} \subset \{1, \dots, T_x\} \times \{1, \dots, T_y\}$,
is finite.
We will write $\mathop{\mathrm{rank}} \mathbb{S}_\infty = r$.
\end{definition}

\begin{remark}
The shaped rank for an infinite array is equal to its rectangular rank (when the maximum is taken over rectangular shapes),
since a sequence of shapes that contain increasing
rectangular shapes and are contained in the rectangular shapes can be easily constructed. Therefore, we will say
about arrays of finite rank omitting ``shaped''.
\end{remark}

In practice, the following remarks are important.
\begin{remark}
\label{rem:suffsquare}
If $\mathbb{S}_\infty$ is an infinite array of finite rank $r$, $\mathbb{S} = (\mathbb{S}_\infty)_\mathfrak{N}$ is a sufficiently large
finite subarray of $\mathbb{S}_\infty$ and $\mathfrak{L}$ is a sufficiently large shape, then
\begin{equation}
\label{eq:rank_equality}
  \mathop{\mathrm{rank}}\nolimits_\mathfrak{L} \mathbb{S} = \mathop{\mathrm{rank}}\nolimits_\mathfrak{L} \mathbb{S}_\infty = \mathop{\mathrm{rank}} \mathbb{S}_\infty = r,
\end{equation}
Let $\mathfrak{K} = \{\kappa \mid \mathfrak{L} \oplus \{\kappa\} \subset \mathfrak{N}\}$.
The sufficient condition for \eqref{eq:rank_equality}
 (see, e.g. \cite{Golyandina.Usevich2010} for the proof)
is that both $\mathfrak{L}$ and $\mathfrak{K}$ shapes
contain at least one $r \times r$ square. More formally, it is sufficient that there exist a 2D index $\alpha=(l, n)$
such that $\{1,\ldots, r\} \times \{1,\ldots, r\} \oplus \{\alpha\} \subset \mathfrak{L}$; the same should be valid for $\mathfrak{K}$.
\end{remark}

\subsection{Infinite arrays of finite rank: planar case}

\begin{proposition}
\label{prop:2dfiniterankmodel}
Let $T_1 = T_2 = \infty$.
Then an infinite array $\mathbb{S}_\infty$ of finite rank has the form
\begin{equation}
  \label{eq:2dfiniterankmodel}
  (\mathbb{S}_\infty)_{l, n} = \sum_{k = 1}^s P_k(l, n) \mu_k^l \nu_k^n,
\end{equation}
where $\mu_k, \nu_k \in \spaceC$, the pairs $(\mu_k, \nu_k)$ are different,
$P_{k}(l, n)$ are complex polynomials of $l$ and $n$;
such representation is unique up to summands order.
\end{proposition}
This important fact is well known starting from  \cite[\S 20.20]{Kurakin.etal95JoMS-Linear}.
Note that the rank of the array $\mathbb{S}_\infty$ given in \eqref{eq:2dfiniterankmodel} does not determined only by degrees of polynomials $P_k$ and is
hardly calculated (see~\cite{Golyandina.Usevich2010}).

An important special case is $(\mathbb{S}_\infty)_{l,n}=\sum_{k=1}^r A_k \mu_k^l \nu_k^n$, which is widely used in signal processing. In this case, $\mathop{\mathrm{rank}} \mathbb{S}_\infty$ is equal to the number of different  $(\mu_k, \nu_k)$-pairs.

\subsection{Infinite arrays of finite rank: circular case}
It follows from the definition that in the $T_x$-circular case only arrays that are $T_x$-periodic by the $x$-axis can be of finite rank.
Thus, we obtain two propositions, which characterize the cylindrical and toroidal cases.

\begin{proposition}
Let $T_x < \infty$, $T_y = \infty$.
Then an infinite array $\mathbb{S}_\infty$ of finite rank $r$  has the form
\begin{equation}
  \label{eq:cylfiniterank}
  (\mathbb{S}_\infty)_{l, n} = \sum_{k = 1}^s P_k(n) \mu_k^l \nu_k^n,
\end{equation}
where $\mu_k, \nu_k \in \spaceC$, the pairs $(\mu_k, \nu_k)$ are different,
$\mu_k^{T_x} = 1$,
$P_{k}(n)$ are polynomials of $n$ of degree $p_k$, $\sum_k (p_k+1)=r$;
such representation is unique up to summands order.
\end{proposition}
\begin{proposition}
Let $T_x < \infty$, $T_y < \infty$.
Then an infinite array $\mathbb{S}_\infty$ of finite rank $r$ has the form
\begin{equation}
  \label{eq:cylfiniterank2}
  (\mathbb{S}_\infty)_{l, n} = \sum_{k = 1}^r A_k \mu_k^l \nu_k^n,
\end{equation}
where $\mu_k, \nu_k \in \spaceC$, the pairs $(\mu_k, \nu_k)$ are different,
$\mu_k^{T_x} = 1$, $\nu_k^{T_y} = 1$, $A_k$ are some constants;
such representation is unique up to summands order.
\end{proposition}

\begin{remark}
\label{rem:cyclic_f}
In the toroidal case, $T_x < \infty$, $T_y < \infty$, an array of finite rank $r$ can be presented in the form
of sum of $r$ components of the Discrete Fourier Transform.
\end{remark}

\section{Shaped ESPRIT}
\label{sec:shaped_esprit}
The well known ESPRIT \cite{Roy.Kailath1989} and 2D-ESPRIT \cite{Rouquette.Najim01Estimation} methods are used for estimation of parametes in the model  $\mathbb{X}=\mathbb{S}+\mathbb{R}$, where $\mathbb{R}$ is noise and
$\mathbb{S}$ is the signal of finite rank, which is of the form $(\mathbb{S})_{l}=\sum_{k=1}^r A_k \mu_k^l$ (for ESPRIT) and $(\mathbb{S})_{l,n}=\sum_{k=1}^r A_k \mu_k^l \nu_k^n$ (for 2D-ESPRIT).

Next, we briefly describe the general scheme of ESPRIT-like methods, on the example of 2D-arrays.
\subsubsection{Estimation of the signal subspace}
We assume that the signal is at least approximately separated from the residual. In this case, $r$ left singular vectors $\{U_{i_1}, \dots, U_{i_r}\}$ of the trajectory matrix (and corresponding shaped arrays $\{\mathbb{U}_{i_1}, \dots, \mathbb{U}_{i_r}\}$, where $\mathbb{U}_{i_k} = \EuScript{J}_\mathfrak{L}^{-1}(U_{i_k})$)  approximate the column signal subspace. (In signal processing, typically, first $r$ components are chosen, i.e. $ \{i_1, \dots, i_r\} = \{1,\ldots,r\}$.)
\subsubsection{Construction of shifted matrices}
The algorithm uses the notion of shifts for shaped arrays from
the space spanned by $\{\mathbb{U}_{i_1}, \dots, \mathbb{U}_{i_r}\}$. Directions of shifts corresponds to the
directions in the initial space. Thus, four matrices $\mathbf{P}_x$, $\mathbf{Q}_x$, $\mathbf{P}_y$, $\mathbf{Q}_y$ are constructed, which columns span the shifted subspaces. For 2D-ESPRIT this step was already considered in
\cite{Rouquette.Najim01Estimation}. For Shaped ESPRIT, we define it in Section~\ref{sec:sh_construct}.
\subsubsection{Construction of shift matrices}
 The relation between the shifted subspaces allows to estimate the so called shift matrices of order $r \times r$ as the approximate solutions of $\mathbf{P}_x \mathbf{M}_x \approx \mathbf{Q}_x$ and $\mathbf{P}_y \mathbf{M}_y \approx \mathbf{Q}_y$. The method of estimation is based on least squares (LS-ESPRIT) or total least squares (TLS-ESPRIT).

\subsubsection{Estimation of the parameters}
The eigenvalues of the shift matrices provide estimates of the parameters: the eigenvalues of the $x$-direction shift matrix provide
estimates of $\mu_k$, while the eigenvalues of the $y$-direction shift matrix produce estimates of $\nu_k$. However, the method should
provide estimates of the pairs $(\mu_k,\nu_k)$.
There are different approaches for pair estimation: simultaneous diagonalization of shift matrices (see \cite{Rouquette.Najim01Estimation})
and a pairing of independently calculated eigenvalues of two shift matrices (the latter method is called 2D-MEMP \cite{Rouquette.Najim01Estimation}
and is improved in \cite{Wang.etal05IToSP-Comments}).

\subsection{Construction of shifted matrices}
\label{sec:sh_construct}
Let us introduce the shaped version of the ESPRIT method and describe its specific features.
A difference of the shaped version from the rectangular planar one consists in a special shift construction for shapes at the second step
of the ESPRIT scheme. (Other steps are exactly the same as in
the planar versions.) Also, the circular topology influences the algorithm.

\begin{algorithm}[Construction of shifted matrices]
\label{alg:ESPRIT}

\noindent
\textit{Input}: \\
1. The topology of the initial 2D array parameterized by $(T_x, T_y)$; \\
2. the 2D shape $\mathfrak{L}$ equipped with the topology of the initial object;\\
3. a basis $\mathbb{U}_1, \dots, \mathbb{U}_r \in \mathsf{R}^\mathfrak{L}$
of the estimated signal subspace.
For example, $\{\mathbb{U}_k\}_{k = 1}^r$ are $\mathfrak{L}$-shaped eigenvectors from the chosen eigentriples
of the trajectory matrix identified as related to an image component that one wants to analyze.

\smallskip
\noindent
\textit{Output}: the matrices $\mathbf{P}_x$, $\mathbf{Q}_x$, $\mathbf{P}_y$, $\mathbf{Q}_y$.

Let us show how to construct the $x$-shifted matrix, that is, the matrix
obtained by the shift along the $x$ axis; the case of the shift along the $y$ axis
is analogous.

1. Find the subset of points from $\mathfrak{L}$ that have adjacent points of $\mathfrak{L}$ from the right in the $x$-direction:
\begin{equation}
\label{eq:Msx}
  \mathfrak{M}_x = \{ \ell \in \mathfrak{L} \mid \ell \oplus (2, 1) \in \mathfrak{L}  \}
\end{equation}
Recall that the sum $\oplus$ with 1 corresponds to no shift.

2. Consider the shaped eigenarrays restricted on $\mathfrak{M}_x$ and on $\mathfrak{M}_x \oplus \{(2, 1)\}$:
\begin{equation}
  \label{eq:tPtQ}
  \mathbb{P}_k = (\mathbb{U}_k)_{\mathfrak{M}_x}, \quad \mathbb{Q}_k = (\mathbb{U}_k)_{\mathfrak{M}_x \oplus \{(2, 1)\}},\  k=1,\ldots,r.
\end{equation}

3. Vectorize
 the arrays $\mathbb{P}_k$ and  $\mathbb{Q}_k$  into $P_k$ and $Q_k$ correspondingly.

4. Stack the obtained vectors to the matrices
\begin{equation*}
  \mathbf{P}_x= [P_1, : \dots : P_r], \quad \mathbf{Q}_x = [Q_1 : \dots : Q_r].
\end{equation*}
\end{algorithm}

\smallskip
Thus, the shifted matrices are constructed and the shift matrices $\mathbf{M}_x$ and $\mathbf{M}_y$ can be estimated from the relations
\begin{equation}
  \label{eq:ESPRITapprox}
  \mathbf{P}_x \mathbf{M}_x \approx \mathbf{Q}_x, \quad \mathbf{P}_y \mathbf{M}_y \approx \mathbf{Q}_y.
\end{equation}

\begin{remark}
Note that for the toroidal rectangular case the application of ESPRIT has little sense, since the estimation
of frequencies of a finite-rank toroidal signal can be performed by Fourier analysis, see
Remark~\ref{rem:cyclic_f}.
\end{remark}

\begin{remark}
We have considered the version of ESPRIT based on the column space.
The version based on the row space can be also considered.
\end{remark}

\subsection{Conditions of the algorithm correctness}
Let a $\mathfrak{N}$-shaped array $\mathbb{S}$ of rank $r$ have the common term in the form $(\mathbb{S})_{l,n} = \sum_{k = 1}^r A_k \mu_k^l \nu_k^n$
for a set of different pairs $\{(\mu_k, \nu_k)\}_{k = 1}^r$,
$\mu_k, \nu_k \in \spaceC$, and noise $\mathbb{R}$ is zero.
The following proposition forms the basis of the ESPRIT-type methods (see e.g. \cite{Rouquette.Najim01Estimation})
and provides the conditions, when Algorithm~\ref{alg:ESPRIT} and the ESPRIT scheme as a whole are correct, that is, the equations~\eqref{eq:ESPRITapprox}
 have exact solutions
and the constructed shift matrices produce exactly the pairs $(\mu_k, \nu_k)$ by exact joint diagonalization at the last step of the scheme.
\begin{proposition}
\label{prop:esprit_exist}
Let a window $\mathfrak{L}$ can be chosen such that the forms $\mathfrak{M}_x$ and
$\mathfrak{M}_y$ given in \eqref{eq:Msx} be nonempty and $\mathop{\mathrm{rank}}_{\mathfrak{M}_x} \mathbb{S} = \mathop{\mathrm{rank}}_{\mathfrak{M}_y} \mathbb{S} = \mathop{\mathrm{rank}}_\mathfrak{L} \mathbb{S} = r$.
Denote $\{\mathbb{U}_k\}_{k=1}^r$,  $\mathbb{U}_k\in \mathsf{R}^\mathfrak{L}$, a basis of $\mathsf{S}^\mathfrak{L}(\mathbb{S})$.
Then \\
(1) the equalities $\mathbf{P}_x \mathbf{M}_x = \mathbf{Q}_x$ and $\mathbf{P}_y \mathbf{M}_y = \mathbf{Q}_y$ (see \eqref{eq:ESPRITapprox})
have unique exact solutions
$\mathbf{M}_x$ and $\mathbf{M}_y$;\\
(2) there exist such a matrix $\mathbf{T} \in \spaceC^{r \times r}$ that
$\mathbf{M}_x = \mathbf{T} \mathop{\mathrm{diag}}(\mu_1, \dots, \mu_r) \mathbf{T}^{-1}$
and $\mathbf{M}_y = \mathbf{T} \mathop{\mathrm{diag}}(\nu_1, \dots, \nu_r) \mathbf{T}^{-1}$.
\end{proposition}
\begin{proof}
Take the matrix $\mathbf{T}$ as a matrix of transformation of a basis $\{ \mathbb{U}_k \}_{k=1}^r$ to the basis
$\{\mathbb{U}'_k\}_{k=1}^r$, where
\begin{equation*}
  (\mathbb{U}'_k)_{l, n} = \mu_k^l\nu_k^n.
\end{equation*}
We prove both statements of the proposition for $\mathbf{M}_x$;
for the other direction the proof is the same.
If we consider the  basis $\mathbb{U}'_k$ instead of $\mathbb{U}_k$ and perform Algorithm~\ref{alg:ESPRIT}, then the corresponding shifted matrices
$\mathbf{P}'_x$ and $\mathbf{Q}'_x$ can be represented as follows:
\begin{equation*}
  \mathbf{P}'_x = \mathbf{P}_x \mathbf{T}, \quad \mathbf{Q}'_x = \mathbf{Q}_x \mathbf{T},
\end{equation*}
since construction of the shifted matrices is a linear procedure.
Obviously, the matrix $\mathbf{M}'_x = \mathop{\mathrm{diag}}(\mu_1, \dots, \mu_r)$ will be an exact solution of~\eqref{eq:ESPRITapprox}, $\mathbf{P}'_x \mathbf{M}'_x = \mathbf{Q}'_x$.
Therefore, $\mathbf{P}_x \mathbf{T} \mathbf{M}'_x = \mathbf{Q}_x \mathbf{T}$ and hence
  $\mathbf{P}_x \mathbf{T} \mathbf{M}'_x \mathbf{T}^{-1}= \mathbf{Q}_x$.

Now let us prove that the solution $\mathbf{M}_x$ is unique.
The condition of uniqueness for the solutions of the equalities~\eqref{eq:ESPRITapprox} is
$\mathop{\mathrm{rank}} \mathbf{P}_x = r$.
As before, $(\mathbb{U})_{\mathfrak{M}_x}$ is the restriction of the shaped array $\mathbb{U}$ to the shape $\mathfrak{M}_x$.
Then
\begin{gather*}
  \mathop{\mathrm{rank}} \mathbf{P}_x = \dim\mathop{\mathrm{span}}\{P_1, \dots, P_r\} = \\ =
  \dim\mathop{\mathrm{span}}\{(\mathbb{U}_1)_{\mathfrak{M}_x}, \dots, (\mathbb{U}_r)_{\mathfrak{M}_x} \} = \\ =
  \dim\mathop{\mathrm{span}}\{(\mathbb{U}'_1)_{\mathfrak{M}_x}, \dots, (\mathbb{U}'_r)_{\mathfrak{M}_x} \} = \\ =
  \dim\mathsf{S}^{\mathfrak{M}_x}(\mathbb{S}) = \mathop{\mathrm{rank}}\nolimits_{\mathfrak{M}_x} \mathbb{S}.
\end{gather*}
\end{proof}

The following proposition provides sufficient conditions
for the correctness of the Shaped 2D ESPRIT algorithm for both
planar and circular versions.

\begin{proposition}
\label{cor:ESPRITsuff}
Let
the shape $\mathfrak{L}$ contain at least one $(r + 1) \times (r + 1)$ square and
the shape $\mathfrak{K} = \{\kappa \mid \mathfrak{L} \oplus \{\kappa\} \subset \mathfrak{N}\}$ contain at least one
$r \times r$ square.
Then the conditions of Proposition \ref{prop:esprit_exist} are valid.
\end{proposition}
\begin{proof}
Obviously, if the shape $\mathfrak{L}$ contains an $(r + 1) \times (r + 1)$ square, then the
shapes $\mathfrak{M}_x$ and $\mathfrak{M}_y$ contain at least one $r \times r$ square, which is (along with the condition on the shape $\mathfrak{K}$)
the sufficient condition
for $\mathop{\mathrm{rank}}_{\mathfrak{M}_x} \mathbb{S} = \mathop{\mathrm{rank}}_{\mathfrak{M}_y} \mathbb{S} = \mathop{\mathrm{rank}}_\mathfrak{L} \mathbb{S} = \mathop{\mathrm{rank}} \mathbb{S} = r$ (see Remark~\ref{rem:suffsquare}).
\end{proof}

\section{Examples}
\label{sec:examples}
In this section we consider two examples, for demonstration of Shaped 2D-ESPRIT and of
cylindric Shaped 2D-SSA. All the examples are reproducible, and the code in \pkg{R} is available on request.
\subsection{Shaped 2D-ESPRIT}
Let us describe the example of application of the shaped version of 2D-ESPRIT.
We take a part of the commonly used image ``Barbara'' $512\times 512$ (see e.g. \cite{McGuire2011Data})
containing the table covered by checkerboard cloth.
However, due to edges of the table, the tablecloth consists of three parts with different views of the texture.
Let us consider this picture to demonstrate how Shaped 2D-SSA together with Shaped 2D-ESPRIT can help to
analyze textures in different complex-shaped areas of the image.

We construct masks for three parts of the tablecloth and consider them separately (see Figure~\ref{fig:barbara_snippets}).
The first two subimages are triangular, while the third one has complex-shaped boundary and is non-convex.
Since the first two subimages are convex, we can choose the window of the same shape as the subimage (that is,
a triangular window), but of smaller size ($4$ times smaller for subimage 1 and $9$ times smaller for subimage 2).
For the third subimage of complex shape with a gap, we choose the circle window shape of radius $10$.
Since the amplitudes of the texture components are not constant, we choose not large windows, which are though sufficient
for extraction of the texture.

Checkerboard form of the texture means that it consists of two sine planar waves that run to different directions.
This means that the main part of texture can be extracted by means of four SSA components, each pair corresponds to a separate angle.
Checkerboard texture is easily seen on the first subimage. It is not evident on the second one and is not seen at all on the third subimage.

Analysis of the eigenarrays helps to indicate which eigenarrays correspond to the texture. For example, the third image contains only one-direction strips
on the eigenarray images.
Therefore we choose four eigentriples ET2--5 for the first and the second subimages and two eigentriples ET2,3 for the third one. Figure~\ref{fig:barbara_eigenvalues} contains the result of
reconstruction (top), where the reconstructions of separate strips are depicted under the full reconstruction (for the subimage 3 we have only
one strip and therefore the picture of the second strip is empty).

\begin{figure}[thpb]
  \centering
  \includegraphics{barbara_example/barbara_123small_blue.png}
  \caption{Barbara's table with enumerated subimages}
  \label{fig:barbara_snippets}
\end{figure}

\begin{figure}[thpb]
  \centering
  \includegraphics[width=\columnwidth]{barbara_example/ESPRIT_recon.pdf}
  \caption{Barbara's table: reconstructions of texture}
  \label{fig:barbara_eigenvalues}
\end{figure}

The frequency estimations obtained by the column LS 2D-ESPRIT are presented in Table~\ref{t:freqs}.
The estimated angles (in degrees) and widths of the strips are also presented. The exponential rates show how rapidly the amplitudes are changed.

The coordinate system is as follows: the $x$ coordinate is measured from top to bottom, while the $y$ coordinate
is measured from left to right.

The relation between periods $t_x$ and $t_y$ allows one to find the angle of strips with the horizontal line by the formula
$\alpha = \arctan(t_x / t_y)$.
The strips on the subimage 1 are approximately orthogonal and have angles near 45$^\circ$. The strips on the subimage 2
are closer to vertical (angles 63$^\circ$ and 70$^\circ$), while that on the image 3 are closer to horizontal.
Note that widths of the strips can be found by the formula $w = t_x \cos\alpha = t_x t_y/\sqrt{t_x^2 + t_y^2}$.

Let us look at the exponential rates. Positive rates correspond to increasing of amplitudes in the direction from the left top corner.
Therefore, for example, the decreasing of the amplitudes of the subimage 3 from left-bottom to right-top means that the $x$-rate is positive,
while the $y$-rate is negative.

\begin{table}[h]
\centering
\caption{Barbara, ESPRIT periods and rates}
\label{t:freqs}
\begin{tabular}{
    ||
    c
    |
    S[table-format=-2.1]
    |
    S[table-format=-2.1]
    |
    S[table-format=-2.1]
    |
    S[table-format=2.0]
    |
    S[table-format=-1.4]
    |
    S[table-format=-1.4]
    ||}
\hline
{$\mathrm{Part}$} & {$t_x$} & {$t_y$} & {$\mathrm{Width}$} & {$\alpha, ^\circ$} & {$\mathrm{Rate}_x$} & {$\mathrm{Rate}_y$} \\
\hline
1    & 8.1      &  6.9     & 5.2   & 50    & -0.0066 & -0.0061 \\
1    & 11.0     &  10.1    & 7.4   & 47    & -0.0022 & -0.0064 \\
2    & 9.7      &  5.0     & 4.4   & 63    & -0.0031 & -0.0075 \\
2    & 7.2      &  2.6     & 2.5   & 70    & 0.0025  & 0.017 \\
3    & 5.1      &  6.8     & 4.1   & 37    & 0.0054  & -0.0046 \\
\hline
\end{tabular}
\end{table}

\subsection{Cylindric Shaped 2D-SSA}
Let us consider the data for expression of gene activity measured on the embryo of the drosophila (fruit fly).
The form of the embryo is similar to ellipsoid and therefore the cylindrical projection is adequate for
a middle part of the embryo. The technique of SSA analysis of such kind of 2D data by planar non-shaped 2D-SSA
was developed in \cite{Golyandina.etal2012a, Holloway.etal2011}.
After projection, the initial data given on 3D embryo surfaces are represented in the form $(x_i, y_i, f_i)$, where $(x_i, y_i)$ are coordinates of nuclei centers in
the cylindric
projection transformed to a planar rectangle,
$f_i$ are the expression values. The aim of the analysis is to decompose the data into the sum of
a pattern and noise and then measure the signal/noise ratio. The data are cropped and
then are interpolated to a regular grid; therefore, we obtain a digital image, which
can be processed by 2D-SSA. After that, the values of the smoothed expression are interpolated backward onto
nuclei centers.

\begin{figure}[thpb]
  \centering
  \includegraphics[width=\columnwidth]{circular_example/Circular_factor.pdf}
  \caption{``Kr\"uppel'': factor vectors}
  \label{fig:drosophila_factor}
\end{figure}

\begin{figure}[thpb]
  \centering
  \includegraphics[width=\columnwidth]{circular_example/Circular_reconstructed.pdf}
  \caption{``Kr\"uppel'': original image, reconstruction and residual}
  \label{fig:drosophila_reconstruction}
\end{figure}

Here we demonstrate that the developed circular version of 2D-SSA can perform smoothing without
artificial transformation of the cylinder to a rectangle. Thus the result of processing
does not depend on the technique of this transformation and has no extra edge effects. We consider the cylindrical topology ($T_x = N_x$ and $T_y = \infty$).

The data are downloaded from the BDTNP archive \cite{BDTNP},
the file \texttt{``v5\_s11643-28no06-04.pca''}, gene ``Kr\"uppel''.
The data on the surface of embryo are given in 3D coordinates, therefore we calculated cylindric coordinates
by finding the principal axis of the data. The interpolation is performed with the step 0.5\% and the
middle part from 20 to 80\% of the embryo lengths were processed.

The result of decomposition (the factor vectors from the $12$ leading eigentriples) is depicted in
Figure~\ref{fig:drosophila_factor}, where contributions of the corresponding eigentriples are indicated in the headers.

The components 1--6 grouped together provide an adequate smoothing; the residual oscillates around zero, see Figure~\ref{fig:drosophila_reconstruction}
(the maximal and minimal values of the reconstructed components are shown in the headers).
Note that the bottom and top edges are coincide, that is, there is no edge effect.

\section{Conclusions}
\label{sec:conclusions}
The shaped extensions of SSA are considerably extend the area of the method applications, including
the digital images of non-rectangular shape, images or series with gaps, planar projections of 3D surfaces.
Complemented by the circular case, Shaped SSA is able to deal with cylindrical projections of spherical or elliptic
surfaces and hence to avoid edge effects.

There is a close relation between SSA-like methods and subspace based methods for frequency estimation.
Therefore, new extensions of SSA should produce new extensions of methods for frequency estimation.
In the paper we consider the corresponding ESPRIT-like methods, formulate the algorithms  and the conditions of their correctness.

The algorithms and their implementation are presented for 2D images in the form that can be easily
extended to higher dimensions.

\section*{Acknowledgment}
The authors are thankful to Konstantin Usevich for useful comments and suggestions, which helped to improve
the paper presentation.

\input{shapedssa_bib.tex}

\end{document}

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Despite the provision of any other Act, if a person, including a person who performs professional or official duties with respect to children, has reasonable grounds to suspect child abuse or neglect, the person shall forthwith report the suspicion and information on which it is based…
The duty to report applies for any child or youth who appears to be under the age of 16. It is an ongoing obligation—which means if someone makes a report and, later, has reasonable grounds to suspect a child or youth still needs protection, they are required to make a further report.
The duty to report applies to all people: professionals, officials and the general public.
In 2017, the Child, Youth and Family Services Act was amended to increase the age of protection to include youth who are 16 or 17 years of age. For youth ages 16 or 17, any person who has reasonable grounds to suspect a youth is in need of protection, may make a report to FCSLLG.
What are "reasonable grounds to suspect" child abuse?
A person does not need to be certain a child or youth needs protection to make a report. "Reasonable grounds" are what an average person suspects based on his or her training, background or life experience, when exercising normal and honest judgment.
Sometimes children will tell an adult they are being or have been abused. They can communicate this directly or indirectly. If a child tells you about being abused or neglected, stay calm. Listen carefully and offer reassurances. Let them know you're sorry this happened and that it's not his or her fault. Do not promise to keep the information a secret or that everything will be fine now—it may take some time before the situation is resolved.
Do contact us as soon as possible: call 1-855-667-2726 any time. Our lines are open 24 hours a day, seven days a week.
All calls are confidential.
Your identity will be kept confidential except as required by law. Individuals who call are protected from legal action for reporting your concerns unless they have deliberately made a false accusation. This is established under section 72. (1.3) of the Child You and Family Services Act
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Processing the Hugh Morton Photographs and Films
Covering the Beat: The University in the WWII Era
Hugh MacRae at Invershiel
Hugh Morton Among the Movers and Shakers
Hugh Morton and North Carolina's Native Plants
The Grandfather Backcountry: A Bridge Between the Past and Preservation
The Tar Heels' 'White House Photographer'
Wilmington: Faded Glory to Fresh Achievement
Wait, wait . . . is that Carl Kasell?
NPR's Carl Kasell and North Carolina Collection Photographic Archivist Stephen Fletcher examine photographs in the Wilson Library Grand Reading Room, University of North Carolina at Chapel Hill. Photograph by Mark Perry.
Last Tuesday was a fun day at the office. In the morning, library staff gave Carl Kasell a tour of Wilson Library. Kassel, a UNC alumnus, returned to Chapel Hill for an evening event sponsored by the library moderated by WUNC radio host Eric Hodge. Kasell was a member of UNC's class of 1956 (although he did not graduate, having been drafted into the United States Army after four years as a student).
Kasell's tenure at National Public Radio began in 1975 as a part-time news announcer for Weekend Edition. Starting in 1979 he was the voice of the network's morning news for the next thirty years. Since retiring from that role at NPR in 2009, Kasell became a "roving ambassador," and continued as the judge and scorekeeper for the "Oddly Informative News Quiz" Wait Wait . . . Don't Tell Me!, which debuted in January 1998.
As you might imagine, Kasell has received several awards during his sonorous career. In 2004 the UNC School of Journalism and Mass Communication inducted Kasell into the North Carolina Journalism Hall of Fame. In 2010 the National Radio Hall of Fame inducted Kasell into its ranks. In March 2013 the North Carolina Press Association named Kasell "North Carolinian of the Year" for 2013, and the association made a wonderful biographical video available on their YouTube site. Despite his stature in journalism, A View to Hugh has not been able to feature Kasell because Hugh Morton hadn't photographed him, even though he been a co-founder of WUNC radio with Morton's long-time friend Charles Kuralt.
Andy Griffith as Sir Walter Raleigh in "The Lost Colony." Carl Kasell, as Wanchese, is in the lower right corner of the photograph.
We featured the above photograph a few years ago in a post about the comeback of The Lost Colony after a fire destroyed the production's costumes and props. Playing the role of Sir Walter Raleigh (right) is Andy Griffith. But wait . . . wait! Who is the fellow in the lower right corner wearing too much face paint? None other than Carl Kasell!
As seen in the opening photograph, I showed Hugh Morton's photograph to Mr. Kasell and he confirmed that that indeed was he in the corner. The reference to too much face paint came from a story Kasell told during Tuesday evening's event, when Andy Griffith told Kasell he had been a bit heavy handed in the makeup room before dress rehearsal. Kasell confided that Griffith later helped him with a more appropriate application of face paint, and that Griffith was "a big, big help" during that season. (Kasell's high school drama teacher was Clifton Britton, not Griffith as is often incorrectly stated on numerous web pages.)
We don't know if Morton made the above photograph before or after that cosmetic lesson, but we now know the year Morton made the photograph: Kasell said it was 1952 after he had graduated from high school, and 1952 is the only year Kasell's name appears in the official program. And because we know what Kasell's costume looked like, we can now identify other Morton photographs of Kasell.
Lillian Prince as Queen Elizabeth and Carl Kasell as Wanchese in "The Lost Colony," 1952.
Kasell played the role of "Wanchese, an Indian chief." I believe as he looked at Morton's photograph he dredged up from his memory a couple of his lines: "Mish-wi aga, Wingina" and "Wanchese no more chief. Wanchese now king."
Wanchese confronted by the character "Old Tom" holding his arquebus. "Get out of here, ye knavish rogues! Scat!" Is this also Carl Kasell? If so, Frederick Young played the part of Old Tom Harris in 1952.
If you couldn't make the evening with Carl Kasell, you can watch a video recording of the event, which includes Kasell's recollections from his performance in The Lost Colony while Morton's photograph is projected on the screen. Below is an image from a color transparency from the Morton collection not previously scanned.
This photograph is remarkably similar to the one that appears on the cover of the 1953 "The Lost Colony" souvenir program (see below).
Cover of the 1953 edition of "The Lost Colony" Souvenir Program.
But least we think that the similarity between the two photographs means that Hugh Morton made the eventual 1953 cover photograph, too, here is a photograph published on page 35 of the 1952 souvenir program:
Lillian Prince and Carl Kasell pose during the 1952 annual press photographers day.
The cover photograph could have been made by any of the photographers above. . . . But wait . . . wait, don't tell me! Is that Hugh Morton (center right) among the press photographers?!
Author Stephen FletcherPosted on April 22, 2013 Categories Biography, Tourism & Development, UNC, Who Am I?
7 thoughts on "Wait, wait . . . is that Carl Kasell?"
Stephen Fletcher says:
A follow up to Clifton Britton. Britton first appears in programs for "The Lost Colony" in 1947 as Stage Manager, and his biographical sketch states that it was his first year with the production. He was Director of the Goldmasquers and head of the Dramatic and Radio Arts at Goldsboro High School, "the largest high school dramatic arts department in the South." By then he had been at Goldsboro High School for five years and had "complete radio broadcasting facilities, stage and dramatic workshops." He had also been made an honorary member of the Carolina Playmakers. Britton became Assistant Director of "The Lost Colony" in 1951, Associate Director in 1952, and Director in 1954.
Jack Hilliard says:
Stephen, the Carl Kasell post is one of the best ever…if not the best ever. Cleverly written with great information plus adding more identifications to the Morton Collection. It doesn't get any better than that.
In that final picture, the photographer on the left, standing on the steps is legendary photographer and Hugh Morton contemporary Aycock Brown. Seeing Brown in this image prompted me to check Editor David Stick's 1976 book "Aycock Brown's Outer Banks," and on page 61 there is an image very similar to the second image in this post. The only difference being a tighter shot, slight costume differences and the little girl standing between Griffith and Prince is not the same. The caption for Brown's photograph identifies the little girl as Page Girl Sylvia Cox, a role she played in 1953. In '52 she played one of the Colonist Children. So, I think we can assume the book image was taken in 1953. There was likely a press photographer's day each year and most likely Brown and Morton were there often.
Thanks, Jack. I enjoyed pulling it together. Here's a few more items. Aycock Brown was the Publicity and News Director for "The Lost Colony starting in 1948. I didn't have time to look through all the programs to see how long he held that position, but it was into the mid 1950s. Ben Dixon MacNeill, whose negative collection is also in the North Carolina Collection Photographic Archives, served in that role beginning in 1937 or 1938. MacNeill's negatives are mostly casual snapshots off stage, but there are several promotional shots of Donald Somers playing "Old Tom," and his photograph "A Modern U. S. 'Flying Fortress" over Fort Raleigh" circa 1940.
Julia Morton says:
Stephen, you continue to amaze me! Regards, Julia
Thanks, Julia!
I have another interesting discovery that has sprung from this post. While looking to see if any Morton photographs appeared in "The Lost Colony" programs, I found that he was on the Roanoke Island Historical Association board of directors from 1952 through 1961. (The RIHA is the outdoor drama's producer.) This may explain why no Morton images won honors in the press photographers competitions, nor appeared in the souvenir programs during these years.
From its inception, RIHA board membership consistently comprised many people of note in North Carolina, and Morton's tenure with the organization likely served as an important component of his early years in public service. For context, the North Carolina Azalea Festival began in 1948 and the birthing of the USS North Carolina in Wilmington occurred 1961. From 1951 to 1961, Morton served as a member of the North Carolina Board of Conservation and Development under governors W. Kerr Scott, William B. Umstead, and Luther H. Hodges. He was Publicity Director for Hodges's 1956 gubernatorial campaign and, under Governor Hodges, served as the chair of the State Advertising Committee. Morton's first year on the RIHA board followed Bill Sharpe's year as board chairmen in 1951, probably no coincidence. Sharpe began his role of editor and publisher of THE STATE, a magazine in which Morton was a frequent contributor of photographs.
Pingback: Remembering the late-night easy listening of Carl Kasell « North Carolina Miscellany
Tar Heel and NPR legend Carl Kasell died earlier this week. He was 84 years old..
http://www.dailytarheel.com/article/2018/04/kasell-0419
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Tag Archives: Indigenous peoples of Colombia
Zuluaga/Uribe win first round of Presidential Elections – What next?
Last Sunday, Oscar Ivan Zuluaga, the candidate of Alvaro Uribe's "Democratic Centre", won the first-round of the 2014 Presidential elections with 29% of the vote.
The President-candidate for the 'National Unity' party, Juan Manuel Santos, came in second place with 25% of the vote.
Over 60% of Colombian electors abstained from voting.
Martha Lucia Ramirez, the candidate for the Conservative Party and Uribe's former Defense Minister got a little over 15% of the vote, as did Clara Lopez Obregon for the Leftist Alternative Democratic Pole. Former Bogota Mayor Enrique Peñalosa of the Green Party came in last place with around 8%.
The option of 'voting in blank', or opting to vote for none of the candidates in protest came last, although for sometime it was Santos' main rival.
Since no candidate received a majority/plurality of votes, the two main contenders (Zuluaga and Santos) will square off in a second round/run-off on June 15th.
A re-election about peace?
The wedge issue between both candidates is the current peace talks with Colombia's largest guerrilla group, the FARC, in Havana. Zuluaga, representing Uribe's hard-line military approach to ending the conflict, vehemently opposes the negotiations and if elected will probably call them off.
Santos in his concession speech on Sunday night again re-iterated that this is a 'historic' election about choosing between more war or peace (meaning to continue the seemingly promising negotiations through his re-election).
The FARC for their part have yet to comment on Sunday's result.
The issue at hand now is whether Santos will be able to convince the Colombian people of both the need for the current peace process, and if he will be able to attract the support of the other parties.
The significance of Zuluaga's victory is that Uribe is still one of the most powerful forces in Colombian electoral politics. Uribe was able to take a candidate with little national prominence six months ago to first place on Sunday. The nearly 3.7 million votes for Zuluaga are no doubt a testament to Uribe's popularity, but are also relatively small compared to Uribe's results in 2002, 2006 and other elections. Zuluaga, who is not particularly charismatic, is understood to be "Uribe's candidate"; during his victory speech the crowd began chanting "Uribe! Uribe!".
On the other hand, it is surprising that Santos lost. Incumbents are typically favoured in elections. Perhaps Sunday's results show that many of the votes Santos won in 2010 (when he was framed as Uribe's natural successor) were actually for Uribe. Moreover, one of the major deficiencies in Colombian democracy is the rampant clientelism. Santos still lost despite having the entire State apparatus at his disposal with some saying that traditional political 'machineries'/establishments will decide the second round/ the run-off.
The name of the game for Zuluaga and Santos now is to try and lure the votes from the other parties. However, discipline in Colombia's political parties is not great, nevertheless these endorsements matter. Zuluaga recently received the endorsement of the Conservative candidate who urged him to be more "flexible" with the peace talks which she conditionally supported. However, the Conservative congressional caucus seems to be rooting for Santos, and the Party as a whole is still open to both candidates.
The Greens are telling their followers that they are 'free' to choose either Zuluaga, Santos, or to vote 'blank'/for none.
Santos, with his flagship initiative being a call to peace, was hoping to attract liberal and progressive voters to his re-election campaign. However, the Alternative Democratic Pole or 'el Polo', the main Leftist party in Colombia, has said that it cannot endorse Santos. Jorge Enrique Robledo of the Pole, and one of the most popular Senators in Colombia, for example, says that he supports the process but that the peace talks cannot overshadow Santos' acceptance of Free Trade Agreements, and what is seen as a harmful economic and social policy.
At the same time, other opinion leaders in the Centre and on the Left like former Senator Piedad Cordoba, Senator-elect Ivan Cepeda, and former mayor of Bogota Antanas Mockus are saying that they will 'vote for peace', a clear nod to Santos. Cepeda has additionally said that he is not a "santista"/Santos supporter, but that he wants his party to understand the high stakes in the election – that breaking the peace process may mean thousands of more dead and a Zuluaga victory a return of Uribe and 'paramilitarization' to Colombia.
The issue on the Left seems to be that, if people accept the credibility of the peace process (which is still an issue in contention), whether or not they are willing to accept a continuing economic liberalization/Santos' neoliberal economic program in exchange for a potentially historic change (peace with the strongest insurgent group).
The different Colombias vote differently…..
Colombia, like most societies, is deeply stratified along lines of class privilege, region/geography, and race. The regions where the FARC are still a force to be reckoned with are rural areas outside the limits of not only Urban Colombia but also the success of Uribe's counterinsurgency. Many of these areas are considered 'peripheral' by urbanites and elites and in places like Cauca have large Afro-descendent/Black and indigenous populations.
It is important to note that these 'peripheral' regions where the active combat with the FARC is still ongoing, Santos and the candidates most in favour of the peace process won by large margins, and little popularity for Zuluaga.
Zuluaga, by contrast won all over the country but also had extremely strong support in urban areas, and among the middle and upper classes.
This means that if indeed Zuluaga's win on Sunday was evidence that people still love Uribe (and his hardline against the FARC), this perhaps maybe a sentiment coming from those who are probably not currently living with the war. The hard-line/war sentiment is therefore something that may be imposed on those who will actually bear the brutal consequences of rejecting a negotiated settlement to the war.
Tagged as Afro-Colombians, Afro-descendent people, Alternative Democratic Pole, Alvaro Uribe, Alvaro Uribe Vélez, Antanas Mockus, Cauca, Centro Democratico, Colombian Conservative Party, Colombian Presidential Elections 2014, costs of war, Counterinsurgency, Democratic Centre, Democratic Security, Enrique Penalosa, FARC, Free Trade Agreements, Indigenous peoples of Colombia, insurgency, Ivan Cepeda, Jorge Enrique Robledo, Juan Manuel Santos, La Alianza Verde, love of Uribe, Martha Lucia Ramirez, neogtiations with the FARC, Oscar Ivan Zuluaga, Peace, Peace talks in Havana, Peace talks with the FARC, Polo Democratico Alteernativo, Regionalism in Colombia, run-off on June 15, Rural Colombia, Santos campaign for peace, Senador electo Ivan Cepeda, Senator Ivan Cepeda, social divisions of Colombia, strategic voting, strategic voting in Colombia, the Green Alliance, the Left in Colombia, The politics of Regionalism in Colombia, Urban Colombia, Uribe, Uribismo, Voting for Peace, War
Putting Profits over People: Extractivism and Human Rights in Colombia
Originally published on Friday, 15 November 2013 12:56 at Upside Down World, and written by Mariel Perez and Dana Brown.
César García, a husband, father, and outspoken leader was assassinated on November 2 by a gunshot wound to the head as he was heading home with his wife and nine-year-old daughter after a day of work in his small-farmer community. Garcia led farm workers in brave and staunch opposition to a large-scale mining project in the Tolima department of central Colombia. Little more than one month prior, a similarly tragic story unfolded. On September 30, 36-year-old Adelinda Gómez left a meeting of her community's women group, part of her countless efforts as a leader and human rights defender in the small agricultural municipality of Almaguer, in the Cauca department of southwestern Colombia. As she was walking home, she was attacked by two unidentified individuals, who shot her to death and left her 16-year-old son in critical condition. Just one month before her death, Adelinda had received an anonymous telephone call in which she was ordered to stop speaking out against mining or she would get herself killed. Adelinda and César's tragic assassinations are sobering examples of the increasingly violent context surrounding large-scale mining and other extractive industry projects in Colombia.
When President Juan Manuel Santos took office in 2010 and declared mining one of the principal locomotoras or engines of the Colombian economy, communities and individuals like Adelinda and César strengthened their mobilization efforts to peacefully protest mining projects because of the serious environmental and human rights issues associated with the largely unregulated industry. Colombian human rights organization CINEP notes an exponential rise since 2008 in the number of social movements protesting extractive industries such as carbon, gold, and petroleum, seemingly in response to the increased economic focus on mining. In a manifestation of civil society's mobilization in response to the serious problems caused by mining, communities and rights groups, organized under the Network of Solidarity and Fraternity with Colombia (Red de Hermandad y Solidaridad con Colombia), recently conducted a Juicio Ético or People's Tribunal against transnational mining corporation AngloGold Ashanti, citing evidence of grave violations of human rights and International Humanitarian Law, including forced displacement, aggressions against community leaders, and lack of consultation of affected communities. This people's tribunal concluded that transnational corporations as well as the Colombian government must be held politically and legally accountable to citizens, given the devastating human rights effects of the largely unregulated mining sector in Colombia. This rising trend in social unrest exposes how mining activities constitute an imminent threat to the livelihoods of local communities; human rights defenders and communities have had to organize in response to recent legal efforts to ease restrictions on mining and to combat the consequences of Free Trade Agreements (10 of which have been signed or negotiated since Santos began his presidential term), which ultimately prioritize transnational companies by imposing restrictions that make it more difficult for the Colombian government to protect its people.
The issues at stake are so pressing that the Colombian government's own oversight institution, the Comptroller's Office, dedicated a 200+ page report to the consequences of large-scale mining. In the document, the Comptroller warns of the serious human rights effects of unbridled and unregulated large-scale mining, using data to show how mining projects reward companies with accumulated wealth while leaving Colombia with only accumulated waste. The institution warns that current laws impose no limits on awarding mining titles for projects, they do not limit environmental licenses that permit mining activity, and they do not employ adequate enforcement mechanisms in terms of environmental impact studies related to mining projects. Even more grave is the lack of appropriate consultation of Indigenous and Afro-Colombian communities in regards to proposed projects, despite the fact that Free, Prior, and Informed Consent of these communities is enshrined in the 1991 Constitution.
Although Colombians are exercising their constitutional rights in mobilizing against these devastating large-scale mining projects, the deaths of brave defenders like Adelinda and César show the high risks involved in confronting the powerful economic and political interests at stake in large-scale extractive projects, as well as the State's failure to protect and defend the rights of its citizens.
Mega-projects and Human Rights
The Comptroller's report underscores the strong links between extractive projects and violations of human rights, underlining concern around the increased militarization and the exacerbation of conflicts that mining causes. The statistics presented in the report seem to justify these worries. For example, 87% of forced displacement originates in areas with mining and energy projects. Other numbers further cement this correlation: 78% of crimes against trade unionists occur in these regions; 89% of violations against indigenous peoples; and 90% of those against Afro-Colombians. In total, 80% of human rights violations in Colombia occur in zones contemplating or already hosting large-scale mining and energy projects. Civil society presented one example of this correlation during its juicio ético against AngloGold Ashanti; human rights defender Alejandro Uribe Chacón was killed by members of the military, who were assigned to the Sur de Bolivar region to protect strategic zones for mining. While this execution took place in 2006, the human rights problems persist in this economically strategic region. Just a few weeks ago, human rights groups warned of a plan to assassinate leaders in Sur de Bolivar who are mobilizing against mining projects in the area. The huge risks to the lives and livelihoods of those opposing mega-projects reaches beyond the mining sector. In the municipality of Ituango in the department of Antioquia, the Movimiento Rios Vivos, a rights group peacefully protesting the construction of a hydroelectric dam, denounces frequent threats and attacks against its leaders. Just last month, Rios Vivos leader Genaro Graciano was nearly killed after a small explosion was intentionally caused just in front of his home. ASOQUIMBO, an organization protesting the construction of the El Quimbo dam in the southwestern Huila department of Colombia warns of a similar situation of violence, denouncing massive forced displacement of communities by the armed forces and violence against those peacefully protesting the dam project.
Colombian human rights organization CODHES also reports a relationship between occurences of forced displacement and regions or municipalities that are in the government's Territorial Consolidation Plan, a plan that foments foreign investment in the extractive industries in rural regions. This correlation underscores the state's support of transnational corporations over its own people. Further evidence of the state's prioritization of transnational interests is the fact that the 2001 Mining Code, which is still in force, classifies mining projects as public utility works. This implies that national development projects will always take precedence over local interests. Given the current reality, this means that the government's locomotora, or economic engine, legally trumps the human rights of its citizens.
The Comptroller's office warns that human rights violations related to mining will become a bigger problem as the government grants more and more land titles to victims claiming their land rights through the 2011 Victims and Land Restitution Law. This is because almost all towns that are at the center of the government's land restitution law are currently developing mining projects.
In a sense, the increased link between the presence of extractive industries, megaprojects and violations of human rights seems to be reminiscent of the "conflict minerals" situation in certain African countries. Though in the Colombian case, it is important to note that mineral wealth not only lines the guerrillas' pockets, but also those of state and para-state actors. While the FARC's role in illegally mining tungsten ore is most visible at the international level, corporations, state agents, and paramilitary groups have also benefitted from a loosely regulated extractives industry. US coal mining company Drummond, for example, is known to have extensive links with paramilitary groups whom they paid to threaten and assassinate those contesting the company's economic interests in Colombia. Furthermore, virtually the entire emerald trade in Colombia (which accounts for a whopping 80-90% of the world market) has long been controlled by paramilitary actors. The military's 2006 assassination of human rights defender Uribe Chacón for the benefit of AngloGold Ashanti exemplifies the state's direct role in fomenting conflict mining. A more recent example involves Colombia's use of legal recourses to protect large-scale mining interests over the rights of Colombian citizens. In the municipality of Piedras in the department of Tolima, citizens held a popular referendum in which 2,791 individuals voted to reject mining projects in the region and only 24 voted in support of large-scale mining. While these mechanisms of participatory democracy are binding according to current law, the government directly undermined these rights in May of this year, enacting a decree that rules that citizens cannot halt the awarding of titles for mining projects, regardless of the degree of popular opposition. In effect, the state is legalizing conflict mining through its economic policies and through the use of legal recourse that benefits large-scale corporations, to the serious detriment of Colombian citizens.
While Santos agreed to put the land issue on the table of negotiation with the FARC, recognizing its role in the exacerbation of the Colombian conflict, victims of the armed conflict are not party to the negotiations and there are no discussions of mineral rights for communities, leaving dangerous room for loopholes that may allow corporations to continue to take lands from their rightful owners. Given Colombia's increased economic aperture and the growing prominence of extractive industries and megaprojects, the government cannot expect to fully address the land issue without talking about natural resources. A true political will for peace must go beyond demobilizing the guerrilla and address all of the factors and actors that exacerbate violence in the country.
A Lasting Peace in Colombia
This week we celebrated news of a new agreement at the negotiating table between the FARC and the Colombian government regarding political participation. This is an important step towards reaching a full agreement on the end of the armed conflict and a huge achievement for the negotiators. Nevertheless, Colombians know that much more than a signed agreement with the FARC is needed in order to bring lasting peace to Colombia.
In addition to the need to dismantle neoparamilitary organizations and negotiate with the other remaining guerrilla groups, a lasting peace in Colombia would require economic and social justice that includes equitable access to land and natural resources.
While the prospects for peace in Colombia seem grim given the increasingly violent conflict surrounding extractive industries and their so-called development projects, the tireless efforts of members of civil society cannot be overlooked. Recently, Afro-Colombian communities succeeded in legal action against the State, which had identified portions of their collectively-held land as strategic mining zones under a 2012 Resolution. The Court declared that the labeling of these areas as strategic mining zones violated Afro-Colombian groups' rights to Free, Prior, and Informed Consent, a success in upholding communities' rights over the rights of transnational corporations. Nonetheless, the Court failed to make a statement on other fundamental rights, including the communities' rights to land and cultural diversity, and a healthy environment. As the shortcomings of the decision show, much work remains in ensuring the rights of Colombian citizens. Adelinda and César's recent deaths are a testament to this fact. They are the devastating manifestations of the dehumanizing effects of uncontrolled large-scale extractivism and neoliberal development in Colombia and of the high costs of putting national and transnational economic interests before the lives and livelihood of the Colombian people.
Dana Brown and Mariel Pérez are human rights activists at the US Office on Colombia (http://www.usofficeoncolombia.org/) where they work to support civil society voices for peace with justice, an end to impunity and respect for human rights in Colombia."
Tagged as 2001 Colombia mining code, 2011 Victim's and Land Restituion Law, Adelinda Gomez, Afro-Colombians, Anglogold Ashanti, BACRIM, Canadian-written Colombia mining code, Cauca, Cesar Garcia, CINEP, Colombia's Territorial Consolidation Plan, Colombian coltan, Conflict minerals, Controloria report on mining, Criminal Bands, Displacement due to mining, FARC, Indigenous peoples of Colombia, Juan Manuel Santos, Locomotera minero-energetica, Mining, mining and violence, Neoparamilitarie, Neoparamilitaries, No limits to mining titles, People's Tribunal, Tolima, Uribe Chacon


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Rumbles: Investment needed to reach cycling target
Scottish Liberal Democrat transport spokesperson Mike Rumbles MSP has today said that additional investment in active travel is needed for the Scottish Government to meet its 2020 cycling target.
The comments come after the Scottish Government announced the creation of a taskforce with the objective to push forward cycling infrastructure across Scotland. Mr Rumbles also commented that plans to cut Air Passenger Duty showed an SNP government prioritising tax cuts for aviation over boosting active travel.
Before the election in May, Scottish Liberal Democrats set out plans for a £20 million investment to boost cycling.
Commenting Mr Rumbles said:
"We saw recently that the proportion of journeys taken by bike fell to just 1.2% in 2015 and now stands lower than what it was in 2011. This shows the challenge facing the Scottish Government if it wants to achieve its 2020 target of 10% of journeys being taken by bike.
"The proposed cut in APD from the SNP is despite the number air passengers increasing by millions in the last five years with a 6% jump in the last year alone.
"It's a shame that they cannot replicate a jump like that in active travel.
"While a taskforce may sound exciting to the Minister it will do little unless it is backed up by significant additional investment. Without strong leadership in this area there is no way on earth that the 2020 target will be met. Even with a taskforce we will not get anywhere near it."


Train: The nearest train station is Pulborough (West Sussex) or Haslemere (Surrey). Taxi from Pulborough takes around 15 minutes, from Haslemere around 30 minutes.
Plane: The nearest airport is Gatwick, North and South Terminals. Heathrow is little further.
Car: We are approx 1 hour 30 mins from London, 1 hour from Brighton and 2 hours from Bristol.
Buying trips - we may be closed or have reduced shop opening hours. So please always check in or contact us before setting out to come and see us - we would hate you to have a wasted journey. Thank you.


text:
Q: sourcetree - enter passphrase for key after restarted I'm using Sourcetree for Bitbucket and Github
I completed setup but when I restart PC, Sourcetree require passphrase for key
How I can save passphrase in Sourcetree and autoload no need enter everytime restart?

A:
How I can save passphrase in Sourcetree and autoload no need enter everytime restart?

That seems expected, as the ssh-agent (which should launch with SourceTree) will cache your passphrase once per session (not forever: that wouldn't be very secure)

redpajama_set_name: RedPajamaStackExchange


text:
Strengthening American Infrastructure (SAI)
Funding Agency:
Strengthening American Infrastructure (SAI) is an NSF program seeking to stimulate human-centered fundamental and potentially transformative research aimed at strengthening America's infrastructure. Effective infrastructure provides a strong foundation for socioeconomic vitality and broad improvement in quality of life. Strong, reliable and effective infrastructure spurs private-sector innovation, grows the economy, creates jobs, makes public-sector service provision more efficient, strengthens communities, promotes equal opportunity, protects the natural environment, enhances national security and fuels American leadership. Achieving these objectives requires the integration of expertise from across all science and engineering disciplines. SAI focuses on how fundamental knowledge about human reasoning and decision-making, governance, and social and cultural processes enables the building and maintenance of effective infrastructure that improves lives and society and builds on advances in technology and engineering. Successful projects will represent a convergence of expertise in one or more social, behavioral or economic sciences, deeply integrated with other disciplines to support substantial and potentially pathbreaking fundamental research applied to strengthening a specific focal infrastructure.
Full Proposal Deadline: March 15, 2023
https://beta.nsf.gov/funding/opportunities/strengthening-american-infrastructure…
Proposals must bring deep expertise in at least one SBE disciplinary program area, and provide details on how such SBE disciplinary expertise and leadership will contribute to strengthening American infrastructure. Proposals must also bring relevant expertise in the focal infrastructure, which is likely to be in one or more research areas represented in the programs supported by the NSF Office of Integrated Activities (OIA) or the other participating NSF directorates:
Proposals must describe how this diverse expertise will be integrated and applied to the specific infrastructure that is the focus of the proposed research. Successful proposals will demonstrate an interdisciplinary and convergent approach beyond that of any single NSF program, division or directorate and will seek to advance both the SBE fields as well as the partnering fields in which they are based.
For the purpose of this solicitation, the focal infrastructure of the proposed research may be of any kind, including physical, cyber, biological, technological, social, economic or educational. However, the proposal must identify a specific, focal and well-defined infrastructure. Proposals must exhibit and build on a deep understanding of at least one SBE science relevant to the design, development or sustainability of the focal infrastructure. These sciences may include those of human cognition, perception, information processing, decision-making processes, social and cultural behavior, legal frameworks, governmental structures and other areas of SBE science supported in the disciplinary programs of the SBE directorate.
Proposals may only be submitted by the following:
Institutions of Higher Education (IHEs) - Two- and four-year IHEs (including community colleges) accredited in, and having a campus located in the US, acting on behalf of their faculty members. Special Instructions for International Branch Campuses of US IHEs: If the proposal includes funding to be provided to an international branch campus of a US institution of higher education (including through use of subawards and consultant arrangements), the proposer must explain the benefit(s) to the project of performance at the international branch campus, and justify why the project activities cannot be performed at the US campus.
Non-profit, non-academic organizations: Independent museums, observatories, research labs, professional societies and similar organizations in the U.S. associated with educational or research activities.
PIs must hold appointments at U.S.-based campuses/offices of eligible organizations (IHEs or Non-profit, non-academic organizations). There are no restrictions on who may serve as co-PIs, Senior Personnel or Consultant.
An individual may appear as PI, co-PI, senior personnel or consultant on only one proposal submitted in response to this solicitation. This eligibility constraint will be strictly enforced. In the event an individual exceeds this limit, the first proposal received prior to the deadline will be accepted and the remainder will be returned without review. This limitation includes proposals submitted by a lead organization, collaborative non-lead proposals, and any subawards included as part of a collaborative proposal involving multiple institutions. No exceptions will be made.
Amount Description
Estimated Number of Awards: 15
The number of awards is dependent upon the proposals received and the degree to which proposals meet the solicitation goals and NSF merit review criteria.
Anticipated Funding Amount: up to $9,600,000 pending the availability of funds and quality of proposals.
Environmental & Life Sciences


Online radio is a great way to listen to music, news and other events from around the world – there's a far greater selection available on the internet than you would be able to find anywhere else. TuneIn Radio earned itself a great reputation and large following with its Android and iOS apps, and the release of Windows 8 sees the release of a new version of the app.
Just as with the mobile version of the software, you can use the app to search for radio stations that match particular specifications. You can also browse through the various categories of station such as rock, sports, news etc and you can choose to filter by country or language. Local radio stations are highlighted so you can easily access those that are going to carrying local news and events information.
When you're listening to a radio station you can view information about the currently playing track, and you can view suggestions that TuneIn Radio thinks that you might be interested in based on your history. Just as with TV stations, you may well find that there are radio stations you return to again and again, and to make it easier to access them you can build up a list of favorites.
The Modern interface is something that works very well for browsing through information of this kind, and it works particularly well when working with a touch screen device. There are currently some problems with streaming that crop up from time to time, but this is still an early release that will hopefully be updated with fixes in the near future.
A promising app that needs some work to match the level of the mobile version, but looking good.